Canard Cycles: From Birth to Transition (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 73) [1st ed. 2021] 3030792323, 9783030792329

Table of contents :
Preface
Part I
Part II
Part III
Guide for the Reader
Acknowledgements
Contents
Part I Basic Notions
1 Basic Definitions and Notions
1.1 Slow–Fast Families of Vector Fields
1.2 Examples
1.3 Examples Where More Than One Admissible Expression Is Needed
1.4 Normally Hyperbolic Versus Contact Points
2 Local Invariants and Normal Forms
2.1 Normal Forms Near Contact Points
2.2 Invariants at Contact Points
2.3 Example
2.4 Remarks About Contact Points
2.5 Invariants at Normally Hyperbolic Points
3 The Slow Vector Field
3.1 Definition
3.2 Differential 1-Forms Along the Critical Curve
3.3 Calculating Slow Vector Fields
3.4 The Slow Vector Field Near Contact Points
3.5 Slow Singularities
4 Slow–Fast Cycles
4.1 Definitions
4.2 Elementary Slow–Fast Segments
4.3 Regular Common Cycles
4.4 Canard Cycles
4.5 Ordinary Canard Cycles
4.6 Transitory Cycles
4.6.1 Singular Points in the Slow Vector Field: Transition to Singular Homoclinic
4.6.2 Loss of Hyperbolicity on One of the Two Branches in a Layer
4.6.3 Birth of Canard Cycles
5 The Slow Divergence Integral
5.1 Preliminaries
5.2 Definition and Intrinsic Nature of Slow Divergence Integral
5.3 Invariance of the Slow Divergence Integral Under Equivalences
5.4 Slow Divergence Integral Near Singularities of the Slow Vector Field
5.5 Slow Divergence Integral Near Contact Points
5.6 Slow Divergence Integral of a Slow–Fast Cycle
5.7 Examples
5.7.1 Van der Pol
5.7.2 Some Canards in Quartic Liénard Systems
5.7.3 Zeros in the Slow Divergence Integral
6 Breaking Mechanisms
6.1 Normal Forms for Generic Jump Points and Generic Turning Points
6.2 Generic Jump Breaking Mechanism
6.3 Generic Hopf Breaking Mechanism
6.4 Formal Power Series Methods for the Generic Hopf Breaking Mechanism
6.5 Generic Breaking Mechanisms
6.6 Other Breaking Mechanisms
6.7 Examples
7 Overview of Known Results
7.1 Periodic Orbits Near Common Cycles
7.1.1 Existence of Periodic Orbits Near Common Cycles
7.1.2 Multiple Periodic Orbits Near Common Cycles
7.2 Unicity of Periodic Orbits Near Unbalanced Canard Cycles
7.3 Existence of Periodic Orbits Near Ordinary Canard Cycles
7.4 Entry–Exit Relations
7.5 Multiple Periodic Orbits in Layers
7.6 Contact Points of Higher Singularity Order Or Contact Order
7.7 Canard Cycles with Singularities in the Slow Vector Field
7.8 Multi-Layer Canard Cycles
7.8.1 Two-Layer Canard Cycles and Their Transitory Boundaries
7.8.2 More Than Two Layers
7.9 Birth of Canard Cycles
7.9.1 Birth of Canards in Liénard Systems
7.9.2 The Conjecture
7.9.3 The Infinite Codimension Case
7.9.4 Birth of Canard Cycles for the Slow–Fast Bogdanov–Takens Singularity
7.9.5 Birth of Canard Cycles for More Degenerate Contact Points
Part II Technical Tools
8 Blow-up of Contact Points
8.1 Blow-up Procedure
8.2 Blow-up of a Generic Jump Point
8.3 Blow-up of Regular Contact Points
8.3.1 The Saddle s-
8.3.2 The Saddle s+
8.3.3 The Semi-Hyperbolic Points
8.4 Blow-up of a Generic Turning Point
8.4.1 Blow-up of the Turning Point for |a|ε
8.4.2 Asymptotic Expansions in the Blow-up of the Hopf Point
8.5 Global Aspects in the Blow-up of Contact Points
8.5.1 Closed Form Expressions for the Orbits on the Blow-Up Locus of the Generic Jump Point
8.5.2 Passage Time in the Blow-up of Jump Points
8.5.3 Separatrices on the Blow-up Locus of the Generic Jump Point
8.5.4 Regular Splitting of Separatrices in the Blow-up of a Generic Turning Point
8.5.5 Hopf Breaking Mechanism Revisited
8.6 From the Hopf Bifurcation to the Polycycle and the Birth of Canards
9 Center Manifolds
9.1 Ck-Invariant Manifolds for Diffeomorphisms
9.2 Ck-Invariant Manifolds for Vector Fields
9.3 Smooth Invariant Manifolds in Slow–Fast Systems
9.3.1 The Case of a Closed Critical Curve
9.3.2 The Case of a Closed Critical Interval
9.3.3 The Case of a Critical Semi-Hyperbolic Point
9.3.4 The Case of Singularities of the Slow Vector Field
10 Normal Forms
10.1 Preliminaries
10.1.1 The Path Method
10.1.2 First Order Differential Equation
10.1.2.1 Generalities
10.1.2.2 Affine First Order Equations
10.1.2.3 Solution for Hyperbolically Attracting Affine Equations
10.2 Regular Points of the Critical Curve
10.2.1 The Formal Solution
10.2.2 The Semi-Formal Solution
10.2.3 The Final Step
10.3 Semi-Hyperbolic Points in the Blow-up Locus
10.3.1 The Formal Solution
10.3.1.1 Solution Along a Line of Zeros
10.3.1.2 The Case αβ= 0
10.3.2 The Semi-Formal Solution
10.3.3 The Final Step
10.4 Construction of Center Manifolds
10.4.1 The Case of a Regular Interval
10.4.2 The Case of a Semi-Hyperbolic Point
10.4.3 Intervals Ending at a Semi-Hyperbolic Point
10.5 Hyperbolic Saddle Points in the Blow-up Locus
10.5.1 Resonant Monomial Vectors uαvβyγ∂y
10.5.2 Formal Normal Form
10.5.3 Reducing to a Differential Equation on Flat Functions
10.5.4 Solving the Differential Equation on Flat Functions
11 Smooth Functions on Admissible Monomials and More
11.1 Admissible Monomials and Functions in Admissible Monomials
11.2 Derivation
11.3 Counting the Number of Roots
11.4 Asymptotically Smooth Functions in Admissible Monomials
11.5 Functions of Exponentially Flat Type
11.5.1 Some Basic Properties of the Exponential Term
11.5.2 Coherence of Definition 11.9
11.5.3 Composition of Families of Diffeomorphisms of Exponentially Flat Type
Part III Results and Open Problems
12 Local Transition Maps
12.1 Transition Along an Arc of Regular Points of the Slow Dynamics
12.1.1 Transition in a Normal Form Chart
12.1.1.1 Changing the Exit Section
12.1.1.2 Restricting to a Starting Section Transverse to the Critical Curve
12.1.2 General Expressions for Regular Transitions
12.1.2.1 Transition Between Two Sections Transverse to the Critical Curve
12.1.2.2 Transition from An Exterior Section
12.1.2.3 Remark Concerning the Choice of the Coordinate z
12.1.3 Properties of Transitions Along Regular Arcs
12.2 Transition Near Semi-Hyperbolic Points
12.2.1 Equation for the Transition Component Z̃
12.2.2 A Simple Case
12.2.3 Preparing the Function G
12.2.4 Estimates for the Integral I in (12.8)
12.2.5 Theorems for the Transition Map
12.2.6 Transitions Near Particular Semi-Hyperbolic Points
12.2.6.1 Transition Near the Semi-Hyperbolic Point q1 (for λ)
12.2.6.2 Transition Near the Semi-Hyperbolic Point q2 (for -λ)
12.2.6.3 Transition Near the Semi-Hyperbolic Point s3 (for X,μ)
12.3 Transition Near Hyperbolic Saddle Points
12.3.1 The Transition Map in the Case p=1
12.3.2 Transition in the General Case (for pN)
12.3.3 The Saddle Points of Chap.8
12.4 Transition at a Jump Point
12.5 Transition Along an Attracting Sequence
12.6 Transition Along a Hopf Attracting Sequence
13 Ordinary Canard Cycles
13.1 Introduction
13.2 Basic Settings
13.2.1 Difference Functions
13.2.1.1 The Case of a Jump Mechanism
13.2.1.2 The Case of a Hopf Mechanism
13.2.1.3 A Common Expression for the Two Difference Functions
13.2.2 Tubular Neighborhood of the Canard Cycle
13.3 Results of Bifurcation
13.3.1 A Mild Preparation for Eq.(13.11)
13.3.2 The Canard Phenomenon
13.3.3 Formal Power Series Expansion of the Canard Surface for Generic Hopf Breaking Mechanisms
13.3.4 Canard Explosion, Flying Canard, and Sitting Canards
13.3.5 Counting the Limit Cycles over a Whole Layer Strip
13.3.5.1 The Non-orientable Case
13.3.5.2 The Orientable Case
13.3.6 Limit Cycles and Bifurcations in a Rescaled Layer
13.3.6.1 Equivalence of Families of Functions
13.3.6.2 Asymptotic Expression for the Difference Function ε,()
13.3.6.3 Bifurcation Results in a Rescaled Layer
13.3.7 Limit Cycles Outside the Rescaled Layer
13.3.7.1 Uniform Result for a Simple Zero of the Slow Divergence Integral
14 Transitory Canard Cycles With Slow–fast Passage Through a Jump Point
14.1 Statement of the Results
14.2 Behavior of the Slow Divergence Integral
14.2.1 The Slow Divergence Integrals J, K, and L
14.2.2 Slow Divergence Integrals of Slow–fast Cycles
14.3 Local Study Near the Jump Point q
14.3.1 Blowing Up the Jump Point
14.3.2 Transition at the Saddle Point s
14.3.3 Transition at the Semi-Hyperbolic Point q1
14.3.4 Transition at the Semi-Hyperbolic Point q2
14.4 Transition Maps Outside the Jump Point q
14.5 Cyclicity of the Transitory Canard Cycle 0
14.5.1 The Displacement Function η
14.5.2 Structure of the Transition Maps Toward T
14.5.3 Covering of the Section C
14.5.4 Unique Maximum Properties
14.5.5 Proof of Theorem 14.1
14.6 Limit Cycles and their Unfoldings
14.6.1 Saddle-Node Bifurcation of Limit Cycles in Case I
14.6.2 Proof of Theorem 14.5
15 Transitory Canard Cycles with Fast–fast Passage Through a Jump Point
15.1 Introduction
15.2 Blow-up of the Jump Point
15.3 Transitions Near the Singular Points of λ
15.3.1 Transition at the Saddle Points s
15.3.2 Transition at the Semi-Hyperbolic Point q
15.4 Regular Transitions for λ Along the Blow-up Locus
15.4.1 Regular Transition Near the Interior of the Blow-up Locus
15.4.2 Regular Transition Near the Boundary of the Blow-up Locus
15.5 Cyclicity of the Canard Cycle
15.5.1 The Displacement Function η
15.5.2 Normal Form for Transitions Toward T
15.5.3 From Global to Local Displacement Functions
15.5.4 Proof of Theorem 15.2 for ηm
15.5.5 Proof of Theorem 15.3 for ηd
15.5.6 Proof of Theorem 15.3 for ηu
15.6 Proof of the Main Theorem
16 Outlook and Open Problems
16.1 Introduction
16.2 Codimension
16.2.1 Codimension of Contact Points
16.2.2 Codimension of Jumps Between Contact Points
16.2.3 Codimension of Singularities of the Slow Vector Field
16.2.4 Codimension of a Slow–fast Unfolding
16.2.5 Codimension of a Canard Cycle
16.3 Desingularization of Unfoldings
16.3.1 Generic Unfoldings
16.3.2 Existence of Versal Unfoldings
16.3.3 Blowing Up of Versal Unfoldings
16.4 Analytic Slow–fast Unfoldings of Infinite Codimension
16.5 The Question of Finite Cyclicity for Canard Cycles
16.6 Disorienting Canard Cycles
16.7 Recapitulation of Open Problems and Questions
16.7.1 Questions About Codimension
16.7.2 Questions About Versal Unfoldings and their Desingularization
16.7.3 Questions About Asymptotic Properties
16.7.4 Questions About Analytic Unfoldings and Canard Cycles
16.7.5 Questions About the Finite Cyclicity Conjecture
16.7.6 Questions About Disorienting Canard Cycles
References
Index

Citation preview

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge A Series of Modern Surveys in Mathematics 73

Peter De Maesschalck Freddy Dumortier Robert Roussarie

Canard Cycles From Birth to Transition

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Volume 73

Series Editors L. Ambrosio, Pisa V. Baladi, Paris G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette G. Huisken, Tübingen J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay U. Tillmann, Oxford J. Tits, Paris D.B. Zagier, Bonn

Ergebnisse der Mathematik und ihrer Grenzgebiete, now in its third sequence, aims to provide summary reports, on a high level, on important topics of mathematical research. Each book is designed as a reliable reference covering a significant area of advanced mathematics, spelling out related open questions, and incorporating a comprehensive, up-to-date bibliography.

More information about this series at http://www.springer.com/series/728

Peter De Maesschalck • Freddy Dumortier • Robert Roussarie

Canard Cycles From Birth to Transition

Peter De Maesschalck Mathematics and Statistics Hasselt University Diepenbeek, Belgium

Freddy Dumortier Mathematics and Statistics Hasselt University Diepenbeek, Belgium

Robert Roussarie Institut de Mathématiques de Bourgogne Université Bourgogne Franche-Comté Dijon, France

ISSN 0071-1136 ISSN 2197-5655 (electronic) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-030-79232-9 ISBN 978-3-030-79233-6 (eBook) https://doi.org/10.1007/978-3-030-79233-6 Mathematics Subject Classification: 34E17, 34E15, 34D15, 34C07 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The term “canard cycle” was coined by Benoit and al in [BCDD81] in the context of the theory of singular perturbations in two-dimensional differential equations. Roughly speaking, a canard cycle is a simple closed curve from which can bifurcate periodic orbits (also called relaxation oscillations in this context) exhibiting, besides fast behaviour, slow behaviour of both attracting and repelling type. We will give precise definitions and explain the essential properties in Part I. We will also recall the reason of this terminology. A singular perturbation is, in its Standard Form, considered to be a (p + q)dimensional differential equation which can be written as 

 dx ds = f (x, y) dy ds = g(x, y),

(1)

where x ∈ Rp and y ∈ Rq and  is a small positive parameter. This equation is called singular because the dimension of the phase space falls from p + q to q when  reaches 0. In this book we will, as is often done, change the so-called slow time s by the so-called fast time t = s . This change of time is singular for  = 0, but it has the advantage of changing (1) into a non singular differential equation  dx dt dy dt

= f (x, y) = g(x, y).

(2)

This is the differential equation of a vector field X . The set of zeros of X0 is given by W = {f (x, y) = 0}. Under the condition that the Jacobian of the matrix ∂f ∂x is not zero along W , this set is a q-dimensional submanifold of Rp+q , called the critical (or slow) manifold of the system (2). The system is no longer singular but the vector field X0 is degenerate in the sense that it has non-isolated zeros. Working with it has the advantage to remain in the context of ordinary differential equations, permitting the use of the traditional techniques from dynamical systems theory. This approach is in contrast to the Non Standard Analysis approach of v

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[BCDD81] where the singular aspects played an essential role. Equations like (2) are called slow-fast vector fields. The reason for using this terminology is that, for  small but different from 0, a typical trajectory consists of a succession of slow phases, when the point on the trajectory is near the critical manifold and of fast phases when this point is at a noticeable distance from the critical manifold. The minimal dimension of the phase space for a slow-fast vector field is 2 when p = q = 1. Apart from the so-called small parameter , we will include an arbitrary finite number of other parameters (for simplicity, these parameters were not written in (1) or (2)). The book is restricted to the slow-fast systems in this minimal dimension 2. There surely exist reasons to also consider slow-fast vector fields in a dimension higher than 2. There, the properties may however be much more complicated than in dimension 2 and may even be completely new. For instance, a canard cycle may occur in a structurally stable way in higher dimensions, which is never the case in dimension 2. We refer the reader to the vast literature on slow-fast vector fields in higher dimension, see for example [SW01, MS01, Guc08, SW04, Liu00, KPK08, BKK13, Wec20, Kue15]. Singular perturbations are quite important for modelling in sciences as physics, chemistry, biology, earth sciences and many more. The first such modelization was made by Van der Pol for electrotechnic systems. A huge literature is devoted to applications of the theory of singular perturbations in various domains. See [Bar88, DGK+ 12, KP18, PA96, BJSW08, HvHM+ 14], . . . . There are nevertheless specific reasons to consider the two-dimensional slow-fast vector fields. Firstly, many important and technical methods and basic facts, which can be relevant in any dimension, appear already in dimension 2 in a non trivial way. But they are much easier to be understood and studied in this low dimension. It already permits to introduce the concept of canard cycle itself and the related socalled canard phenomenon. We can present the importance of centre manifolds, as first used by Fenichel [Fen79], in the least degenerate situations, but also the blowup technique introduced in [DR96], extending the use of centre manifolds to more degenerate situations. It all helps to get a very good understanding of the dynamics near the critical manifold, including the so called turning points. Moreover, it is only in dimension 2 that we can expect achieving a quite general theory of limit cycles (isolated periodic orbits) bifurcating from canard cycles, in families of slowfast vector fields depending on a finite but arbitrary number of parameters. On the other hand, the dynamics in a higher dimensional slow-fast system may be very complicated (including existence of chaos) and there is no reasonable expectation to get results on the bifurcations in such generality as we will get in the twodimensional case. The two-dimensional case is not only interesting as a paradigm for the study of the multi-dimensional case, but there is more. It also has a direct interest for twodimensional differential equations and in particular for two famous problems: the 16th Hilbert problem about polynomial vector fields in the plane [Hil00] and the more recent Smale problem for classical polynomial Liénard equations [Sma00]. Recall that these problems deal with the existence of finite bounds, depending on the degree, for the number of limit cycles. Slow-fast vector fields occur in a systematic

Preface

vii

way in treating these questions. Of course, the family of all vector fields of a given degree n ≥ 2 contains vector fields with non-isolated zeros, and such family can be reduced locally to families of slow-fast vector fields. In a more hidden way, slowfast vector fields may occur when one uses a compactification of the phase space to study the behaviour near infinity or when one uses blowing-up to unfold degenerate singularities. For instance in families Ln of Liénard equations of a fixed degree n, starting already with the classical Liénard equations, an interesting idea consists not only of extending the vector fields to an appropriate Poincaré-Lyapunov disk, but also to compactify the family itself in a related way. This method leads unavoidably to study canard cycles. For a precise description of the process we refer to [Rou07] and [Dum06]. This method already lead to disprove the conjecture of Lins et al. [LdMP77] on the number of limit cycles for classical Liénard equations. The families of slow-fast systems, that we deal with in the book, will be of class C ∞ , although some results can easily be extended to systems of class C r , with r sufficiently large. We will even impose extra conditions on the smooth systems in order to avoid situations of infinite codimension. One might hence wonder why not to restrict to analytic systems, since the results will mostly be applied to polynomial systems. Restricting to analytic systems would however not be a good idea, since it is impossible to stay in the analytic class for long: the centre manifolds, needed for a good understanding but also largely used in calculations, are not analytic in general. In Chap. 9 we will prove that, under certain restrictions, the centre manifolds are smooth, which is already a strong result that can surely not be extended to higher dimensional systems. It justifies the choice to work with smooth systems from the start. We could expect intermediary results in a Gevrey context (see [DM07, FS13], . . . ), both for centre manifolds and possibly extending the Gevrey-characterization to some extent in combination with the blow-up technique, but it does not seem possible to obtain general results about limit cycles in a Gevrey context, similar to the smooth results in this book. We decided not to mention Gevrey type results in the book, in order to avoid the introduction of notions and techniques that are not necessary for the rest of the presentation. This book focuses essentially on questions about canard cycles. A precise definition of canard cycle will be given in Chap. 4, along with the more general notion of slow-fast cycle. Roughly speaking, a slow-fast cycle is a curve defined for  = 0, along which may bifurcate limit cycles for  > 0. Along a slow-fast cycle one can have repelling and attracting behaviour and a canard cycle is a slow-fast cycle simultaneously containing the two types of behaviour. The difficulties in the study of the dynamics near a canard cycle are related to the fact that the contractions near the attracting parts may be compensated by the dilatation near the repelling parts. Starting with the article [DR96], we have treated a lot of partial problems about bifurcations of canard cycles in a series of papers ([DR01b, DMD08, DMDR11] to name just a few). The more simple ones concern the cyclicity, that is the upper bound on the number of bifurcating limit cycles but we have also treated bifurcation problems. These studies gave us the opportunity to introduce and develop in successive steps different technical tools. The most important one is the (family) blow-up, a technique already used in the first paper of the series [DR96]. Using

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this technique of blow-up, results have been obtained to study in a detailed way the dynamics near degenerate points, like jump points and turning points. Two remarks can be made about these papers. First, the results were often presented under restrictive conditions, like for equations of Liénard type, even if later it became clear that such restrictions were not necessary. Second, many results, like for example those concerning the transition maps, are spread over several papers, often not stated and proved in an optimal way. These observations motivated us for writing this book. In this book a lot is new with respect to the currently existing literature, including our former papers on the subject. The main object we work with, the smooth family of slow-fast vector fields, is a new and quite general notion, largely extending the specific vector fields expressed by a traditional standard form as in (2). In Part I there is a systematic introduction of the most important basic notions, stressing their intrinsic nature and their invariance under smooth equivalences. Everything is explained in the general framework of smooth slow-fast families of vector fields on smooth manifolds. Part II is a technical and theoretical part, self contained and almost totally new. Besides an exhaustive description of the blow-up construction, it contains results on centre manifolds and local normal forms, stressing their smoothness, it presents an interesting algebra of functions that are smooth on specific monomials and finally contains a proof that the local transition maps, that will play an essential role in Part III, belong to this algebra. The smoothness property of these maps and of subsequently derived notions is an important result, inducing among other things, the precise form of their asymptotic expansion. Such an expression can be used as a practical ansatz for accurate calculations. Part III deals with theorems that rely on the results obtained in Part II. Among them we present the study of the generic transitory canard cycles. These proofs haven’t been published yet. We end this part by describing a number of interesting open problems that will hopefully attract the attention of a next generation of mathematicians to singular perturbation problems. In writing this book we had two purposes in mind: on one hand providing precise definitions and detailed proofs, on the other hand presenting precise results and practical tools that can be relied on. Although essentially restricting to periodic orbits and their bifurcations it is clear that many results in the book can serve to the treatment of other problems too. We now present a detailed plan of the book.

Part I Part I contains an overview of definitions and results that can be obtained with standard techniques from geometric singular perturbation theory. Many of the results have been described with alternative methods as well, albeit not in the generality from this book and not with the amount of detail as in here. The focus in

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Part I lies on a general definition of the slow-fast vector fields, on the determination of periodic orbits of slow-fast type and their limits as  → 0, the slow-fast cycles. Geometric singular perturbation theory is used only in a naive sense by assuming that orbits for small  > 0 are close to objects derived from two limiting systems: the fast vector field and the slow vector field. In this part of the book, no blow-up is necessary, and the results are confined to proving either the absence of periodic orbits, the presence of at least or at most one periodic orbit near a given slow-fast cycle. In Chap. 1 we define the notion of a family slow-fast vector fields (shortly called slow-fast systems), and of a first type of reduced system, that is the fast system. We focus on deriving properties of slow-fast systems, such as normal hyperbolicity, contact points etc, in an intrinsic way. Contact points (jump points, turning points or more degenerate) can be distinguished from each other using intrinsic invariants, which is the subject of Chap. 2. We then proceed to discuss the second reduced system, that is . the family of slow vector fields, in Chap. 3. In these chapters we clearly indicate how to derive all intrinsic notions directly from the equations, or using local normal forms. Having defined the two limiting systems, the fast system and the slow system, we can now combine information from both, in the spirit of geometric singular perturbation theory, to present slow-fast cycles and the nearby orbits for  > 0, the relaxation oscillations. This is done in Chap. 4. In this chapter we distinguish canard cycles from common cycles and announce a first result on existence of relaxation oscillations near so-called strongly common cycles. The unicity of nearby relaxation oscillations is less trivial, and for that the notion of slow divergence integral becomes crucial. It is defined in an intrinsic way in Chap. 5, and its basic properties are proved in detail. In Chap. 6 we discuss two so-called generic breaking mechanisms. Breaking mechanisms lead to a way of proving the presence of one or more canard cycles, by matching orbits along attracting branches of the critical curve to orbits along repelling branches of the critical curve. The presence of a breaking parameter is essential in that case. We finish Part I by presenting some known results on canard cycles in Chap. 7 that relate to the topic of this book and that can be stated without introducing the blow-up. The presentation in that chapter will be done without proofs, since many of the results will be proven in a unified way in Part III, and whenever it is not the case, a reference to the literature is made.

Part II While the material and techniques in Part I are rather elementary, more sophisticated techniques are necessary when it comes to treating slow-fast cycles of canard type passing near singular contact points, and slow-fast cycles that undergo a change of stability (leading to multiple nearby relaxation oscillations). In Part II, no results on slow-fast cycles will be presented, but the necessary ingredients, needed for fully understanding the dynamics near canard cycles, are derived in this part. The reader who is only interested in applying the results might very well skip this part and

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continue reading Part III. The reader who wants to learn the fine details of geometric singular perturbation theory in the plane will benefit from reading this part. An essential ingredient that was added to the standard geometric singular perturbation theory is the technique of family blow-up. While in essence it is just a change of coordinates to a weighted spherical coordinate system around contact points, it has been a breakthrough in the treatment of slow-fast systems. We review the technique in Chap. 8. From Part II on, we will adapt a three-dimensional vision on slow-fast systems on a surface: the vector field is considered on a lifted phase space, where the -axis adds an extra dimension. As such, -families of orbits are gathered in invariant manifolds in the extended phase space. At points where the invariant manifolds are close to the critical curve, they will actually form centre manifolds of the singular points of the critical curve. This is shown in Chap. 9. In fact, in the spirit of presenting a text that is largely self-contained, we present and prove a version of the centre manifold theorem adapted to our context. It gives in fact slightly smoother results than the general centre manifold theorem. Along centre manifolds we can now proceed to defining simple local normal forms near all points of the slow-fast cycles, both in the original phase space and in the blow-up space. A unified proof for the existence of all these smooth local normal forms is given in Chap. 10. The local normal forms will be used to define local transition maps (which will form the building blocks for the later study of the first return map near slow-fast cycles). In the expressions of the local transition maps, some nonsmooth elements (like  log ) will appear. Instead of reducing the smoothness in the statements by saying that such terms are just C0 , we prefer to keep a higher degree of detail and we will show that functions appearing in the expressions of the transition maps are smooth in (,  log , . . . ). The algebra of functions that are smooth w.r.t. (generalized) monomials (like  log ) is discussed in detail in Chap. 11, paving the way to discussing the local transition maps in Chap. 12.

Part III Using the techniques presented in Parts I and II, shorter and more unified proofs can be given to many of the known results on canard cycles. While it is impossible to review all results, we do present a large class of results in Chap. 13. We essentially restrict to canard cycles with one layer using a generic breaking mechanism, either a Hopf breaking mechanism or a jump breaking mechanism. We discuss the unicity or multiplicity of the limit cycles, together with their stability. The key idea is to show that bifurcation diagrams of the number and stability of limit cycles can be derived from bifurcation diagrams related to the zeros of the slow divergence integral. From the information in this section, one can explain the well-known canard explosion in a generic situation: it is the fast growth of amplitude of the periodic orbits as a parameter is varied. In the Van der Pol system for example, the cycles are “born” in a Hopf bifurcation near the contact point, and grow as canards without head until they make the transition to canards with head, they continue growing

Preface

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until a full-sized relaxation oscillation is reached. The canard explosion in the Van der Pol system is the most generic one we can think of, and this for several reasons. First, the contact point is a slow-fast Hopf point of codimension 1; the topic of birth of canard cycles for more degenerate contact points is done in [DR09]. Second, the slow divergence integrals along the involved slow-fast cycles are never zero, implying the absence of saddle-nodes of limit cycles or changes of stability. Third, the transition from canard without head to canard with head involves a passage through a generic jump point and the slow divergence is nonzero. In this book, “Canard Cycles from Birth to Transition”, we describe in a very general context what happens during the transition from canard without head to canard with head. We in fact want to know what happens at the boundary of a layer of canards. As the canard cycle shrinks, it reaches one end of the layer and the study of the boundary at this end amounts in the Hopf case to studying the birth of canard cycles. On the other hand, as the canard cycle increases in amplitude, it can ultimately experience a transition from one shape to another shape. A difficulty arising during this transition is the piecewise-smooth nature of the slow divergence integral. As this integral is non-smooth at the transition, it is no longer obvious how bifurcation diagrams for the number of zeros of this function relate to bifurcation diagrams of limit cycles. Chapter 14 deals with a first kind of transition. In essence, it contains as a subcase the “Van der Pol-like” transition from canard without head to canard with head, though this time under the assumption that the slow divergence integral changes sign exactly at the transition point. We prove that up to 2 or 3 limit cycles can appear near the transition (depending on the subcase), and that the slow divergence integral does predict the correct answer on the number of limit cycles despite of its non-smooth behaviour. The obtained results are new and a deep understanding of the blow-up procedure is necessary to deal with this situation. Indeed, in many of the previous topics, the blow-up can be seen as a rescaling where the most important phenomena appear in one chart of the rescaling, the family chart. The other charts, called phase-directional rescaling or matching charts, simply connect the information from the system in rescaled coordinates to the information from the system in original coordinates. In the study of transitions, the most delicate behaviour occurs exactly at the boundary between the two coordinate systems. In other words, an in-depth study of the phase-directional rescaling coordinates is necessary. Chapter 15 deals with a second kind of transition. It may occur for example in a layer of headless canards. At some height in the layer of canards, the fast orbit hits a generic jump point in its interior. The analysis is more difficult: while codimension k canard cycles may appear in any layer, in this case, a codimension k canard cycle can also occur at the transition point. This phenomenon complicates the study. We present an upper bound for the number of limit cycles near the transition point. We end the book by discussing some open problems in Chap. 16. Taking into account the unification, generalization and simplification worked out in this book, its title also wants to emphasize the passage to greater accessibility to the different results and techniques that we present.

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Guide for the Reader 1. All along the text we will use indifferently the notations G(u, w) and Gw (u) for a function of (u, w). The second notation is used when we want to emphasize that u is considered as the true variable and w as a parameter. The first notation will be preferred when we want to treat equally u and w (for instance, in order to compute partial derivatives. In general, u will be a phase variable as (x, y) ∈ R2 or z ∈ R and w will be a parameter as (, λ). 2. We will mostly write the complete list of variables entering in braces: f (x, y, λ, ) and so on, making our notations perhaps a bit heavy. The justification for this “apparent rigor” is that our method relies on a lot of changes of variables, based on the use of normal forms, blowing-up, and other operations. Nevertheless, it is crucial to keep control of the intermediate variables, in order to be able at the end, to return to the initial variables in which we want to state the results. Of course, the reader is not obliged to attach too much attention at the precise variables used during the successive computations. It might suffice to just look at the general form of the expressions. Diepenbeek, Belgium; Dijon, France April 2021

Peter De Maesschalck Freddy Dumortier Robert Roussarie

Acknowledgements

It took us a long time to write the book. It all started in 2012 when we got the possibility to intensively work together on transitory canard cycles at the Flemish Academic Centre, organized by the Royal Flemish Academy of Belgium for Science and the Arts (KVAB). We want to thank the KVAB for this opportunity. We also want to thank the University of Hasselt for the many times that we could meet during the preparation of the book.

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Part I 1

Basic Notions

Basic Definitions and Notions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Slow–Fast Families of Vector Fields . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Examples Where More Than One Admissible Expression Is Needed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Normally Hyperbolic Versus Contact Points. . . .. . . . . . . . . . . . . . . . . . . .

3 3 5 9 10

2

Local Invariants and Normal Forms . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Normal Forms Near Contact Points . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Invariants at Contact Points . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Example.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Remarks About Contact Points . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Invariants at Normally Hyperbolic Points . . . . . . .. . . . . . . . . . . . . . . . . . . .

15 15 17 20 22 25

3

The Slow Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Differential 1-Forms Along the Critical Curve .. . . . . . . . . . . . . . . . . . . . 3.3 Calculating Slow Vector Fields . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 The Slow Vector Field Near Contact Points . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Slow Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

27 27 31 32 34 37

4

Slow–Fast Cycles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Elementary Slow–Fast Segments .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Regular Common Cycles . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Canard Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Ordinary Canard Cycles . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Transitory Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.1 Singular Points in the Slow Vector Field: Transition to Singular Homoclinic .. . . .. . . . . . . . . . . . . . . . . . . .

39 39 41 44 46 48 50 50

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4.6.2 4.6.3 5

6

7

Loss of Hyperbolicity on One of the Two Branches in a Layer . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Birth of Canard Cycles . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 52

The Slow Divergence Integral .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Definition and Intrinsic Nature of Slow Divergence Integral . . . . . . 5.3 Invariance of the Slow Divergence Integral Under Equivalences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Slow Divergence Integral Near Singularities of the Slow Vector Field .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Slow Divergence Integral Near Contact Points .. . . . . . . . . . . . . . . . . . . . 5.6 Slow Divergence Integral of a Slow–Fast Cycle . . . . . . . . . . . . . . . . . . . . 5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7.1 Van der Pol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7.2 Some Canards in Quartic Liénard Systems. . . . . . . . . . . . . . . . 5.7.3 Zeros in the Slow Divergence Integral.. . . . . . . . . . . . . . . . . . . .

53 53 55

Breaking Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Normal Forms for Generic Jump Points and Generic Turning Points .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Generic Jump Breaking Mechanism . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Generic Hopf Breaking Mechanism .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Formal Power Series Methods for the Generic Hopf Breaking Mechanism.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Generic Breaking Mechanisms . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Other Breaking Mechanisms.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63

Overview of Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Periodic Orbits Near Common Cycles . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.1 Existence of Periodic Orbits Near Common Cycles . . . . . . 7.1.2 Multiple Periodic Orbits Near Common Cycles . . . . . . . . . . 7.2 Unicity of Periodic Orbits Near Unbalanced Canard Cycles.. . . . . . 7.3 Existence of Periodic Orbits Near Ordinary Canard Cycles . . . . . . . 7.4 Entry–Exit Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Multiple Periodic Orbits in Layers . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Contact Points of Higher Singularity Order Or Contact Order.. . . . 7.7 Canard Cycles with Singularities in the Slow Vector Field . . . . . . . . 7.8 Multi-Layer Canard Cycles . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.8.1 Two-Layer Canard Cycles and Their Transitory Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.8.2 More Than Two Layers.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.9 Birth of Canard Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.9.1 Birth of Canards in Liénard Systems . .. . . . . . . . . . . . . . . . . . . . 7.9.2 The Conjecture.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

56 57 58 59 60 60 61 62

63 64 65 67 68 69 70 73 73 75 76 76 78 81 84 86 87 88 88 90 90 91 92

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7.9.3 7.9.4 7.9.5

Part II 8

9

The Infinite Codimension Case . . . . . . . .. . . . . . . . . . . . . . . . . . . . Birth of Canard Cycles for the Slow–Fast Bogdanov–Takens Singularity . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Birth of Canard Cycles for More Degenerate Contact Points . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 94

Technical Tools

Blow-up of Contact Points .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Blow-up Procedure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Blow-up of a Generic Jump Point . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Blow-up of Regular Contact Points . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 The Saddle s− . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 The Saddle s+ . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.3 The Semi-Hyperbolic Points . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Blow-up of a Generic Turning Point . . . . . . . . . . . .. .√ .................. 8.4.1 Blow-up of the Turning Point for |a|   . . . . . . . . . . . . . . 8.4.2 Asymptotic Expansions in the Blow-up of the Hopf Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Global Aspects in the Blow-up of Contact Points . . . . . . . . . . . . . . . . . . 8.5.1 Closed Form Expressions for the Orbits on the Blow-Up Locus of the Generic Jump Point . . . . . . . . . . . . . . . 8.5.2 Passage Time in the Blow-up of Jump Points .. . . . . . . . . . . . 8.5.3 Separatrices on the Blow-up Locus of the Generic Jump Point .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5.4 Regular Splitting of Separatrices in the Blow-up of a Generic Turning Point . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5.5 Hopf Breaking Mechanism Revisited .. . . . . . . . . . . . . . . . . . . . 8.6 From the Hopf Bifurcation to the Polycycle and the Birth of Canards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

97 97 101 106 108 109 111 113 117 118 119 119 123 126 127 128 129

Center Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Ck -Invariant Manifolds for Diffeomorphisms . .. . . . . . . . . . . . . . . . . . . . 9.2 Ck -Invariant Manifolds for Vector Fields . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Smooth Invariant Manifolds in Slow–Fast Systems . . . . . . . . . . . . . . . . 9.3.1 The Case of a Closed Critical Curve. . .. . . . . . . . . . . . . . . . . . . . 9.3.2 The Case of a Closed Critical Interval .. . . . . . . . . . . . . . . . . . . . 9.3.3 The Case of a Critical Semi-Hyperbolic Point . . . . . . . . . . . . 9.3.4 The Case of Singularities of the Slow Vector Field . . . . . . .

131 131 141 144 144 146 147 151

10 Normal Forms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.1 The Path Method . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.2 First Order Differential Equation . . . . . .. . . . . . . . . . . . . . . . . . . .

153 154 154 155

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10.2 Regular Points of the Critical Curve.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 The Formal Solution .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 The Semi-Formal Solution . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.3 The Final Step . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Semi-Hyperbolic Points in the Blow-up Locus .. . . . . . . . . . . . . . . . . . . . 10.3.1 The Formal Solution .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.2 The Semi-Formal Solution . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.3 The Final Step . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Construction of Center Manifolds .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.1 The Case of a Regular Interval .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.2 The Case of a Semi-Hyperbolic Point .. . . . . . . . . . . . . . . . . . . . 10.4.3 Intervals Ending at a Semi-Hyperbolic Point . . . . . . . . . . . . . 10.5 Hyperbolic Saddle Points in the Blow-up Locus . . . . . . . . . . . . . . . . . . . 10.5.1 Resonant Monomial Vectors uα v β y γ ∂y . . . . . . . . . . . . . . . . . . . 10.5.2 Formal Normal Form .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5.3 Reducing to a Differential Equation on Flat Functions .. . 10.5.4 Solving the Differential Equation on Flat Functions.. . . . .

159 161 164 167 168 169 172 177 177 177 180 186 187 188 189 191 192

11 Smooth Functions on Admissible Monomials and More .. . . . . . . . . . . . . . 11.1 Admissible Monomials and Functions in Admissible Monomials .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Derivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Counting the Number of Roots . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Asymptotically Smooth Functions in Admissible Monomials .. . . . 11.5 Functions of Exponentially Flat Type . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.5.1 Some Basic Properties of the Exponential Term .. . . . . . . . . 11.5.2 Coherence of Definition 11.9 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.5.3 Composition of Families of Diffeomorphisms of Exponentially Flat Type .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

193

Part III

193 196 198 202 203 204 208 210

Results and Open Problems

12 Local Transition Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 Transition Along an Arc of Regular Points of the Slow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1.1 Transition in a Normal Form Chart . . . .. . . . . . . . . . . . . . . . . . . . 12.1.2 General Expressions for Regular Transitions . . . . . . . . . . . . . 12.1.3 Properties of Transitions Along Regular Arcs . . . . . . . . . . . . 12.2 Transition Near Semi-Hyperbolic Points. . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.1 Equation for the Transition Component Z˜ . . . . . . . . . . . . . . . . 12.2.2 A Simple Case . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.3 Preparing the Function G . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.4 Estimates for the Integral I in (12.8) . .. . . . . . . . . . . . . . . . . . . . 12.2.5 Theorems for the Transition Map . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.6 Transitions Near Particular Semi-Hyperbolic Points . . . . .

215 216 217 220 224 226 227 228 229 232 235 236

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12.3 Transition Near Hyperbolic Saddle Points . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3.1 The Transition Map in the Case p = 1 . . . . . . . . . . . . . . . . . . . . 12.3.2 Transition in the General Case (for p ∈ N) . . . . . . . . . . . . . . . 12.3.3 The Saddle Points of Chap. 8 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.4 Transition at a Jump Point . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.5 Transition Along an Attracting Sequence .. . . . . .. . . . . . . . . . . . . . . . . . . . 12.6 Transition Along a Hopf Attracting Sequence ... . . . . . . . . . . . . . . . . . . .

239 240 242 243 246 254 258

13 Ordinary Canard Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 Basic Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2.1 Difference Functions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2.2 Tubular Neighborhood of the Canard Cycle . . . . . . . . . . . . . . 13.3 Results of Bifurcation .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.1 A Mild Preparation for Eq. (13.11) . . . .. . . . . . . . . . . . . . . . . . . . 13.3.2 The Canard Phenomenon.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.3 Formal Power Series Expansion of the Canard Surface for Generic Hopf Breaking Mechanisms .. . . . . . . . 13.3.4 Canard Explosion, Flying Canard, and Sitting Canards.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.5 Counting the Limit Cycles over a Whole Layer Strip . . . . 13.3.6 Limit Cycles and Bifurcations in a Rescaled Layer .. . . . . . 13.3.7 Limit Cycles Outside the Rescaled Layer . . . . . . . . . . . . . . . . .

267 267 268 268 273 274 275 276

278 282 285 298

14 Transitory Canard Cycles With Slow–fast Passage Through a Jump Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Statement of the Results . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2 Behavior of the Slow Divergence Integral . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.1 The Slow Divergence Integrals J , K, and L . . . . . . . . . . . . . . 14.2.2 Slow Divergence Integrals of Slow–fast Cycles . . . . . . . . . . 14.3 Local Study Near the Jump Point q . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3.1 Blowing Up the Jump Point .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3.2 Transition at the Saddle Point s . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3.3 Transition at the Semi-Hyperbolic Point q1 . . . . . . . . . . . . . . . 14.3.4 Transition at the Semi-Hyperbolic Point q2 . . . . . . . . . . . . . . . 14.4 Transition Maps Outside the Jump Point q . . . . .. . . . . . . . . . . . . . . . . . . . 14.5 Cyclicity of the Transitory Canard Cycle 0 . . .. . . . . . . . . . . . . . . . . . . . 14.5.1 The Displacement Function η . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.2 Structure of the Transition Maps Toward T . . . . . . . . . . . . . . . 14.5.3 Covering of the Section C . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.4 Unique Maximum Properties . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.5 Proof of Theorem 14.1 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.6 Limit Cycles and their Unfoldings . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.6.1 Saddle-Node Bifurcation of Limit Cycles in Case I . . . . . . 14.6.2 Proof of Theorem 14.5 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

305 306 308 308 310 311 312 314 315 316 318 320 320 321 324 325 336 337 339 340

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15 Transitory Canard Cycles with Fast–fast Passage Through a Jump Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Blow-up of the Jump Point . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Transitions Near the Singular Points of X¯ λ . . . . .. . . . . . . . . . . . . . . . . . . . 15.3.1 Transition at the Saddle Points s± . . . . .. . . . . . . . . . . . . . . . . . . . 15.3.2 Transition at the Semi-Hyperbolic Point q . . . . . . . . . . . . . . . . 15.4 Regular Transitions for X¯ λ Along the Blow-up Locus . . . . . . . . . . . . . 15.4.1 Regular Transition Near the Interior of the Blow-up Locus.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4.2 Regular Transition Near the Boundary of the Blow-up Locus.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5 Cyclicity of the Canard Cycle . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.1 The Displacement Function η . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.2 Normal Form for Transitions Toward T . . . . . . . . . . . . . . . . . . . 15.5.3 From Global to Local Displacement Functions . . . . . . . . . . . 15.5.4 Proof of Theorem 15.2 for m η . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.5 Proof of Theorem 15.3 for dη . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.6 Proof of Theorem 15.3 for uη . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.6 Proof of the Main Theorem .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16 Outlook and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2 Codimension.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2.1 Codimension of Contact Points . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2.2 Codimension of Jumps Between Contact Points .. . . . . . . . . 16.2.3 Codimension of Singularities of the Slow Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2.4 Codimension of a Slow–fast Unfolding . . . . . . . . . . . . . . . . . . . 16.2.5 Codimension of a Canard Cycle . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 Desingularization of Unfoldings.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3.1 Generic Unfoldings .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3.2 Existence of Versal Unfoldings . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3.3 Blowing Up of Versal Unfoldings . . . . .. . . . . . . . . . . . . . . . . . . . 16.4 Analytic Slow–fast Unfoldings of Infinite Codimension .. . . . . . . . . . 16.5 The Question of Finite Cyclicity for Canard Cycles. . . . . . . . . . . . . . . . 16.6 Disorienting Canard Cycles. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7 Recapitulation of Open Problems and Questions .. . . . . . . . . . . . . . . . . . 16.7.1 Questions About Codimension . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7.2 Questions About Versal Unfoldings and their Desingularization . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7.3 Questions About Asymptotic Properties . . . . . . . . . . . . . . . . . .

343 343 348 350 351 353 354 354 358 360 360 362 363 366 369 375 379 381 381 383 383 385 385 386 386 389 389 392 393 395 396 397 398 398 399 399

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16.7.4 Questions About Analytic Unfoldings and Canard Cycles . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 400 16.7.5 Questions About the Finite Cyclicity Conjecture . . . . . . . . . 400 16.7.6 Questions About Disorienting Canard Cycles. . . . . . . . . . . . . 400 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 401 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 407

Part I

Basic Notions

Chapter 1

Basic Definitions and Notions

1.1 Slow–Fast Families of Vector Fields The results we want to present hold on an arbitrary smooth orientable surface M and some results even on non-orientable surfaces. For sure the basic definitions can be given on general smooth surfaces. Since we only obtain results that are valid in the neighborhood of some piecewise smooth simple closed curve, one can as well suppose to work on an annulus lying in a plane and exceptionally on a Möbius band. On M we consider a smooth family of vector fields X,λ , defined for  ∈ [0, 1 ] (for a given 1 > 0) and for λ ∈ , with  a subset of an Euclidean space Rp . The parameter  is hence a small one-dimensional parameter, while λ is a p-dimensional parameter, which will often be taken in the neighborhood of some specific value λ0 ∈ Rp . Definition 1.1 (Slow–Fast Family of Vector Fields) A Slow fast family of vector fields@slow–fast family of vector fields with singular parameter , or shortly called an -Slow fast system@slow–fast system, on a smooth surface M is a smooth family of vector fields X,λ , with X,λ = X0,λ + Qλ + O( 2 ).

(1.1)

The λ-family of Limiting vector fields X0,λ is supposed to have a set Sλ of (nonisolated) singularities (called the Critical set) and each singularity has an open neighborhood U on which X0,λ |U = Fλ Z λ for some smooth family of functions Fλ and a smooth family of vector fields Zλ , where for each λ the following properties hold: 1. Zλ is a vector field without singularities, and 2. Fλ is a function with a regular set of zeros: dFλ (p) = 0 for p ∈ {Fλ = 0}.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_1

3

4

1 Basic Definitions and Notions

Each such triple {U, Zλ , Fλ } is called an admissible (local) expression for X0,λ at a singularity. Remarks 1.1 1. In Definition 1.1, the open sets are not necessarily charts of the manifold M. In fact, it could happen that U = M. 2. Related to each orbit of the family of vector fields X,λ , there is a natural notion of time, defined up to an ((, λ)-dependent) additive constant. In general one makes a definite choice, independent of the orbit, that one represents by t and calls the fast time. This is also what we are going to do in this book. In any case the additive constant, as is usual in working with vector fields, will be irrelevant. 3. In many examples of slow–fast systems, we can work with a unique admissible expression {M, Zλ , Fλ } in the sense that we can write X0,λ = Fλ Z λ for a family of functions Fλ and a family of regular vector fields Zλ defined globally on M. In the general case, even the general orientable case, it is not always possible to restrict to one expression; see Sect. 1.3 for counterexamples. As a direct consequence of the existence of an open covering of M by admissible local expressions, the limiting vector fields X0,λ have, for each λ, the following global properties, which are intrinsic, i.e. are defined in terms of X0,λ alone and in particular have a coordinate-free definition: 1. The set of zeros, Sλ = {p ∈ M | X0,λ (p) = 0}, is a one-dimensional submanifold of M that is called the critical curve. We assume of course that Sλ = ∅. The critical curve does not need to be connected. If {U, Zλ , Fλ } is an admissible expression, then Sλ ∩ U = {Fλ = 0}. 2. There is a one-dimensional foliation on M, smoothly depending on λ, which is tangent to the vector field Zλ in each admissible expression {U, Zλ , Fλ }. We will f call it the fast foliation of X0,λ and denote it by Fλ ; we denote by λ,p the leaf through any p ∈ M. The flow of X0,λ on M \ Sλ is called the fast flow or also f the fast dynamics. The orbits of the fast flow are contained in the leaves of Fλ . The techniques that we will present permit to study generalizations of the slow– fast systems considered in Definition 1.1. We could accept the presence of critical curves Sλ with transverse crossings or even with more degenerate critical points. We will however not do it in this book. We will, on the contrary, even impose some further restrictions that we will specify in a number of assumptions, starting in Sect. 1.4. We will impose extra conditions as well on X0,λ as on Qλ in order to avoid situations of “infinite codimension.” For analytic systems, the extra conditions f merely express that {Fλ = 0} is not a leaf of the foliation Fλ and that Qλ is not identically zero on {Fλ = 0}. In a more general setting, including slow–fast systems beyond the ones described in Definition 1.1, one sometimes uses the name “Tame slow–fast family of vector fields” for the systems fulfilling Definition 1.1. Since the book only deals with such (tame) systems, we prefer not to add the adjective “tame.”

1.2 Examples

5

In Sects. 1.2 and 1.3, we show examples of systems that fulfill Definition 1.1 as well as simple examples of non-tame systems.

1.2 Examples (1) Standard form Slow–fast families of vector fields are commonly presented in a “Standard form”  x˙ = Fλ (x, y) + Fλ1 (x, y) + O( 2 ), X,λ : (1.2) y˙ = Gλ (x, y) + O( 2 ). The variable x can be considered as the fast variable, while y reacts slowly when   1, see Fig. 1.1. In this form, one possible vector field Zλ in Definition 1.1 is simply the horizontal flow box {x˙ = 1, y˙ = 0}. (Note that Zλ is not unique.) (2) Slow formulation Sometimes in the literature, slow–fast systems of differential equations are presented in the form  dx  ds = Fλ (x, y) + Fλ1 (x, y) + O( 2 ), (1.3) dy ds = Gλ (x, y) + O(). Such equations are obtained from (1.2) after rescaling time using s = t. From this expression it is clear that x is called the fast variable: the speed of evolution in time s is O( 1 ). We refer to Chap. 3 for the determination of a so-called slow vector field using the “Slow time” s. (3) Liénard systems As a particular case of (1.2), we will often illustrate definitions and results for families of (generalized) slow–fast Liénard equations  x˙ = y − fλ (x), X,λ : (1.4) y˙ = gλ (x). The critical curve is a graph y = fλ (x), and like above the fast foliation is given by horizontal lines. Observe that the extremes of fλ play a special role in two ways: geometrically, at those points the critical curve is tangent to the fast foliation. Second, algebraically, the Jacobian of the linearization of X0,λ at such y

y

x Fig. 1.1 Dynamics of (1.2) for  = 0 (left) and 0 <   1 (right)

x

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1 Basic Definitions and Notions

points,  Jλ (x) =

 −fλ (x) 1 , 0 0

has two zero eigenvalues (and is nilpotent). More about this is given in the next section. We recall that (1.4) is called a classical slow–fast Liénard equation when gλ (x) = −x. (3) Closed critical curve Slow–fast families of vector fields need not be presented in a standard form (see also [Wec20] and [JW20]). In fact it is not always possible to give a global expression in a standard form. Consider for example  X,λ :

x˙ = −(y + λx)Fλ (x, y) + g(x, y) + O( 2 ), y˙ = (x + λy)Fλ (x, y) + h(x, y) + O( 2 ).

(1.5)

The zero set of Fλ is the critical curve (consider for example an ellipse like in Fig. 1.2), and the fast vector field Zλ from Definition 1.1 is the linear vector field x˙ = −y + λx, y˙ = x + λy. The origin is a singular point of Zλ , so X,λ is a well-defined slow–fast family of vector fields on M = R2 \ {(0, 0)}. (4) Fast oscillations on the torus We consider a torus parameterized by θ, ϕ ∈ S 1 and also consider the following system that satisfies Definition 1.1:

Fig. 1.2 Nontrivial fast dynamics

ϕ

ϕ

2π

2π

θ 0

2π

θ 0

2π

Fig. 1.3 An example of a slow–fast system on a torus, dynamics for  = 0 (left) and a typical periodic orbit for 0 <   1 (right)

1.2 Examples

7



θ˙ = O(), ϕ˙ = Fλ (θ, ϕ) + O(),

see for example in Fig. 1.3. The critical curve is the zero set of Fλ , and the fast dynamics corresponds to oscillations along circles. While the local passage near any of the points of the critical curve can be studied using the techniques exposed in this book, we will no further consider this situation. More information on canards on tori can for example be found in [GI01, Sch11], and references therein. Aside from the above examples, we also present several singularly perturbed situations that will not be studied in this work, but that can be considered of slow– fast type. (5) Non-tame limiting vector fields We could consider slow–fast family with a more general limiting vector field X0,λ . Consider for example 

x˙ = y 2 − x 2 + O(), y˙ = O().

(1.6)

The critical curve is the union of two lines y = ±x, crossing each other at the origin. In terms of the notations of Definition 1.1, an admissible expression would be formed by a horizontal flow-box vector field, together with the multiplicative function Fλ = y 2 − x 2 . Condition (2), stating that dFλ (x, y) = 0 at points of the curve, is not satisfied at the origin. For many such systems, the techniques developed in this book still work, but there is no general desingularization theorem for slow–fast systems in the plane. f We could also consider cases where the fast foliation Fλ has isolated critical points located on Sλ or outside this set. Notice that these different possibilities, critical points for the set Sλ and singular points of Zλ , may occur simultaneously. Examples of such non-tame limiting vector fields are presented in Fig. 1.4. For a non-tame limiting vector field, it is always possible to continue working on f the complement of the singularities of Sλ and Fλ , as far as on this complement,

Fig. 1.4 Examples of non-tame limiting vector fields

8

1 Basic Definitions and Notions

the requirements of Definition 1.1 are satisfied. However, studying orbits that come close to both an isolated singular point and a branch of singular points is very interesting. But it requires both techniques from the study of slow–fast systems and techniques from regular perturbation theory. In this book, we have chosen not to explicitly deal with these cases. (6) Singularly perturbed differential equations A considerable amount of articles (see e.g. [Was65, Wal94, Ver05, CDRSS00, KC96], and so on) is devoted to the study of 

dy = Fλ (x, y, ). dx

(1.7)

While such systems do give rise to slow–fast systems in a general sense, we will not consider these equations in this book because the model does not allow periodic orbits. On top of that, the most interesting phenomena appear when ∂Fλ ∂y (x0 , 0, 0) = 0 for one or more points (x0 , 0). It can be verified that in such a case, condition (2) of Definition 1.1 cannot be satisfied. (It potentially corresponds to a normal crossing of two critical curves, in the non-generic setting where one of the critical curves is tangent to the fast vector field.) Despite the lack of attention to such systems here, many of the presented techniques can be and have been used in the study of canard solutions appearing in (1.7), see for example [DM08]. (7) Slow–fast systems on a Möbius band There is no need for the manifold to be orientable and in case of a non-orientable surface M, for example, a Möbius band (see Fig. 1.5). The critical curve in fact lies in an orientable part of M. One could study slow–fast cycles where a fast orbit connects the critical curve to itself via an orientation-reversing tour along the Möbius band. (8) Continuous slow–fast systems Systems like (1.4) could also be studied under the assumptions that f and g are only continuous (see e.g. [PPZ09]), thereby relying on topological techniques rather than techniques of invariant manifolds and blow-up. At the same time, the literature gives some attention to piecewise smooth slow–fast systems (see e.g. [RCG12]). In this book, we assume that all vector fields are smooth, and hence (non-smooth) continuous or piecewise smooth systems have been excluded from Definition 1.1. Fig. 1.5 Slow–fast system on a Möbius band

1.3 Examples Where More Than One Admissible Expression Is Needed

9

1.3 Examples Where More Than One Admissible Expression Is Needed Reviewing Definition 1.1 of a slow–fast family of vector fields, it can be observed that by using the context of manifolds, one allows changes of coordinates in the expressions of the differential equations by changing charts. One might wonder whether there is really a need of choosing admissible expressions locally. In other words, it might not be true that for every slow–fast family of vector fields, there exist a function Fλ and a vector field Zλ on the whole of the manifold M such that X0,λ = Fλ .Zλ , where Fλ and Zλ have the properties like in Definition 1.1. The answer is negative, even in case the surface M is orientable, as will be clear from the counterexamples in this section. We have included this section to show the need for generality in the definition, while the counterexamples themselves will not be referenced further in the book. In the language of foliations, we present examples of non-orientable foliations on an orientable manifold. As a first counterexample, we consider, on the space M = R2 \ {(0, 0)}, the smooth family of vector fields  X,λ :

x˙ = −x  + f (x, y, , λ), y˙ = x 2 + y 2 − y + g(x, y, , λ),

(1.8)

see Fig. 1.6. Suppose there exists a global admissible expression (M, Z, F ). The zero set of F should correspond to the critical curve, which is the half line H : {x = 0, y > 0}. In that case, y/F ˙ will experience a sign change when passing through H . The fibers of X0,λ through H fill the entire plane except for the negative y-axis, so this would imply that y/F ˙ has a fixed sign on the whole of {x > 0} and the opposite sign on the whole of {x < 0}. This leads to an inconsistency along the negative y-axis, where y/F ˙ is not supposed to change sign. The example does meet the conditions of Definition 1.1. Locally near all points of M, there exists an admissible expression. We remark that the inconsistency comes from the observation that a function on the cylinder cannot have a half line as the set of zeros (cylinder minus half line is homeomorphic to open disk). A second counterexample, i.e. a slow–fast family of vector fields where a single admissible expression is not possible, will not be described analytically, but only Fig. 1.6 Slow–fast family of vector fields (1.8)

y

x

10

1 Basic Definitions and Notions

?

?

Fig. 1.7 Slow–fast family of vector fields on a cylinder. The left and right boundaries are identified to form a cylinder in both pictures. Left: slow–fast dynamics for  = 0. Right: determination of the vector field Zλ between the dashed lines

geometrically, explaining the ideas on the figure. The benefit of this counterexample over the previous one is that slow–fast cycles are present here, while (1.8) does not have periodic orbits. Consider an open cylinder (annulus) minus a closed disc D, where the dynamics of the fast vector field together with the dynamics of the vector field Zλ in one choice of an admissible expression is shown in Fig. 1.7. The disc D is centered in the picture and is excluded because Zλ is not allowed to have a singular point. It is clear that an admissible expression is possible in a region U bounded by the dashed lines, and however it cannot be extended smoothly on the whole of the cylinder.

1.4 Normally Hyperbolic Versus Contact Points As made clear in the Liénard example (1.4), the linearization of the vector field at points of the critical curve can have one or two zero eigenvalues. (One zero eigenvalue with an eigenvector tangent to the critical curve is always present.) This leads to the following definition: Definition 1.2 (Normally Hyperbolic Point and Contact Point) A point p ∈ Sλ is called a Normally hyperbolic (respectively, Normally attracting or Normally repelling) point of X0,λ if the linear part of X0,λ at p has a nonzero (respectively, negative or positive) eigenvalue. We denote this eigenvalue by Vλ (p) = V (p, λ) and call it the transverse eigenvalue at p (the second eigenvalue is equal to 0 with eigenspace Tp Sλ ). A point p ∈ Sλ is called a Contact point when the linear part has two zero eigenvalues. We denote the set of contact points by Cλ . In the example shown in Fig. 1.1, all points are normally attracting. In the example in Fig. 1.2, there are both normally attracting and normally repelling parts of the ellipse, with four contact points in between (two of which are clearly visualized in the figure). For Liénard systems (1.4), contact points are found at (x0 , fλ (x0 )) with fλ (x0 ) = 0. As indicated in Sect. 1.2, it can be seen geometrically as points of contact between the leaves of the fast foliation and the critical curve, see Fig. 1.8.

1.4 Normally Hyperbolic Versus Contact Points

11

y

r.

p.

rep.

t at

re

t at

r.

.

r att

x x1

x2

x3 x4

Fig. 1.8 A Liénard system with critical curve y = fλ (x) and contact points (xj , fλ (xj )), j = 1, . . . , 4

Preliminary Assumption on Contact Points In this book we suppose that all contact points are isolated. (Because of Definition 1.1, each contact point is nilpotent and non-degenerate.) The full assumption on contact points will be formulated in Sect. 2.2. Remarks 1. If p ∈ Sλ \ Cλ , then the eigenspace of Vλ (p) is the tangent line Tp λ,p = δλ,p to the leaf λ,p . The hyperbolic invariant manifold at p ∈ Sλ \ Cλ is the union of p with the two orbits of the fast dynamics, which have p as limit point. It is f contained in the leaf λ,p of Fλ . 2. At a point p ∈ Sλ \ Cλ , the transverse eigenvalue is equal to the trace of D(X0,λ )(p). It is hence clearly an intrinsic notion. 3. The set Cλ could also be defined as the set of contact points between the curve f Sλ and the foliation Fλ . By the assumption, we hence suppose that such points of contact are isolated. 4. In the literature, alternative names for contact points are in use: jump points, turning points, singular Hopf points, and so on. We will review their meanings in Sect. 2.4. In the next lemma, we will show that in an admissible expression {U, Zλ , Fλ }, the transverse eigenvalue Vλ (p) is equal to dFλ [Zλ ](p). Before doing this, we first calculate what happens to the last quantity when we can replace the pair (Fλ , Zλ ) by a pair (Fλ , Zλ ) with Fλ = aλ Fλ and Zλ = Zaλλ , where each aλ is a nowhere-zero function. We have dFλ [Zλ ] =

daλ 1 1 d(aλ Fλ )[Zλ ] = (Fλ daλ +aλdFλ )[Zλ ] = Fλ [Zλ ]+dFλ [Zλ ]. aλ aλ aλ

Then, on Sλ ∩ U , we obtain that dFλ [Zλ ] = dFλ [Zλ ], so that this function does not depend on the choice of the pair (Zλ , Fλ ).

12

1 Basic Definitions and Notions

Lemma 1.1 Let {U, Zλ , Fλ } be an admissible expression and p ∈ (Sλ \ Cλ ) ∩ U . Then the nonzero eigenvalue is given by Vλ (p) = dFλ [Zλ ](p) = dFλ (p)[Zλ (p)].

(1.9)

Furthermore, the corresponding eigenspace is generated by the vector Zλ (p). Proof As all expressions in (1.9) are coordinate-free, we can choose any admissible expression around p to verify it. We choose local coordinates (x, y) around p where ∂ and p = (0, 0). We have that Zλ = ∂x ∂  ∂F ∂Fλ λ X0,λ = x (0, 0) + y (0, 0) + O(|x|2 + |y|2 )). ∂x ∂y ∂x Then, δp is the x-axis and Vλ (p) =

∂Fλ ∂x (0, 0)

= dFλ [Zλ ](p).



Choose now an admissible expression {U, Zλ , Fλ } containing a contact point p0 . At p0 , and for any choice of λ, the function Vλ is equal to 0, but the regularity properties of the function Vλ are not yet clear at p0 , at least not from the original definition. However, in U , the function p → dFλ [Zλ ](p) is defined and smooth on Sλ ∩ U , being zero at p0 (in fact, Cλ ∩ U = {p ∈ Sλ ∩ U | dFλ [Zλ ](p) = 0}). Then, a direct consequence of Lemma 1.1 is the following corollary: Lemma 1.2 The transverse eigenvalue function Vλ , which is smooth on Sλ \ Cλ , extends smoothly by 0 at the points of Cλ .  Before making further assumptions on the contact points and linking nice invariants to a slow–fast family at a contact point, we will now first check some ∂X invariance result on the family of vector fields Qλ = ∂,λ |=0 from Definition 1.1. Of course, for each λ, Qλ behaves as a vector field on M under action of diffeomorphisms of M. But the natural conjugacy for slow–fast systems is an action of -families of diffeomorphisms and in general Qλ is not invariant by a general action of that kind. We have just the following result, which fortunately is sufficient to give invariance of some forthcoming definitions: Lemma 1.3 Under action of an (, λ)-family of diffeomorphisms G,λ , and for any admissible local expression {U, Zλ , Fλ }, the vector field Qλ is transformed along Sλ into a vector field Qλ + gλ Zλ , where gλ is a smooth family of functions (more precisely, Qλ is transformed into (G0,λ )∗ (Qλ ) + gλ (G0,λ )∗ (Zλ ) along G0,λ (Sλ )). Proof If G,λ does not depend on , i.e. if G,λ ≡ G0,λ , we have that X,λ = Fλ Zλ + Qλ + O( 2 ) is transformed into (Fλ ◦ Gλ )(G0,λ )∗ (Zλ ) + (G0,λ )∗ (Qλ ) + O( 2 ), i.e. behaves as a vector field of M. Then, if G,λ is a general family of

1.4 Normally Hyperbolic Versus Contact Points

13

diffeomorphisms, we can replace it by the family G,λ ◦ (G0,λ )−1 , i.e. we can suppose that G0,λ = Id. To simplify the computation (which is a local one around each point of p), we will work in an arbitrary chart W of M: in such a chart, we can consider a vector field Y as a local map of M into R2 and the linear part dY (p) is well defined at each p ∈ W . On W , we can write G,λ = Id +Hλ + O( 2 ), where Hλ is a family of maps W → R2 . Taking into account that (G,λ )−1 = Id −Hλ + O( 2 ), we can expand (G,λ )∗ (X,λ ) at order one in : (G,λ )∗ (Fλ Zλ + Qλ + O( 2 )) = Fλ (Id +dHλ)[Zλ ](Id −Hλ ) + Qλ + O( 2 ). This gives (G,λ )∗ (Fλ Zλ + Qλ + O( 2 )) = Fλ Zλ + (Qλ + dFλ [Hλ ]Zλ − Fλ dZλ [Hλ ]) + O( 2 ).

As Fλ dZλ [Hλ ]|Sλ = 0, we obtain that Qλ is transformed into Qλ + dFλ [Hλ ]Zλ along Sλ . 

In conclusion of this section, we can notice that we have defined several Geometric invariants associated with X,λ (they are transformed by (, λ)-families of diffeomorphisms like X,λ and they are given in a coordinate-free way): the f critical one-dimensional manifold Sλ , the fast foliation Fλ , the set of contact points Cλ ⊂ Sλ , the transverse eigenvalue Vλ , the field δλ of transverse eigenspaces p ∈ Sλ \ Cλ → δλ (p), and the vector field Qλ |Sλ modulo Zλ , in any admissible local expression {U, Zλ , Fλ }. All the subsequent properties of the slow–fast system, and in particular some new geometric invariants, will be expressed through these first geometric invariants.

Chapter 2

Local Invariants and Normal Forms

2.1 Normal Forms Near Contact Points Before defining the local invariants, we recall, from [DMDR11], a normal form that can be used near a contact point. A nice property of this normal form is to be close to the general expression of a Liénard equation. (In fact in some cases, one can actually reduce to Liénard form, see [DM14].) We add the proof, for the sake of completeness. Since the definition of a slow–fast family of vector fields in Definition 1.1 is stated in terms of vector fields on manifolds, a change of coordinates to normal form does not change the vector field; it is just about choosing a different chart of the manifold. On top of a suitable change of coordinates, a time reparameterization is needed to obtain the normal form; we call it a normal form for equivalence (also known as orbital normal form). Proposition 2.1 Consider a smooth slow–fast system X,λ on a smooth surface M. Let p be a nilpotent contact point for a parameter value λ = λ0 . There exist smooth local coordinates (x, y) such that p = (0, 0), and in which, up to multiplication by a smooth strictly positive function, the system X,λ , for (, λ) ∼ (0, λ0 ) is written in the following normal form: 

x˙ = y− fλ (x) 

y˙ =  g(x, , λ) + y − fλ (x) h(x, y, , λ) ,

for smooth functions f, g, h and fλ0 (0) =

∂f ∂x (0, λ0 )

(2.1)

= 0.

Proof We will obtain the normal form (2.1) by a succession of several steps. We will assume that each successive coordinate system is chosen such that p = (0, 0). (1) Suppose that X0,λ has an expression as in (1.1), let us say X0,λ = Fλ (x, y)∂/∂x, and suppose that p = (0, 0). The point p is a nilpotent contact point, so © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_2

15

16

2 Local Invariants and Normal Forms ∂Fλ

∂Fλ

Fλ0 (0, 0) = 0, ∂x 0 (0, 0) = 0, and ∂y 0 (0, 0) = 0. Then, the critical curve is locally a graph {y = fλ (x)} for a smooth family of functions fλ (with dfλ0 dx (0) = 0), and one can write Fλ (x) = Uλ (x, y)(y − fλ (x)) for a smooth function U such that Uλ0 (0, 0) > 0. We will choose a neighborhood in which Uλ (x, y) > 0 for every (x, y, λ). After division by the function U , the system is locally written as  x˙ = y − fλ (x) + G1 (x, y, , λ) y˙ = G2 (x, y, , λ), for smooth functions G1 and G2 . (2) The map (x, y) → (x, y + G1 (x, y, , λ)) is a smooth family of diffeomorphisms for (x, y, , λ) ∼ (0, 0, 0, λ0 ). It defines a conjugacy, which brings the previous expression into  x˙ = y − fλ (x) 2 (x, y, , λ), y˙ =  G 2 . for a new smooth function G (3) As x and y − fλ (x) are independent functions at (x, y) = (0, 0), one can write

2 (x, y, , λ) = g(x, , λ) + y − fλ (x) h(x, y, , λ) for smooth functions g G and h. 

Remarks 2.1 (a) From the proof, it is clear that starting with whatever curve of zeros given by y = fλ (x) (e.g. y = x 2 ), the fλ (x) in the normal form is still the one we started with. (b) A similar proof implies that near a normally hyperbolically attracting point, X,λ is C∞ -equivalent to  x˙ = y− x 

(2.2) y˙ =  g(x, , λ) + y − x h(x, y, , λ) . A similar normal form with x˙ = y + x is obtained at repelling points. (c) In a normal form (2.1), the limiting vector field is equal to  x˙ = y − fλ (x) y˙ = 0. This system of equations, in which y can be considered as a parameter, is essentially called a layer equation, since the layers {y = const} constitute f the foliation Fλ . The eigenspaces δλ are tangent to it. We also have Sλ = {(x, y)|y = fλ (x)} and Cλ = {(0, 0)}. 

2.2 Invariants at Contact Points

17

In this book we will merely consider problems related to the number of limit cycles and the bifurcations they undergo. Time will hence not be an essential ingredient so that we can work with C∞ -equivalences when making a local study. Of course it is essential that all the notions we use and all the results we obtain be invariant for (smooth) equivalence, although we might define them in well chosen local coordinates, like the ones leading to the expressions (2.1) and (2.2). We will pay attention to this when necessary.

2.2 Invariants at Contact Points We can already observe that some geometric invariants from Chap. 1 clearly remain f invariant under smooth equivalences, like Sλ , Cλ , Fλ , and δλ . This is however not the case for Vλ and Qλ . We will now introduce some new invariants that remain invariant under equivalences. We start by presenting them in a simple but a priori not intrinsic way. Consider any normal form (2.1) near a contact point p of a slow–fast system X,λ for the value λ = λ0 . In [DMDR11], the order nλ0 (p) = Ord |0 (fλ0 ) (≥ 2) is called the contact order, and the order sλ0 (p) = Ord |0 (g0,λ0 ) (≥ 0) is called the singularity order. In [DMDR11] can be found a proof that the two notions of order do not depend on the chosen normal form; the proof is obtained by direct calculation. We will not repeat this proof here, but we will introduce alternative expressions for nλ (p) and sλ (p), which are clearly invariant under equivalences and that will form the basis of an intrinsic definition. In order to do so, we will use the following trivial result. Lemma 2.1 Let X,λ = X0,λ + Qλ + O( 2 ), and let {U, Zλ , Fλ } be an admissible expression. Any area form  on U defines a function (Qλ , Zλ )|Sλ ∩U : p ∈ Sλ ∩ U → (Qλ , Zλ )(p). This function is well defined up to a smooth multiplicative nowhere-zero function, when we change the triple (Zλ , Fλ , ) by another triple (Zλ , Fλ ,  ) with X0,λ |U = Fλ Zλ = Fλ Zλ . As a consequence, the number of zeros of (Qλ , Zλ )|Sλ ∩U , counted with their multiplicity, is a geometric invariant. Proof Suppose that  = α and Z = bZ, where α and β are smooth nowherezero functions. Then, on U ,  (Qλ , Zλ ) = αβ(Qλ , Zλ ), and αβ is a smooth nowhere-zero function. Now, if g and g are two smooth functions on an interval I ⊂ R such that g = hg with h a smooth nowhere-zero

18

2 Local Invariants and Normal Forms

function, it is an easy exercise to verify that g and g have on I , the same zeros counted with their multiplicity. The claim in the statement follows. 

We will now present the intrinsic definition of singularity order and contact order of a contact point; after the definition, we will verify that it coincides with the definition through the use of the normal form (2.1) from [DMDR11]. A practical example will be outlined in Sect. 2.3. Definition 2.1 (Invariants at a Contact Point) Let X,λ be a slow–fast family of vector fields on M, and let p be a contact point for the value λ0 . Then, 1. The Contact order nλ0 (p) of p is equal to the contact at p between Sλ0 and the f leaf λ0 ,p through p of the foliation Fλ0 . 2. Let {U, Zλ , Fλ } be an admissible expression of X,λ near p, and let  be any area form on U . Then, the Singularity order sλ0 (p) of p is equal to the order at p of the function (Qλ0 , Zλ0 )|Sλ0 . The contact point p is said to be a Regular contact point if the singularity order is zero (i.e. if the vectors Qλ (p) and Zλ (p) are independent) and a Singular contact point otherwise. It is easy to see that the definition leads to the expected values in normal form coordinates (2.1). Such a normal form admits for example the admissible expression ∂ and Fλ (x, y) = y − fλ (x). {U, Zλ , Fλ } with Zλ = ∂x 1. By definition, nλ (p) is the contact order of the curve Sλ ∩ U with the x-axis at x = 0 (which is the leaf λ,p ). Hence, nλ (p) = Ord |0 (fλ ). 2. It follows from Lemma 2.1 that we can choose any area form , for example, we choose  = dx ∧ dy. We compute   ∂ ∂ (x, y) → (dx ∧ dy) [g(x, 0, λ) + (y − fλ (x))h(x, y, 0, λ)] , ∂y ∂x = −g(x, 0, λ) − (y − fλ (x))h(x, y, 0, λ) = −g(x, 0, λ), where the last equality follows because we restrict to Sλ . By definition, sλ (p) is the order of the function g at 0, in agreement with the definition used in [DMDR11]. We can also state some extra properties of sλ (p) and nλ (p): 3. It is also not hard to see that nλ (p) is equal to the multiplicity at p of the zero of the eigenvalue function Vλ , plus 1. It suffices to check it in a normal form (2.1), in which we see that Vλ (x) = dFλ [Zλ ]|Sλ (x) = −fλ (x), for x = 0.

2.2 Invariants at Contact Points

19

4. If  is a local area form and ϕ(θ ) is local parameterization of Sλ , for some λ, we can also see that nλ (p) is the multiplicity at p of the zero of the function (Zλ , dϕ dθ )|Sλ plus one. 5. Introducing a local Riemannian metric in a neighborhood W of p0 ∈ Sλ , we can define the angle function Angle(u, v) between two vectors u, v ∈ Tp (M), p ∈ W . Then the singularity order sλ (p) is equal to the order of zero at p of the function Angle(Zλ , Qλ )|Sλ , which differs from (Zλ , Qλ )|Sλ by a nowhere-zero multiplicative function. 6. The singularity order, if finite, represents the algebraic multiplicity of the (isolated) singularities that we will encounter near the contact point for  > 0.

Assumption on Contact Points In this book we suppose that all contact points are of finite contact order.

Such contact points are clearly isolated, implying that the current assumption is more restrictive than the preliminary one in Chap. 1 for smooth slow–fast families in general. It agrees with the preliminary assumption in case of analytic slow–fast families. For a contact point of singularity order 1, we also define an index, conveniently using the normal form (2.1): Definition 2.2 (Index of a Contact Point of Singularity Order 1) Consider any normal form (2.1) near a singular contact point p of singularity order 1. The point is said to be a singular contact point of Singularity index ±1 when sign

∂g (0, 0, λ0 ) = ∓1. ∂x

For a contact point with singularity order 1, it is clear, from any normal form (2.1), that the family of vector fields X,λ has, for (, λ) close to (0, λ0 ) and  > 0, a singular point (x, y) = (x0 (, λ)), y0 (, λ)) tending to (0, 0) as (, λ) → (0, λ0 ). This singularity is non-degenerate. It is of saddle type when it is a singularity of index −1 and of center/focus type when it is a singularity of index +1. The index is hence invariant under equivalences. Lemma 2.2 A singular contact point p of singularity order 1 for λ = λ0 persists: there exists a smooth family of points p,λ defined near (, λ) = (0, λ0 ) and with p0,λ0 = p, such that X,λ has a singularity at p,λ . For  > 0, it is a hyperbolic saddle in case p has singularity index −1, or a focus, node, or center in case p has singularity index +1.

20

2 Local Invariants and Normal Forms

2.3 Example We focus on the nilpotent contact point at (1, 0) of the slow–fast family of vector fields  x˙ =  (α + βx + γ y) (2.3) X,α,β,γ : y˙ = x 2 + y 2 − 1, with (α, β, γ ) = (0, 0, 0). It is clear that a reduction to normal form will reveal the contact order and singularity √ order. In fact, here it is quite easy to derive the normal form. Applying (x, y) = ( 1 − Y , −X), we find 

X˙ = Y − X2

Y˙ =  g(X) + O(Y − X2 ) ,

with g(X) = −2(α + β) + 2γ X + (α + 2β)X2 + O(X3 ). So when α = −β, the singularity order is 0, when α = −β and γ = 0, the singularity order is 1, and when α = −β and γ = 0, the singularity order is 2. In all cases the contact order is 2 (Fig. 2.1). This example is included however to show how to derive the contact order and singularity order without going into the normal form, making use of the intrinsic nature of these notions, since in general it can be more difficult to derive an explicit normal form. From Definition 1.1, we can use Fλ = x 2 + y 2 − 1,

Zλ =

∂ , ∂y

Qλ = (α + βx + γ y)

∂ . ∂x

Using the standard area form, we find (Qλ , Zλ ) = −(α+βx+γ y)|x 2+y 2 =1 . Using y to parameterize the circle, we need to examine the order of 0 of the function β y → −α − β 1 − y 2 − γ y = −(α + β) − γ y + y 2 + O(y 3 ). 2 Fig. 2.1 Limiting vector field of (2.3)

2.3 Example

21

The conclusions with respect to the singularity order agree with those found using the normal form. We leave it up to the reader to use whatever method is found convenient. In fact, for testing contact points up to singularity order 1, we present the following result that is even easier to apply and without the use of area forms. Lemma 2.3 Let (U, Zλ , Fλ ) be an admissible expression of X,λ around a contact point p (at λ0 ). (i) Then the value p ∈ Sλ0 → Qλ0 (Fλ0 )(p) is well defined up to a nonzero multiplicative constant. It is zero for a singular contact point and nonzero for a regular contact point. (ii) For a singular contact point, the value [Zλ0 , Qλ0 ](Fλ0 )(p), i.e. the Lie bracket of Zλ0 and Qλ0 acting as a differential operator on the function Fλ0 evaluated at p, is intrinsically defined. When it is nonzero, p has singularity order 1 and the sign’s inverse value is the singularity index of p at λ0 . When it is zero, p has at least singularity order 2. Proof As λ = λ0 is fixed, we will not keep the dependence of λ in the notation. (i) First, let (U, Z , F ) be another admissible expression, with Z = bZ and F = b−1 F . Then Q(F ) = Q(bF ) = Q(b)F + Q(F )b. As F = 0 along S, we have Q(F ) = bQ(F ). Next, recall that when changing charts, the vector field Q might be replaced by Q + gZ for some smooth function g. We have (Q + gZ)(F ) = Q(F ) + hZ(F ) for some function h (see Lemma 1.3). Since p is a contact point Z(F ) = 0 at p, which implies (Q + gZ)(F )(p) = Q(F )(p). The relation to the singularity/regularity of the contact point can hence be checked in any coordinate system and with any admissible expression. Using the normal form (2.1), it is an elementary exercise. (ii) We first check the dependence on the choice of admissible expression: [Z , Q ](F ) = [Z , Q ](b−1 F ) = b−1 [Z , Q ](F ) + F [Z , Q ](b−1) = Z(bQ(F )) − Q(bZ(F )) = [Z, Q](F ) + Z(b)Q(F ) − Q(b)Z(F ) = [Z, Q](F ). We have used in the first line that F (p) = 0; the last equality follows from the fact that p is a contact point (Z(F ) = 0) and that it is a singular contact point (part (i)). Like in part (i), we check to what extent the value is coordinate-free.

22

2 Local Invariants and Normal Forms

At a coordinate change, the vector field Q potentially changes to Q + gZ for some g. Then, [Z, Q + gZ](F ) = [Z, Q](F ) + [Z, gZ](F ) = [Z, Q](F ) + Z(F )Z(g). As Z(F ) = 0 at the contact point, the intrinsic property follows. In the normal ∂g (0, 0, λ0 ), so the inverse sign of form (2.1), we find that [Z, Q](F ) equals ∂x [Z, Q](F ) is in agreement with the definition of the singularity index. 

Applying the lemma to example (2.3) gives

∂ 2 2 = 2(α + β), Qλ (Fλ )(1, 0) = (α + βx + γ y) (x + y − 1)

∂x (x,y)=(1,0) and [Zλ , Qλ ] = γ

∂ ⇒ [Zλ, Qλ ](Fλ )(1, 0) = 2γ . ∂x

From the lemma, it is very easy to conclude that when α + β = 0, the point (1, 0) is a regular contact point and that when α + β = 0 and γ < 0 (respectively, γ > 0), the point has singularity index +1 (respectively, −1).

2.4 Remarks About Contact Points In the literature, contact points are sometimes called jump points, turning points, slow–fast Hopf points, or singular Hopf points. In some cases, like in the case of turning points, the notion in the literature may even be wider than the contact points that we consider in this book (as one sometimes studies non-nilpotent contact points). For the sake of clarity, we will fix the notions used here (Fig. 2.2). Definition 2.3 A Jump point is a regular contact point of even contact order. A jump point is called a Generic jump point when the contact order is 2. Regular contact points can be of odd contact order or of even contact order, see Fig. 2.3 for the possible cases (not yet making use of the information on the slow Fig. 2.2 Generic jump point. Single arrows show the direction of the slow vector field (see Chap. 3)

jump point

2.4 Remarks About Contact Points

(a)

23

(b1 )

(b2 )

(c)

Fig. 2.3 Different shapes of the critical curve at a contact point in normal form: (a) a contact of order 2, (b1 ) and (b2 ) a contact of odd order, and (c) a contact of even order larger than 2

unfolding

Fig. 2.4 Unfolding of a non-generic jump point

Fig. 2.5 A slow–fast Hopf point. Single arrows show the direction of the slow vector field (see Chap. 3)

dynamics, which we will introduce in the next chapter). Typically, a regular contact point of odd contact order is not called a jump point. When pλ0 is a generic jump point for X,λ at λ = λ0 , then pλ0 perturbs smoothly to a generic jump point pλ of X,λ for λ close enough to λ0 . This property justifies the name of a generic jump point. When a jump point is not generic, bifurcations may take place as λ varies, for example, in 

x˙ = y − (λ1 x + λ2 x 2 + x 4 ), y˙ = −.

The contact points are located at x-coordinates given by the roots of λ1 + 2λ2 x + 4x 3 = 0, which unfolds near λ = 0 in an elementary catastrophe. What remains true is that all contact points of even order appearing in the unfolding of a non-generic jump point are all jump points (Fig. 2.4). Definition 2.4 A Slow fast Hopf point@slow–fast Hopf point (or a singular Hopf point) is a contact point of contact order 2 and singularity index +1. At a slow–fast Hopf point (Fig. 2.5), the possible presence of a Hopf bifurcation is encoded. We stress however that the definition of slow–fast Hopf point does not

24

2 Local Invariants and Normal Forms

include any information on the unfolding. In the literature, the notion Turning point is often used as a synonym in the above definition or even as a synonym of a singular contact point in general (the name of turning point comes from the fact that the slow dynamics, defined in Chap. 3, does not change its direction when passing through the point); see also [O’M74]. In this book, we will use the notion of turning point to indicate a property of the family of vector fields X,λ at λ = λ0 , and we just work with the generic case. Definition 2.5 The family of vector fields X,λ is said to have a Generic turning point at p for λ = λ0 when p is a slow–fast Hopf point for λ = λ0 and if there exists a coordinate system bringing X,λ in the form (2.1), putting p to the origin, and for which the map λ → g(0, 0, λ) is a submersion at λ0 . In [DMD11d], generic turning points are called slow–fast unfoldings of Hopf points. The genericity comes from the following property: suppose that X,λ,μ is a slow–fast family of vector fields for which the subfamily X,λ,μ0 has a generic turning point at p = pμ0 for λ = λμ0 , then the generic turning point perturbs smoothly to a generic turning point p = pμ for λ = λμ , for the full slow–fast family of vector fields. Generic turning points and generic jump points will be further used in Chap. 6. We remark that Definition 2.5 makes use of normal forms, but in view of Lemma 2.3, the submersion property can equally well be checked on the map λ → Qλ (Fλ )(p) at λ = λ0 . As an involved exercise to the reader, one can derive coordinatefree expressions for detecting slow–fast unfoldings of a Bogdanov–Takens point (see [DMD11d]) directly from the equations. Remark 2.2 From the submersion property in Definition 2.5, it is clear that there exists a locally defined map ψ : (a, μ) → λ, so that ψ(0, μ0 ) = λ0 and that g(0, 0, ψ(a, μ)) = a. Studying the new family Y,a,μ := X,ψ(a,μ) near (a, μ) = (0, μ0 ) is equivalent to studying X,λ near λ0 . This allows to treat a = g(0, 0, λ) as an individual parameter. In fact, sometimes we will even consider a family ∂ X˜ ,a,λ = X,λ + (a − g(0, 0, λ)) . ∂y

2.5 Invariants at Normally Hyperbolic Points

25

It is clear that the family X˜ ,a,λ is larger than X,λ , containing the old family at a = g(0, 0, λ). The benefit of the larger family is that in the expression of the normal form, the zero-order coefficient of the g-function is an independent parameter a that nowhere else appears in the expression of the vector field. Definition 2.6 A Contact point of Morse type is a contact point of contact order 2. Just like generic jump points, contact points of Morse type are persistent: the presence of a contact point of Morse type for λ = λ0 implies that also for λ close to λ0 , there is a contact point of Morse type. The singularity order of the contact point might bifurcate to lower values upon varying λ though. The qualitative study of the different kinds of contact points will be continued in Sect. 3.4, extending the dynamic information from Fig. 2.3 with information from the slow dynamics. Let us finish by stating a simple lemma without proof: Lemma 2.4 When X,λ has at p and for λ = λ0 a singular contact point of order n and of singularity order m, then near λ = λ0 , there can only appear contact points of X,λ with contact order ≤ n and singularity order ≤ m.

2.5 Invariants at Normally Hyperbolic Points We can also introduce the notion of singularity order of normally hyperbolic points. Definition 2.7 (Invariants at Normally Hyperbolic Points) Let p be a normally hyperbolic point of a slow–fast system X,λ for the value λ0 . Let {U, Zλ , Fλ } be an admissible expression of X,λ near p, and let  be any area form on U . Then the Singularity order sλ0 (p) of p is equal to the order at p of the function (Qλ0 , Zλ0 )|Sλ0 . The point p is said to be regular if the singularity order is zero and singular otherwise. The singularity order can be calculated in any coordinate system: the invariance of the order of zero of (Qλ0 , Zλ0 )|Sλ0 follows from Lemma 2.1. In the normal form (2.2), the singularity order sλ0 (p) is just the order of zero of the function g0,λ0 (x) at x = xp , and so the point p is regular if g0,λ0 (0) = 0. As an example, consider again (2.3). When the line L : α+βx+γ y = 0 intersects the critical curve C : x 2 + y 2 = 1 in a point p outside the two contact points, then p has singularity order 1 or 2 depending on the order of contact between L and C, otherwise it is regular.

Chapter 3

The Slow Vector Field

3.1 Definition On Sλ \ Cλ , i.e. on the set of normally hyperbolic singularities, we can define the notion of slow vector field and related slow dynamics. One way to do this is by considering center manifolds, as we previously did. We will recall it in moment. Based on the previous sections, we will however first introduce it in a different and intrinsic way, before showing the link with the more traditional way of definition. As usual, we will from now on also work with the so-called slow time s = t, where t is the fast time as introduced in Chap. 1. Definition 3.1 (Slow Vector Field) For each p ∈ Sλ \ Cλ , let Q˜ λ (p) ∈ Tp Sλ (tangent line at Sλ through p) be the linear projection of Qλ (p) on Tp Sλ , in the direction parallel to the eigenspace δλ (p) (well defined, as δλ (p), the eigenspace of the nonzero eigenvalue, is transverse to Tp Sλ , the eigenspace of the zero eigenvalue). This defines a family of vector fields Q˜ λ on Sλ \ Cλ ; it is called the Slow vector field (or a family of slow vector fields) (see Fig. 3.1). The time variable of its flow is the slow time used in Eq. (1.3). This flow is called the Slow dynamics. Remarks 3.1 1. We know from Lemma 1.3 that, at each point p ∈ Sλ \ Cλ , Qλ is well defined, up a multiple of Zλ . As such, Q˜ λ is a well-defined vector field along Sλ \ Cλ . This means that if G,λ is an (, λ)-family of diffeomorphisms of M, then the slow vector field Q˜ λ associated with (G,λ )∗ (X,λ ) is equal to (G0,λ )∗ (Q˜ λ ). By definition, Q˜ λ is a smooth family of vector fields.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_3

27

28

3 The Slow Vector Field

M

T p Sλ Tp (M ) ˜λ Q

Qλ δλ (p)

Sλ ˜λ Fig. 3.1 Projection of Qλ onto Tp Sλ leads to the slow vector field Q

2. Recall Example 1.2 from Chap. 1. In that example, the slow formulation was determined as  dx  ds = Fλ (x, y) + Fλ1 (x, y) + O( 2 ), dy ds = Gλ (x, y) + O(). Setting  = 0 in this formulation yields 0 = Fλ (x, y),

dy = Gλ (x, y). ds

At normally hyperbolic points of Sλ , we can implicitly solve the equation Fλ (x, y) = 0 in terms of x, allowing us to obtain the slow dynamics directly as dy = Gλ (xλ (y), y). ds We will later prove that this differential equation agrees with the slow vector field of Definition 3.1, see Example 1 in Sect. 3.3. 

We will show that this definition agrees with the more traditional one based on the use of center manifolds. Before recalling this definition, let us first recall the center manifold theorem (or, to be more precise, a version of the center manifold theorem adapted to slow–fast systems in two variables). For a simple statement of the theorem, we suppose to work in local coordinates (x, y) in which a curve γλ0 ⊂ Sλ0 \ Cλ0 is given by {y = 0}. Theorem 3.1 (Center Manifold Theorem) Let p = (x0 , λ0 ) ∈ γλ0 ⊂ Sλ0 \ Cλ0 , with γλ0 = {y = 0}. Then, for each k ∈ N, there exists a Ck family of manifolds W,λ , with X,λ tangent to W,λ , which is given by the graph of a Ck function {y = w(x, , λ)}, defined on (x0 − δ, x0 + δ) × [0, 1 [ × 0 , for some δ > 0, 0 < 1 ≤ 0 and 0 ⊂  a neighborhood of λ0 , and with the property that w(x, 0, λ) = 0.

3.1 Definition

29

In this chapter we rely on the general expression stated in Theorem 3.1. In a proof, the traditional center manifold theorem is used near a singular point (x, y, , λ) = (x0 , 0, 0, λ0 ) of an extended vector field. In Chap. 9 of this book, we will however prove a stronger result on which we will rely in the subsequent parts. A family of invariant manifolds W,λ like in Theorem 3.1 will be conveniently called a center manifold. A center manifold is not uniquely defined. Later on we will see how to benefit from this. Usually the slow vector field of X,λ along Sλ \ Cλ is defined to be 1 Yλ = lim ( X,λ |W,λ ). →0 

(3.1)

We will now prove that Yλ is indeed an intrinsically defined smooth λ-family of vector fields on Sλ \ Cλ , independent of the choice of W , and agreeing with Definition 3.1, by showing that Yλ = Q˜ λ . We can even do better. Proposition 3.1 Consider a segment γλ ⊂ Sλ \ Cλ , and let W,λ be a center manifold of class Cr+1 containing γλ . Then, in sufficiently small neighborhoods ˜ λ for  = 0, of γλ , the family of vector fields 1 X,λ |W,λ , for  > 0, together with Q r forms a C family of vector fields on W,λ (see Fig. 3.2). The family will be smooth if W,λ is. We remark that the slow vector field can change under smooth equivalence. ˜ λ is defined in an intrinsic way (a coordinate-free way), it suffices to Proof As Q verify the claim in any admissible expression around any given point p ∈ γλ . In a neighborhood of p, we choose coordinates (x, y) s.t. p = (0, 0), Sλ = {y = ∂ 0}, Fλ (x, y) = yfλ (x, y) with fλ (0, 0) = 0 and Zλ = ∂y . Let y = z(x, , λ), with r z of class C , be the graph of a center manifold containing p. We take r sufficiently big. We have X,λ = yfλ

∂ ∂ ∂ + Qλ + O( 2 ) = Aλ + (Q1λ + O()) , ∂y ∂y ∂x

Fig. 3.2 Slow vector field seen as a limit

ε Wε,λ

Xε,λ {ε = 0} X0,λ

Σλ \ Cλ

{ε = 0}

30

3 The Slow Vector Field

∂ ∂ with Aλ = yfλ + Q2λ + O( 2 ) and Qλ = Q1λ ∂x + Q2λ ∂y . At the point (x, z(x, , λ)), the vector field X,λ is tangent to the curve x → z(x, , λ). This ∂z gives, at the point (x, z(x, , λ)) (using z for ∂x ),

(Q1 + O()) 1 λ

Aλ z

λ



= 0,

so that Aλ = O( 2 ). Bringing this in the expression of X,λ , we obtain, for  > 0, that 1 1 ∂ X,λ |W,λ = X,λ (x, z(x, , λ)) = Q1λ (x, 0) + O().   ∂x The O()-terms in this expression are Cr (as obtained by division by  from a Cr+1 ∂ vector field). Also the slow vector field is given locally, near x = 0, by Q1λ (x, 0) ∂x , ∂ ˜ since it is the projection Qλ (x, 0) of Qλ (x, 0) in the direction of Zλ (x, 0) = ∂y . This proves, locally around each point p, that  lim

→0

1 X,λ |W,λ 



= Q˜ λ

and that the regularity properties hold as announced.



In fact, later we will work with center manifolds that are not local at a point p ∈ Sλ \ Cλ , but instead contain a segment γλ , which can be chosen arbitrarily large in Sλ \ Cλ . Definition 3.2 We denote by λ = {p ∈ Sλ \ Cλ : Q˜ λ (p) = 0} the set of zeros of the slow vector field. (By definition of Q˜ λ , λ consists of those points p where Qλ is tangent to δλ (p).) Zeros of the slow vector field will often be called slow singularities. The points of λ are the singular points defined in Definition 2.7, and the singularity order sλ (p) agrees with the order of zero of Q˜ λ as will be proved in Lemma 3.1 below. From Proposition 3.1, it easily follows that, with the notation of that proposition, in a sufficiently small neighborhood of a compact segment γλ ⊂ Sλ \ (Cλ ∪ λ ), the orbits of X,λ |W,λ , for  > 0, together with Sλ , form a Cr foliation. The foliation will be smooth if W,λ is smooth. In Sect. 3.3, we will consider some examples on how to compute the slow vector field, relating it to the more traditional way of defining it. First we will however define 1-at regular points of the critical curve.

3.2 Differential 1-Forms Along the Critical Curve

31

3.2 Differential 1-Forms Along the Critical Curve In coming chapters, we will often need to integrate along the critical curve, for example, if one wants to compute divergence integrals. It is also needed if one would like to calculate the period function along periodic orbits in the limit as  → 0. It reveals necessary to integrate along the critical curve “with respect to the slow time.” This section is devoted to making this statement precise, i.e. we will define a 1-form at regular points of the critical curve in an intrinsic way. Let us consider any closed interval γλ ⊂ Sλ \Cλ , parameterized by r ∈ [r1 , r2 ] → ∂ , ϕλ (r) ∈ γλ = ϕλ ([r1 , r2 ]). On [r1 , r2 ], the slow vector field Q˜ λ is equal to qλ (r) ∂r for some smooth family of functions qλ , or equivalently, dϕλ . Q˜ λ (ϕλ (r)) = qλ (r) dr

(3.2)

The differential equation of Q˜ λ in the coordinate r is dr = qλ (r). ds On γλ , the zeros of Q˜ λ correspond to the zeros of qλ . If we suppose now that γλ does not contain zeros of Q˜ λ , then we can write on γλ : ds =

dr . qλ (r)

(3.3)

This means that the differential ds of slow time can be seen as a differential 1-form on γλ . We clearly have the following pairing between this form and Q˜ λ : ds(Q˜ λ ) ≡ 1.

(3.4)

The pairing (3.4) shows that ds, seen as a differential 1-form on γλ , depends on Q˜ λ but not on a choice of coordinate r (as it could be suggested by (3.3)). The choice of slow time s itself is hence intrinsically related to X,λ up to an additive (λ-dependent) constant. The strength of (3.4) is that it not only characterizes ds as a global differential 1-form on Sλ \ (Cλ ∪ λ ) but also could be used as an intrinsic definition of ds. Remark that ds is not invariant under smooth equivalences but, evidently, ds(Q˜ λ ) is.

32

3 The Slow Vector Field

3.3 Calculating Slow Vector Fields In this section we will calculate the slow vector field Q˜ λ in a number of examples and see its relation with the traditional way of defining it. Example 1 Consider a smooth slow–fast family of vector fields 

x˙ = F (x, y, λ) y˙ = G(x, y, λ)

at some point p = (x0 , y0 ) and λ = λ0 , with F (x0 , y0 , λ0 ) = 0. We have Qλ = (0, Gλ ), and we may take Zλ = (1, 0). (i) In case

∂F ∂y (x0 , y0 , λ0 )

= 0, we represent Sλ near (p, λ0 ) as y = (x, λ) for

some smooth function so that (x , λ (x)x ) with x = generator of T Sλ . To find Q˜ λ from Qλ , we solve 

x λ (x)x



= x

dx ds

can be used as

    1 0 + G(x, λ (x), λ) , 0 1

inducing λ (x)x = G(x, λ (x), λ) as equation for the slow vector field Q˜ λ . (ii) In case ∂F ∂x (x0 , y0 , λ0 ) = 0, we represent Sλ near (p, λ0 ) as x = ρ(y, λ) for some smooth function so that (ρλ (y)y , y ) with y = dy ds can be used as generator of T Sλ . It leads to solving 

ρλ (y)y y



= ρλ (y)y

    0 1 , + G(ρλ (y), y, λ) 1 0

inducing y = G(ρλ (y), y, λ) as equation for the slow vector field Q˜ λ . (iii) In generality we can use a regular parameterization (x, y) = (aλ (r), bλ (r)) for Sλ near (p, λ0 ) and use (aλ (r)r , bλ (r)r ) as generator of T Sλ . It leads to solving 

aλ (r)r bλ (r)r



= aλ (r)r

    1 0 + G(aλ (r), bλ (r), λ) , 0 1

3.3 Calculating Slow Vector Fields

33

inducing bλ (r)r = G(aλ (r), bλ (r), λ) as equation for the slow vector field Q˜ λ . Example 2 Next, we treat an expression that is often encountered in the literature, and which, with the traditional definition (3.1), requires an approximation of the center manifolds. There is in fact no need to do this as we will see now. Consider the slow–fast family of vector fields 

x˙ = y y˙ = yH (x, y, λ) + (k(x) + O(y) + O()).

∂ , and for calculating Q˜ λ , In this case Sλ = {y = 0}, Qλ = (k(x) + O(y) + O()) ∂y we use       x 1 0 + k(x) , =x H (x, 0, λ) 1 0

inducing x = −

k(x) H (x, 0, λ)

as equation for the slow vector field Q˜ λ . Example 3 We present an example, inspired by (1.5): 

x˙ = −(y + λx)(y − 1) + g(x, y) y˙ = (x + λy)(y − 1) + h(x, y).

The critical curve y = 1 has a slow–fast Hopf point at x = −λ when g(−λ, 1) > 0 and h(−λ, 1) = 0. Note that it is possible to put the vector field near the Hopf point in the normal form, as prescribed in Proposition 2.1. The computations are weary though, since following the steps in the proof of the proposition, one needs to put ∂ ∂ + (x + λy) ∂y in flow-box coordinates near (−λ, 0). We fortunately −(y + λx) ∂x do not need to do so, and we can use Lemma 2.3 to check the nature of the contact point, it is an exercise (Fig. 3.3).

34

3 The Slow Vector Field y

Fig. 3.3 Critical curve with slow–fast Hopf point

(−λ, 1) {y = 1}

O

x

It is readily checked that slow manifolds have the asymptotic expression y = 2 1 −  h(x,1) x+λ + O( ) outside the Hopf point. As a consequence, one trivially finds that x =

(1 + λx)h(x, 1) + (x + λ)g(x, 1) x+λ

is the slow vector field along y = 1.

3.4 The Slow Vector Field Near Contact Points Although the slow vector field Q˜ λ is not invariant under smooth equivalences, let us nevertheless, as a special case of Example 1 from the previous section, see how the slow vector field Q˜ λ looks like in a normal form (2.1), permitting to draw some interesting conclusions on the dynamics near the contact points. Given a contact point of a slow–fast family of vector fields X,λ in the normal form 

x˙ = y − f (x, λ) y˙ = (g(x, .λ) + (y − f (x, λ))h(x, y, , λ)).

Recall that this normal form admits an admissible expression with Zλ =

∂ , ∂x

Fλ (x, y) = y −fλ (x),

and

Qλ = (gλ (x)+hλ (x, y)(y −fλ (x)))

∂ , ∂y

with fλ (x) = f (x, λ), gλ (x) = g(x, 0, λ), and hλ (x, y) = h(x, y, 0, λ). We can easily obtain that gλ (x) ∂ ∂ Q˜ λ = + gλ (x) . fλ (x) ∂x ∂y

3.4 The Slow Vector Field Near Contact Points

35

Any interval γλ ⊂ Sλ \ Cλ = {y = f (x, λ)} \ {(0, 0)} can be parameterized by either x or y. Using the above formula for Q˜ λ , we have that the differential equation for the slow dynamics, in the x-parameterization, is fλ (x)

dx = gλ (x). ds

(3.5)

In the y-parameterization, we have dy = gλ (xλ (y)), ds where xλ (y) is the implicit solution of y = fλ (xλ (y)). From (3.5), we see that Q˜ λ0 , for some choice λ = λ0 , cannot be extended to a vector field at x = 0 if x = 0 represents, for λ = λ0 , a regular contact point (i.e. g(0, 0, λ0 ) = 0), but it can be extended when gλ0 is zero at the contact point (i.e. g(0, 0, λ0 ) = 0), depending on the singularity order and contact order of the contact points. We will come back to this issue in Sect. 5.5. Next, we aim at providing a classification of the possible phase portraits of the slow and fast limiting systems of p, depending on the singularity order and contact order of the contact point. In the following description, we write fλ0 (x) = σf x n + O(x n+1 ),

gλ0 (x) = σg x m + O(x m+1 ),

with σf .σg = 0. Recall that m ≥ 0 and n ≥ 2 (n ≥ 1 if we include normally hyperbolic points). It reveals that elementary qualitative information can be obtained purely based on the parities of n and m and on the signs of σf and σg . Note that when n is even (respectively, m is even), a change of coordinates (x, y) → (−x, −y) changes the sign of σf (respectively, σg ), while it remains the same in case of opposite parity. Similarly, when n is odd (respectively, m is even), a time-reversing symmetry (x, t) → (−x, −t) changes the sign of σf (respectively, σg ). These symmetries will be used in the classification described below and visualized in Fig. 3.4. 1. The singularity order m is even. After a change of coordinates, it is always possible to assume that σg > 0; in other words, and the slow dynamics is upward in this coordinate system. Four subcases appear, depending on the parity of n and the sign of σf : (1a+ )

(1a− )

n even and σf > 0: the contact point is surrounded by an attracting and a repelling branch, and the slow dynamics points away from the contact point on both sides. n even and σf < 0: the contact point is surrounded by an attracting and a repelling branch, and the slow dynamics points toward the contact point on both sides. Case (1a− ) is a time-reversed version of (1a+ ).

36

3 The Slow Vector Field

(1a+ )

(1b+ )

(1a− )

(1b− )

(2a− )

(2a+ )

(2b+ )

(2b− )

(2c+ )

(2c− )

Fig. 3.4 Slow and fast dynamics near contact points. The slow dynamics and fast dynamics are conveniently shown in the same picture, indicating the fast dynamics with double arrows and the slow dynamics with single arrows

(1b+ ) (1b− )

n odd and σf > 0: the contact point is surrounded by attracting branches on both sides. n odd and σf < 0: the contact point is surrounded by repelling branches on both sides. Case (1b− ) is a time-reversed version of (1b+ ).

2. The singularity order m is odd. We distinguish the following: (2a− )

(2a+ ) (2b+ )

(2b− ) (2c+ )

(2c− )

n is even and σg < 0: after a change of coordinates, we can ensure that σf > 0. The contact point is surrounded by an attracting and a repelling branch, and the slow dynamics points from the attracting toward the repelling branch. n is even and σg > 0: this is not a time-reversed version of (2a− ): the slow dynamics points from repelling to attracting. n is odd, σg < 0, and σf > 0: the contact point is surrounded by attracting branches on both sides and the slow dynamics points toward p. n is odd, σg < 0, and σf < 0: this is a reflected and time-reversed version of (2b+ ). n is odd, σg > 0, and σf > 0: the contact point is surrounded by attracting branches on both sides, and the slow dynamics points away from p. n is odd, σg > 0, and σf < 0: this is a reflected and time-reversed version of (2c+ ).

Remark 3.2 Jump points belong to the category (1a± ) and turning points to the category (2a− ). Contact points of singularity index +1 belong to category (2a− ) or (2b± ), and contact points of singularity index −1 belong to category (2a+ ) or (2c± ). Contact points of type (2b± ) will be without interest in this book, as only small-amplitude limit cycles can come close to such points.

3.5 Slow Singularities

37

3.5 Slow Singularities ˜λ Let us recall that slow singularities are singularities of the slow vector field. Let Q be the slow vector field near p ∈ Sλ \ Cλ . In Sect. 2.5, the singularity order of a normally hyperbolic point was defined (Definition 2.7). It should now be clear that the singularity order at such points can equally well be defined intrinsically through the slow vector field. Lemma 3.1 Consider an admissible expression {U, Zλ, Fλ },  an area form on U , and γλ a closed interval in U ∩ (Sλ \ Cλ ), parameterized by ϕλ (r). Then   

dϕλ  (r) (qλ (r)) =  Zλ ϕλ (r) , Q(ϕλ (r)) ,  Zλ (ϕλ (r)), dr

(3.6)

with qλ (r) as in (3.2). The zeros of the slow vector field on U ∩(Sλ \Cλ ), counted with their multiplicity, correspond to the zeros of the function  (Zλ (ϕλ (r)) , Qλ (ϕλ (r))), counted with their multiplicity. Proof At any point of U ∩ (Sλ \ Cλ ), we can write Qλ = ρ(λ)Zλ + Q˜ λ (for some ρ(λ)).Then, as (Zλ , Zλ ) = 0, we have (Zλ , Qλ ) = (Zλ , Q˜ λ ), on U ∩ (Sλ \ Cλ ). We canevaluate this equation  at ϕλ (r) and then use (3.2) to obtain (3.6). We dϕλ have that  Zλ (ϕλ (r)), dr (r) is nowhere zero on γλ , as Zλ is independent of dϕλ dr

at each point of γλ . The second claim is consequence of the fact that qλ (r) and 

 Zλ ϕλ (r) , Qλ (ϕλ (r)) differ by a nowhere-zero multiplicative function. 

As already observed, the slow vector field Q˜ λ itself is not invariant under smooth equivalence. However, λ is invariant, including the multiplicity of its points. As an example, we recall Example (2.3):  X,α,β,γ : The slow dynamics can be seen as

x˙ =  (α + βx + γ y) y˙ = x 2 + y 2 − 1.

dx ds

= α + βx + γ y|

the slow dynamics are points (x, y) with x = 1 for

√

y=± 1−x 2 which dx ds = 0.

. Singularities in

Lemma 3.2 Singularities of the slow vector field of singularity order 1 are persistent in the sense that when p ∈ Sλ0 \ Cλ0 is a singularity of order 1, then there exists a smooth family of points p,λ . It is hyperbolic for  = 0. Proof The proof is elementary.



38

3 The Slow Vector Field

Remarks 3.3 1. Already in the well-known Van der Pol system 

2

3

x˙ = y − x2 − x3 y˙ = (a − x),

singularities of the slow vector field appear: when a > 0, a singularity appears on the attracting slow arc to the right, and it perturbs to a hyperbolic stable node of the full system. When −1 < a < 0, a singularity appears on the repelling slow arc, and it perturbs to a hyperbolic unstable node of the full system. At a = 0, a change of stability takes place, and the origin is called a slow–fast Hopf point (turning point). To see the Hopf bifurcation, one must understand that before the change of stability occurs as a proceeds from negative to positive, there is first a change of node to focus in an O()-neighborhood of a = 0. So it is not the node that changes stability, but rather the focus that changes stability. To see this, no advanced techniques like blow-up are needed. It can just be analyzed by linearization. 2. The center manifold theorem yields finitely smooth invariant manifolds near normally hyperbolic points of the critical curve. Later in this book, we will give detailed proofs, not relying on known center manifold theorems, showing that these invariant manifolds can be chosen C∞ under the condition of absence of singularities in the slow dynamics. For precise smoothness information of invariant manifolds near singularities in the slow dynamics, we refer to [DMD08] (Theorem 3.1). 

Chapter 4

Slow–Fast Cycles

4.1 Definitions In this book we are principally interested in limit cycles that appear in slow–fast systems, for  > 0 but near the limit value  = 0. It is hence of a natural interest to consider how periodic orbits, or more precisely families of periodic orbits, behave as  → 0. We will see that the periodic orbits tend toward specific shapes, called limit periodic sets. The following definition is inspired by the general definition of limit periodic set as given in [Rou98], adapted to the context of slow–fast systems (Fig. 4.1). Definition 4.1 (Limit Periodic Set) A Limit periodic set  of a slow–fast family of vector fields X,λ is a compact non-empty subset of M for which there exists a sequence (n , λn )n≥0 tending toward (λ∗ , 0) such that Xn ,λn has a periodic orbit γn with the property that γn tends toward  in Hausdorff sense. Limit periodic sets can for example be periodic orbits itself, saddle loops, or more complicated curves. However, in the context of slow–fast systems the most common shape of an l.p.s. is a slow–fast cycle that we are about to introduce. It consists, for some X0,λ0 , of fast orbits, with their natural orientation, and of pieces of critical curve that are oriented according to the orientation induced by the slow vector field (Fig. 4.2). Definition 4.2 (Slow–Fast Cycles) Given λ0 ∈ . A subset , diffeomorphic to a piecewise smooth circle and consisting of a finite number of fast orbits of X0,λ0 and a finite number of slow arcs of X0,λ0 , is called a Slow fast cycle@slow–fast cycle of X0,λ0 if it contains at least one slow arc and if moreover it is possible to orient the circle in a way that the orientation is compatible to the orientation on the fast orbits and such that on all slow arcs it agrees with the orientation of the slow vector field.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_4

39

40

4 Slow–Fast Cycles

Fig. 4.1 A limit periodic set with 3 isolated singular points

Fig. 4.2 Examples of slow–fast cycles

Remarks 4.1 1. We recall that in this book we only work with contact points that are nilpotent and of finite contact order. Moreover, we will essentially only consider contact points that are either regular or have singularity order 1. While the developed techniques can be used to study more singular contact points (see for example [MR12, DMD11d, HDMD13, DMD10, DMD11c]), such contact points are not the central topic of this book. 2. Given this definition of slow–fast cycle, it does not include all possible limit periodic sets found in families of slow–fast systems. Cycles without any slow arc are possible, single contact points could be a limit periodic set, and periodic orbits could converge to objects with an infinite number of components. Also the case of oscillations on a torus is not covered through this definition (see Fig. 1.3); there some limit periodic sets are not even curves. 3. The periodic orbits for n > 0 near slow–fast cycles that are limit periodic sets are commonly called Relaxation oscillations. The pronoun “relaxation” refers to the part of the periodic orbit that is close to the slow arcs of the slow–fast cycle, where the dynamics is more or less relaxed in contrast to parts where it jumps from one arc to another. 

4.2 Elementary Slow–Fast Segments

41

Definition 4.3 (Cyclicity of a Slow–Fast Cycle) Let M be orientable. Let  be a slow–fast cycle of X,λ for (, λ) = (0, λ0 ). Then: (1)  is said to have finite cyclicity in X,λ if there exists a finite N such that any X,λ with (, λ) ∼ (0, λ0 ) has at most N limit cycles Hausdorff close to ; if not  is said to have infinite cyclicity. (2) The minimum of such N is called the Cyclicity of  in X,λ .

Main Problem Let X,λ be a smooth slow–fast family of vector fields on an arbitrary orientable smooth surface. Suppose that for (, λ) = (0, λ0 ) we have a slow–fast cycle . What is the cyclicity of  in X,λ , for (, λ) ∼ (0, λ0 )?

In the whole book we will suppose that the slow vector field has at most isolated zeros. Common slow–fast cycles have been studied in the literature (see for example [Gra87, MKKR94], . . . ), often restricting to the cubic-shaped ones, and not considering the more general forms (but on the other hand also in higher dimensions). The central idea is that orbits near a common (attracting respectively repelling) slow–fast cycle can do nothing else but get attracted (in positive respectively negative time) toward each other and ultimately converge to a limit cycle. However, looking into more general shapes of common slow–fast cycle, canard phenomena can be present and it reveals not to be guaranteed that periodic orbits are present near any common slow–fast cycle. We will report on it in Chap. 7.

4.2 Elementary Slow–Fast Segments In this book we will work with both common slow–fast cycles and canard slow– fast cycles, shortly called canard cycles. We will also permit to have contact points of singularity order 1. Once in a while we will also accept isolated zeros (of finite order) of the slow vector field on Sλ \ Cλ . To study all these slow–fast cycles we will decompose them into elementary slow–fast segments. This decomposition is based on the fact that at all points of a slow–fast cycle one has convenient normal forms. Along fast parts (at points away from Sλ ) the flow-box theorem yields local normal forms; at points on Sλ \ Cλ , i.e. at normally hyperbolic points of the critical curve, the normal form (2.2) (for equivalence) can be used; at contact points normal form (2.1) (for equivalence) can be used. Later, we will even use stronger normal forms at different parts of the cycle. Since a slow–fast cycle is compact, we can cover it by a finite number of neighborhoods where normal forms are valid. The intersection of the slow–fast cycle with such sufficiently small neighborhoods is called an elementary slow–fast segment. In Figs. 4.3, 4.4, 4.5, 4.6, 4.7, and 4.8, we

42 Fig. 4.3 Regular fast and normally hyperbolic slow and slow–fast segments

Fig. 4.4 Normally hyperbolic slow dynamics with a zero of finite singularity order, occurring in a slow–fast cycle

Fig. 4.5 Regular contact points of even contact order, occurring in a slow–fast cycle

Fig. 4.6 Regular contact points of odd contact order, occurring in a slow–fast cycle

4 Slow–Fast Cycles

4.2 Elementary Slow–Fast Segments

43

Fig. 4.7 Contact points of even contact order with singularity order 1 and index −1 (left) +1 (right), occurring in a slow–fast cycle

Fig. 4.8 Contact points of odd contact order with singularity order 1 and index −1, occurring in a slow–fast cycle

present a list of all possibilities that we aim to treat; without claiming that this list is exhaustive, we make the following assumption:

Assumption on Slow–Fast Cycles In this book we only consider slow–fast cycles that are a (finite) succession of elementary slow–fast segments as represented in Figs. 4.3, 4.4, 4.5, 4.6, 4.7, and 4.8.

Segments Away from Contact Points Away from the critical curve, the dynamics is locally as shown in Fig. 4.3(left). On a normally hyperbolic part of the critical curve without singularities in the slow dynamics, we can have the attracting and the repelling case. All possible passages are shown in the rest of Fig. 4.3. Away from contact points, these are the only segments allowed in regular common or canard cycles. When isolated singularities in the slow dynamics are allowed, like in arbitrary slow–fast cycles, the slow dynamics, near λ = λ0 , is locally given by dr = β0 (r − r0 )m + O((r − r0 )m+1 ) + O(λ − λ0 ), ds

44

4 Slow–Fast Cycles

where m is the singularity order of the point at r = r0 and where β0 = 0. Depending on the sign of β0 and the signature (odd or even) of m, we distinguish different possibilities of slow–fast segments containing the singularity, see Fig. 4.4. Segments Near Regular Contact Points Regular contact points can be of odd contact order or even contact order. For regular contact points of even contact order, we distinguish the two cases (1a+ ) and (1a− ) from Fig. 3.4. In the first case, one branch can be followed to approach the contact point and three possible branches can be followed to leave; the situation is vice versa in the second case. This amounts to six different cases, shown in Fig. 4.5. Regular contact points of odd contact order were categorized as cases (1b+ ) and (1b− ) in Fig. 3.4. In the first case, three branches can be followed to approach the contact point, and there is only one possibility to leave, vice versa in the second case. The six cases are shown in Fig. 4.6. Segments Near Singular Contact Points We describe slow–fast segments for even contact orders and odd contact orders, and for singularities with index +1 or −1. We will restrict to listing the possible slow– fast segments near contact points of singularity order 1. The cases of even contact order are listed as cases (2a− ) and (2a+ ) in Fig. 3.4. Case (2a− ) respectively (2a+ ) corresponds to singularity order 1 and singularity index +1 respectively −1 (and more generally to odd singularity order). In both cases, a slow–fast cycle can approach the contact points along two possible branches and can leave along two possible branches. Hence, both for singularity index +1 and for singularity index −1, there are 4 possible slow–fast segments, listed in Fig. 4.7. Singular contact points of odd contact order and odd singularity order are distinguished in Fig. 3.4 as cases (2b± ) and (2c± ) and (1a± ) (1b± ) may occur for an odd contact order and even singularity order. It is easily seen that in case (2b+ ) there is no way for a slow–fast cycle to leave the contact point, and similarly, in case (2b− ) there is no way for a slow–fast cycle to approach the contact point. Hence, with these two cases there will correspond to slow–fast segments. The segments appearing in cases (2c± ) are listed in Fig. 4.8.

4.3 Regular Common Cycles We recall that in a common slow–fast cycle all arcs are attracting or all are repelling. Since repelling common cycles are the time-reversed equivalents of attracting common cycles, we will focus only on the latter, and at a first level we will discuss regular common slow–fast cycles: Definition 4.4 A Regular common cycle is a common cycle where all involved contact points are regular and where the slow vector field on all slow arcs is without singular points.

4.3 Regular Common Cycles

45

regular fast regular slow

hyperbolic fast-slow

slow-fast (jump)

funnel fast-fast

odd fast-slow regular type

canard fast-slow

canard fast-fast

odd fast-slow cuspidal type

odd slow-slow

Fig. 4.9 Elementary attracting regular slow–fast segments

The list of possible elementary slow–fast segments contained in a regular common cycle (an attracting one) is represented in Fig. 4.9. A repelling regular common slow–fast cycle will consist of the same segments in which time is reversed. It may appear obvious to the reader that common cycles are in a sense very robust objects, but it is not true in general: it is for example easy to prove that a common cycle containing a segment from Fig. 4.9 of type “canard fast-slow” or “canard fastfast” cannot be guaranteed to develop nearby periodic orbits for  > 0. We now proceed by defining a more restrictive class of slow–fast systems that are always approached by nearby orbits, see Theorem 7.2. Definition 4.5 A Strongly common attracting cycle is a regular common attracting cycle that is composed of slow–fast segments in Fig. 4.9 except the two canardtype segments labeled “canard fast-slow” and “canard fast-fast.” A Strongly common repelling cycle is a slow–fast cycle that is strongly common attracting after time reversal. Results concerning common cycles and strongly common cycles are given in Chap. 7. The canard fast–slow segment and the canard fast–fast segment, after reversing time, will play an essential role in, respectively, Chaps. 14 and 15.

46

4 Slow–Fast Cycles

4.4 Canard Cycles Per definition, a (slow–fast) canard cycle contains both attracting and repelling parts of the critical curve. As one travels (in positive direction of time) through the cycle, there must hence be two or more changes of stability that we will be precisely described now. From Repelling to Attracting Through Layers Browsing through the different slow–fast cycles, it is easy to spot several ways to connect an elementary slow–fast segment containing a repelling arc to one containing an attracting arc. (In Fig. 4.7(left) there is even an elementary segment with such a connection within.) The connection that usually appears is a connection between two “hyperbolic slow–fast” segments (see Fig. 4.9), i.e. the case in which between the repelling and attracting parts there is a direct fast connection. The observation to be made here is that slow–fast cycles containing such a fast connection appear in a family. The related family of connections is usually called a “layer” of fast connections (see Fig. 4.10 where two layers are present). In order to study cyclicity problems we consider in each layer a regular section transverse to the flow in the layer, and the section can be parameterized in a smooth way by a so-called layer variable. Definition 4.6 (Layers) Given a slow–fast family of vector fields X,λ . A layer section σ at λ = λ0 is a connected open section transverse to the flow of X0,λ0 and so that all orbits through σ cut σ exactly one time have their ω-limit on a hyperbolically attracting (regular) arc and their α-limit on a hyperbolically repelling (regular) arc. See Fig. 4.11. The Layer associated to σ is the open part of the phase portrait formed by the union of all orbits through σ . A Layer variable v is a regular variable parameterizing σ , i.e. there is an implied chart function (v0 , v1 ) → σ . Two types of layer are distinguished, see Fig. 4.11: a Terminal layer and a Dodging layer. In the formulation of the results we will rarely have to distinguish between the two types of layers, but in at least one section, their difference is important (see Sect. 13.3.4). Fig. 4.10 Canard cycles: decomposed in attracting/repelling parts, layers, and canard connections

canard connection layer

layer canard connection

4.4 Canard Cycles

47

σ

γr

σ

γa

γa

γr

Fig. 4.11 Examples of layers with relevant parts from a slow–fast cycle. The section σ is parameterized by a layer variable v and the hatched region is the actual layer. Increasing v (going upward on σ ) means an increase of the repulsion and decrease of attraction in the left case, or an increase in both repulsion and attraction in the right case. Two additional scenarios are timereversed counterparts. The left situation is called a dodging layer, and to the right a terminal layer is found

Remark 4.2 Upon optimally choosing the layer section, one can assume the layer to be maximal: at the boundary of the layer, one encounters an extra degeneracy in the form of a singularity in the fast or slow vector field, loss of hyperbolicity at one or both branches of the critical curve, or other problems. See Fig. 4.10 for a vector field containing different layers. Clearly, using the boundary of a layer may sometimes still give the possibility to connect repelling segments to attracting segments. Canard cycles positioned in such a way will be called transitory because beyond the boundary the shape of the canard cycle generally changes. Later in this book we will pay a lot of attention to transitory canard cycles. From Attracting to Repelling Through Canard Connections In one of the pictures of Fig. 4.7 we see an elementary slow–fast segment representing a direct transition from an attracting branch to a repelling one. One can also connect an attracting branch to a repelling arc via a fast connection. In this case, the fast connection should connect at least two contact points. The most typical scenario is to connect two jump points, but many other connections are possible. In general, a canard connection could be defined as a part of a slow–fast cycle where, following the orientation of the slow–fast cycle imposed by the slow vector field and fast vector field, the initial part is a slow arc of attracting type and the final part is a slow arc of repelling type. In this book we will restrict to two types of canard connections that we will call elementary canard connections; they are clearly the least degenerate possibilities: Definition 4.7 An Elementary canard connection is either a connection from attracting to repelling within a single elementary slow–fast segment around a slow– fast Hopf point (called a Hopf canard connection), or a union of two slow–fast segments consisting of a regular fast segment that connects two regular jump points (called a Jump canard connection). (Both are seen in Fig. 4.10.)

48

4 Slow–Fast Cycles

Fig. 4.12 Attracting sequences. The left picture satisfies Definition 4.8. The situation to the right is considered more degenerate and will be considered only partially in this book (see Theorem 7.3 later on)

We stress that in the definition of a jump canard connection both contact points involved are supposed to be generic. We will come back to the issue in Chap. 6 when we deal with the so-called generic canard breaking mechanisms, where the notion “elementary canard connection” is extended with a breaking parameter. The Attracting Part and the Repelling Part In principle, attracting parts could be formed of any kind of combination of slow– fast segments containing only attracting slow arcs. In order to be able to state general results, we will, however, need to restrict the possibilities. Definition 4.8 (Attracting Sequence and Repelling Sequence) Given λ0 ∈ . A subset , diffeomorphic to a piecewise smooth segment, is an Attracting sequence if it consists of a (finite) succession of attracting slow–fast segments of jump type, regular fast, regular slow, and hyperbolic fast–slow types as in Fig. 4.9. The jumps must occur at a generic jump point (and must not be of “funnel fast-fast, canard fast-slow, canard fast-fast” types). A Repelling sequence is what we get by taking an attracting sequence and reversing time. We refer to Fig. 4.12 for an attracting sequence that satisfies Definition 4.8 (left situation) and for an example of a sequence that consists of nothing but attracting sequences but that does not satisfy our definition.

4.5 Ordinary Canard Cycles Recall that canard cycles are slow–fast cycles that contain both repelling and attracting slow arcs. Sect. 4.4 is devoted to describing topological possibilities for canard cycles in general and is based on the observation that if a canard cycle contains both attracting and repelling arcs, there should at least be one part of the cycle where a change of stability from repelling to attracting occurs. Most often, the slow–fast segment that connects a repelling arc to an attracting arc is simply a fast orbit between two normally hyperbolic points of those arcs. Such a connection is persistent in the sense that it is not an isolated situation but occurs in a layer of fast connections (Fig. 4.13).

4.5 Ordinary Canard Cycles

49

Fig. 4.13 An ordinary canard cycle

C

− γC

γ−

γ

+ γC

γ+

Definition 4.9 ((1-Layer) Ordinary Canard Cycle) A slow–fast cycle  of X0,λ0 is called an “Ordinary canard cycle” if it is the union of: (1) An elementary canard connection C between an attracting slow arc γC− and a repelling slow arc γC+ (2) A fast orbit γ inside a layer of X0,λ0 , the layer being situated between a repelling arc γ + and an attracting arc γ − (3) An attracting sequence starting with γ − and ending with some fast orbit tending to γC− (unless γ − = γC− ) (4) A repelling sequence starting in reverse time with γC+ and ending with some fast orbit tending to γ + (unless γ + = γC+ ) Remark 4.3 In view of building a general theory the chosen names are maybe too general. Since the attracting and repelling sequences only contain generic jump points, it might be better to add the adjective generic. We will, however, not do this since we do not intend to work with more degenerate types of attracting and repelling sequences in this book. Also the ordinary canard cycles might better be called 1-layer ordinary canard cycles. In [DR07b], canard cycles have been studied that pass through two fast layers and contain two elementary canard connections. They could be called 2layer ordinary canard cycles. Similarly in [DR08], and for N ≥ 3, results, even rather unexpected results, have been given on canard cycles that could be called Nlayer ordinary canard cycles, since they pass through N fast layers and contain N elementary canard connections. Since the fast orbit γ of a 1-layer ordinary canard cycle is part of a layer, it crosses the layer section σ at a point corresponding to the value v0 of the layer variable v. It is important to notice that by definition, such ordinary canard cycles appear in v-families. We will discuss this more in detail, together with their dependence on λ, in Chap. 6.

50

4 Slow–Fast Cycles

4.6 Transitory Cycles Canard cycles that are not ordinary are quite abundant, as there are many reasons why a canard cycle may fail to be ordinary. Instead of giving an exhaustive list of possibilities, we will describe several scenarios that will be studied in detail in this book or that could be studied with the methods presented in this book. We will not include a strict definition of transitory cycle but simply label the non-ordinary canard cycles that we present by Transitory canard cycles. Noticing that one-layer canard cycles are parameterized by a layer variable v ∈ σ , transitory canard cycles may be present at the boundary of σ . Ordinary canards are typically studied in compact subsets of open layers, the compactness giving the uniformity required in the proofs of the results. In order to obtain control of the full open layer, it is natural to look at the boundary of these layers. Quite often one finds at the boundary of a layer a non-ordinary slow–fast cycle that marks the end of this layer but possibly at the same time the beginning of another layer. As such these non-ordinary cycles are transitory. Figure 4.14 shows an example of transitory canard cycle whose study will be made in detail in Chap. 14. We will now discuss some possibilities in detail.

4.6.1 Singular Points in the Slow Vector Field: Transition to Singular Homoclinic The presence of a singular point in the (1-dimensional) slow vector field, at an attracting or a repelling branch, marks the boundary of a layer. By reversing time if necessary we may assume the singular point is located on the attracting branch. In the most generic case, the singular point is a hyperbolic one and, hence, is a slow–fast hyperbolic saddle or a slow–fast hyperbolic node (the notion slow–fast hyperbolic point refers to hyperbolicity for  = 0 small enough). We refer to Fig. 4.15 for the saddle cases and the possible slow–fast segments that include the singular point. We do not discuss the slow–fast node case since it cannot appear as part of a limit periodic set. Clearly the notion slow–fast cycle could be generalized to a slow–fast homoclinic. We refer to [DMD08] for cyclicity results of such slow– fast homoclinics. In fact the finite cyclicity can be proven as an exercise, using the detailed results in Part II. Fig. 4.14 An example of transitory canard cycle

4.6 Transitory Cycles

51

σ1

γa s

γr

σ2

Fig. 4.15 Singular points in the slow vector field delimit layers defined through the layer sections σ1 and σ2 . A saddle s located on γa and a slow–fast segment that includes s

s

s γa γr

γr

σ

s

γa γr

σ

γa

σ

Fig. 4.16 Singular points in the fast vector field delimit the layer defined through the layer section σ

Similarly a saddle may appear in the fast dynamics, see Fig. 4.16. We recall that our notion of slow–fast vector field excludes the possibility of isolated singular points outside the critical set, but it should nevertheless be clear that the techniques in this book suffice to deduce cyclicity of slow–fast homoclinics (for example in the generic case when the ratio of eigenvalues is not equal to −1).

4.6.2 Loss of Hyperbolicity on One of the Two Branches in a Layer Given a layer between γr and γa , it is possible that the fast vector field looses normal hyperbolicity either on γr or on γa , at the boundary of the layer. We first distinguish loss of normal hyperbolicity at γa . Generically, the boundary point ∂γa will be a jump point, denoted J . By possibly reversing time we may assume the slow vector field on the slow arcs around J point toward J , and we choose local coordinates of J so that the critical curve locally is an upward parabola. Generically, the layer between a repelling branch and the attracting branch of J has at the boundary a trajectory that tends toward J . The trajectory naturally glues together with a trajectory from J possibly and most interestingly toward another attracting branch, see Figs. 4.17 and 5.2(2). These transitory cycles will be treated later in this book, namely in Chaps. 14 and 15.

52

4 Slow–Fast Cycles

σ1

γr1

J

σ1

γa1

σ2 σ1

γr1

γa2 γr2

γr1

γa2 γr2

J σ2

J σ2 σ1

γa1

γr1

γa2 γr2

γa2 γr2

J σ2

γa1

γa1

Fig. 4.17 Generic loss of normal hyperbolicity on γa2 in the boundary of the layer section σ1

ε=0

0 0, a family of segments of regular orbits of X,λ with the property that γ,λ tends uniformly to a segment γλ ⊂ Sλ \ (Cλ ∪ λ ) as  → 0. (In other words, the orbits approach a compact part of the critical curve away from contact points and singularities in the slow vector field.) Then, 

 div X,λ dt =

lim 

→0

γ,λ

Vλ ds.

(5.1)

γλ

From Proposition 5.2, we clearly see the relevance of the integral in the right hand side of (5.1). Definition 5.1 We consider an interval γλ ⊂ Sλ \ (Cλ ∪ λ ). Then the Slow divergence integral along γλ is defined as  I (γλ ) =

Vλ ds.

(5.2)

γλ

Remarks 5.1 1. Seen that the definition of the slow divergence integral I (γλ ) does not really require the notion of divergence, it might have been better to have called it differently, like e.g. slow trace integral. We will however continue using the commonly known name. 2. Definition 5.1 is obviously intrinsic, i.e. coordinate-free and without reference to an area form. In this formula, each Vλ is a function and ds is a differential 1-

56

5 The Slow Divergence Integral

form. Then we can integrate the differential 1-form Vλ ds, i.e. we can write (5.2), without mentioning any parameterization of γλ . Of course, by choosing a point p0 in the interior of γλ , we can use the trajectory ϕλ (s) of the slow vector field Q˜ λ through p0 (ϕλ (0) = p0 ) to parameterize γλ by s. If s1 and s2 are the values of s corresponding to the end points of γλ , we have a more intuitive formula, written in the parameterization s:  I (γλ ) =

s2

Vλ (ϕλ (s))ds. s1

3. We also refer to Remark 12.2, where the intrinsic nature of the slow divergence integral is proved by relating it to transition maps. 

5.3 Invariance of the Slow Divergence Integral Under Equivalences In the following lemma, we will see that the differential 1-form Vλ ds is even invariant under smooth (, λ)-equivalences on X,λ . Recall that this was not the case for Vλ , neither for ds. In view of permitting working with Cr normal forms for Cr -equivalence, we consider in the next lemma Cr -equivalence, with r ≥ 1. Lemma 5.2 Let g,λ be a Cr family of functions, with r ≥ 1, defined on a neighborhood of a point p ∈ Sλ \ (Cλ ∪ λ ), with g0,λ (p) = 0. Let σλ be the differential 1-form Vλ ds associated with X,λ and σ¯ λ be the differential 1-form Vλ ds associated with g,λ X,λ . Then σλ = σ¯ λ in a neighborhood  of p on Sλ \ (Cλ ∪ λ ). As a consequence, the slow divergence integral γλ Vλ ds is invariant if we replace X,λ by a Cr -equivalent family in a neighborhood of γλ . Proof Let p ∈ Sλ \ (Cλ ∪ λ ) be a point where g0,λ (p) = 0; recall that also ∂X Q˜ λ (p) = 0. Referring to Eq. (1.1), let Qλ = ∂,λ |=0 . At p, we see that

 

X ) ∂(g ∂ ,λ ,λ 2

Q¯ λ = g,λ (X0,λ + Qλ + O( )

=

∂ ∂ =0 =0 =

∂g,λ |=0 .X0,λ + g0,λ Qλ ∂

so that, if Q˜ λ is the slow vector field of X,λ , then g0,λ Q˜ λ is the slow vector field of ∂ g,λ X,λ (as ∂ g,λ |=0 X0,λ is parallel to Z0,λ ).

5.4 Slow Divergence Integral Near Singularities of the Slow Vector Field

57

˜ λ )(p) = 1 has to be preserved, we deduce that ds(p) is Since the relation ds(Q ds(p) replaced by g0,λ (p) . Now, the value Vλ (p) is replaced by g0,λ (p)Vλ (p), and we can conclude that σ¯ λ (p) = g0,λ (p)Vλ (p)

ds(p) = Vλ (p)ds(p) = σλ (p). g0,λ (p)

The consequence for the slow divergence integral is direct.



To explicitly compute I (γλ ), we introduce a parameterization ϕλ (r) of γλ . If the interval γλ is parameterized by [r1 , r2 ], we will often write Iλ (r1 , r2 ) instead of I (γλ ). Proposition 5.3 Let ϕλ : [r1 , r2 ] → M be a parameterization of the interval γλ ⊂ Sλ \ (Cλ ∪ λ ), and let qλ (r) be the expression of Q˜ λ given by (3.2). Then, in the variable r, the slow divergence integral is given by 

r2

I (γλ ) =

r1

  dr . Vλ ϕλ (r) qλ (r)

(5.3)

Recall that qλ is nowhere zero on γλ . Proof It follows from (3.2) that dϕλ (r). Q˜ λ (ϕλ (r)) = qλ (r) dr On the other hand, we have that 

 dϕλ dr dϕλ = = Q˜ λ ϕλ r(s) . ds ds dr Comparing these two equations, we obtain that dr ds = qλ (r), which is the differential equation of the slow dynamics. To obtain the expression (5.3), it suffices to substitute ds by qλdr(r) and to compose the parameterization ϕλ (r) with the function Vλ . 

5.4 Slow Divergence Integral Near Singularities of the Slow Vector Field If we admit that the slow vector field has zeros on γλ ⊂ Sλ \ Cλ , then the slow divergence integral along the segment is improper, which is coherent with (5.3). We could say that it is equal to +∞ (respectively, −∞) if the vector field is normally repelling (respectively, attracting) along the segment.

58

5 The Slow Divergence Integral

5.5 Slow Divergence Integral Near Contact Points Knowing that the slow divergence integral is invariant under smooth equivalences, we can consider its expression in a normal form (2.1), in which we use t as time, with s = t, and which we repeat here for the sake of convenience:  dx dt dy dt

= y− f (x, λ) 

=  g(x, , λ) + y − f (x, λ) h(x, y, , λ) .

(5.4)

In this normal form, Sλ is equal to {(x, y)|y = f (x, λ)}, and we can parameterize it by the variable x. If we now restrict γλ to [x1 , x2 ], with [x1 , x2 ] not containing the contact point and not containing zeros of the slow vector field, then on [x1 , x2 ] we get the expression for ds from Eq. (3.5), implying that  Iλ (x1 , x2 ) = −

x2

x1

1  ∂fλ 2 (x) dx, gλ (x) ∂x

(5.5)

where gλ (x) = g(x, 0, λ). On such a normal form (5.4), we can equally compute the slow divergence integral along a segment γλ when one of the end points is a contact point, while along the rest of the segment, the vector field is normally hyperbolic and the slow vector field is regular. We suppose that the segment is parameterized by [x1 , 0], with x1 < 0. According to Eq. (5.5), we now define the slow divergence along γλ as I (γλ ) = lim

x2 →0−



 −

x2 x1

2  ( ∂f ∂x (x, λ)) dx , g(x, 0, λ)

(5.6)

keeping x2 ∈ [x1 , 0[. If g(0, 0, λ) = 0, and hence if the contact point is regular, we already know that the slow vector field is infinite at x = 0, but we see that the slow divergence integral I (γλ ) is finite. The slow divergence integral I (γλ ) is also finite when g(0, 0, λ) = 0 under the condition that the multiplicity of the zero of ∂f ∂x (x, λ) at x = 0 is sufficiently high. For a Morse-type contact point, the slow divergence integral I (γλ ) is finite as long as the multiplicity of the zero of g(x, 0, λ) at 0 does not exceed 2. In case the multiplicity is 1, the slow vector field will also extend in a regular way at x = 0. These contact points have been studied in [DMDR11]. In case the multiplicity is 2, the slow vector field has a simple zero at x = 0. For a study of this case, we refer to [DMD08]. In any case we will say that a contact has finite slow divergence if the limit in Eq. (5.6) is finite. Seen the intrinsic nature of

5.6 Slow Divergence Integral of a Slow–Fast Cycle

59

the slow divergence integral, as well as of the contact order and the singularity order of a contact point, we can state a definition as follows: Definition 5.2 (Contact Point with Finite Slow Divergence) We say that a contact point p ∈ Cλ has Finite slow divergence if sλ (p) ≤ 2(nλ (p) − 1),

(5.7)

where, as in Definition 2.1, sλ (p) stands for the singularity order and nλ (p) for the contact order of the contact point. Remark 5.2 Fulfilling equation (5.7) in a normal form (2.1) is equivalent to saying that the limit in (5.6) is finite. The finiteness of the slow divergence integral applies to any of the two slow arcs adherent to p.

5.6 Slow Divergence Integral of a Slow–Fast Cycle Definition 5.3 (Slow Divergence Integral of a Slow–Fast Cycle) Let X,λ be a smooth slow–fast family of vector fields on some smooth surface M, and let  be a slow–fast cycle of X0,λ0 . We suppose that all the contact points of  are nilpotent, of finite contact order and having finite slow divergence. We also suppose that the slow vector field on Sλ \ Cλ is regular. In that case we define the slow divergence integral of the slow–fast cycle  to be the finite number that we obtain by adding the respective slow divergence integrals of the (maximal) normally hyperbolic slow segments of . We denote it by I () or  div X0,λ0 ds. Cases in which the slow divergence integral is infinite at a contact point are much more difficult to be studied (see e.g. [DMD08, DMD10, DMD11c]). In any case, also when the slow divergence integral is finite at a contact point, a further study near the contact point is needed for understanding the precise relation between the slow divergence integral and the genuine divergence integral. Slow Divergence Integral of Cycles in Layers If the slow–fast cycle crosses a layer section σ parameterized by v, it is part of a family of slow–fast cycles v . In that case, it is clear that the slow divergence integral is depending smoothly on v and can be denoted I (v). Depending on the type of layer, the behavior of I may be different. Referring to Fig. 4.11 for two types of layers, the following is clear: Lemma 5.3 If a family of slow–fast cycles is parameterized by v, the intersection point on a layer section in a dodging layer, then I (v) is nonzero everywhere, implying that I is strictly monotonous. As a consequence of this lemma, we will infer that bifurcations of limit cycles crossing a dodging layer will be at most of codimension 1, i.e. of saddle-node type.

60

5 The Slow Divergence Integral

In a terminal layer, on the other hand, limit cycle bifurcations equivalent to any kind of elementary catastrophe are possible.

5.7 Examples 5.7.1 Van der Pol The Van der Pol system 

2

x˙ = y − x2 − y˙ = (a − x)

x3 3

has a slow–fast Hopf point at (x, y) = (0, 0) for a = 0. The system has two layers of canards, one layer for the “Canards without head” v1 , parameterized by v, the y-coordinate of the intersection point of the canard with the line x = 0, and one layer for the canards with head v2 , parameterized using the same symbol v, which is now the y-intersection point  the canard with the line x = −1, see Fig. 5.2. The  of 1 parameter v is kept inside 0, 6 .

The slow vector field is given by x = −1/(x + 1), and the divergence is −(x + As a consequence, the slow divergence integral of v1 and v2 is, respectively, given by

x 2 ).

 Iv1 =

α 1 (v) ω1 (v)

 x(x + 1)2 dx,

Iv2 =

−1

ω2 (v)

 x(x + 1)2 dx +

0

x(x + 1)2 dx.

1/2

In these expressions, the x-coordinates of the α and ω limits of the related fast orbits are being used. It is shown in [DR96] that Iv2 ≤ Iv1 < 0,

(1)

(2)

Fig. 5.2 Canard cycles in Van der Pol’s equation

  ∀v ∈ 0, 16 .

(3)

(4)

5.7 Examples

61

The first inequality is obtained by direct comparison of the integrals; it is an exercise d 1 to prove the second inequality by showing that dv Iv < 0.

5.7.2 Some Canards in Quartic Liénard Systems The two-parameter slow–fast family 

x˙ = y + x 2 − cx 3 − x 4 y˙ = (a − x)

 √ √  has a jump connection for c = 0 and a ∈ − 2/2, 2/2 . There are three Morse √ critical points, located at x = 0, and x = ± 2/2. We choose a vertical √ layer section on top of the critical point to the left, i.e. we choose σ = {x = − 2/2, y > 14 }. Instead of parameterizing σ by the y-coordinate, we parameterize it by the leftmost x-coordinate of the horizontal line cutting σ , denoted by v. The family of slow–fast cycles v is hence parameterized by its leftmost x-coordinate v. We refer to Fig. 5.3 for the different shapes of the slow–fast cycles under consideration, for a > 0. The case a < 0 is similar after changing (x, a, t) → (−x, −a, −t). On these slow–fast cycles, there is always a canard connection between the two local minima, followed by a slow arc on the leftmost branch. √  When v ∈ −1, − 2/2 , the slow–fast cycle jumps to the arc that leads it to the local maximum, where it jumps again to the rightmost branch. When v > −1, the slow–fast cycle immediately jumps at height F (v) to the rightmost branch, with F (v) = v 4 − v 2 . If we write I (v, a) for the slow divergence integral of v , then I (v, a) ≤ I (−1, a) if v ≥ −1. It hence suffices to restrict to {v ≤ −1} in showing that I (v, a) < 0. We have 

√ 1/ 2

 v F (x)2 F (x)2 I (v, a) = dx + dx √ x−a −v −1/ 2 x − a    1/√2  |v| 1 F (x)2 1 − = F (x)2 dx. dx = −2a √ 2 2 x−a x+a −v 1/ 2 x − a

Fig. 5.3 Slow fast cycles in a quartic system

62

5 The Slow Divergence Integral

We obtain three properties: we have not only I (v, 0) ≡ 0 and that I (v, a) < 0 for all a > 0.

∂I ∂a (v, 0)

< 0 but also

5.7.3 Zeros in the Slow Divergence Integral It is fairly easy to provide an example where the slow divergence integral does not have a fixed sign: 

for a = 0, where  F (x) =

x

x˙ = y − F (x) y˙ = (a − x)

f (s) ds, with f (x) = x((x − 1)2 + λ)((x + 2)2 + μ).

0

The geometry of the critical curve is similar to that of a parabola: there is one contact point, of slow–fast Hopf type, separating an attracting slow arc (x > 0) from a repelling one (x < 0). Let v > 0 parameterize the canards in the single layer, by identifying v as the y-coordinate of the intersection point of the slow–fast cycle with x = 0. Let v1 = F (1), v2 = F (−2). For λ = 0, the slow vector field has a saddle-node singularity at x = 1, and for μ = 0, the slow vector field has a saddle-node singularity at x = −2. It means that I (v1 ) tends to −∞ as λ → 0, uniformly in terms of μ, but on the other hand for fixed λ close enough to 0 when I (v1 ) < 0, the integral I (v2 ) tends to +∞ as μ → 0. So for sufficiently small μ > 0, the intermediate value theorem guarantees a zero of the slow divergence integral. We refer for example to [DMD11b] and [LZ13] for more elaborate models with zeros in the slow divergence integral and the implication it has in terms of limit cycles. This implication is also treated in detail in Part III.

Chapter 6

Breaking Mechanisms

We will now give a short description of the generic breaking mechanisms that can produce closed orbits. There are two such generic breaking mechanisms, one called Hopf breaking mechanism, which has been studied in [DR01a], and the other called jump breaking mechanism that has been studied in [DR01b]. We obtain them by starting with an elementary canard connection (see Definition 4.7) and adding some genericity condition on the perturbation. For the sake of convenience, we first recall the notions of generic jump point and generic turning point as given in Sect. 2.4, followed by some extra information on the elementary canard connections.

6.1 Normal Forms for Generic Jump Points and Generic Turning Points In Definition 2.3, we have defined a generic jump point and in Definition 2.4 a slow–fast Hopf point. Both are contact points of order 2. Knowing that the singular curve can be written as y = fλ (x) with fλ0 (x) = α(λ)x 2 (1 + O(x)), we infer that translations in x and y, together with some rescaling in (x, y), permit to write the normal form (2.1) as 

x˙ = y− x 2 + x 3 h1 (x, λ) 

y˙ =  g(x, , λ) + y − fλ (x) h(x, y, , λ) ,

(6.1)

where g and h are smooth. Recall that it is a normal form for C∞ -equivalence. Without loss of generality, we can suppose that h1 √≡ 0. This follows from Remark 2.1(a) and the fact that x 2 − x 3 h1 (x, λ) = (x 1 − xh1 (x, λ))2 , showing that a λ-dependent choice of coordinates (x, y) permits to write the family of singular curves simply as y = x 2 . We prefer however to keep the expression of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_6

63

64

6 Breaking Mechanisms

the critical curve as general as in (6.1). For the generic jump case, we require that g(0, 0, λ0 ) = 0, permitting, with an extra (, λ)-dependent rescaling in (x, y, t) to reduce (6.1), to 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (E + xh2 (x, , λ) + yh3 (x, y, , λ)),

(6.2)

where E = ±1 and all hi are C∞ functions. In case E = +1, the slow vector field along the singular curve points away from the origin, and it is case (1a+ ) in Fig. 3.4. In case E = −1, the slow vector field points toward the origin (case (1a− ) in Fig. 3.4, rotated). E = −1 can be changed into E = +1 if we also admit a time reversal, e.g. using (x, y, t) → (−x, y, −t). In case of a slow–fast Hopf point and starting with (6.1), we now suppose (see ∂g Definition 2.5) that g(0, 0, λ0 ) = 0 and ∂x (0, 0, λ0 ) < 0. It permits, after an extra rescaling in (x, y, t), to write (6.1) as 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (g(0, 0, λ) − x + x 2 h2 (x, , λ) + yh3 (x, y, , λ)).

(6.3)

As in the jump case, we could suppose that h1 ≡ 0 without loss of generality. The system locally corresponds to case (2a− ) in Fig. 3.4.

6.2 Generic Jump Breaking Mechanism We get a Generic jump breaking mechanism when adding a breaking parameter to an elementary jump connection. We therefore suppose to have two jump points p− and p+ that are connected by a fast orbit γ on a common leaf λ0 ,p− = λ0 ,p+ of the fast foliation. Suppose that γ has p− as α-limit and p+ as ω-limit. There are two possibilities, as represented in Fig. 6.1. If we want γ to be part of a canard cycle, then the slow dynamics have to point toward for the jump point p− and away from the jump point p+ .

Fig. 6.1 Two possible elementary jump connections

6.3 Generic Hopf Breaking Mechanism

65

Changing the parameter λ ∼ λ0 , the jump points p− and p+ perturb smoothly to jump points p− (λ) and p+ (λ), with p− (λ0 ) = p− and p+ (λ0 ) = p+ . The connection γ is typically not retained: it is broken. Transverse to the fast orbit γ we choose a segment τ on which we choose a regular parameter. For  = 0 and for λ ∼ λ0 , the leaves λ,p± of the fast foliation cut τ at a single point h± (λ). By definition, h− (λ0 ) = h+ (λ0 ), or in other words, h− (λ0 ) − h+ (λ0 ) = 0. Generically, we can assume that λ → h− (λ) − h+ (λ) is a submersion at λ = λ0 , and we will assume this property in studying closed orbits that pass near γ for λ ∼ λ0 . By a smooth reparameterization, we suppose that λ = (a, μ), where a, defined as a = h1 (λ) − h2 (λ), is called a breaking parameter. We will then write the slow–fast system as X,a,μ .

6.3 Generic Hopf Breaking Mechanism We get a Generic Hopf breaking mechanism by adding a breaking parameter to a slow–fast Hopf point. Working with the normal form (6.3), we can now impose Definition 2.5 to get a generic turning point, i.e. we suppose that λ → g(0, 0, λ) is a submersion, permitting to choose a = g(0, 0, λ) as a new independent parameter. By a smooth reparameterization, we can suppose that λ = (a, μ), where a will be called a breaking parameter. Expression (6.3) changes into 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (a − x + x 2 h2 (x, , λ) + yh3 (x, y, , λ)),

(6.4)

where all hi are smooth and where we can suppose, without loss of generality, that h1 ≡ 0. Remark that for a = 0, the unique contact point in (6.4) is a jump point: the singular contact point has unfolded in a regular contact point and a slow singularity near that contact point. The slow singularity is of attracting node type for a > 0 and

66

6 Breaking Mechanisms

of repelling node type for a < 0, which explains that for a = 0, there is no canard passage. The transition from repelling node to attracting node as a crosses the origin from left to right is in fact part of a more complete transition Hopf

repelling node −→ repelling focus −→ attracting focus −→ attracting node that the singular point √ undergoes when  > 0. The middle part of the transition occurs for |a| < K  for some K > 1, as can be seen using an elementary linear stability analysis of the singular point in (6.4). We are therefore motivated to rescale a in order to better study the Hopf bifurcation: a=

√  a. ¯

However√doing so, we lose information from the part of the parameter set where |a| ≥ K . As a remedy, the proposition below actually implies that there is no canard passage outside this parameter region, a fact that will allow us to use the above rescaling without loss of generality in the results on periodic orbits. With a Canard passage, we mean the existence, for  > 0 small enough, of trajectories passing nearby the turning point, along the critical curve. The following result has been proven in [DR96] using the blow-up construction. We will repeat the argument in Chap. 8. We will now provide an argument that does not require blow-up. Proposition 6.1 There exist a K > 1, 0 > 0, a0 > 0 and a neighborhood U of a sufficiently  close point p on the attracting slow arc such that for  ∈]0, 0 ],  √ a ∈ K , a0 , the orbits from U are all attracted to the unique singular point, which is of attracting node type. Furthermore, there is a neighborhood W of the slow–fast Hopf point in which no periodic orbits exist. A similar statement can be formulated for negative a, reversing time. Proof The main idea consists in constructing a trapping region. We will use h1 ≡ 0. Using the Implicit Function Theorem, we find a singular point (x0 , x02 ) with x0 = x0 (, a, μ) = a + O(a 2 ) > 0. It allows to rephrase y˙ as   y˙ =  (x0 − x)g(x, , λ) + O(y − x 2 ) . Let p = (xp , yp ) be near the origin, on the attracting branch of the critical curve such that near p we have y˙ < 0. We now consider level sets of V = y − Cx 2 + (C − 1)x02, where C is a parameter. Clearly {V = 0} contains the singular point (x0 , x02 ). The region delimited by two such parabolas {V = 0}, one with C > 1 and one with 0 < C < 1, has the singular point as vertex and contains the attracting branch of the critical curve. We delimit the region further by imposing y < yp . Note that for

6.4 Formal Power Series Methods for the Generic Hopf Breaking Mechanism

67

the C values close enough to 1, y˙ < 0 on y = yp in between the two parabolas. Let us now show that the region is a trapping region: V˙ = (−2Cx + O())(y − x 2 ) + (x0 − x)g ⇒ V˙ |V =0 = (−2Cx + O())(C − 1)(x 2 − x02 ) + (x0 − x)g = (x − x0 ) ((2Cx + O())(1 − C)(x + x0 ) − g) . √ When 0 < C < 1 is fixed and x > x0 ≥ K , we have √ √ (2Cx + O())(1 − C)(x + x0 ) − g > (2CK  + O())(1 − C)2K  − g √ = (4C(1 − C)K 2 − g + O( )). It shows that V˙ |V =0 > 0 for large enough K in that case. Similarly, when C > 1 is fixed, we find (easier) that V˙ |V =0 < 0 in that case. Hence on all boundaries of the considered region, the dynamics points inward except at the vertex which is the singular point. The statement concerning W is a direct consequence since any such periodic orbit would have to encircle the singular point and cross the trapping region. 

6.4 Formal Power Series Methods for the Generic Hopf Breaking Mechanism For use in Chap. 13, we provide now an interesting formal result for a generic Hopf breaking mechanism. We consider (6.4), for the sake of convenience with h1 ≡ 0, and in a slightly different form: 

x˙ = y − x 2 y˙ = (a − x + x 2 h2 (x, , λ) + (y − x 2 )h3 (x, y − x 2 , , λ)).

Associated with the vector field, we consider the ordinary differential equation (a − x + x 2 h2 (x, , λ) + (y − x 2 )h3 (x, y − x 2 , , λ)) − (y − x 2 )

dy = 0. dx

We formally solve this differential equation. To that end, we denote by hˆ 2 (x, , a, μ) the formal power series expansion of h2 w.r.t. (a, ), with smooth coefficients in (x, μ). We also denote by hˆ 3 (x, z, , a, μ) the formal power series expansion of h3 w.r.t. (z, a, ), with smooth coefficients in (x, μ).

68

6 Breaking Mechanisms

Proposition 6.2 There exist a unique formal power series ˆ , μ) = x 2 + ∞ ∞ y = ϕ(x, k k ˆ k=1 ϕk (x, μ) and a unique series a = A(, μ) = n=1 Ak (μ) so that dy (a − x + x 2 hˆ 2 (x, , a, μ) + (y − x 2 )hˆ 3 (x, y − x 2 , , a, μ)) − (y − x 2 ) , dx ˆ is identically 0 as a power series in . evaluated for y = φˆ and a = A, Proof The formal expression in the proposition is clearly O() since ϕˆ = x 2 + O(). Its  1 coefficient on the other hand is given by   −x + x 2 hˆ 2 (x, 0, 0, μ) − 2xϕ1 (x, μ), given that Aˆ = O(). It allows to determine ϕ1 (x, μ) = −1+xh22(x,0,0,μ) = − 12 + O(x). Observe now that  n coefficient, n ≥ 2, of the expression in the statement of the proposition only depends on A1 , . . . , An−1 and on ϕ1 , . . . , ϕn (together with the dependencies of the coefficients of hˆ 2 and hˆ 3 of course). More precisely, the  n coefficient is of the form An−1 (μ) + Hn (x, μ) − 2xϕn (x, μ), for some smooth expression Hn depending on A1 , . . . , An−2 and on ϕ1 , . . . , ϕn−1 . This equation allows to determine both An−1 and ϕn uniquely. First we determine An−1 by putting x = 0 in the equation: An−1 (μ) = −Hn (x, μ). Doing so allows to divide the equation by x and consequently solve for ϕn (x, μ) without introducing a pole. 

6.5 Generic Breaking Mechanisms In this book we speak of a Generic breaking mechanism when we do not specify whether or not we work with a generic jump breaking mechanism or a generic Hopf breaking mechanism. They in fact only differ in the canard connection they use. In the first case we use a jump connection as presented in Sect. 6.2, and in the second case we use a turning point as presented in Sect. 6.3. In both cases there is an adapted (breaking) parameter a, and by a smooth change in parameter space, we can suppose that λ = (a, μ). In the jump case we denote by p the union of the fast orbit together with the two jump points, while in the Hopf case p stands for the contact point itself. We denote

6.6 Other Breaking Mechanisms

69

σ

σ α(v)

α(v)

v

v

ω(v)

ω(v)

p

p

Fig. 6.2 Generic breaking mechanisms and associated layer with a family of canard cycles parameterized by the layer variable v

by γp− the attracting critical curve adherent to p (adherent to p− in the jump case) and by γp+ the repelling critical curve adherent to p (adherent to p+ in the jump case). In any case we suppose that the slow vector field is pointing toward p on γp− and away from p on γp+ . In order to create canard cycles containing these connections, we could suppose that there, like in Fig. 6.2, exists a layer of fast orbits that is adherent to both γp− and γp+ , We will however also consider the case in which the fast orbits of the later do not have their ω-limits on γp− but on another piece of critical curve γ − from which starts an attracting sequence (see Definition 4.8) tending to γp− . The fast orbits of the layer also do not need to have their α-limit on γp+ but on some γ + in a way that there is a repelling sequence (see Definition 4.8) from γp+ to γ + . On the layer we will use a layer variable v as explained in Sect. 4.4. The canard cycles that for a = 0 pass through the layer can be parameterized by (v, μ) and will be denoted v (μ). We speak of a generic layer of canard cycles in considering the situation just described, consisting of a layer, a generic breaking mechanism p, a generic attracting sequence tending from the layer to p, and a generic repelling sequence tending from p to the layer. Besides the configuration shown in Fig. 6.2, there are surely other configurations with more involved attracting and repelling sequences, both on orientable and on non-orientable surfaces.

6.6 Other Breaking Mechanisms Besides the Hopf and jump breaking mechanism, there are surely other mechanisms that could be used for creating closed orbits from canard cycles. One mechanism uses transcritical intersections of critical curves as connection, see for example [FPV08, KS01a, DM15]. Because transcritical intersection points are degenerate singular points, we will not treat them in this book. In [DM15], it is however shown

70

6 Breaking Mechanisms

that the methods explained in this book can be largely extended to the case of canard cycles containing transcritical intersection points. Note that in the literature, one also studies cuspidal contact points, see for example [Kue13] and slow–fast Bogdanov– Takens points [DMD11d, DMW15]. Similarly, the torus example in Fig. 1.3 could introduce another breaking mechanism: a north passage on the attracting branch of the ellipsoid would be followed by a fast oscillating part until it reaches the ellipsoid from the other side, possibly connecting to the repelling branch. The two involved contact points could be as simple as jump points, but the passage would still be different from a jump type canard connection, because of two reasons: (1) the fast connection between the jump points has infinite length, and (2) the fast connection passes through any neighborhood of the jump point an unbounded number of times. Additional analysis, beyond the scope of this book, is necessary to treat these types of connections.

6.7 Examples Let us provide two simple and well-known examples of these generic breaking mechanisms. We also show the different canard cycles that occur in these examples. They will play an important role in the rest of the book. We give additionally a third example where multiple independent breaking mechanisms appear. Example with Hopf Breaking Mechanism: Van der Pol’s Equation 

2

2

x˙ = y − x2 − y˙ = (a − x).

x3 3

(6.5)

3

The critical curve {y = x2 + x3 } does not depend on a, but the slow vector field does: x(1 + x)x = a − x. For −1 < a < 0, the unique singular point is a repelling node, while for a > 0 it is an attracting node. For a = 0, the origin is a slow– fast Hopf point, and the family X,a,λ has a generic turning point at the origin. We recall that Van der Pol’s equation shows two layers of canards, see Fig. 6.3. The first layer, the canards without head, is bounded between two special limit periodic sets:

(1)

(2)

Fig. 6.3 Canard cycles in Van der Pol’s equation

(3)

(4)

6.7 Examples

71

the Hopf point as a singleton, and the so-called maximal canard (see (2) in Fig. 6.3). The cyclicity of the Hopf point as a singleton is a problem that is referred to as “birth of canard cycles,” see [DR09]. Canard cycle (2) is called a transitory canard, see Chap. 4. The second layer, the canards with head, is bounded between the transitory canard on one hand and the common cycle (4) in Fig. 6.3. It is the ambition of this book to fully treat all types of canard cycles from birth to transition of course in a framework that goes well beyond the specific Van der Pol case. As can be expected a more general setting should be more complicated than the Van der Pol case itself, since it has been proven in [LdMP77] that cubic classical Liénard equations can have at most 1 limit cycle. Example with Jump Breaking Mechanism: A Liénard Equation with Generic Crossing of Two Maxima The need to blow up the slow–fast Hopf point in order to do a proper analysis can cause difficulties in the understanding of the canard phenomenon, and to some extent, these difficulties appear in a lesser extent when dealing with the jump breaking mechanism. Consider 

x˙ = y − 3 + (x 2 − 1)2 (x + 3) − ax y˙ = −x.

It is an a-family of slow–fast systems, of Liénard type of degree 5. Defining Fa (x) = 0 3 − (x 2 − 1)2 (x + 3) + ax, we have, for a = 0, critical points at x = x1,2 = ±1, a 0 0 third one x3 close to the origin and a fourth one x4 to the left of x = −1. Asymptotic analysis shows that x1,2 perturb to critical points for Fa x1 = −1 +

1 a + O(a 2 ), 16

x2 = 1 +

1 a + O(a 2 ), 32

and as a consequence Fa (x1 ) − Fa (x2 ) = −2a + O(a 2), in other words, a is a regular parameter for the jump breaking mechanism. Close to a = 0, the levels of x3 and x4 do not match, see Fig. 6.4. In Fig. 6.4 the cases (1) and (7) are common, although not strongly common, slow–fast cycles. (For the notion “strongly common” we refer to Chap. 4). The cases (2), (4), and (6) occur in 1-parameter families and are examples of ordinary canard cycles. The cases (3) and (5) are two examples of transitory canard cycles. The cycles in Fig. 6.4 are ordered from large to small amplitude; in that order, common → layer 1 → transitory → layer 2 → transitory → layer 3 → common. Again, we see that transitory canard cycles are found on the boundary of layers.

72

6 Breaking Mechanisms

Fig. 6.4 Canard cycles in Liénard equation with generic crossing of two maxima

Chapter 7

Overview of Known Results

We will now discuss some results concerning planar slow–fast vector fields and their periodic orbits. Of course, this is a very partial overview of results on slow– fast vector fields. Partial in the sense that results are mentioned when they have been proved using the techniques that we will describe in Part II and when they are relevant for the description of canard phenomena from birth to transition. Several things should be kept in mind throughout this whole chapter. Firstly, we use the definitions in earlier chapters, meaning that any slow–fast family of vector fields is a smooth family of vector fields on a smooth surface M, and it has only nilpotent contact points of at most finite order. Note that Definition 1.1 implies that the limiting vector field does not have singular points outside the critical curve. Secondly, we will assume that M is orientable throughout this chapter, occasionally giving a remark on the possible difficulties that may be encountered should one wish to generalize the results to the non-orientable case. Thirdly, we heavily make use of the terminology in the previous chapters and assume the reader can distinguish a limiting object like a common cycle, slow–fast cycle, canard cycle from the possible relaxation oscillations nearby for  > 0. Finally, we focus on results concerning limit cycles in particular. Nevertheless the same techniques can be used to deal with other questions, like transition time, existence of homoclinic and/or heteroclinic connections, and so on.

7.1 Periodic Orbits Near Common Cycles One of the most basic slow–fast systems is Van der Pol’s system  x˙ = y − x 2 /2 − x 3 /3 X,λ : y˙ = (λ − x).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_7

(7.1)

73

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7 Overview of Known Results

It has a cubic critical curve with contact points at (0, 0) and (−1, 16 ), and for   λ ∈ 0, 16 , the contact points are generic jump points and a unique common cycle  is shown. It is a very well-known fact that  perturbs, for  > 0, to a relaxation oscillation. This is actually the generic behavior for slow–fast vector fields: generically, slow–fast vector fields will have the following properties: • all contact points are of generic jump type, and • none of the contact points are connected to each other through a fast segment. Generically, the slow vector fields might be regular or exhibit hyperbolic zeros. The latter case cannot occur in a slow–fast cycle as defined in Definition 4.2. We will hence restrict to the former case, in which it is elementary to see that the slow– fast vector field only has regular common slow–fast cycles (given our definition of slow–fast cycles, examples like in Fig. 1.3 are excluded), and the global cyclicity of the system can be derived from the topology of the slow and the fast vector fields. One could for example deduce global cyclicity results for generic slow–fast classical Liénard systems. The Van der Pol system has received a lot of attention. It has for example been treated with matched asymptotics in [Gra87], non-standard analysis [BCDD81], Gevrey asymptotics [CD91], Gevrey summability [FS99], geometric singular perturbation theory [DR96], and so on. Theorem 7.1 [DMDR11] For any regular common cycle  of X0,λ0 , there are an 0 > 0, a neighborhood 0 of λ0 , and a tubular neighborhood W of  so that for (, λ) ∈ ]0, 0 ] × , the vector field X,λ has at most one periodic orbit in . Furthermore, in case a periodic orbit is present, it is a hyperbolic limit cycle. Remarks 7.1 (i) Excluding the singular points on slow arcs is necessary for example to eliminate the possibility of a common cycle producing homoclinic loops. (ii) Excluding singular contact points is necessary because a perturbation of such a singular contact point might have a singular point on one of its slow arcs. (iii) In generic circumstances, for example, for common cycles with generic contact points (jump points), Theorem 7.1 has been proved before by several authors. The generality in the statement lies in the fact that regular non-generic contact points of arbitrary order n ≥ 2 are allowed. Such contact points may bifurcate into one or more contact points. 

The method of proof consists of computing the derivative of the Poincaré map by tracking the slow divergence integral along compact parts of slow arcs and by showing that the parts outside the slow arcs are negligible in view of this computation. Passage near regular contact points, of high contact order or not, is elementary: the passage near regular contact points has a negligible contribution to the computation of the derivative of the Poincaré map. We refer to [DMDR11] for a complete proof. Theorem 7.1 can be generalized, as we will see in Theorem 7.3. It deals with unbalanced canards, a notion that we will define in Definition 7.1.

7.1 Periodic Orbits Near Common Cycles

75

7.1.1 Existence of Periodic Orbits Near Common Cycles In order to outline a frame in which a slow–fast cycle for certain generates a periodic orbit, it is clear that one has to restrict to regular cycles (otherwise, in case of singularities, the slow–fast cycle might perturb to homoclinic cycles for example). In the framework of a regular common cycle, one might at first sight believe that any sufficiently small regular perturbation will for sure exhibit a unique attracting limit cycle, Hausdorff close to the slow–fast cycle. In case of a strongly common cycle (see Definition 4.5), this is indeed the case. Theorem 7.2 For any strongly common cycle  of X0,λ0 , there are an 0 > 0, a neighborhood  of λ0 , and a tubular neighborhood W of  so that for (, λ) ∈ ]0, 0 ] × , the vector field X,λ has exactly one periodic orbit in . Furthermore, the periodic orbit is a hyperbolic limit cycle tending in Hausdorff sense to  as (, λ) → (0, λ0 ). We refer to [DMDR11] for a proof. Essentially it consists of constructing a chain of flow-box-like neighborhoods forming a circular funnel. The occurrence of a periodic orbit near general regular common cycles is however not guaranteed as we will show by an example. We consider a system X,a where the vector fields X0,a look like in Fig. 7.1. For a ≥ 0, there is the presence of a slow–fast cycle a , as shown in Fig. 7.1, but not for a < 0. We can impose conditions on X,a to guarantee that all a are regular, including 0 . 0 is not strongly common due to the presence at the point p of a canard fast-slow segment as shown in Fig. 4.9. The presence of that segment permits to approach the common cycle by canard cycles, as we see in Fig. 7.2. Let W be an arbitrarily small Hausdorff neighborhood of 0 . For a ∼ 0, the cycle a lays in W . For a > 0, the a are strongly common so that Theorem 7.2 0

=0

Fig. 7.1 Possible disappearance of a common slow–fast cycle

Fig. 7.2 A regular common cycles that can be approached by canard cycles

0

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7 Overview of Known Results

guarantees, for  > 0 small enough, the existence of an attracting hyperbolic limit cycle C,a , close to a , inside W . For W sufficiently small, C,a must be the unique periodic orbit of X,a inside W , due to Theorem 7.1. It is hyperbolic and hence structurally stable. For a < 0 and  > 0 sufficiently small, it is clear that no periodic orbits can be found inside W . It implies that when we fix  > 0 and decrease a, until reaching negative values, C,a cannot remain completely inside W , implying that for a < 0 and  sufficiently small there is no limit cycle close to 0 . What happens is a typical canard phenomenon and it will be explained in full detail in Part III of the book, especially in Chap. 14.

7.1.2 Multiple Periodic Orbits Near Common Cycles In case singular contact points appear on the common cycle, there are known examples where the conclusions of Theorem 7.1 are not correct, even in case the involved slow divergence integral is well defined and nonzero; see also [DMDR11]. A specific example can be found if one amends the equations in the Van der Pol system to an extent that the critical curve is still cubic but having a contact point of slow–fast Bogdanov–Takens type instead of slow–fast Hopf type; see [DMD11d].

7.2 Unicity of Periodic Orbits Near Unbalanced Canard Cycles Theorem 7.1 has been generalized to the context of canard cycles, insofar that the canard cycle has nonzero slow divergence integral. Definition 7.1 Given the slow–fast vector field X,λ . A canard cycle  for λ = λ0 is called an Unbalanced canard cycle when the slow divergence integral Iλ0 () is nonzero. When Iλ0 () = 0, we speak of a Balanced canard cycle. Canard cycles can contain passages through singular contact points. We will allow singular contact points of Hopf type (see Definition 2.4) and remind the reader that both regular contact points and slow–fast Hopf points satisfy Definition 5.2 implying that slow divergence integrals computed up to the contact points converge to a finite value. Theorem 7.3 [DMDR11] Let  be a canard cycle of X0,λ0 that is regular (no singular contact points and no singularities in the slow vector field) or for which the only singular contact points are of slow–fast Hopf type (Def. 2.4), and the singularities of the slow vector field are either all on the attracting branches or all on the repelling branches of .

7.2 Unicity of Periodic Orbits Near Unbalanced Canard Cycles

77

Assume, in the absence of singularities of the slow vector field, that the slow divergence integral I () is nonzero and hence that we deal with an unbalanced canard cycle. Let q1 , . . . , q be the involved singular contact points ( ≥ 0), and let W1 , . . . , W be mutually disjoint open sets with qi ∈ Wi , for i = 1, . . . , . Then there exist a tubular neighborhood W of , an 0 > 0, and a neighborhood  of λ such that for (, λ) ∈]0, 0 ] × , there is at most one periodic orbit of X,λ inside W that does not lie entirely in one of the Wi ; if it appears, this periodic orbit is hyperbolically attracting (in case I (λ0 ) < 0) or repelling (in case I (λ0 ) > 0). We highlight the fact that Theorem 7.3 works both for ordinary canard cycles and for transitory cycles. The proof essentially has two ingredients: first one defines a covering  by a well-chosen chain of neighborhoods that do not necessarily form a funnel like in the strongly common case. However, the neighborhoods Vk , k = 1, . . . , K, will be of a flow-box-like shape, having an inset Ak and an outset Bk , and will be chosen so that a connected part of Ak reaches the inset of Ak+1 before Bk is encountered. It allows to define the first return map on a (parameter-dependent) connected section. Near singular contact points, one considers a so-called whirlpool neighborhood instead of a flow-box-like neighborhood; such neighborhoods could contain small-amplitude limit cycles and at the same time permit passage of a large-amplitude periodic orbit (hence the necessity of introducing the Wi neighborhoods in the statement of the theorem). Second, one relates the nonzero slow divergence integral to the actual divergence integral along orbits close to  (for  > 0 small enough). Then, the chain of neighborhoods defines an annulus around  along which an attracting or repelling dynamics is seen. We have not considered the non-orientable case, but we could have done so without any problem, working with the second return map instead of the first return map. Corollary: Global Unicity in Layers When  is a canard cycle part of a layer of canard cycles, and if all canard cycles in the layer satisfy the assumptions of Theorem 7.3, then a compactness argument allows to conclude that the upper bound of at most one periodic orbit is valid on a tubular neighborhood of the whole layer (or more specifically on compact subsets of the layer). Let us illustrate this on the Van der Pol equation (7.1). For λ ∈ 0, 16 , there is only one possible slow–fast cycle, which is a common cycle. When λ = 0 on the other hand, there are two layers of canards, one corresponding to canards with head and one corresponding to canards without head, recall  Sect.  5.7.1. There two layers σ1 and σ2 are identified, both parameterized by 1 v ∈ 0, 6 .   If I ⊂ 0, 16 is a compact interval, then Theorem 7.3 can be applied for all canard cycles v1 and v2 , v ∈ I , and in the union of the obtained tubular neighborhoods, we then obtain the existence of at most one periodic orbit. In this case, the boundary between the two layers is a transitory canard (sometimes referred

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7 Overview of Known Results

to as the maximal canard of the Van der Pol equation). For the Van der Pol system, this canard also has negative slow divergence integral, implying that one can join the tubular neighborhoods of the headless canards and the tubular neighborhoods of the canards with head to find the global upper bound of at most one periodic orbit. We want to stress though that the unicity statement in Theorem 7.3 excludes a neighborhood of the slow–fast Hopf point. Treatment of this small-amplitude cycles is the topic of the “birth of canards” and will be discussed later in this chapter. The global unicity of the limit cycle in the Van der Pol system can be shown by other methods (using the Liénard shape and taking advantage of the low degree), as has been done in [DR90].

7.3 Existence of Periodic Orbits Near Ordinary Canard Cycles Ordinary canard cycles will be treated in full detail in Chap. 13, in a generality and depth that is higher than with other presentations in the literature. We will state an interpretation of previously obtained results in the literature, writing them in the language and using the notations introduced in previous chapters. This allows the reader to compare the different achievements and to understand their limits. Chapter 13 is devoted to giving a unified proof of many of these results. We start with a discussion on the well-known Van der Pol system and gradually extend and generalize the setting. Theorem 7.4 ([DR96]) Let  i (v) be a headless canard (i = 1) or canard with head (i = 2) of the Van der Pol system, with v ∈ ]0, 1/6[. There exists a v-family of smooth canard curves λ = λi (v, ) so that the family of vector fields X,λi (v,) has a periodic orbit crossing σi at v. The periodic orbit is an attracting limit cycle and it is unique. The result in [DR96] actually shows the existence of canard curves λi (v, ) that √ are smooth w.r.t. ; the smoothness w.r.t.  was later obtained in [DMD06] and was already known before by the theory of matched asymptotics. We will provide an argument in Chap. 13. Remark 7.2 (Cubic-Shaped Critical Curves) From the point of view of applications, cubic-shaped critical curves, like the one appearing in the Van der Pol system, are quite abundant. Popular models like the Fitzhugh–Nagumo model, the Oregonator model, etc. have such a cubic-shaped critical curve, i.e. the curve is a graph y = f (x) with two extremes and the fast vector field is horizontal, marking two contact points at the two extremes of f . In [KS01a], the Van der Pol model is generalized to systems with a cubic-shaped critical curve, i.e. a graph y = f (x) with two simple extremes, under the assumption that the fast vector field is horizontal (a horizontal fast flow is often referred to as the standard slow–fast splitting). Furthermore, the authors discuss two scenarios: the supercritical Hopf bifurcation

7.3 Existence of Periodic Orbits Near Ordinary Canard Cycles

79

(like Van der Pol’s scenario) and the subcritical Hopf bifurcation. In a cubic-shaped geometry, a subcritical Hopf bifurcation producing a repelling limit cycle will be surrounded by a bigger attracting cycle. In this context, the canard that is born out of the Hopf bifurcation will end in a saddle-node of limit cycle bifurcation instead of growing toward a full-sized relaxation oscillation that is seen in a supercritical scenario. Ordinary Canard Cycles with One Attracting and One Repelling arc and a Turning Point (“FSTS” Cycles) In [DMD06], the authors dropped the standard slow–fast splitting and described canard cycles geometrically on a 2-manifold M. The involved canard cycles were composed of a single attracting arc and a single repelling arc, with a turning point in between, the so-called Fast–Slow–Turning Point–Slow cycles, or short FSTS cycles (see also [Dum13]). In [DMD06], the turning point need not be generic; the singularity order and/or contact order could be different from that of a generic turning point; however, the results in [DMD06] are restricted to those families of vector fields where one simple blow-up operation suffices to analyze the system. In particular, this entails that no bifurcations of the contact order and/or singularity order of the contact point are allowed. Recall that at a generic turning point, a family of slow–fast vector fields X,λ = X,a,μ can be put in the form (6.4). The so-called simple passage turning points in [DMD06] are more general. In particular, the following class of vector fields is simple passage turning points: 

x˙ = y − x 2n + x 2n+1 h1 (x, λ) y˙ = (a − x 2n−1 + x 2n h2 (x, , λ) + yh3 (x, y, , λ)).

(Besides this model, there are other kinds of simple passage turning points such as those related to transcritical intersections of critical curves, see [KS01a] and [DM15].) Theorem 7.5 Let X,λ be a family of slow–fast vector fields with λ = (a, μ) and such that at p a simple passage turning point is found (in the sense that locally around p the slow–fast system admits above the normal form). Let σ be a layer section, parameterized by v. There exists a (v, μ)-family of canard curves a = a0 (, v, μ) so that the family of vector fields X,a0 (,v,μ),μ has a periodic orbit crossing σ at v. The family of canard curves is smooth w.r.t. ( 1/2n , v, μ). Theorem 7.5 is also shown independently in [Pan02] (generalizing the allowed turning points). FSTS cycles of Theorem 7.5, with a generic turning point, are particular kinds of ordinary 1-layer canard cycles covered in this book. Remark 7.3 Theorem 1 in [DMD06] is formulated more generally and allows to prove the existence of canard-type orbits that are not necessarily periodic. The canard curves are in general not smooth in , but only smooth in  1/2n . Only when n = 1, smoothness in terms of  can be obtained; we will provide an argument in Chap. 13, essentially based on the existence of the formal power series expansions

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7 Overview of Known Results

shown in Proposition 6.2. The limited smoothness for n > 1 can be explained in a complex analytic setting; then, such general turning points connect, besides the attracting and repelling branches in the real directions, also several other branches in complex directions (one speaks of valleys and mountains, see [FS13, BFSW98]). Roughly speaking, the canard curve in Theorem 7.5 only regulates the dynamics along the real axes and fails to match the complex directions, which is a necessary condition for a0 (, v, λ) to have a Taylor series w.r.t. . Remark 7.4 In Part III, we will introduce the displacement function μ (v, b, ) whose zeros in v for  > 0 are in one-to-one correspondence to the bifurcating limit ∂ cycles. If one considers a regular FSTS (i.e. for n = 1), one has that ∂b ( −1/2 μ ) = 0, and so one can solve the equation μ = 0 in order to obtain the implicit function b(v, μ, ), which, taking into account Proposition 12.4 below, is a priori smooth in ( 1/2, v, μ). Clearly we have that a(v, μ, ) =  1/2b(v, μ, ) is the function defining the (v, μ)-family of canard curves appearing in Theorem 7.5. It is rather remarkable that this function is in fact smooth in (v, μ, ), as mentioned above in Remark 7.3. Remark 7.5 The results in [DMD06] are restricted to orientable manifolds. Ordinary Canard Cycles with One Attracting and One Repelling arc and a Jump Breaking Mechanism (“FSJS” Cycles) FSJS cycles are similar to FSTS cycles, and the Hopf canard breaking mechanism is replaced by a jump breaking mechanism. The terminology of FSJS and FSTS cycles was introduced in [Dum13], but canard cycles with a jump breaking mechanism were first reported in [DR07a], there restricting to a Liénard setting. The setting was later generalized in [Dum11]. Theorem 7.6 Let X,λ be a family of slow–fast vector fields with λ = (a, μ) and such that at p a jump canard connection is found with jump breaking parameter a. Let σ be a layer section, parameterized by v, so that for each v a FSJS cycle is identified. There exists a (v, λ)-family of canard curves a = a0 (, v, λ) so that the family of vector fields X,a0 (,v,λ),λ has a periodic orbit crossing σ at v. The function a0 (, v, λ) can be written as an expression that is smooth w.r.t. ( 1/3 ,  1/3 ln , v, λ). FSJS cycles of Theorem 7.6 are particular kinds of ordinary 1-layer canard cycles covered in this book. Expressions that are smooth in terms of monomials like  1/3 ,  1/3 log  are dealt with in Chap. 11. Note that this is should be clear that the two jump points in the canard connection could be replaced by regular contact points of higher contact order, as long as they do not bifurcate; the smoothness of the canard curve will need to be adapted for jump points of higher contact order though. (Up to the knowledge of the authors, [DMDR11] is the only reference in the literature treating slow–fast cycles that pass along bifurcating jump points.) Remark 7.6 The results in [Dum11, DR07a], and [Dum13] are restricted to orientable manifolds and to a jump canard connection connecting jump points with bumps on the same side (see Fig. 6.1 for the opposite case).

7.4 Entry–Exit Relations

81

7.4 Entry–Exit Relations We will work in the context of an FSTS cycle, because this is the setting where these statements have been proved in the literature [DMD04, Ben81]. We assume that the slow–fast family of vector fields X,λ is such that λ = (a, μ) and that p is a slow–fast Hopf point, where a is a Hopf breaking parameter at a = 0. We choose two sections σ+ and σ− transverse to the orbits of X0,(0,μ). In applications where periodic orbits are studied, the two sections will be the same and will be a layer section, but in view of describing the entry–exit relation, it is not necessary to make this assumption. We will assume however that the two sections σ+ and σ− lie on the same side of the critical curve (in a small disc around p, the parabolically shaped critical curve separates the disc in two components, identifying the two sides, see Fig. 7.3). We assume that for each v ∈ σ− , the fast orbit in positive time reaches a point ωμ (v) on an attracting slow arc, and we assume that the slow vector field has a regular dynamics pointing toward p. We assume that for each v ∈ σ+ , the fast orbit in negative time reaches a point αμ (v) on a repelling slow arc, with slow dynamics pointing away from p. (When σ− = σ+ , we could hence simply assume that through the layer section an FSTS cycle is drawn.) Inverting v → ωμ (v) and v → αμ (v), we find v = vμ− (ω),

v = vμ+ (α).

funnel

inverse funnel σ+

σ− σ− −

ω

v (ω)

α

v + (α)

tunnel

ωb

v − (ω)

p

Fig. 7.3 Entry–exit relation, identifying a tunnel section, a funnel, and an inverse funnel. This figure is under the assumption that a canard curve is chosen so that a canard orbit connects v − (ω) to v + (α)

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7 Overview of Known Results

Theorem 7.7 ([DMD05b]) There exists a C∞ -smooth (ω, α, μ)-family of curves a = a0 (, ω, α, μ) in the (a, )-plane so that the family of vector fields X,a0 (,ω,α,μ),μ has an orbit crossing σ− at vμ− (ω) and σ+ at vμ+ (α) and tending in positive time from vμ− (ω) to vμ+ (α). Remarks 7.7 1. In Theorem 7.7, we speak about curves, since a = a0 (, ω, α, μ) is often only considered in the (a, )-plane, depending on parameters (ω, α, μ). Such a curve is called a “Control curve” or a “Canard curve.” 2. Theorem 7.5 is the special case where v + coincides with v − on coinciding sections σ+ and σ− . In this context, the name “canard curve” looked like a natural name since along such a curve, the vector field X,a,μ exhibits a periodic orbit tending to a canard cycle through α and ω for  → 0. Associated with a pair (ω, α), we consider the slow arc from ω, along p, toward α and denote the slow divergence integral of that part by Iμ (ω, α). We can take ω in an open subset aμ of the attracting critical curve and α in an open subset rμ of the repelling critical curve, both aμ and rμ having p in their boundary. Note that on 2-manifolds like we consider here, a critical curve near a generic turning point has two well-identified sides (the critical set divides M ∩ B(p, r) in two connected components for r sufficiently small, and the two components could be called an inside and an outside). In the setup, it does not need to be the case that σ− and σ+ lie in the same connected component. Remark 7.8 A similar theorem can be proven for a FSJS cycle, except that it can only be guaranteed for a0 to be C∞ in ( 1/3,  1/3 ln , ω, α, ν), as we will see in Chap. 13. It may happen that a0 is smooth in (, ω, α, ν) but this is an exceptional situation. Proposition 7.1 Let a = a0 (, ω, α, μ) be a control curve as in Theorem 7.7. Suppose that Iμ (ω, α) < 0. Depending on α, we define in aμ two subsets: = {ω˜ : Iμ (ω, ˜ α) > 0}, tunnel μ funnel = {ω˜ : Iμ (ω, ˜ α) < 0}, μ and we consider the orbit through vμ− (ω) ˜ in positive time.

7.4 Entry–Exit Relations

83

(i) If ω˜ ∈ tunnel , then for  sufficiently small the orbit intersects σ+ at a location μ v = vμ+ (α) ˜ + o(1),  → 0, with ˜ α) ˜ = 0. Iμ (ω, funnel

(ii) If ω˜ ∈ μ , then for  sufficiently small the orbit intersects σ+ at a location v = vμ+ (α) + o(1),  → 0. Depending on ω, we define in rμ two subsets: tunnel Ainverse = {α˜ : Iμ (ω, α) ˜ < Iμ (ω, α)}, μ funnel = {α˜ : Iμ (ω, α) ˜ > Iμ (ω, α)}. Ainverse μ

˜ in negative time, then If we consider the orbit through vμ+ (α) tunnel , then for  sufficiently small the orbit intersects σ at a (i) If α˜ ∈ Ainverse − μ location v = vμ− (ω) ˜ + o(1),  → 0, with

˜ α) ˜ = 0. Iμ (ω, , we define a section σ− transverse to the fast fiber on the (ii) For α˜ ∈ Aμ other side of the attracting arc. The orbit intersects the section σ− at a location that is o(1)-close to the fast fiber of ωb , with Iμ (ωb , α) = 0. inverse funnel

Tunnel behavior is presented in Fig. 7.4 (left) and Funnel behavior in Fig. 7.4 (right). In case there exists a unique point ωb (μ) ∈ aμ , laying between the tunnel and the funnel sections, then (ωb (μ), α) is called a pair of “Buffer points.” It is

α

σ−

σ+

ωb

σ−

σ+

ω

α ˜

ω ˜

α=α ˜

ω ˜ ωb

p

p

Fig. 7.4 Tunnel behavior (left) and funnel behavior (right) under the assumptions of Proposition 7.1. Orbits are shown for  > 0 sufficiently small. The left shows orbits through points on σ− that correspond to some ω˜ “below” the buffer point ωb . Orbits reach the section σ+ at asymptotically distinct positions. Right: ω˜ lies “above” the buffer point. Orbits reach the section σ+ at a position asymptotically dictated by α

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7 Overview of Known Results

defined by the relation Iμ (ωb (μ), α) = 0. In many situations, there is indeed a unique solution to this equation. A time-reversed version of the proposition takes care of the case where the slow divergence integral is positive; in that case the buffer point is defined by the relation Iμ (ω, αb (μ)) = 0. In the tunnel subset of the attracting critical curve, we can consider a map S˜μ , implicitly defined by Iμ (ω, S˜μ (ω)) = 0. It leads to a map Sμ : σ− → σ+ , defined by Sμ (ν) = νμ+ (S˜μ (ωμ (ν))). In case we have σ− = σ+ , then the tunnel section is particularly interesting because the first return map ν → Sμ, (ν) is reduced to an o(1)-perturbation of this map Sμ . It is clear that in this case, simple zeros of the map v → Iμ (αμ (v), ωμ (v)) indicate the presence of periodic orbits in the tunnel section. The proposition was first proved by Benoit, generalized in [DMD04], and generalized to an FSJS context in [Dum11]. The function Sμ : σ− → σ+ is called the Slow relation function in [Dum11]. There, also, a Fast relation function is defined: Fμ : σ− → σ+ whenever v and Fμ (v) lie on the same orbit of X0,(0,μ) .

7.5 Multiple Periodic Orbits in Layers Following on the discussion of the entry–exit relation, we now assume σ− and σ+ are connected by a family of fast orbits (and both qualify as layer sections). It means that both the slow relation function Sμ and the fast relation functions Fμ are well defined. A slow–fast cycle cutting σ− at some v has a slow divergence integral given by Iμ (v) = Iμ (αμ (Fμ (v)), ωμ (v)).

7.5 Multiple Periodic Orbits in Layers

85

Under the assumptions of Proposition 7.1, for values v in the tunnel section of σ− , a sign change of Iμ (v), or equivalently, a sign change of Fμ − Sμ , marks the presence of a periodic orbit cutting σ− o(1)-close at a value o(1)-close to a zero of Iμ . It is a direct consequence of the tunnel behavior. Multiple zeros of Iμ hence mark the presence of multiple periodic orbits. Using this fact, saddle-nodes of limit cycles and more degenerate bifurcations can occur near zeros of the slow divergence integral. A first treatment of such bifurcations was done in the context of Liénard systems in [DR01b] for FSTS cycles, similar ideas in FSJS cycles were studied in [DR07a], and an overview can also be found in [Dum11]. In this book, we give a unified proof for the far more general one-layer ordinary canard cycles introduced here. We refer to Chap. 13 for detailed statements. Remark 7.9 In polynomial vector fields, the slow and fast relation functions are defined algebraically, which makes the study of zeros of Iμ amenable. For example, for generalized polynomial Liénard systems 

x˙ = y − F (x, μ) y˙ = G(x, μ),

the multiplicity of the zero of a slow divergence integral at a given height Y can be formulated as the intersection number of two algebraically defined curves: the slow relation function and the fast relation function. This algebraic property has been successful in producing vector fields with a high number of limit cycles, which is relevant in understanding Hilbert’s 16th problem and Smale’s 13th problem. Upper bounds for the number of limit cycles of a given class of vector fields, i.e. for polynomial systems of some degree d in the case of Hilbert’s 16th problem or classical Liénard systems in the case of Smale’s 13th problem, are very hard to obtain, and up to now it has been impossible to obtain any result in a nontrivial class of systems. The only noteworthy result is the result by Chenghzi and Llibre [LL12], stating that classical Liénard equations of degree four can have at most 1 limit cycle. Other results, by Ilyashenko and Llibre, are partial in the sense that they avoid the singular perturbations in the formulation of their results. Indeed, for example, in the class of classical Liénard systems, one can rescale coefficients to make them bounded (so that the whole polynomial family becomes compact) at the price of introducing slow–fast systems on the boundary of the compact set, see [Rou07] for example. It shows that difficulties in bounding limit cycles accumulate near the slow–fast systems, a thought that is strengthened by the fact that in some classes the best known lower bounds are found near the slow–fast limit. In the Liénard setting, lower bounds are found in [DPR07, DMD11a, DMD11b, DMH15, HDM14], and [CDP13]. For general polynomial systems, a recent work [ACDMP20] shows that also there, the best known lower bound is found near the singular limit.

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7 Overview of Known Results

Global Bound on the Number of Periodic Orbits in Layers of FSTS and FSJS Cycles in Liénard Systems Let σ be a layer section so that each v ∈ σ defines a canard cycle of FSTS type or FSJS type (i.e. they are composed of one fast orbit, one attracting arc, one repelling arc, and a canard connection that is a slow–fast Hopf point or a fast connection between the jump points). We assume the family of vector fields X,λ is such that λ = (a, μ), where a is a breaking parameter. Let W be the annular neighborhood of the union of all canard orbits through a compact subset of the layer section, and let W1 be a small neighborhood of the slow–fast Hopf point (and W1 = ∅ in the FSJS case). If the slow divergence integral Iμ (v) has exactly k simple zeros for μ = μ0 on the compact subset of σ (counted with multiplicity), then for (, a, μ) close enough to (0, 0, μ0 ), the vector field X,(a,μ) has at most k + 1 limit cycles in W that do not lie entirely in W1 . This statement is a direct consequence of Theorem 1 of [DR01b] in case of Liénard equations and for the Hopf scenario and of Proposition 4 of [DR07a] in case of the jump scenario. The more general case will be discussed in Chap. 13; the results are identical. Remark 7.10 For more involved slow–fast cycles, e.g. where the cycle crosses more than one layer section, it is not true that a simple zero of Iμ (v) generates at most two limit cycles nearby; to study such cycles, it is absolutely necessary to take into account all involved layer variables, see the section below on multiple layer canards.

7.6 Contact Points of Higher Singularity Order Or Contact Order Most results have been obtained for canard cycles of FSTS type, and this book restricts the new results to contact points of contact order 2. For the sake of completeness, we mention that FSTS cycles with a higher order turning point have been considered in [DMD05a] and [DMD06]. In these papers, families of vector fields are allowed to have higher order contact points, though they are not supposed to undergo bifurcations. It is an open question whether or not the results in this book can be extended to the context of ordinary canard cycles where the canard connection contains a higher order, non-bifurcating, contact point. A stronger challenge lies in extending the results on ordinary canard cycles containing higher order contact points that undergo a versal unfolding; this is completely open. In [DMDR11], it is shown that it is possible to recursively blow up contact points of higher order. In that paper the recursive procedure restricts to regular contact points and points of singularity index 1 at most. Additional information on recursive blowups is discussed in [Rou07] and [Pan02].

7.7 Canard Cycles with Singularities in the Slow Vector Field

87

A thorough study of the desingularization of a specific higher order turning point is done in [DMD11d], where the slow–fast Bodanov–Takens point is examined: after a primary blow-up, several charts need to be studied, and among these charts there is one chart where a slow–fast Hopf singularity is found that requires an additional blow-up. Results on entry–exit relations and limit cycle bifurcations involving higher order contact points are not elementary. One cause of difficulty lies in the fact that the slow divergence integral is possibly not defined (see Definition 5.2). Nevertheless, it is shown in [DMD10] for contact points of singularity order 2 and in [DMD11c] for contact points of singularity n ≥ 3 that it is possible to study bifurcations of canard cycles in a layer by considering the derivative of the slow divergence integral w.r.t. the layer variable instead of the layer variable itself. Let us give a bit more detail. If v is a canard cycle parameterized by a layer variable v, then the slow divergence integral is clearly v-dependent. If, instead of computing the full slow divergence integral, we cut away a fixed piece around the singular contact points, then one removes the cause of unboundedness of the integrals. Call the result I0 (v). Since the cut away piece is independent of v, the notion ∂I0 ∂v is clearly meaningful. Where in previous section, a zero of multiplicity k of the slow divergence integral I gives rise to potentially k + 1 canard cycles, and the results in [DMD10] and [DMD11c] show that a zero of multiplicity k of ∂I ∂v gives rise to potentially k + 2 canard cycles. For more information, we refer to these papers.

7.7 Canard Cycles with Singularities in the Slow Vector Field Singular points in the slow vector field typically reverse the orientation of the slow dynamics. If such a singular point appears in a layer of canard cycles, then the point marks the boundary of the layer; the canard cycles inside the layer then accumulate toward what could be called a canard homoclinic. The study of the behavior near such canard homoclinics is done in [DMD08]. It is typically not so hard in case one singular point appears on the slow vector field, because one such point causes the slow divergence integral to tend to +∞ (when it is located on the repelling branches) or to −∞ (when it is located on the attracting branches). Since canard cycles with nonzero slow divergence integrals are the most easy ones to study, such canard homoclinics follow the same principle. It becomes challenging when multiple zeros of the slow vector field are present on opposite branches of the critical curve. In that case, the slow divergence integral becomes unbounded but without a fixed sign and bifurcations are possible. Several scenarios are discussed and treated in [DMD08].

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7 Overview of Known Results

7.8 Multi-Layer Canard Cycles An essential difficulty in dealing with canard cycles is the lack of a good expression of the first return map. The best approach for dealing with such cycles is through a difference map: along the part of the cycle near attracting branches of the critical curve, one integrates in positive time, and along the part of the cycle near repelling branches, one integrates in negative time. This method works well for one-layer canards. Multi-layer canards on the other hand have multiple iterations of attracting and repelling passages, so a more involved method is needed. Let us remark that multi-layer canard cycles can appear in slow–fast Liénard equations, although not in classical ones.

7.8.1 Two-Layer Canard Cycles and Their Transitory Boundaries The results in this section have been taken from [DR07b, Dum11], and [MR12]. Two of these papers present their work in the setting of slow–fast Liénard systems and the third to planar systems. We can in general talk about a concatenation of two ordinary (1-layer) canard cycles in the following sense: Definition 7.2 ((2-Layer) Ordinary Canard Cycle) A slow–fast cycle  of X0,λ0 is called a “2-layer ordinary canard cycle” if it is composed of (1) an attracting sequence γ1− , (2) an elementary canard connection between γ1− and a repelling sequence γ1+ , (3) a fast orbit γ1 inside a layer of X0,λ0 , the layer being situated between a repelling arc of γ1+ and an attracting arc of an attracting sequence γ2− , (4) an attracting sequence γ2− , (5) an elementary canard connection between γ2− and a repelling sequence γ2+ , and (6) a fast orbit γ2 inside a layer of X0,λ0 , the layer being situated between a repelling arc of γ2+ and an attracting arc of the attracting sequence γ1− , An example is shown in Fig. 7.5. Here, the first canard connection p1 is of Hopf type, and the second one p2 is of jump type (P1 → P2 ). For each attracting and repelling sequence, we identify a slow divergence integral, reversing time during the repelling parts so that all related slow divergence integrals are negative. The total slow divergence integral is then D(u, v) := I (u) + L(v) − J (v) − K(u). The two parameters u and v are two-layer variables uniquely identifying the twolayer canard cycle; in fact the need to use two variables is a reflection of the fact that we deal with a two-layer cycle. The papers [DR08] and [MR12] provide results on

7.8 Multi-Layer Canard Cycles

89 p2

Fig. 7.5 Example of a 2-layer ordinary canard cycle

J

I v u L

K σ1

p1

σ2

the situation presented in Fig. 7.5, restricted to Liénard equations. The proofs can easily be extended to the same configuration in general smooth slow–fast families. We recall these results in the next two theorems. Theorem 7.8 ([DR08]) When D(u, v) = 0, at most one cycle can bifurcate from the two-layer cycle. When D(u, v) = 0 and I (u) = J (v), at most two cycles can bifurcate from the two-layer cycle. When D(u, v) = 0 and I (u) = J (v) and I (u)L (v) = K (u)J (v), at most three cycles can bifurcate from the two-layer cycle. Furthermore, under the condition of presence of breaking parameters for both canard connections, the unfolding of the breaking parameters contains a full bifurcation diagram in all cases. The last two conditions can be formulated in terms of intersection numbers of two curves I (u) − J (v) = 0

and

K(u) − L(v) = 0.

Indeed it means that the intersection number is 0 in the second case of the theorem and 1 in the third case. In [MR12], the theorem has been generalized to any intersection number. Theorem 7.9 ([MR12]) If the above intersection number is k, then at most k + 2 limit cycles can bifurcate from the two-layer canard cycle. It is important to again point out that both results are not valid for transitory canards. Here, transitory canards mark the boundary between 1-layer and 2-layer canards and have not been studied before, see Fig. 7.6.

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7 Overview of Known Results

Fig. 7.6 Transitory cycles at the boundary between one-layer and two-layer cycles

7.8.2 More Than Two Layers In [DR08], a treatment of multi-layer canards is initiated: the periodic orbits are investigated in a small neighborhood around a given cycle. It is shown that the fixed point equation is related to an equation involving a finite iteration of translated power functions. In [CR16], the case of 3 layers is investigated in detail where it is shown under a specific genericity condition on the slow divergence integrals that 3-layer balanced canard cycles (non-transitory) lead to at most 5 periodic orbits nearby.

7.9 Birth of Canard Cycles The birth of canard cycles refers to the analysis of periodic orbits that are found in a small neighborhood of a generic turning point. Using a blow-up it will be possible to identify two kinds of -families of periodic orbits in such a small neighborhood: (1) -families of periodic orbits of slow–fast (canard) type; these periodic orbits have a period that is asymptotic to 1 (T + o(1)) for some T > 0. As  → 0, the orbits tend in Hausdorff sense toward an actual canard cycle (which is a limit periodic set). (2) -families of periodic orbits of regular type; These -families of periodic orbits have a period that is asymptotic to √1 (T + o(1)) and will stay in a o(1)-neighborhood of the turning point as  → 0. These orbits tend toward the turning point in Hausdorff sense as  → 0. It should be clear that there is no strict separation between the two scenarios. Nevertheless, we will see in Chap. 8 that by blowing up the turning point, we will be able to identify a specific limit periodic set that forms the separation between on one hand the limit periodic sets of canard type and on the other hand the limit periodic sets of regular type. This limit periodic set, that will be strictly identified in Chap. 8, will be referred to as the Birth cycle. Analytic tools implemented in [DR96] and later also in [KS01a] were useful in studying both the periodic orbits of type (1) and the periodic orbits of type (2), but have not been a sufficient answer in controlling periodic orbits near the birth cycle. In [DR96] there was no need in doing so because of the specific nature of the Van der Pol system, which is a cubic system of classical Liénard type. The birth cycle was

7.9 Birth of Canard Cycles

91

studied first in detail in [DR09]. The rest of this section is devoted to describing the results of this paper, avoiding at this stage the knowledge needed to do the blow-up. Recall the normal form for a generic turning point given in (6.4): 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (a − x + x 2 h2 (x, , λ) + yh3 (x, y, , λ)).

In [DM14, DMD17]) a study has been made to reduce such analytic families to Liénard form up to exponentially small terms, and a follow-up result in [Huz17] actually shows the possibility to do this without exponentially small errors. Motivated by these results, we restrict the discussion on the birth of canard cycles to the context of Liénard equations, as is done in [DR09]:

7.9.1 Birth of Canards in Liénard Systems Restricting to the Liénard setting, a slow–fast family with generic turning point is written as  x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (a − x + x 2 h2 (x, , λ)). Recall from Chap. 6 that one could assume h1 = 0. It reveals more convenient to assume h2 = 0 instead, and it was shown in [DR09] that this can be done without any problem. This further reduction to classical Liénard context allows a more easy determination of the codimension of the slow–fast Hopf point. Also when h2 = 0, the system is a center if and only if a = 0 and x 3 h1 (x, 0, μ) only contains even terms. We say that the slow–fast Hopf point, written in the above form and assuming h2 = 0, has finite codimension q + 1 ≥ 1 when h1 (x, λ) + h1 (−x, λ) = Cλ x 2q + O(x 2q+2),

for some Cλ = 0.

The Van der Pol system has codimension 1. Theorem 7.10 ([DR09]) If the slow–fast Hopf point has finite codimension q +1 ≥ 1, then the family of vector fields has finite cyclicity at the generic turning point. If moreover the conjecture stated in the next subsection holds for all values k ≤ q + 1, then the cyclicity is bounded by q + 1, e.g. no more than q + 1 limit cycles can bifurcate from the turning point. Remark 7.11 Though the reduction of a general normal form of a generic turning point as in (6.4) to Liénard form can only be done for analytic systems, it could certainly be done for smooth systems as well up to any finite order. A determination of the finite codimension of general smooth systems can hence be done using such a

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7 Overview of Known Results

reduction up to sufficiently high order. The above theorem, which avoids the setting of infinite codimension, is hence also valid for smooth systems.

7.9.2 The Conjecture The conjecture mentioned in Theorem 7.10 is formulated in terms of a sequence of integrals over well-defined ovals in the plane. Let   1 H (x, y) = e−2y y − x 2 + , 2 and let γh = {(x, y)  : H (x, y) = h} denote the level sets, which are known to be 1 ovals for h ∈ 0, 2 . At h = 12 , the level set is a single point, the origin. At h = 0, the level set is a parabola   1 2 γ0 = (x, y) : y = x − . 2 The conjecture concerns the integrals  Jk = x 2k−1 dy,

k = 0, 1, 2, . . . ,

γh

  where the integration is done counterclockwise. (It is well defined for all h ∈ 0, 12 , and all k ≥ 0.)   Conjecture 7.1 Given k ∈ N and h0 ∈ 0, 12 , the set of functions J := (J0 , J1 , . . . , Jk )   forms a strict Chebyshev system on the interval h0 , 12 . It means that J is a sequence of smooth functions on the given interval with J0 = 0 on the interval and so that the sequence inductively gives rise to additional sequences of smooth functions J1 = (J11 , J21 , . . . , Jk1 ),

with Ji1 :=

Ji J0

with the condition that (J11 ) = 0 on the interval (the prime denotes derivation w.r.t. h), and , . . . , Jk ), J = (J , J +1

with Ji :=

(Ji −1 ) −1 (J −1 )

for = 2, . . . , k, with the condition that (J ) = 0 for all = 2, . . . , k.

7.9 Birth of Canard Cycles

93

Remark 7.12 The definition of a strict Chebyshev system above is equivalent to Definition 13.9. Recently the conjecture was proven using computer-assisted proofs for k ≤ 2 (see [FTV13]).

7.9.3 The Infinite Codimension Case Theorem 7.11 ([DR09]) Let X,(a,μ) be an analytic slow–fast family with a generic turning point at p, with respect to breaking parameter a at a = 0. Suppose the family of vector field is written in classical Liénard form. Then the family has finite cyclicity at p. This theorem includes the center case, i.e. the case where for some parameter value λ = (0, μ0 ), the turning point does not have finite codimension. The above result also has a finer variant formulated in [DR09], under the condition that the conjecture holds. In that case, the concept of finite codimension can be replaced by the concept of finite order, where the order is the number of generators of a well-defined ideal formed by coefficient functions of h1 ; we refer to that paper for details.

7.9.4 Birth of Canard Cycles for the Slow–Fast Bogdanov–Takens Singularity Bounding the local cyclicity of slow–fast Hopf points turned out to be quite difficult. Seemingly contradictory, the similar analysis for the more degenerate contact point with contact order 2 and singularity order 2 is easier: such a point, called a slow–fast Bogdanov–Takens point, has been considered in [DMD11d] (and later in [DMW15]). The local normal form 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (a + bx ± x 2 + x 3 h2 (x, , λ) + yh3 (x, y, , λ))

shows a contact point of singularity order 2 for a = b = 0. Observe that in the unfolding, slow–fast Hopf points are encountered for b < 0, b ∼ 0. However, the ±x 2 term actually ensures that for these values of b the codimension is 1. In this specific context, one was able to show (without referring to the conjecture) the following: Theorem 7.12 The cyclicity of the slow–fast Bogdanov–Takens point at the origin is 2.

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7 Overview of Known Results

7.9.5 Birth of Canard Cycles for More Degenerate Contact Points When the contact point is singular with singularity order m and contact order n, with (m, n) different from (1, 2) and (2, 2), then a desingularization procedure is proposed in [Rou07]. At this moment, it is completely open, even for contact points of classical Liénard type, i.e. of type (m, n) = (1, n) with n ≥ 3. It is observed in [Rou07] that any hope of finding bounds on the cyclicity of a contact point of order (1, n) will rely on global cyclicity bounds of Liénard families of degree less than n.

Part II

Technical Tools

Chapter 8

Blow-up of Contact Points

To make a thorough study of the behavior of regular X,λ -orbits, with  > 0, near a contact point we will blow up the contact point, transforming it into a sphere. The method heavily relies on a three-dimensional point of view in the sense that besides the planar coordinates (x, y), we add  as a third variable. We did something similar when discussing the center manifold theorem (see Theorem 3.1). We will essentially present the blow-up procedure for a single vector field in (x, y, )-space, keeping in mind that we need to control that the whole procedure works in a smooth way with respect to λ ∈ . Let us, for simplicity and motivated by the fact that the blow-up procedure is local, suppose that we work in a local chart of M × [0, 0 ] placing the contact point at (x, y, ) = (0, 0, 0), for all λ ∈ . (It is legitimate to assume without loss of generality that the contact point is located at the origin, independently of λ near some λ0 in case the contact order is 2.)

8.1 Blow-up Procedure We first change the (, λ)-family of two-dimensional vector fields X,λ into a ∂ λ-family of three-dimensional vector fields X,λ + 0 ∂ . To blow up the origin (x, y, ) = (0, 0, 0), we can use ⎧ ⎨ x = up x¯ y = uq y¯ ⎩  = um ¯ , 2 = {(x, with (x, ¯ y, ¯ ) ¯ ∈ S+ ¯ y, ¯ ) ¯ : x¯ 2 + y¯ 2 + ¯ 2 = 1 and ¯ ≥ 0}, and + u ∈ R . The coefficients p, q, and m are natural numbers, which have to be chosen well, depending on the specific equations. In fact, as we will see, in the blow-up

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_8

97

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8 Blow-up of Contact Points

procedure, one can often, but not always, choose m = 1, merely by adapting in an appropriate way the parameter (, λ). Let us denote by  the corresponding blow-up map 2  : ((x, ¯ y, ¯ ¯ ), u) ∈ S+ × R+ → (up x, ¯ uq y, ¯ um ) ¯ ∈ R3 .

We note by Xˆ λ the λ-family of three-dimensional vector fields defined by ∗ (Xˆ λ ) = X,λ + 0

∂ . ∂

∂ In blowing up a λ-family of three-dimensional vector fields X,λ + 0 ∂ at some ∞ point, which we have put at the origin, we get a new C λ-family of vector fields defined in the neighborhood of a half-sphere (see Fig. 8.1). In most situations, the new vector field Xˆ λ is identically zero on the sphere {u = 0}, but since we are only ∂ interested in the behavior of X,λ + 0 ∂ outside the blow-up locus, and hence in the behavior of Xˆ λ outside {u = 0}, we will divide Xˆ λ by some well-chosen power k of u, introducing X¯ λ = u−k Xˆ λ . We will choose k in a way that X¯ λ , defined for u > 0, extends in a smooth way to {u = 0} and is the largest integer such that u−k Xˆ λ extends in a smooth way to the blow-up locus {u = 0} (it implies that this smooth extension is not identically zero). As we will see, this procedure proves to be useful in many cases. On the 2-sphere, we have the relation x¯ 2 + y¯ 2 + ¯ 2 = 1, often implying the need of working in charts. A first  chart is a chart covering the top of the half-sphere. One could eliminate ¯ by ¯ = 1 − x¯ 2 − y¯ 2 and use a chart (x, ¯ y, ¯ u) where x¯ 2 + y¯ 2 < 1

u y



The chart {ε = 1} x

y ε

ε

y

y u

The chart {x = −1}

Fig. 8.1 Blowing up a singular point.

x

u The chart {x = 1}

8.1 Blow-up Procedure

99

to cover the open half-sphere. We prefer a projective chart however, given by {¯ = 1}, commonly known as the family chart. Working in this chart amounts to use ⎧ p ⎨ x = u1 x¯1 q y = u1 y¯1 ⎩ m  = u1 . The chart is valid for (x¯1 , y¯1 ) in a bounded set. Notice that d = 0 is replaced by du1 = 0. This is the traditional chart where people do “rescaling” in. Under the map p q (x¯1 , y¯1 , u1 ) → (u1 x¯1 , u1 y¯1 , um 1 ), the (, λ)-family of vector fields X,λ is transformed into a (u1 , λ)-family, with u1 =  1/m . f f ˆf We could denote this family by Xˆ u1 ,λ and relate to it X¯ u1 ,λ = u−k 1 Xu1 ,λ , where k has the same value as above. By adding “u˙ 1 = 0” as third equation, we find 3D f f vector fields Xˆ λ and X¯ λ . In relation to the spherical blow-up, we easily see that ⎧ ⎨ u1 = u(1 − x¯ 2 − y¯ 2 )1/2m x¯ = x(1 ¯ − x¯ 2 − y¯ 2 )−p/2m ⎩ 1 y¯1 = y(1 ¯ − x¯ 2 − y¯ 2 )−q/2m , with (x, ¯ y) ¯ in a compact set of the disk x¯ 2 + y¯ 2 < 1, corresponding to a compact set f in the (x¯1 , y¯1 )-space. Xˆ u1 ,λ + 0 ∂u∂ 1 is just Xˆ λ written in other coordinates. But as u1 f is just a function proportional to u (see above), X¯ λ is C∞ -equivalent (not conjugate) to X¯ λ . Since 1− x¯ 2 − y¯ 2 = 0, this equivalence also holds between the limiting vector f fields X¯ λ |u=0 and X¯ λ |u1 =0 . This remark is valid for all the coordinate changes we will consider and will not be repeated in detail. Suppose we want to look in the chart given by {x¯ = −1}. We then use ⎧ p ⎨ x = −u2 q y = u2 y¯2 ⎩  = um 2 ¯2 . This chart is also valid for (y¯2 , ¯2 ) in a bounded set and is called a phase-directional chart (or matching chart) or simply the {x¯ = −1} chart. Remark 8.1 Previously we called it a phase-directional chart of phase-directional rescaling chart; the term matching chart is now chosen to highlight the fact that such a chart matches the “interior space” covered by the blow-up space with the “exterior space” outside the blow-up locus {p}.

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8 Blow-up of Contact Points

Notice that d = 0 is replaced by d(um 2 ¯2 ) = 0. Under the map p

q

: (u2 , y¯2 , ¯2 ) → (−u2 , u2 y¯2 , um {x=−1} ¯ 2 ¯2 ), ∂ the vector field X,λ + 0 ∂ is lifted into a C∞ λ-family of three-dimensional vector {x=−1} ¯ {x=−1} ¯ ∂ with ({x=−1} )∗ (Xˆ λ ) = X,λ + 0 ∂ . The blown-up vector field fields Xˆ λ ¯ { x=−1} ¯ { x=−1} ¯ −k ¯ ˆ is given by Xλ = u2 Xλ . The new three-dimensional vector field does not have an invariant foliation defined by planes but has an invariant foliation defined by {um 2 ¯2 = constant}. The constant is given by , and for  = 0, the “leaf” of the foliation is not a regular surface but is given by {u2 = 0} ∪ {¯2 = 0}. {x=−1} ¯ On the domain on which it is defined, the vector field Xˆ λ is not equal to Xˆ λ , ∞ but it is C conjugated to it, or, in other words, it is equal to Xˆ λ up to a coordinate {x=−1} ¯ ¯ ˆ {x=−1} change. Since u2 is different from u, X¯ λ = u−k and X¯ λ = u−k Xˆ λ are 2 Xλ ∞ merely C -equivalent. This will not create problems as long as we aim at studying ∂ properties of X,λ + 0 ∂ that are invariant under C∞ -equivalence. The property that really matters is the fact that locally there exist mutual C∞ equivalences between {x=−1} ¯ ∂ X,λ + 0 ∂ outside the origin, X¯ λ outside {u = 0}, and X¯ λ outside {u2 = 0}. Similarly as for the family rescaling, it can be proven that the C∞ -equivalence {x=−1} ¯ between X¯ λ outside {u = 0} and X¯ λ outside {u2 = 0} extends to, respectively, {u = 0} and {u2 = 0} on their common domain of definition. {x=−1} ¯

|u2 =0 , in terms of (y¯2 , w) Remark 8.2 Also interesting to observe is that X¯ λ with ¯2 = w−m , is C∞ -equivalent to a (p, q)-Poincaré–Lyapunov compactification f of X¯ 0,λ (see [DLA06]). This is a direct consequence of the relations ⎧ p p ⎨ u1 x¯1 = −u2 q q u y¯ = u2 y¯2 ⎩ 1 m1 u1 = um 2 ¯2 ,

(8.1)

which are valid on domains related to { > 0, x < 0}. From (8.1), it follows that 

−1/m

x¯1 = −wp y¯1 = wq y¯2

with w = ¯2 . The transformation is indeed the one that can be used as chart for the PL-compactification on the half plane {x¯1 < 0}. The relation of the multiplying factors is equal to (u1 /u2 )k = w−k , which corresponds to the factor that would be used in the PL-compactification to get a meaningful compactification.

8.2 Blow-up of a Generic Jump Point

101

A similar construction and similar observations can be made in the other charts. On overlapping charts, it is important to make use of the correct coordinate changes, to see the exact relation. For example, on the overlap of the family chart {¯ = 1} and the matching chart {x¯ = −1}, the coordinate change is given by (8.1), which, taking into account that x¯1 < 0, leads to ⎧ ⎨ u2 = u1 (−x¯1 )1/p y¯ = y¯1 (−x¯1 )−q/p ⎩ 2 ¯2 = (−x¯1 )−m/p . To avoid overloaded notation, we will mostly simply use (u, x, ¯ y, ¯ ¯ ) in whatsoever chart that we make the calculation in. This does not cause problems in working with a single chart but requires attention when changing from one chart to the other. It is important always to have the overall geometric picture in mind. Let us now treat some examples that are important in view of the remainder of the book, in order to see how it works.

8.2 Blow-up of a Generic Jump Point Recall that the normal form (2.1) for slow–fast contact points has been refined in Chap. 6 for generic jump points: 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (E + xh2 (x, , λ) + yh3 (x, y, , λ)),

where E = ±1 and all hi are C∞ functions (see (6.2)). Furthermore, the case E = +1 can be reduced to the case E = −1 after reversing the time; we will blow up the generic jump point with E = −1. We use the blow-up: ¯ u3 ¯ ), (x, y, ) = (ux, ¯ u2 y,

(8.2)

with x¯ 2 + y¯ 2 + ¯ 2 = 1, and we divide the vector field by u, see Figs. 8.2 and 8.3. The family rescaling (¯ = 1) leads, after division by u, to ⎧ ¯ λ) ⎨ x˙¯ = y¯ − x¯ 2 + ux¯ 3 h1 (ux, y˙¯ = −1 + O(u) ⎩ u˙ = 0.

(8.3)

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8 Blow-up of Contact Points

ε ε

x

p

s−

s+ q1

x

q2

y y

Fig. 8.2 The jump point, before and after the blow-up. The relative positions for the separatrices of q1 and the strong stable separatrix are proved for the case n = 2 in Sect. 8.5.3.

q2

q1

s+

s−

Fig. 8.3 Bird’s-eye view of the blown-up jump point. The dynamics drawn inside the circle is the dynamics on the spherical surface from Fig. 8.2.

The vector field for u = 0 does not depend on λ and is given by 

x˙¯ = y¯ − x¯ 2 y˙¯ = −1.

(8.4)

Observe that we use here the letters (x, ¯ y) ¯ instead of, e.g. (x¯1 , y¯1 ) (this mild abuse of notation will also be used in other charts) and that we also denote derivation by time by means of a dot, although the time is not the original one since we have changed it by a factor u. System (8.4) is a very simple system, without singularities, that can easily be solved. For u ∼ 0, the orbits of (8.3) stay close to orbits of (8.4).

8.2 Blow-up of a Generic Jump Point

103

In the matching chart {x¯ = 1}, we find ⎧ ⎨ u˙ = u(y¯ − 1 + uh1 (u, λ)) ˙¯ = −3¯ (y¯ − 1 + uh1 (u, λ)) ⎩˙ y¯ = −2y( ¯ y¯ − 1 + uh1 (u, λ)) − ¯ (1 + O(u)).

(8.5)

We clearly see that  = u3 ¯ remains unchanged or, in other words, that it is a first integral. Important in this chart is what happens near the line {u = 0, ¯ = 0}. In fact the situation near {u = 0, ¯ ≥ ¯0 }, for some 0 > 0, can better be studied in the family rescaling, while the situation near {u ≥ u0 , ¯ = 0}, for some u0 > 0, can better be studied in the original coordinates, without blowing up. These matching charts (x¯ = ±1 and y¯ = ±1) exactly serve to study the passage from the original coordinates to the traditional rescaling coordinates; a reason that we could also call them matching charts instead of phase-directional charts. On {u = 0, ¯ = 0}, we have y˙¯ = −2y( ¯ y¯ − 1), and hence we find two singularities, at, respectively, y¯ = 0 and y¯ = 1. The singularity at (0, 0, 0) will be denoted s− and the one at (0, 0, 1) will be denoted q1 , see Fig. 8.3. At s− the linear part of (8.5) has (−1, 3, 2) as eigenvalues, while at y¯ = 1 the eigenvalues are (0, 0, −2). So the singularity s− is a (resonant) hyperbolic saddle, while the singularity q1 at (u, ¯ , y) ¯ = (0, 0, 1) is semi-hyperbolic with a onedimensional stable manifold exactly given by {u = ¯ = 0}, and the linear center space is given by {y¯ = 0}; it has two-dimensional center behavior, of which every center manifold is transverse to {u = ¯ = 0} and intersects {¯ = 0} at the critical curve γ or more precisely at its transform  γ by means of (8.2). Near s− (8.5) is equivalent to ⎧ ⎪ ⎨ u˙ = −u ˙¯ = 3¯ ⎪ ⎩ y˙¯ = 2y¯ − ¯ 1+O(u) . 1−y−uh ¯ 1 (u,λ)

(8.6)

In fact, we will show in Sect. 8.3.1 for the jump point as well as for more general regular contact points that an additional linear change of coordinates changes the system into ⎧ ⎨ u˙ = −u (jump point near s− ) ˙¯ = 3¯ ⎩˙ ¯ λ), Y = 2Y + ¯ G− (Y, u, ,

(8.7)

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8 Blow-up of Contact Points

where G− (0, 0, 0, λ) = 0. It is this local representation that turns out to be important in the later analysis, more particularly in Chap. 12. The advantage of the blow-up procedure becomes clear when studying the singular point q1 at (0, 0, 1), since the normal hyperbolicity that we did not have on γ at (0, 0) is now found on the limiting point of  γ , i.e. at (0, 0, 1). To study system (8.5) near q1 , it is better to introduce z¯ = y¯ − 1, changing (8.5) into ⎧ ⎨ u˙ = u(¯z + uh1 (u, λ)) (8.8) ˙¯ = −3¯ (¯z + uh1 (u, λ)) ⎩˙ z¯ = −2(1 + z¯ )(¯z + uh1 (u, λ)) − ¯ (1 + O(u)). On ¯ = 0, system (8.8) is given by 

u˙ = u(¯z + uh1 (u, λ)) z˙¯ = −2(1 + z¯ )(¯z + uh1 (u, λ)),

(8.9)

which clearly has a curve of singularities, defined by z¯ = −uh1 (u, λ).

(8.10)

Along that curve system (8.9) is normally attracting, including at (u, z¯ ) = (0, 0). This curve  γ is the blow-up of γ . On u = 0, system (8.8) is given by 

¯˙ = −3¯ z¯ z˙¯ = −2(1 + z¯ )¯z − . ¯

(8.11)

System (8.11) has a unique center manifold in {¯ ≥ 0} and it is smooth (see [DLA06]). Its asymptotic development is given by 1 1 z¯ = − ¯ + ¯ 2 + O(¯ 3 ), 2 8 and the center behavior, in terms of ¯ , is given by 3 ˙¯ = ¯ 2 + O(¯ 3 ). 2

(8.12)

We can also calculate the behavior inside a two-dimensional center manifold of (8.8), and we easily get that there are no singularities outside the curve of singularities given by (8.9). In (8.8), as we know, for any k ∈ N \ {0} (and for any λ ∈ ), there is a Ck+1 center manifold W , depending on a Ck+1 way on λ, that can be written as z¯ =

8.2 Blow-up of a Generic Jump Point

105

w(u, , ¯ λ), with w(u, 0, λ) defined by (8.10) and w(0, ¯ , λ) as a center manifold of (8.11). If we introduce the new variable Z = z¯ − w(u, , ¯ λ), without changing (u, ¯ ), then we get the following expression for the center behavior on {Z = 0}: 

u˙ = u¯ w(u, ¯ ¯ , λ) ˙¯ = −3¯ 2w(u, ¯ ¯ , λ).

(8.13)

We have used w(u, , ¯ λ) = −uh1 (u, λ) + ¯ w(u, ¯ , ¯ λ) for some Ck function w(u, ¯ , ¯ λ). Because of (8.12) we know that w(0, ¯ 0, λ) > 0, proving our claim. We hence see that the behavior on a center manifold is Ck -equivalent to 

u˙ = u¯ ˙¯ = −3¯ 2 .

If we make the calculations in the {x¯ = −1} matching chart, we will also find a resonant saddle s+ and a (repelling) semi-hyperbolic singularity q2 . In fact there is no need to do this, because of the fact that in the blow-up procedure (8.2), the exponent of u in front of x¯ is odd. So, instead of putting x¯ = −1, we can as well keep x¯ = 1 and look on the side u ≤ 0. Of course the time will be reversed for ¯ since we have divided X  by u. For the sake of future the resulting vector field X, reference, we write down the specific form of the vector field near s+ , obtained in the same way as the one near s− (see Sect. 8.3.2): ⎧ ⎨ u˙ = u (jump point near s+ ) ˙¯ = −3¯ ⎩˙ ¯ λ), Y = −2Y + ¯ G+ (Y, u, ,

(8.14)

where G+ (0, 0, 0, λ) = 0. If we work out the matching chart {y¯ = −1}, we find no singularities along {u = 0 = ¯ }. In the matching chart {y¯ = +1}, we only have the two semi-hyperbolic singularities q1 and q2 that we already found in the respective charts {x¯ = +1} and {x¯ = −1}. Putting all the information together, we end up with a blown-up situation as represented in Fig. 8.3. In Figs. 8.2 and 8.3, the circle C represents {u = ¯ = 0}, and the region inside the circle represents the blow-up locus, on which we have drawn the system (8.4). The region outside the circle represents the two-dimensional blow-up of the singularity. Figure 8.3 represents a bird’s eye view of a three-dimensional situation, in (x, y, )-variables, in which the inner part of the circle is like a blister on the (x, y)plane with the circle as edge. The planes { = constant} are represented in this three-dimensional picture like sticking plasters above the blister. On this picture it is now clear that, close to the generic jump point, solutions for  > 0 can best be studied after blow-up. The reason is that the contact point has been

106

8 Blow-up of Contact Points

desingularized, in the sense that we now only encounter singularities that are either hyperbolic or partially hyperbolic. Of course we have to work in three dimensions and study transition maps near these three-dimensional situations. This will be done in Chap. 12. The phase portrait in Fig. 8.3 is easily justified, using the Poincaré–Bendixson theorem. The relative position of the strong separatrices of the saddles s± and of the center separatrix of q1 is justified in Sect. 8.5.3.

8.3 Blow-up of Regular Contact Points In this section we generalize the analysis made in Sect. 8.2. Based on the normal form (2.1) and after an extra rescaling in (x, y, t), we can write a regular contact point as 

x˙ = y − fλ (x) y˙ = (±1 + xh2 (x, , λ) + (y − fλ (x))h3 (x, y, , λ)),

(8.15)

where fλ (x) = ±x n (1 − xh1 (x, λ)), h2 (x, , λ) and h3 (x, y, , λ) are smooth functions, and the ± signs in y˙ and in fλ are independent from one another. It is easy to see that the four cases (±, ±) reduce to just one, up to linear diffeomorphism, change of time, and adaptation of the functions h1 , h2 , and h3 . It will hence suffice to consider the following case (−, +):  n : X,λ

x˙ = y − x n (1 − xh1 (x, λ)) y˙ =  [−1 + xh2 (x, , λ) + h3 (x, y, , λ) (y − x n (1 − xh1 (x, λ)))] .

As blow-up, we use x = ux, ¯ y = un y, ¯  = u2n−1 ¯ , with the corresponding blow-up map 2 × R+ → (ux, ¯ un y, ¯ u2n−1 ¯ ) ∈ R3 ,  : ((x, ¯ y, ¯ ¯ ), u) ∈ S+ 2 = {x¯ 2 + y¯ 2 + ¯ 2 = 1, ¯ ≥ 0}. We denote by X ˆ n the three-dimensional with S+ λ λ-family of vector fields defined by n ∗ (Xˆ λn ) = X,λ

and the blown-up family (of vector fields) as X¯ λn =

1 ˆn X . un−1 λ

8.3 Blow-up of Regular Contact Points

107

2 ∩ {¯ To study X¯ λn on S+  > 0}, we use the family chart {¯ = 1}. We find, like in the study of the generic jump point, a vector field without singular points:

⎧ ⎨ x˙¯ = y¯ − x¯ n + O(u) y˙¯ = −1 + O(u) ⎩ u˙ = 0. More qualitative information can be deduced from the matching charts. To study X¯ λn near the equator {¯ = 0}, we use the four charts {x¯ = 1}, {x¯ = −1}, {y¯ = 1}, {y¯ = −1}. On the blow-up sphere, we will find four singular points for X¯ λn , all located on the circle u = ¯ = 0: (i) Two hyperbolic saddles s− and s+ . (ii) Two semi-hyperbolic points q1 and q2 . The saddles s− and s+ can best be studied in the respective charts {x¯ = 1} and {x¯ = −1}. If n is even, the y-value ¯ of both q1 and q2 is strictly positive and if n is odd, the y-value ¯ of q1 is strictly positive while the y-value ¯ of q2 is strictly negative, see Fig. 8.4. The qualitative information in this figure will follow from the analysis in the following subsections. We will study these points in the charts {y¯ = ±1} where it will be easier to establish a normal form.

q2

q1 q1

s+

s−

s+

s− q2

Fig. 8.4 Left: blow-up of regular contact point of even degree. Right: blow-up of regular contact point of odd degree

108

8 Blow-up of Contact Points

8.3.1 The Saddle s− In the matching chart {x¯ = 1}, the blow-up map is given by x = u, y = un y, ¯ and  = u2n−1 . ¯ We obtain ⎧ y˙¯ = −nun−1 H (y, ¯ u, , ¯ λ)y¯ ⎪ ⎪

⎨ n−1 ¯ −1 + uh (u, u2n−1 ¯ , λ) + O(un ) +u 2 n ˆ Xλ : ⎪ u˙ = un−1 H (y, ¯ u, , ¯ λ)u ⎪ ⎩ ¯ u, , ¯ λ)¯ , ˙¯ = −(2n − 1)un−1 H (y, with H (y, ¯ u, , ¯ λ) = y¯ − 1 + uh1 (u, λ). 1 ˆn Xλ . Let us recall that on its The blown-up family we consider is X¯ λn = un−1 ∂ n + 0 ∂ , but domain of definition, there might not hold that ∗ (un−1 X¯ λn ) = X,λ ∂ n ∞ n 2 ¯ in any case we know that Xλ on R × ]0, ∞[ is C -equivalent to X,λ + 0 ∂ on R3 ∩{x > 0}. We will also not repeat this observation in the sequel of this definition, when working in other charts. The family X¯ λn has two singular points on the blow-up sphere in this chart: the point s− = (0, 0, 0) and the point q1 = (1, 0, 0). As we just want to study the point s− in this chart, we can restrict to a compact neighborhood W− of s− in which H (y, ¯ u, ¯ , λ) < 0, since H (0, 0, 0, λ) = −1. In this neighborhood, the blown-up 1 ¯ ˆ family is equivalent to the family −1 H Xλ = − un−1 H Xλ : ⎧ n) −1+uh2 (u,u2n−1 ,λ)+O(u ¯ ⎪ ¯ ⎨ y˙¯ = ny¯ + 1−y−uh ¯ 1 (u,λ) − H −1 X¯ λ : u˙ = −u ⎪ ⎩˙ ¯ = (2n − 1)¯ . The linear part of

−1 ¯ H Xλ

(8.16)

at s− is equal to ⎧ ⎨ y˙¯ = ny¯ − ¯ u˙ = −u ⎩˙ ¯ = (2n − 1)¯ .

(8.17)

As the three eigenvalues are pairwise distinct (a consequence of n ≥ 2), we can diagonalize this linear part. Explicitly, a diagonalization is given by (y, ¯ u, ) ¯ → (Y, u, ) ¯ = ((n − 1)y¯ + ¯ , u, ) ¯ . If we apply this coordinate change to (8.16), we obtain ⎧ ⎨ Y˙ = nY + (n − 1)¯ G− (Y, u, u2n−1 ¯ , λ) u˙ = −u ⎩ ˙ ¯ = (2n − 1)¯ ,

(8.18)

8.3 Blow-up of Regular Contact Points

109

where G− (Y, u, u2n−1 ¯ , λ) =

(n − 1)u(h2 (u, u2n−1 ¯ , λ) − h1 (u, λ)) − Y + ¯ + O(un ) . (n − 1)(1 − uh1 (u, λ)) − Y + ¯

We notice that G− (0, 0, 0, λ) = 0 so that the term (n − 1)¯ G− is of order 2. This explains the expression for the case n = 2, formulated in (8.7).

8.3.2 The Saddle s+ In the matching chart {x¯ = −1}, the blow-up map is given by x = −u, y = u2 y, ¯  = u2n−1 ¯ . We obtain ⎧ y˙¯ = −nun−1 H (y, ¯ u, , ¯ λ)y¯ ⎪ ⎪

⎨ n−1 ¯ 1 + uh (−u, u2n−1 ¯ , λ) + O(un ) −u 2 n ˆ Xλ : ⎪ u˙ = un−1 H (y, ¯ u, , ¯ λ)u ⎪ ⎩ ˙¯ = −(2n − 1)un−1 H (y, ¯ u, ¯ , λ)¯ ,

(8.19)

with H (y, ¯ u, , ¯ λ) = −y¯ + (−1)n (1 + uh1 (−u, λ)). The point s+ is located at the origin. There are now two different cases: (a) Case n Even In this case we have H (y, ¯ u, , ¯ λ) = 1 + uh1 (u, λ) − y¯ . Then H (0, 0, 0, λ) = 1, and we can find a compact neighborhood W+ of s+ in which H (y, ¯ u, , ¯ λ) > 0. The 1 ˆ blown-up family is defined on W+ as X¯ λ = n−1 Xλ , but we consider u

⎧ 2n−1 ¯ n) ⎪ y˙¯ = −ny¯ + −1−uh2 (−u,u ,λ)+O(u ¯ 1+uh1 (−u,λ)−y¯ 1 ¯n ⎨ Xλ : u˙ = u ⎪ H ⎩˙ ¯ = −(2n − 1)¯ . The linear part of

1 ¯ H Xλ

(8.20)

is given by ⎧ ⎨ y˙¯ = −ny¯ − ¯ u˙ = u ⎩˙ ¯ = −(2n − 1)¯ .

As the three eigenvalues are pairwise distinct (a consequence of n ≥ 2), we can diagonalize. Explicitly, a diagonalization is given by (y, ¯ u, ) ¯ → (Y, u, ) ¯ = ((n − 1)y¯ − , ¯ u, ). ¯

110

8 Blow-up of Contact Points

If we apply the coordinate change to (8.20), we obtain ⎧ ⎨ Y˙ = −nY + (n − 1)¯ G+ (Y, u, u2n−1 ¯ , λ) u˙ = u ⎩ ˙ ¯ = −(2n − 1)¯ , where G+ (Y, u, u2n−1 ¯ , λ) =

−(n − 1)u(h2 − h1 ) − Y − ¯ + O(un ) , (n − 1)(1 + uh1 (−u, λ)) − Y − ¯

where h2 = h2 (−u, u2n−1 ¯ , λ) and h1 = h1 (−u, λ). We notice that G− (0, 0, 0, λ) = 0 and that the term (n − 1)¯ G+ is of order 2. This explains the expression for the case n = 2, formulated in (8.14). (b) Case n Odd In that case we have H (y, ¯ u, , ¯ λ) = −1 − uh1 (−u, λ) − y. ¯ Then H (0, 0, 0, λ) = −1, and we can find a compact neighborhood W+ of s+ in which H (y, ¯ u, , ¯ λ) < 0. 1 ˆ The blown-up family is defined on W+ as X¯ λn = un−1 Xλ , but we consider ⎧ 2n−1 n ⎪ y˙¯ = ny¯ − 1+uh2 (−u,u ¯ ,λ)+O(u ) ¯ 1+uh1 (−u,λ)+y¯ −1 ¯ n ⎨ X u˙ = −u H λ⎪ ⎩˙ ¯ = (2n − 1)¯ . The linear part of

−1 ¯ H Xλ

(8.21)

is given by ⎧ ⎨ y˙¯ = ny¯ − ¯ u˙ = −u ⎩˙ ¯ = (2n − 1)¯ .

As the three eigenvalues are pairwise distinct (a consequence of n ≥ 2), we can diagonalize. Explicitly, a diagonalization is given by (y, ¯ u, ) ¯ → (Y, u, ) ¯ = ((n − 1)y¯ + , ¯ u, ). ¯ If we apply the coordinate change to (8.21), we obtain ⎧ ⎨ Y˙ = nY + (n − 1)¯ G+ (Y, u, u2n−1 ¯ , λ) u˙ = −u ⎩ ˙ ¯ = (2n − 1)¯ ,

(8.22)

8.3 Blow-up of Regular Contact Points

111

where G+ (Y, u, u2n−1 , ¯ λ) =

(1 − n)u(h2 (−u, u2n−1 ¯ , λ) − h1 (−u, λ)) + Y − ¯ + O(un ) . (n − 1)(1 + uh1 (−u, λ)) + Y − ¯

We notice that G+ (0, 0, 0, λ) = 0 and that the term (n − 1)¯ G+ is of order 2.

8.3.3 The Semi-Hyperbolic Points In the Matching Chart {y¯ = 1} In this chart, the blow-up map is given by x = ux, ¯ y = un ,  = u2n−1 . ¯ We have   ⎧ n−1 1 − x¯ n (1 − uxh ⎪ ¯ 1 ) + n1 ¯ x(1 ¯ − uxh ¯ 2 + O(un )) ⎨ x˙¯ = u Xˆ λn : u˙ = − 1 un (1 ¯ − uxh ¯ 2 + O(un )) ⎪ n ⎩˙ u˙ n−1 ¯ 2 (1 − uxh ¯ = −(2n − 1) u ¯ = 2n−1 ¯ 2 + O(un )), n u

(8.23)

where h1 = h1 (ux, ¯ λ) and h2 = h2 (ux, ¯ u2n−1 ¯ , λ). As blown-up family of vector 1 n ¯ Xˆ n . If we choose any interval fields we can choose Xλ = un−1 (1−uxh ¯ 2 +O(un )) λ [−x¯1 , x¯1 ], the family is defined in a neighborhood of [−x¯1 , x¯1 ] × {0} × {0}, if u and ¯ are small enough. ⎧ 1−x¯ n (1−uxh ¯ 1) 1 ⎪ ⎨ x˙¯ = 1−uxh ¯ 2 +O(un ) + n ¯ x¯ n 1 ¯ Xλ : u˙ = − n u ¯ ⎪ ⎩ ˙¯ = 2n−1 ¯ 2 . n

(8.24)

Observe that we simply write X¯ λn instead of H1 X¯ λn , for some strictly positive function H . We will often do this in the sequel if we are only interested in results that are invariant under C∞ -equivalences. In restriction to the axis {u = ¯ = 0}, Eq. (8.24) reduces to x˙¯ = 1 − x¯ n . There are two cases as follows: (a) n Even The vector field X¯ λ has two singular points q1 = (1, 0, 0) and q2 = (−1, 0, 0). At q1 , one chooses a chart with coordinates X = x¯ − 1, u, ¯ . In this coordinate system, X¯ λn is given by ⎧ 1 ⎨ X˙ = −nX + h1 (0, λ)u + n ¯ + O((X, u, ¯ )2 ) n 1 X¯ λ : u˙ = − n ¯ u ⎩ ˙ 2 ¯ = 2n−1 n ¯ .

(8.25)

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8 Blow-up of Contact Points

Then the vector field for ¯ = 0 is normally hyperbolic along X = 0, and X¯ λn has a two-dimensional center direction transverse to the X-axis. At q2 , we choose a chart with coordinate X = x¯ + 1 and in the chart ⎧ 1 ¯ 2) ⎨ X˙ = nX − h1 (0, λ)u − n ¯ + O((X, u, ) n 1 X¯ λ : u˙ = − n u ¯ ⎩ ˙ 2n−1 2 ¯ = n ¯ .

(8.26)

At this point q2 , the vector field is also normally hyperbolic along X = 0, with a two-dimensional center direction transverse to X-axis. For future reference, we will now write down detailed expressions of (8.25) and (8.26) in case n = 2, which we obtain after rescaling time by a factor of 2 and an additional appropriate linear change of coordinates X → X + ci (λ)u + di (λ)¯ , i = 1, 2: ⎧ ¯ λ) ⎨ X˙ = −4X + G1 (X, u, , (jump point near q1 ) u˙ = −¯ u ⎩ ˙ ¯ = 3¯ 2 ,

(8.27)

¯ λ) is of order 2 at (0, 0, 0); where (X, u, ) ¯ → G1 (X, u, , ⎧ ⎨ X˙ = 4X + G2 (X, u, ¯ , λ) (jump point near q2 ) u˙ = −¯ u ⎩ ˙ ¯ = 3¯ 2 ,

(8.28)

¯ λ) is of order 2 at (0, 0, 0). where (X, u, ) ¯ → G2 (X, u, , (b) n Odd The vector field X¯ λ has only one singular point q1 = (1, 0, 0). At q1 , one chooses a chart with coordinates X = x¯ − 1, u, . ¯ In these coordinates, X¯ λ is given by (8.25). In the matching chart {y¯ = −1}, the blow-up map is given by x = ux, ¯ y = −un , 2n−1 and  = u ¯ . By derivation, we have   ⎧ 1 n−1 −1 − x¯ n (1 − uxh n )) ˙ ⎪ x ¯ = u ¯ ) −  ¯ x(1 ¯ − u xh ¯ + O(u 1 2 ⎨ n n )) Xˆ λn : u˙ = 1 un ¯ (1 − uxh ¯ + O(u 2 n ⎪ ⎩˙ n−1 ¯ 2 (1 − uxh ¯ = −(2n − 1) uu˙ ¯ = − 2n−1 ¯ 2 + O(un )), n u

(8.29)

where h1 = h1 (ux, ¯ λ) and h2 = h2 (ux, ¯ u2n−1 ¯ , λ). An appropriate blown-up 1 ¯ Xˆ . This family is defined at family of vector fields is Xλ = un−1 (1−uxh ¯ 2 +O(un )) λ n the points where 1 − uxh ¯ 2 + O(u ) = 0. If we choose any interval [−x¯1 , x¯1 ], the

8.4 Blow-up of a Generic Turning Point

113

family is defined in a neighborhood of [−x¯1 , x¯1 ] × {0} × {0}, if u and ¯ are small enough. ⎧ −1−x¯ n (1−uxh ¯ 1) 1 ⎪ ⎨ x˙¯ = 1−uxh ¯ 2 +O(un ) − n ¯ x¯ X¯ λ : u˙ = n1 ¯ u ⎪ ⎩ ˙¯ = − 2n−1 ¯ 2 . n

(8.30)

In restriction to the axis {u = ¯ = 0}, Eq. (8.30) reduces to x¯˙ = −1 − x¯ n . Then, the family X¯ λ has no singular points along the axis when n is even and one singular point q2 = (−1, 0, 0) when n is odd. In the local coordinates X = x¯ + 1, u, ¯ , we have ⎧ 1 ¯ 2) ⎨ X˙ = −nX + h1 (0, λ)u + n ¯ + O((X, u, ) 1 X¯ λ : (8.31) u˙ = n ¯ u ⎩ ˙ 2n−1 2 ¯ = − n ¯ , see also Fig. 8.4 (to the right). Conclusion When n is even, the two semi-hyperbolic points are in the matching chart {y¯ = 1}. When n is odd, the point q1 belongs to the matching chart {y¯ = 1} and the point q2 to the matching chart {y¯ = −1}. This point q2 can be studied in a similar way as the point q1 .

8.4 Blow-up of a Generic Turning Point Recall that the normal form (2.1) for slow–fast contact points has been refined in Chap. 6 for generic turning points: 

x˙ = y − x 2 + x 3 h1 (x, λ) y˙ = (a − x + x 2 h2 (x, , λ) + yh3 (x, y, , λ)),

(8.32)

with h1 , h2 , and h3 C∞ functions (see (6.4)); recall that λ = (a, μ). If we want to blow up the turning point, we first change the (μ-dependent) two-parameter family X,a of two-dimensional vector fields into a (μ-dependent) 1-parameter family of three-dimensional vector fields, i.e. we write (, a) = (ν 2 ¯ , ν a), ¯ with ¯ 2 + a¯ 2 = 1 and ν ∈ [0, ν0 [. Working directly with a circular parameter (¯ , a) ¯ is not so convenient, hence we make use of the different possibilities ¯ = 1 and a¯ = ±1.

114

8 Blow-up of Contact Points

√ It reveals that the last two ones are not so interesting: it is where |a| ≥ K  for some large K, which is a region in parameter space that does not allow a canardtype passage, see Proposition 6.1 for an argument without blow-up or Sect. 8.4.1 for an argument with blow-up. We will not pay further attention to these charts in this book. We will hence only consider (, a) = (ν 2 , ν a), ¯

(8.33)

with a¯ belonging to some (arbitrarily) large interval [−a¯ 1 , a¯ 1 ] in R. By using the parameters (ν, a), ¯ we actually will work with a family of vector fields, induced by the map (8.33):  = Xν 2 ,(ν a¯ ,μ) Xν, a,μ ¯  or equivalently with the (nondifferentiable) X√ . It means that we restrict the ,a,μ ¯ initial family to in the (, a)-space to a cone

√ |a| ≤ a¯ 1 , see Fig. 8.5. Remark 8.3 During considerable parts of the next chapters of this book, we will restrict our study to the family of vector fields in the parameter set (8.33). In fact we will do so whenever it is clear that results on the induced family of vector fields  := X,(√ a,μ) are sufficient to directly deduce the similar results on the full X, ¯ a,μ ¯ family X,λ in view of Proposition 6.1. Furthermore, since parts of the analysis near the blow-up locus of the Hopf point are similar to the analysis of a jump point, we will introduce a new notation (b, η) = (a, ¯ μ) for the parameters of the blown-up vector field at a Hopf point. This way, and by renaming (b, η) = (a, μ) for vector fields that do not require a blow-up of a Hopf point, we will be able to present a lot of results in a unified way.

a ¯

a

ν

Fig. 8.5 Rescaling of the parameter domain (8.33)

ε

8.4 Blow-up of a Generic Turning Point

115

We hence change (8.32) into the (a, ¯ μ)-family  X¯ a,μ ¯

⎧ ¯ μ) ⎨ x˙ = y − x 2 + x 3 h1 (x, ν a, : y˙ = ν 2 (ν a¯ − x + x 2 h2 (x, ν 2 , ν a, ¯ μ) + yh3 (x, y, ν 2 , ν a, ¯ μ)) ⎩ ν˙ = 0,

(8.34)

with h1 , h2 , and h3 C∞ functions. The parameter a¯ is called a breaking parameter of the Hopf breaking mechanism, for a reason that will become clear during the construction. To blow up (8.34), we use (x, y, ν) = (ux, ¯ u2 y, ¯ u¯ν ). The family rescaling (¯ν = 1) leads, after division by u, to ⎧ ¯ ua, ¯ μ) ⎨ x˙¯ = y¯ − x¯ 2 + ux¯ 3 h1 (ux, ˙y¯ = a¯ − x¯ + O(u) ⎩ u˙ = 0.

(8.35)

The vector field for u = 0 does not depend on μ and is given by 

x˙¯ = y¯ − x¯ 2 y˙¯ = a¯ − x. ¯

(8.36)

In Fig. 8.6, we draw the different phase portraits, depending on a, ¯ of the systems (8.36). We also draw their behavior at infinity, which can be calculated in the different matching charts. It reveals that there are four singularities at infinity, of which s1 and s4 are hyperbolic saddles, while s2 and s3 are semi-hyperbolic points. In Fig. 8.6, we represent the phase portraits of the blown-up vector fields on the blow-up locus (i.e. on the upper half-sphere). The pictures indicate that with changing a, ¯ near a¯ = 0, a connection between s2 and s3 breaks, explaining why we call a¯ a breaking

0

=0

Fig. 8.6 Blow-up pictures of a generic turning point.

0

116

8 Blow-up of Contact Points

parameter. We will prove this explicitly in Sect. 8.5.4, including the fact that the splitting of separatrices occurs in a regular way. The study near s1 , s2 , s3 , s4 is made in the phase-directional charts. We start with the matching chart {x¯ = 1} and divide by u: ⎧ ¯ μ)) ⎨ u˙ = u(y¯ − 1 + uh1 (u, u¯ν a, ν˙¯ = −¯ν (y¯ − 1 + uh1 (u, u¯ν a, ¯ μ)) ⎩˙ ¯ μ)) − ν¯ 2 (1 − ν¯ a¯ + O(u)). y¯ = −2y( ¯ y¯ − 1 + uh1 (u, u¯ν a,

(8.37)

We clearly see that ν = u¯ν remains unchanged or, in other words, that it is a first integral. Expression (8.37) is very similar to the one that we found in (8.5) for the generic jump point. We can study it in an analogous way. s2 and s3 can best be studied in the matching chart {y¯ = 1}. In this chart, the blown-up vector field is given by ⎧ 1 ¯ u, ν, ¯ a, ¯ μ) ⎨ u˙ = − 2 ν¯ 2 uH (x,  1 3 ¯ ˙ Xa,μ ¯ u, ν, ¯ a, ¯ μ) ¯ : ⎩ ν¯ = 2 ν¯ H (x, x˙¯ = 1 − x¯ 2 + 12 x¯ 2 ν¯ 2 − 12 a¯ x¯ ν¯ 3 + O(u),

(8.38)

where H (x, ¯ u, ν, ¯ a, ¯ μ) = (x¯ − a¯ ν¯ + O(u)). The point s2 is located at x¯ = −1, and the point s3 is located at x¯ = 1. To study the point s2 , we use the coordinates (X = x¯ +1, u, ν¯ ). The function H is changed into −H2 (X, u, ν¯ , a, ¯ μ) = H (−1 + X, u, ν¯ , a, ¯ μ) = −1 + X − a¯ ν¯ + O(u). Up to a smooth equivalence, we will consider in a neighborhood of s3 the field given by ⎧ X˙ = +4X + G2 (X, u, ν, ¯ a, ¯ μ), 2 ¯ ⎨ 2u Xa,μ : u ˙ = ν ¯ ⎩ ˙ H2 ¯ ν¯ = −¯ν 3

(8.39)

where G2 (X, u, ν¯ , a, ¯ μ) = ν¯ 2 + X.O((X, ν¯ )) + O(u) is smooth. To study the point s3 , we use the coordinates (X = x¯ − 1, u, ν¯ ). The function H is changed into H3 (X, u, ν¯ , a, ¯ μ) = H (1 + X, u, ν¯ , a, ¯ μ) = 1 + X − a¯ ν¯ + O(u) up to a smooth equivalence we will consider in a neighborhood of s2 the field given by ⎧ ¯ a, ¯ μ) X˙ = −4X + G3 (X, u, ν, 2 ¯ ⎨ 2u : Xa,μ u ˙ = −¯ ν ⎩ ˙ H3 ¯ ν¯ = ν¯ 3

(8.40)

where G3 (X, u, ν¯ , a, ¯ μ) = ν¯ 2 + X.O((X, ν¯ )) + O(u) is smooth. The only delicate point concerning the phase portraits in Fig. 8.6 deals with the breaking of the existing connection between the semi-hyperbolic singularities s2 and s3 when a¯ = 0. we can not only show that it exists for a¯ = 0 and only for a¯ = 0

8.4 Blow-up of a Generic Turning Point

117

but also show that it breaks in a regular way, explaining why we call a¯ a “breaking parameter.” For the sake of completeness, we recall the proof in Sect. 8.5.4.

8.4.1 Blow-up of the Turning Point for |a| 

√ 

In Proposition 6.1, we have seen that orbits for  > 0 sufficiently√small cannot follow a canard segment through a turning point whenever |a|  . The proof was based on an argument with a trapping region. Using blow-up, we can give a more insightful argument, see also [DR96]. We will use the parameter charts (, a) = (ν 2 E, σ ν),

for σ = ±1,

(8.41)

√ with ν ≥√0 and E ∈ [0, E1 [. The restriction E ≤ E0 translates to |a| ≥ K  with K = 1/ E1 . Together with (8.33), a full neighborhood of (, a) = (0, 0) is covered when choosing E1 a¯ 12 > 1. We will prove, for E ∈ [0, E1 [ and for E1 sufficiently small, no canard-type passages are possible, hereby restricting the study of relevant orbits to the parameter chart (8.33), taking a¯ 1 sufficiently large (Fig. 8.7). A full analysis would require us to examine both the family chart and all the matching charts in the parameter regime (8.41), but a geometrically clear argument can already be given by examining the family chart alone. The vector field in the family chart changes from (8.35) into ⎧ ⎨ x˙¯ = y¯ − x¯ 2 + O(u) y˙¯ = E(σ − x¯ + O(u)) ⎩ u˙ = 0. This is a (u, μ)-family of slow–fast systems expressed in the singular parameter E. Its analysis is quite simple: the critical curve is given by y = x¯ 2 + O(u). It has a contact point at O(u)-distance from the origin, which separates the attracting branch (to the right) from the repelling branch. The critical curve possesses a so-called slow singularity at x¯ = σ + O(u), which is an attracting node located on the attracting

Fig. 8.7 Blow-up of a slow–fast Hopf point in all parameter regions

118

8 Blow-up of Contact Points

branch when σ = +1 and which is a repelling node located on the repelling branch when σ = −1. The contact point is a generic jump point; the fast fiber passing through this point connects s4 to s1 . From the presence of the slow singularity, it is now immediately clear that, for sufficiently small values of E, it is impossible for orbits to pass from a neighborhood of s3 to a neighborhood of s2 . This again proves Proposition 6.1.

8.4.2 Asymptotic Expansions in the Blow-up of the Hopf Point Recall from Sect. 6.4 that, for a slow–fast Hopf point expressed in normal form coordinates like in (8.32), there exists an asymptotic expansion, which might be called an “outer expansion.” One can blow up this outer expansion and study it in rescaled coordinates. Alternatively, one can derive the so-called inner expansion directly from the vector field in the family chart. Relating the blown-up outer expansion to the inner expansion will not be necessary to prove our results; instead, we adopt a geometric point of view in this book, working with invariant manifolds instead of asymptotic series, and we will trace the invariant manifolds through the different blow-up charts using traditional techniques from dynamical systems. There is one point in the analysis though where the presence of an outer expansion reveals useful; in Chap. 13, we will use it to improve the smoothness√of the canard curve (or more precisely, μ-family of canard curves) from smooth in  to smooth in  (see Proposition 13.3). Let us now describe how the inner expansion can be derived. Associated with the blown-up vector field (8.35) in the family chart, we consider the (formal) ordinary differential equation (y¯ − x¯ 2 + ux¯ 3 h1 (ux, ¯ ua, ¯ μ)) ˆ x, where y¯ = ϕ( ¯ , μ) = x¯ 2 − 12 + are the unknowns.

d y¯ = a¯ − x¯ + O(u)

, d x¯ ˆ a= ¯ b(u,μ)

∞

¯ μ)u k=1 ϕ k (x,

n

ˆ μ) = and b(u,

(8.42) ∞

k=1 bk (μ)u

k

Proposition 8.1 Equation (8.42), together with the requirement that all coefficient functions ϕk have at most polynomial growth toward ±∞, identifies the couple ˆ b) ˆ in a unique way. (ϕ, Proof Equation (8.42) at level uk , k ≥ 1, is given by −

1 dϕ k + 2xϕ ¯ k = bk + Fk (x, ¯ μ), 2 d x¯

where Fk is an expression involving ϕ 1 , . . . , ϕ k−1 , b1 , . . . , bk−1 and the expression of the vector field; it is by induction supposed to be of at most polynomial growth. It is now elementary to see that only for a unique choice of bk , there is a solution

8.5 Global Aspects in the Blow-up of Contact Points

119

of the above equation that is of at most polynomial growth toward ±∞, and it is a unique solution. Indeed, all solutions of the above equation can be written in the form    x¯ 2 2 ϕk = C − 2 (bk + Fk (s, μ))e−2s ds e2x¯ −∞

  = C −2

∞

−∞

(bk + Fk (s, μ))e

−2s 2

 ds + 2

x¯

∞

(bk + Fk (s, μ))e

−2s 2



ds e2x¯ , 2

∞ 2 2 where C is an integration constant. Since | x¯ (bk + Fk (s, μ))e2x¯ −2s ds| ≤ Mk |x| ¯ Nk for some Mk , Nk , due to the growth properties of Fk , and similarly for the integral toward −∞, it implies that ϕ k can only be of at most polynomial growth when C = 0 and when  ∞ 2 (bk + Fk (s, μ))e−2s ds = 0. −∞



Remark that the blown-up outer expansion from Sect. 6.4 satisfies the requirement stated in Proposition 8.1 and solves (8.42).

8.5 Global Aspects in the Blow-up of Contact Points In the blow-up of the generic jump point and the generic turning point, the vector field in the family chart is remarkably simple when reduced to the blow-up locus: in both cases, they are quadratic vector fields (with a parameter in the case of the generic turning point), whose qualitative study is quite easy. In this section, we want to emphasize some analytic properties of these vector fields that reveal to be important in the study of Chaps. 13, 14, and 15.

8.5.1 Closed Form Expressions for the Orbits on the Blow-Up Locus of the Generic Jump Point We deal with (8.4), which we repeat here using (x, y) instead of (x, ¯ y): ¯ 

x˙ = y − x 2 y˙ = −1.

(8.43)

120

8 Blow-up of Contact Points

The orbits are graphs of y, and more in particular, the orbit through (x, y) = (0, y0 ) can be written as x=−

ϕ (y; y0 ) , ϕ(y; y0)

where ϕ(y; y0) is a solution of the Airy equation ϕ (y) = yϕ(y) with Indeed,

ϕ (y0 ;y0 ) ϕ(y0 ;y0 )

= 0.

ϕ 2 − ϕϕ dx ϕ 2 − yϕ 2 = = = x 2 − y. 2 dy ϕ ϕ2 We know that the space of solutions of the Airy equation is spanned by two particular solutions A(y) and B(y), known as the Airy A function and the Airy B function. So we can take ϕ(y; y0) = A(y)B (y0 ) − B(y)A (y0 ). (The solution is unique up to a multiplicative constant, which we have normalized to 1.) The Airy functions are in particular well defined on the whole real line, but combinations like in the above expressions will have zeros. These zeros determine the domain of the trajectory: we denote by L(y0 ) and R(y0 ) the two zeros of the function ϕ(y; y0) closest to y0 , respecting L(y0 ) < y0 < R(y0 ) (or R(y0 ) = +∞ if there is no solution larger than y0 ). Hence the domain for y of ϕ(y; y0) is given by ]L(y0 ), R(y0 )[ ,

with L(y0 ) < y0 < R(y0 ).

Let us explain this geometrically using Fig. 8.8. In the figure, the pair of two Airy functions is used as coordinates to draw a planar curve, image of y → γy = (A(y), B(y)). As y → −∞, the curve spirals increasingly fast along the origin; as y → +∞, the curve is asymptotically tangent to the B-axis. Given a point γy0 , marked in Fig. 8.8, its tangent vector (A (y0 ), B (y0 )) is visualized in a dashed fashion. Equation A(y)B (y0 ) − B(y)A (y0 ) = 0, which is solved at L(y0 ) and R(y0 ), can be interpreted as saying that the vectors (A(y), B(y)) and (A (y0 ), B (y0 )) are parallel. Hence, as seen in Fig. 8.8, the two end points of L(y0 ) and R(y0 ) of the domain of ϕ(y; y0) are diametrically located on the curve. Let us now check how these expressions are seen in the matching rescaling charts. If we assume that the blow-up was done with weights (1, 2, 3) for the original

8.5 Global Aspects in the Blow-up of Contact Points

121

B

Fig. 8.8 The Airy parameter curve γy := (A(y), B(y)). The labels in the marked points are the parameters y corresponding to the points γy .

L(y0 ) ↓

y0

y0max A

R(y0 ) → 0 y+

coordinates that we write as (x , y , ), then the matching chart in the positive x direction is accessed after a change of coordinates −1/3

(x, y) = (¯−

−2/3

, y¯− ¯−

),

writing (y¯− , ¯− ) for the more traditional (y¯2 , ¯2 ) in the matching chart. (We use the − subscript because the coordinate transformation is used near the saddle s− . The convention of denoting the saddle in the matching chart in the positive x -direction by s− and the saddle in the chart in the negative X-direction by s+ is inspired from the fact that the orbits will go from s− to s+ .) This is almost a transformation as used in a Poincaré–Lyapunov compactification, but with fractional exponents in the radial variable, see Remark 8.2. Similarly, accessing the matching chart in the negative x direction is done using −1/3

(x, y) = (−¯+

−2/3

, y¯+ ¯+

),

using (y¯− , ¯− ) as coordinates in the matching chart. The vector fields in the two 1/3 1/3 charts (after division by the positive factors − and + ) have the respective saddles s− and s+ as singular points at the origin and are given by 

¯˙− = 3¯− (1 − y¯− ) 2 − ¯ , y˙¯− = 2y¯− − 2y¯− −

 and

¯˙+ = −3¯+ (1 − y¯+ ) 2 − ¯ . y˙¯+ = −2y¯+ + 2y¯+ +

122

8 Blow-up of Contact Points

Both systems have a hyperbolic singular point at the origin, of non-resonant node type. Following for example [DLA06], there exist analytic (near-identity) changes of coordinates linearizing both systems: 



˙˜− = 3˜− y˙˜− = 2y˜− − ˜− ,

and

˙˜+ = −3˜+ y˙˜+ = −2y˜+ − ˜+ .

(8.44)

In these linearized coordinates, the orbits are given explicitly by the families 2/3

y˜− = Y− ˜− − ˜− ,

2/3

y˜+ = Y+ ˜+ + ˜+ ,

where Y± are coordinates on the respective transverse sections ± := {˜± = 1}. (In fact Y− = y˜− (1) + 1 and Y+ = y˜+ (1) − 1.) Since the linearization is obtained after an analytic near-identity transformation, these non-smooth families of graphs are transformed back to equally non-smooth 2/3

y¯− = Y− ¯− + O(¯− ),

2/3

y¯+ = Y+ ¯+ + O(¯+ ).

Changing back to family chart coordinates gives y = Y− + O(x −1/2 ),

x → +∞,

y = Y+ + O((−x)−1/2),

x → −∞.

This allows to match the expressions of the orbits in normal form coordinates with the expressions of the orbits in the family chart using Airy functions: the orbit that (y;y ) 0 has matches y = Y− + o(1) is that specific orbit for which the graph x = − ϕϕ(y;y 0) a vertical asymptote at y = Y− , in other words for which ϕ(Y− ; y0 ) = A(Y− )B (y0 ) − B(Y− )A (y0 ) = 0. Similarly, orbits that tend toward the saddle in the left chart are such that ϕ(Y+ ; y0) = A(Y+ )B (y0 ) − B(Y+ )A (y0 ) = 0 for some Y+ ∈ R. It implies that orbits that connect s− to s+ satisfy Y− = R(y0 ), for some y0 .

Y+ = L(y0 )

8.5 Global Aspects in the Blow-up of Contact Points

123

8.5.2 Passage Time in the Blow-up of Jump Points The fact that y˙ = −1 implies that orbits connecting s− to s+ will do in finite time R(y0 ) − L(y0 ) = Y− − Y+ . The next proposition states some analytic properties of this time function. In order to give an intrinsically meaningful statement, we will not parameterize the orbits using y0 but instead will use Y− as a parameter: any orbit intersecting the transverse section {˜− = 1} at a coordinate y˜− = Y− will reach the section {x = 0} in finite time and will intersect it at a coordinate y0 = R −1 (Y− ). To see that such a y0 exists, we state a property of the Airy functions [AS64]: (A(y), B(y)) = (M(y) cos θ (y), M(y) sin θ (y)) for some M, θ . We know that θ (y) > 0 and that θ → ∞ as y → −∞ (see Remark 8.4 below). As a consequence, both the vector (A, B) and the vector (A , B ) spiral around the origin with increasing speed as y → −∞. It guarantees the existence of solutions of A(Y− )B (y) − B(Y− )A (y) = 0 for y < Y− . Denote by y0 = R −1 (Y− ) the solution closest to Y− but smaller than Y− . Remark 8.4 The properties of θ are derived from [AS64]. The asymptotic property of θ can be seen directly from this source and the monotonicity of θ for y < 0. The monotonicity for y ≥ 0 was not found directly, but for y ≥ 0, we see that A (y) < 0 < B (y), which can easily be proved from the definition and the facts that A (0) < 0 < B (0) and B(y) = 0 = A(y). So also there, θ = arctan(B/A) is strictly monotonous. In a similar way, denote by Y+ the rightmost solution of A(y)B (y0 ) − B(y)A (y0 ) strictly smaller than y0 ; it defines L(y0 ). This way, we define the intrinsic time function T (Y− ) = Y− − L(R −1 (Y− )). Proposition 8.2 Let Y− parameterize the section {˜− = 1} in normal form coordinates presented by (8.44) near the singularity s− , as in previous subsection. Each Y− identifies an orbit that reaches s+ in finite positive time and that reaches s− in finite negative time. The total time function Y− → T (Y− ) is convex in the d 2T sense that dY 2 (Y− ) > 0 for any Y− ∈ R. Furthermore, the range of the function dT dY−

−

is in the interval ]0, 1[.

Lemma 8.1 The convexity claim in Proposition 8.2 is proven when



L (y ) R (y0 )

α(y0 ) :=

0 L (y0 ) R (y0 ) is strictly positive.

124

8 Blow-up of Contact Points

Proof We have to show that L ◦ R −1 has negative second order derivative (because T = Id −L ◦ R −1 ). Since (R −1 ) > 0 (clear because R −1 maps {˜ = 1} diffeomorphically to an interval in {X = 0} in the family chart), it suffices to assume that α > 0. 

Before proving the convexity, let us quickly prove a lemma that together with the convexity suffices to finish the proof of the proposition. Lemma 8.2 Under the condition of Lemma 8.1, we have limY− →−∞ and

dT limY− →+∞ dY (Y− ) −

= 1.

dT dY− (Y− )

=0

Proof The convexity implies that T is monotonously increasing, and therefore both limits exist on [−∞, ∞]. Since the transition time is given by Y− − Y+ , where Y+ is the rightmost solution less than Y− of A(Y )B(Y− ) − B(Y )A(Y− ) = 0, and since the planar curve (A(Y ), B(Y )), close to Y = −∞, oscillates with monotone angle around the origin and with frequency tending to +∞ as Y → ∞, it follows that Y− − Y+ → 0 as Y− → −∞. The graph of T has hence a horizontal asymptote, and therefore the proper limit of T at Y = −∞ must be zero. The planar curve (A(Y ), B(Y )) has the vertical coordinate axis as asymptote, for Y → +∞, showing that the phase of (A(Y ), B(Y )) tends to π/2 as Y → ∞. As a consequence, limY− →+∞ L(R −1 (Y− )) should be defined by the rightmost solution of A(Y ) = 0, which numerically evaluates to Y+0 ≈ −2.338107. The graph of T hence has an asymptote given by the graph of Y− → Y− + Y+0 , so that T can only tend toward 1. 

The required property on α was first verified numerically and below is shown rigorously using specific properties on Airy functions. The coordinates L(y0 ) and R(y0 ) both satisfy the equation ϕ(y; y0) = A(y)B (y0 ) − B(y)A (y0 ) = 0. (8.45)



A(L) B(L) A(R) B(R) Observe that (8.45) can be rewritten as A (y0 ) B (y0 ) = A (y0 ) B (y0 ) = 0. We derive these expressions w.r.t. y0 . Since the calculation for R is the same as for L, we will only do it for the first equation:



A (L) B (L) A(L) B(L)

L +

A (y0 ) B (y0 ) = 0.

A (y0 ) B (y0 )

8.5 Global Aspects in the Blow-up of Contact Points

125

Recall that A (y0 ) = y0 A(y0 ) and B (y0 ) = y0 B(y0 ). Introducing the notations



A (L) B (L)

,

PL = A (y0 ) B (y0 )



A(L) B(L)

,

QL = A(y0 ) B(y0 )



A (L) B (L)

,

ZL = A(y0) B(y0 )

one easily obtains ⎧ L ⎪ ⎪ ⎨ PL ⎪ Q L ⎪ ⎩ ZL

= = = =

−y0 QL /PL y0 ZL −y0 ZL QL /PL LQL + PL .

2 Q L ZL L Using these rules, we obtain L = − Q PL + 2y0 PL , and additionally

α(y0 ) =

QL QR −2y03 PL PR



1 1

ZL /PL ZR /PR .

We claim that PL and PR have a fixed sign and that QL and QR have a fixed sign, implying that the sign of α is determined by the sign of α(y ˜ 0 ) := ZL PR −ZR PL . Let us for a moment assume that this claim is true. Expanding all involved determinants and simplifying give us



A(y0 ) B(y0 ) A (L) B (L) 1

A (L) B (L)



α(y ˜ 0) = · = , A (y0 ) B (y0 ) A (R) B (R) π A (R) B (R) where we have used a Wronskian property for the Airy functions. Now following Abramowitz and Stegun, we write (A(y), B(y)) = (M(y) cos θ (y), M(y) sin θ (y)), and (A (y), B (y)) = (N(y) cos ϕ(y), N(y) sin ϕ(y)). We find that [AS64] MN sin(ϕ − θ ) =

1 . π

First it follows that M = 0 = N and also that sin(ϕ −θ ) = 0. Suppose θL = θR + π ˜ for some integers and . ˜ Then ϕL − θL = ϕR − θR + ( ˜ − )π, and ϕL = ϕR + π so sin(ϕL − θL ) = sin(ϕR − θR ). It would imply that ML NL = MR NR . Now the function MN is strictly monotone (this is a known property), so this is a contradiction. We conclude that α˜ has a fixed sign. Let us come back to the claims regarding PL , PR , QL , and QR . Suppose PL is zero at some point. This means that the vectors (A (L), B (L)) and (A(L), B(L)) are parallel. It would imply that θL = 0, which is in contradiction with [AS64]: θ < 0. A similar argument can be used for PR .

126

8 Blow-up of Contact Points



0 )) B(L(y0 )) Suppose QL is zero at some point. Then the function κ : → A(L(y A(y) B(y) has a zero at y0 . Furthermore, per definition of L, we have κ (y0 ) = 0. Also, κ = y0 κ, so all derivatives of κ are 0 at y0 . Analyticity implies κ = 0 everywhere. This is clearly a contradiction (it would imply L ≡ 0). Proposition 8.2 can also be formulated in terms of coordinates near s+ : Proposition 8.3 Let Y+ parameterize the section {˜ = 1} in normal form coordinates presented by (8.44) near the singularity s− . Let Y∗ denote the Y -coordinate of the intersection point of the center separatrix of q1 with ˜ = 1. Each Y+ < Y∗ identifies an orbit that reaches s+ in finite positive time and that reaches s− in finite negative time. Furthermore, the total time function Y+ → T+ (Y+ ) is convex in the 2 sense that d T2+ (Y+ ) > 0 for any Y+ < Y∗ . dY+

Proof As all orbits from near s− (for all values of Y− tend toward s+ ), this family of orbits reaches the section {˜ = 1} in an interval. Since Y− −Y+ → 0 as Y− → −∞, this interval is unbounded from below. Since at least one orbit connects to q1 , it is an interval of the form ]−∞, Y∗ [ for some Y∗ ∈ R. On this domain, we find T+ (Y+ ) := T (L−1 (Y+ )) = (R ◦ L−1 )(Y+ ) − Y+ . The concavity of L ◦ R −1 , proved above, implies the convexity of R ◦ L−1 .



The main difference between the orbits in Proposition 8.3 and Proposition 8.2 is that in Proposition 8.3 not all orbits near s+ reach s− in finite time: one orbit connects to q1 and the others to q2 . Note also that T+ takes values in ]0, ∞[, unlike T which took values in ]0, 1[. It is our goal in the next section to study the location of the orbit that connects to q1 , i.e. we find properties of Y∗ . In particular we will prove that Y∗ < 0, a feature that is relevant in Chap. 12.

8.5.3 Separatrices on the Blow-up Locus of the Generic Jump Point There are three notable separatrices on the blow-up locus of generic jump points: the strong separatrix at s− , the one at s+ , and the center separatrix at q1 . Proposition 8.4 The center separatrix at q1 and the strong separatrix of s− are orbits that connect to s+ below the strong separatrix of s+ , i.e. intersecting {˜ = 1} in a coordinate y˜ < 0, while the strong separatrix of s+ is an orbit coming from q1 (see Fig. 8.3). The proof is elementary, given the insight in the orbits given in the previous sections. We only have to prove the statement on the strong separatrix of q1 . To that end, notice that this amounts to studying the location of the rightmost solution of A(y)B(Y− ) = A(Y− )B(y) as Y− → +∞. As already observed before, the vector

8.5 Global Aspects in the Blow-up of Contact Points

127

(A(Y− ), B(Y− )) is asymptotically tangent to (0, 1), and therefore we are looking for the rightmost solution of A(y) = 0. It is known that the Airy A function only has strictly negative roots. Remark 8.5 For general regular contact points of even degree, the phase portrait is similar, but it is an open problem to see the position of the strong separatrix connections of s− and s+ for contact orders at least 4. The reason is the lack of a reduction to known special functions in that case.

8.5.4 Regular Splitting of Separatrices in the Blow-up of a Generic Turning Point This section deals with (the compactification of) (8.36), which we repeat using capital letters: 

X˙ = Y − X2 Y˙ = a¯ − X.

At infinity, using the transformation {X = x/¯ ¯ ν , Y = 1/¯ν 2 }, we find back the vector field (8.38) in the matching chart {y¯ = 1}, restricted to the blow-up locus {u = 0}: 

x˙¯ = 1 − x¯ 2 + 12 x¯ 2 ν¯ 2 − 12 a¯ x¯ ν¯ 3 ν˙¯ = 12 ν¯ 3 (x¯ − a¯ ν¯ ).

In this chart, the point s2 has coordinates (−1, 0), while s3 = (1, 0). Proposition 8.5 Given the (unique) a-family ¯ of center separatrices γa¯− at s2 (at (x, ¯ ν) ¯ = (−1, 0)) and the a-family ¯ of center separatrices γa¯+ at s3 (at (x, ¯ ν¯ ) = (1, 0)). For a¯ ≈ 0, the separatrices γa¯± meet the section {X = 0} in the family chart in points (0, Ya¯± ), and (Ya¯+ − Ya¯− )|a=0 = 0, ¯

∂ (Y + − Ya¯− )|a=0 > 0. ¯ ∂ a¯ a¯

The separatrices γa¯+ and γa¯− coincide if and only if a¯ = 0. Proof The system for a¯ = 0 is integrable, with integrating factor e−2Y : the equivalent system 

X˙ = (Y − X2 )e−2Y Y˙ = −Xe−2Y

128

8 Blow-up of Contact Points

is conservative with Hamiltonian H (X, Y ) = 12 e−2Y (Y − X2 + 12 ). Note that H = 0 corresponds to the connection between s2 and s3 , given by the algebraic curve Y =  1 1 2 X − 2 , whereas H ∈ 0, 4 correspond to periodic orbits, getting closer to the singular point at the origin as H → 14 . The most elegant way to proceed is to use the dual 1-form formulation of the vector field   ωa¯ = (−a¯ + X)dX + (Y − X2 )dY = −e2Y dH + e−2Y adX ¯ . From that point on, it is an exercise, following [DR96], to prove that   − h+ − h = lim a¯ a¯

h→0 H =h

 =

Y =X 2 − 12

 e−2Y dX a¯ + O(a¯ 2) 

e−2Y dX a¯ + O(a¯ 2),

where ha± ¯ are the H -values of the intersection of the two separatrices with {X = 0}. + − The first part of the proof follows since ∂H ∂Y (0, −1/2) > 0, so that also Ya¯ −Ya¯ > 0 for a¯ > 0 sufficiently small. It shows that close to a¯ = 0, the separatrices meet if and only if a¯ = 0. Let us finish the proof by showing that the separatrices will never meet for a¯ = 0. Consider to that end V = Y − X2 + 12 , then V˙ |V =0 = a, ¯ implying that for a¯ > 0 {V > 0} is positively invariant. Since the compactified {V = 0} contains s2 and s3 , it implies near s3 (where there is topologically saddle behavior) the center separatrix of s3 lies on the side {V > 0}, and similarly, near s2 the center separatrix of s2 lies on the side {V < 0}. So the two separatrices cannot coincide when a¯ > 0. A similar argument can be used for a¯ < 0. 

Alternatively to the proof of the above proposition, one might use the technique of rotating parameters, similar to what has been done in [DR96] (see also [DMW15]).

8.5.5 Hopf Breaking Mechanism Revisited In Sect. 6.3, we have initiated the study of a generic Hopf breaking mechanism. We have seen in the previous section that {X = 0} is a section τ transverse to the separatrices γa¯± , for a¯ sufficiently close to 0. By Proposition 8.5, the function ζ : (y, ¯ a) ¯ →

y¯ − Ya¯−

Ya¯+ − Ya¯−

a¯

8.6 From the Hopf Bifurcation to the Polycycle and the Birth of Canards

129

defines a smooth function, smoothly extensible to a¯ = 0, and for which ∂ζ ∂ y¯ (0, 0) > 0. The a-family ¯ of coordinate functions ζa¯ : τ → R defines a new coordinate z = ζa¯ (y), ¯ and using this coordinate, the separation distance will be trivial. Proposition 8.6 Given the (unique) a-family ¯ of center separatrices from Proposition 8.5. Parameterizing the section τ : {X = 0} by z as above, we have that the separatrices γa¯− and, γa¯+ meet the section τ in the family chart in the point parameterized by respectively z = 0 and z = a. ¯ It is easy to see that the section {X = 0} can be replaced by any section transverse to the joint connection γ0± . So analogous to the case of a jump breaking mechanism, the parameter a¯ plays the role of a generic breaking parameter.

8.6 From the Hopf Bifurcation to the Polycycle and the Birth of Canards Recalling the local setup of a generic turning point in Sect. 8.4, we again consider (8.34), which we repeat here for the sake of convenience.  X¯ a,μ ¯

⎧ ¯ μ) ⎨ x˙ = y − x 2 + x 3 h1 (x, ν a, : y˙ = ν 2 (ν a¯ − x − x 2 h2 (x, ν 2 , ν a, ¯ μ) + yh3 (x, y, ν 2 , ν a, ¯ μ)) ⎩ ν˙ = 0,

(8.46)

with h1 , h2 , and h3 C∞ functions. The blown-up vector field in the family chart gives (see (8.36)) 

x˙¯ = y¯ − x¯ 2 y˙¯ = a¯ − x¯

on the blow-up locus. The singular point at (a, ¯ a¯ 2 ) undergoes a Hopf bifurcation at a¯ = 0. The system restricted to the blow-up locus undergoes a degenerate Hopf bifurcation though, as for a¯ = 0 the system is a center (hence all Lyapunov coefficients vanish). In general, under the condition that h1 ≡ 0, the perturbed system has a nonzero first Lyapunov coefficient, which can be used to show the emergence of a unique hyperbolic cycle for u = 0, see [DR09]. Because the system for u = a¯ = 0 is a center (see Fig. 8.9), the origin as a singleton is not the only set from which limit cycles may bifurcate; noting that H (x, ¯ y) ¯ =

1 −y¯ 1 e (y¯ − x¯ 2 + ) 2 2

is a first integral for a¯ = 0, all level sets H = h, which correspond to closed loops when h ∈ 0, 14 , are potential sets from which limit cycles may bifurcate

130

8 Blow-up of Contact Points

Fig. 8.9 The blown-up turning point for a¯ = 0 and level sets of H with the polycycle marking the transition from small-amplitude cycles to canard cycles

the polycycle

s2

s1

s3

s4

(“limit periodic sets”). These loops are of course not canard cycles, since they are of bounded amplitude in the blow-up space and hence of o(1)-amplitude in original coordinates. However, as h → 0, the set H = h tends toward the polycycle shown in Fig. 8.9, which lies on the boundary of the layer of canards. The polycycle H = 0 is hence to be considered as a transitory cycle.

Chapter 9

Center Manifolds

In this chapter we will give a short proof of a particular case of the Fenichel Theorem [Fen79] for the existence of center manifolds for slow–fast systems in dimension two. It is particular in the sense that the normally hyperbolic direction is one-dimensional and is invariant by the dynamics. We also suppose that the slow dynamics has no zeros. We will give precise statements adapted to slow–fast systems, needed in the remainder of the book, in Sect. 9.3. These results will be deduced from more general ones that are presented in Sects. 9.1 and 9.2. For results dealing with zeros in the slow dynamics, we refer to Fenichel’s general statement, or to the refinements shown in [DMD08]. In Chap. 10, we will again discuss center manifolds (Sect. 10.4), based on the knowledge of smooth normal forms for families of vector fields.

9.1 Ck -Invariant Manifolds for Diffeomorphisms In this section, we present a result on the existence of invariant manifolds of diffeomorphism in n + 1 variables verifying some conditions. We have in mind diffeomorphisms F : U × [z0 , z1 ] → Rn+1 : (x, z) → (g(x), f (x, z)), where z0 < 0 < z1 , n ≥ 1, and with the following assumptions: (H1 ) (H2 )

U is a compact n-submanifold with boundary of Rn . g : U → Rn is a smooth diffeomorphism defined on a neighborhood of U into Rn such that h = g −1 is also defined on a neighborhood of U . We assume that h(U ) ⊂ U .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_9

131

132

(H3 ) (H4 )

9 Center Manifolds

f is smooth, and for any x ∈ U we have that f (x, ·) maps [z0 , z1 ] into the open interval ]z0 , z1 [. There exists a positive constant K < 1 such that 0 < ∂f ∂z (x, z) < K for all (x, z) ∈ U × [z0 , z1 ].

Under conditions (H1 –H4 ) the map F (x, z) = (g(x), f (x, z)) is a smooth diffeomorphism from U × [z0 , z1 ] into Rn × R. We are looking for an F -invariant manifold Wϕ contained in U × R and defined as a graph of a function ϕ : U → R : x → ϕ(x). Clearly Wϕ = {(x, z) : z = ϕ(x)} is F -invariant if and only if ϕ satisfies ϕ(x) = f (h(x), ϕ ◦ h(x)),

∀x ∈ U,

(9.1)

see Fig. 9.1. We will simply say that ϕ is F -invariant. In order to prove the existence of an F -invariant manifold, which is the topic of this section, we extend the domain of f to U × R keeping its properties and with bounded partial derivatives; this can easily be proved: Lemma 9.1 We can extend f smoothly on U × R such that: 1. f keeps its value on U × [z0 , z1 ]. 2. We have 0 < ∂f ∂z (x, z) < K. 3. Each partial derivative of f is bounded.



˜ To simplify, we introduce the function  ˜ f (x, z) ≡ f (h(x), z). We see that

∂f ∂f ≡ ∂z (h(x), z) and then sup ∂z ((x, z) ≤ K.

∂ f˜ ∂z (x, z)

Fig. 9.1 F -invariant graph Wϕ

z

z1

(x, ϕ(x)) F

z0

x U

g

(h(x), ϕ(h(x)))

h(x)

9.1 Ck -Invariant Manifolds for Diffeomorphisms

133

As U is compact, each partial derivative of f as well as of a component of h is also bounded on U . It is also true for f˜ and hence we can assume that: There exists an infinite sequence of positive numbers M0 < M1 < M2 < · · · < +∞

(9.2)

such that each partial derivative of f˜ or of a component of h of order at most k is bounded in absolute value by Mk , for any (x, z) ∈ U × R. The invariance condition (9.1) in ϕ is written as ϕ(x) = f˜(x, ϕ ◦ h(x)).

(9.3)

We define the graph transform (associated to F ) to be the map T : C0 (U ) → C0 (U ) given by T (ϕ)(x) = f˜(x, ϕ ◦ h(x)). As h(U ) ⊂ U , this map is well defined. The function ϕ is F -invariant if and only ϕ is a fixed point of the graph transform: T (ϕ) = ϕ. We recall that the norm on C0 (U ) is defined by ψ0 = sup{|ψ(x)| | x ∈ U }. It is rather trivial to verify that T is a Lipschitz contraction and then has a unique fixed point in the C0 -topology: Lemma 9.2 There is a unique F -invariant function ϕ0 ∈ C0 (U ). Moreover, if 0 0 f0 (x) = f (x, 0), we have that: ϕ0 0 ≤ f 1−K and also, for any x ∈ U , we have that z0 < ϕ0 (x) < z1 : the function ϕ0 is invariant for the given initial map F |U ×[z0 ,z1 ] . Proof Take any pair of functions ϕ1 , ϕ2 ∈ C0 (U ). We have T (ϕ2 )(x) − T (ϕ1 )(x) = f˜(x, ϕ2 ◦ h(x)) − f˜(x, ϕ1 ◦ h(x)) =

∂ f˜ (x, θ(ϕ1 ◦ h)(x) + (1 − θ)(ϕ2 ◦ h)(x)(ϕ2 − ϕ1 ) ◦ h(x), ∂z

for some function θ : U → ]0, 1[. Then we find T (ϕ2 ) − T (ϕ1 )0 ≤ Kϕ2 − ϕ1 0 .

(9.4)

As a consequence, T is a Lipschitz contraction and has a unique fixed point ϕ0 . To prove the inequality, we apply the inequality to 0 and ϕ0 . Writing f˜0 (x) = ˜ f (x, 0) = T (0), we have that ϕ0 0 − f˜0 0 ≤ ϕ0 − f˜0 0 = T (ϕ0 ) − T (0)0 ≤ Kϕ0 0 . Then it follows that (1 − K)ϕ0 0 ≤ f˜0 0 ≤ f0 0 , where we used the property f˜0 0 = sup{|f0 (x)| : x ∈ h(U )}) in the last step.

134

9 Center Manifolds

To prove the last claim, we notice that the condition (H3) above implies that, if ϕ(x) ∈ [z0 , z1 ] for any x ∈ U , the same is true for T (ϕ). Now we have that T n (ϕ) → ϕ0 in the C0 -topology for any ϕ ∈ C0 (U ). If we choose ϕ with the property that ϕ(x) ∈ [z0 , z1 ] for any x ∈ U , we have the same for the limit ϕ0 .  Remark Due to its definition the invariant function ϕ0 does not clearly depend on the chosen extension of f |U . Differentiability of ϕ0 We define the operator norm of a n × n real matrix A by A = sup A(u), u=1

where u is the Euclidean norm in Rn ), and we define A0 = supx∈U A(x) for a continuous map A(x) from U into the space of n × n real matrices. Then we have the following result, the proof of which can be found in the rest of this section: Theorem 9.1 Let F (x, z) = (g(x), f (x, z)) be a diffeomorphism as above. Put Ng = sup{dg −1 0 , 1}. Then if KNgk < 1, the invariant function ϕ0 is Ck . Remark 9.1 We can replace the condition (H2 ) by the condition g(U ) ⊂ U . The results in this case are completely similar to the ones proved below by interchanging g and h. For instance the invariance condition (9.1) is replaced by ϕ(x) = f (g(x), ϕ ◦ g(x)) and the constant Ng in Theorem 9.1 by N˜ g = sup{dg0 , 1}. The same remark is also valid for families of diffeomorphisms. Remark 9.2 If dg −1 0 ≤ 1 (for instance if g is an hyperbolic expansion or even an isometry), then we can apply Theorem 9.1 for any k, implying that the invariant function ϕ0 is smooth. If dg −1 0 > 1, the condition in the statement reduces to K −1 . k < ln ln dg −1  0

To prove Theorem 9.1 we will prove that the graph transform T is a Lipschitz contraction in a ball of Ck (U ), for a well-chosen norm. We first recall some definitions. A norm of a k-multilinear map A(u1 , . . . , uk ) from Rn to Rp is defined by A = sup{A(u1 , . . . , uk ) | u1  = · · · = uk  = 1} with the Euclidean norms  ·  on Rn and Rp . If A is a continuous map from U to the space of k-multilinear maps from Rn to Rp , we write A0 = sup{A(x) | x ∈ U }. Recall that d k ϕ, inductively defined by d k ϕ(x) = d[d k−1 ϕ](x), is a k-multilinear form (multilinear maps from Rn to R): d k ϕ(x)[u1 , . . . , uk ] =

! (σ1 ,...,σk )∈k

∂xσ1 ,...,xσk ϕ(x)u1σ1 · · · ukσk ,

9.1 Ck -Invariant Manifolds for Diffeomorphisms

135

where k is the permutation group on k elements. Let Ck (U ) be the space of Ck functions on U . It is a Banach space with the norm: ϕk = ϕ0 + dϕ0 + · · · + d k ϕ0 . Let us notice that we have the induction formula ϕ +1 = ϕ + d +1 ϕ0 . In the same way, replacing the absolute value in R by the Euclidean norm in Rp we can define the k-differential and its norm for a map from U to Rp . If ψ = T (ϕ) = f˜(·, ϕ ◦ h), we have to compute the differentials of ψ in terms of the differentials of ϕ. Let us look at the first orders: dψ =

∂ f˜ dϕ ◦ dh + dx f˜ ∂z

(9.5)

and d 2ψ =

∂ f˜ 2 ∂ f˜ d ϕ(h)[dh, dh] + dϕ(h) ◦ d 2 h ∂z ∂z ∂ f˜ ∂ 2 f˜ + dx2 f˜ + 2 dx f˜ ⊗ dϕ(h) ◦ dh + 2 dϕ(h) ◦ dh ⊗ dϕ(h) ◦ dh, ∂z ∂z (9.6)

where the application f˜ and its differentials are evaluated at the point (x, ϕ(h)(x)). More generally, we can make the following observation: Lemma 9.3 For any ≥ 1, we have that d ψ =

∂ f˜ d ϕ(h)[dh, . . . , dh] + · · · , ∂z

(9.7)

where + · · · is a finite sum of -multilinear form fields whose components are product of partial derivatives of f˜, h and ϕ. For ≥ 2, each of these components is a product of partial derivatives of f˜ or h whose total order is at least 2 and of at least one partial derivative of ϕ, each of them of total order between 1 and − 1.  Remark 9.3 The first term written on the right is an -multilinear form field whose norm verifies " " " ∂ f˜ " " ∂z d ϕ(h)[dh, . . . , dh]" ≤ KNg d ϕ0 . 0

For = k the coefficient KNgk is the one that appears in Theorem 9.1.

136

9 Center Manifolds ˜

A consequence of Lemma 9.3 is that the term ∂∂zf d ϕ(h)[dh, . . . , dh] is dominant in some sense. This is already clear in the proof the following result: Proposition 9.1 Let k ≥ 1 satisfy the condition in Theorem 9.1 and let ϕ0 be the C0 -solution found in Lemma 9.2. For any sequence R = (R1 , . . . , Rk ), such that 0 < R1 < R2 < · · · < Rk , one defines a closed set: DRk = {ϕ ∈ Ck (U ) | ϕ − ϕ0  ≤ 1, dϕ0 ≤ R1 , . . . , d k ϕ0 ≤ Rk }. Then, there exists a sequence R such that DRk = ∅ and such that this set is invariant by T : T (DRk ) ⊂ DRk . Proof (1) We first look for a condition on a sequence R to have that DRk = ∅. For this, we choose an approximation of ϕ0 by a ϕ¯ ∈ Ck (U ) such that ϕ¯ − ϕ0  ≤ 1. To have that ϕ¯ ∈ DRk , it suffices to choose R such that: R1 > d ϕ, ¯ R2 > d 2 ϕ, ¯ . . . , Rk > d k ϕ. ¯

(9.8)

(2) We now look for sufficient conditions on the Ri to fulfill the T -invariance. We notice that, as Ng ≥ 1, the condition KNgk < 1 implies that KNg < 1 for any ≤ k. Using the constants Mi from (9.2), we will find sufficient conditions on the Ri by recurrence on i: (i) First, using Eq. (9.5) we see that dψ0 ≤ KNg dϕ0 + nM1 . Then the nM1 condition on R1 is that R1 ≥ KNg R1 + nM1 , i.e. R1 ≥ 1−N . gK (ii) We similarly estimate each term in the right hand side of (9.6) and find an equality d 2 ψ0 ≤ KNg2 d 2 ϕ + P1 (dϕ20 ), for some quadratic polynomial P1 with positive coefficients (expressed in terms of the Mi and n). Taking into account that we intend to keep dϕ0 ≤ R1 , we find d 2 ψ0 ≤ KNg2 d 2 ϕ0 + P1 (R1 ). Then a sufficient condition on R2 is that R2 ≥ KNg2 R2 + P1 (R1 ), P1 (R1 ) equivalently written as R2 ≥ 1−N 2K . g

(iii) Finally we see by induction on that for any < k there exists a polynomial P (R1 , . . . , R ) with positive coefficients, such that d +1 ψ0 ≤ KNg +1 d +1 dϕ0 + P (dψ0 , . . . , d ψ0 ) ≤ KNg +1 d +1 dϕ0 + P (R1 , . . . , R ).

9.1 Ck -Invariant Manifolds for Diffeomorphisms

137

Then, for any < k, a sufficient condition on R +1 is that R +1 ≥

P (R1 , . . . , R ) 1 − Ng +1 K

(9.9)

.

To finish the proof it suffices to observe that it is trivially possible to find sequences R that simultaneously fulfill both conditions (9.8) and (9.9). 

Let us now prove that T is a Lipschitz contraction on DR . This is not evident if we use the norm  · k introduced above. Fortunately, the estimates we can obtain on partial derivatives depend in general on the choice of coordinates. Now, the idea is to obtain the Lipschitz property by making the following linear change of coordinates: (x, z) = (ρ 2 X, ρZ),

(9.10)

where ρ is a parameter in ]0, 1] to be chosen later, while the new coordinates (X, Z) are taken in ρ −2 U × R. The function z = ϕ(x) is replaced by the function Z = ϕ ρ (X) = ρ −1 ϕ(ρ 2 X). To avoid confusion we add a superscript ρ to the norms computed with the coordinates (X, Z). With this convention we have for instance ρ that d ϕ ρ 0 = ρ 2 −1 d ϕ ρ 0 . We will denote by Ck (ρ −2 U ) the space of Ck functions Z = ϕ(X) with X ∈ ρ −2 U. In the X-coordinate the set DRk becomes k DR,ρ = {ϕ ∈ Ck (ρ −2 U ) |

ϕ − ϕ0 0 ≤ ρ −1 , dϕ0 ≤ ρR1 , . . . , d k ϕ0 ≤ ρ 2k−1 Rk }. ρ

ρ

ρ

Writing f˜ρ (X, Z) = ρ −1 f˜(ρ 2 X, ρZ) and hρ (X) = ρ −2 h(ρ 2 X), the graph transform in the coordinates (X, Z) takes the form: T ρ (ϕ)(X) = f˜ρ (X, ϕ ◦ hρ (X)).

(9.11)

Mind that in (9.11), ϕ(X) is supposed to be a function of X with values on the Z-axis and not a function z = ϕ(x), written in the new coordinates (X, Z) by the k ) ⊂ Dk change (9.10). The transcription of Proposition 9.1 is that T ρ (DR,ρ R,ρ for a well-chosen sequence R and any ρ ∈ ]0, 1]. The benefit of making this coordinate change is that we have, for any (α, β) ∈ Nn × N, the following estimates: |∂xα ∂zβ f˜ρ (X, Z)| ≤ ρ 2|α|+β−1 M|α|+β and |∂xα hρ (X)| ≤ ρ 2|α|−2 M|α| ,

(9.12)

for (X, Z) ∈ ρ −2 U ×R. As a consequence, we can say that in the coordinate change ˜ (9.10), the first order derivatives ∂ f = ∂z f˜ and ∂xi hj are preserved and all others ∂z

138

9 Center Manifolds

are multiplied by a strictly positive power of ρ. It is even the case for the first order derivatives ∂xi f˜ (it is to obtain this effect that we have chosen to use x = ρ 2 X and not x = ρX). On the other hand, the function f˜ itself is divided by ρ. We use these properties to prove the following result: ρ

k , T ρ and ·k , where Proposition 9.2 With the notations introduced above for DR,ρ R is a sequence given by Proposition 9.1, we have that

  ρ ρ T ρ (ϕ2 ) − T ρ (ϕ1 )k ≤ KNgk + O(ρ) ϕ2 − ϕ1 k ,

(9.13)

k for any functions ϕ1 , ϕ2 ∈ DR,ρ (the symbol O(ρ) depends on f, g, R but not on k ρ ϕ1 , ϕ2 ). As a consequence, T is a Lipschitz contraction of the closed set DR,ρ if ρ is small enough.

Proof We write ψ = T ρ (ϕ). Take any , 1 ≤ ≤ k. To compute d ψ we can use (9.7), replacing f˜, h by f˜ρ , hρ . Write  for the remainder term + · · · in (9.7). It follows from (9.12) and Lemma 9.3 that (f˜, h, ϕ) = ρρ (f˜ρ , hρ , ϕ) for a new k -multilinear form field ρ . Now, take ϕ1 , ϕ2 ∈ DR,ρ and write ψ1 = T ρ (ϕ1 ) and ρ ψ2 = T (ϕ2 ). We have that   d ψ2 − d ψ1 = ∂z f˜ρ (x, ϕ2 (h))d ϕ2 − ∂z f˜ρ (x, ϕ1 (h))d ϕ1 [dh, . . . , dh]   + ρ ρ (f˜ρ , hρ , ϕ2 ) − ρ (f˜ρ , hρ , ϕ1 ) . (9.14) We begin by estimating the first term in the right hand side of (9.14), which we denote by T1 [dh, . . . , dh], with T1 := ∂z f˜ρ (x, ϕ2 (h))d ϕ2 − ∂z f˜ρ (x, ϕ1 (h))d ϕ1 = (∂z f˜ρ (x, ϕ2 (h)) − ∂z f˜ρ (x, ϕ1 (h)))d ϕ2 + ∂z f˜ρ (x, ϕ1 (h))(d ϕ2 − d ϕ1 ). As ∂z f˜ρ (x, ϕ2 (h)) − ∂z f˜ρ (x, ϕ1 (h)) = ∂z2 f˜ρ (x, θ ϕ2 (h) + (1 − θ )ϕ1 (h)), for a θ ∈ [0, 1] we obtain   ρ ρ ρ T1 [dh, . . . , dh]0 ≤ ρM2 R2 ϕ2 − ϕ1 0 + Kd ϕ2 − d ϕ1 0 Ng . It follows from this inequality that ρ

ρ

ρ

T1 [dh, . . . , dh]0 ≤ KNg d ϕ2 − d ϕ1 0 + O(ρ)ϕ2 − ϕ1  .

(9.15)

9.1 Ck -Invariant Manifolds for Diffeomorphisms

139

We now look at the second term T2 := ρ(ρ (f˜ρ , hρ , ϕ2 ) − ρ (f˜ρ , hρ , ϕ1 )) of the right hand side of (9.14). Recall that each component of ρ (f˜ρ , hρ , ϕ) is a ρ monomial in partial derivatives of f˜ρ , hij and partial derivatives of ϕ of order at most − 1. Then, using the same trick as above, and the fact that each partial derivative of order at most of ϕ1 , ϕ2 is bounded by R we obtain that ρ

ρ

ρ

T2 0 ≤ O(ρ)ϕ2 − ϕ1  −1 ≤ O(ρ)ϕ2 − ϕ1  .

(9.16)

Putting together (9.15) and (9.16) we obtain, for 1 ≤ ≤ k: ρ

ρ

ρ

d ψ2 − d ψ1 0 ≤ KNg d ϕ2 − d ϕ1 0 + O(ρ)ϕ2 − ϕ1  .

(9.17)

By adding together the inequality (9.17) with the inequality ρ

ρ

ψ2 − ψ1 0 ≤ Kϕ2 − ϕ1 0

coming from (9.4), the fact that KNg ≤ KNgk , for 0 ≤ ≤ k, and the definition ρ

ρ

ρ

ϕ2 − ϕ1 0 + · · · + d k ϕ2 − d k ϕ1 0 = ϕ2 − ϕ1 k we obtain that

ρ ρ ρ ρ

ψ2 − ψ1 k ≤ KNgk ϕ2 − ϕ1 k + O(ρ) ϕ2 − ϕ1 0 + · · · + ϕ2 − ϕ1 k   ρ ≤ KNgk + O(ρ) ϕ2 − ϕ1 k , which is the desired inequality (9.13).



We can now finish the proof of Theorem 9.1. Proof of Theorem 9.1 As KNgk < 1, we can take ρ small enough in order to get a Lipschitz coefficient Kρ = KNgk + O(ρ) inside inequality (9.13) that is strictly k , which is the image of DRk smaller than 1. On the other side, the closed set DR,ρ by the coordinate change (X, Z) = (ρ −2 x, ρ −1 z), is invariant by T ρ . Then T ρ is a k Lipschitz contraction on DR,ρ and has a unique fixed point. This corresponds to an invariant function ϕk for T , of class Ck . But, as the function ϕ0 given by Lemma 9.2 is the unique C0 -T -invariant function, ϕ0 and ϕk must coincide. This proves that the function ϕ0 is of class Ck . 

Condition to Have ϕ0 Asymptotically Smooth Definition 9.1 Let  ⊂ U be a non-empty closed subset. We say that ϕ : U → R is Asymptotically smooth along  if there exists a sequence of open neighborhoods U = U0 ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uk ⊃ · · · of , such that ϕ|Uk is of class Ck for all k.

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Remark 9.4 It may happen that U˜ := ∩k Uk is a neighborhood of  (for instance if we can choose all the Uk to be the same neighborhood). In this case ϕ is smooth on U˜ and then ϕ is locally smooth along . If it is not the case, it is not a restriction to suppose that ∩k Uk =  and in general ϕ is not smooth on any neighborhood of . Theorem 9.2 Let f, g, with h = g −1 be as above. We moreover suppose that there exist a non-empty closed subset  ⊂ U and a sequence of open neighborhoods U = U0 ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uk ⊃ · · · of , such that for each k: 1. h(Uk ) ⊂ Uk . 2. For gk = g|Uk , putting Ngk = sup{1, dgk−1 0 } for a well-chosen system of coordinates on Uk , we have KNgkk < 1. Then the unique invariant function ϕ0 is asymptotically smooth along . Proof Let ϕ0 : U → R be the invariant function on U . The two conditions in the statement allow to apply Theorem 9.1 on Uk , in differentiability class Ck . As a consequence the function ϕk = ϕ0 |Uk , which is the unique invariant function on Uk , is of class Ck . This means that ϕ0 is asymptotically smooth along . 

Remark 9.5 The constant K does not depend on the coordinate system chosen on U . It is not the case for dg −1  and as we have seen in the proof of Theorem 9.1, where it was interesting to choose a well-adapted coordinate system. This is the reason why, in the statement of Theorem 9.2, we permit choosing a different system of coordinates on each Uk . The Case of Families of Diffeomorphisms We can also consider parameter families of diffeomorphisms Fλ (x, z) = (gλ (x), fλ (x, z)), with λ = (λ1 , . . . , λk ) ∈ P , some p-submanifold of Rp . We can apply the above study to the diffeomorphism F¯ (x, z, λ) = (Fλ (x, z), λ). An invariant manifold given by a graph of a function ϕ¯ : U × P → R of class Ck gives a Ck -family of invariant manifolds Wk , graph of the Ck -family of functions ϕλ (x, z) = ϕ(x, ¯ λ) (the fact that U × P is a manifold with corners does not give problems). The constant Ng entering in the statement of Theorem 9.2 must a priori be replaced by the constant NG , where G(x, λ) = (gλ (x), λ): if H (x, λ) = (hλ (x), λ), with hλ (x) = gλ−1 (x), then NG = sup{dH 0 , 1}. The partial derivatives of the components of gλ−1 in terms of the parameters λj enter in the computation of dH 0 . But these derivatives are inessential. We can see this in the following way: rather than using the linear change of variables (x, z, λ) = (ρ 2 X, ρZ, ρ 2 λ¯ ) we consider the change ¯ (x, z, λ) = (ρ 2 X, ρZ, ρ 3 λ).

9.2 Ck -Invariant Manifolds for Vector Fields

141

With this change, it is easy to see that ρ

dH ρ 0 = sup dhλ 0 + O(ρ) λ∈P

ρ

and then we can replace NG by NG = supλ∈P dhλ 0 . We finally obtain the following version with parameters of Theorems 9.1 and 9.2: Theorem 9.3 Let Fλ (x, z) = (gλ (x), fλ (x, z)) be a λ-family of diffeomorphisms ρ that verify the conditions (H ) for any λ ∈ P , as above. Put NG = supλ∈P dhλ 0 , where hλ = gλ−1 . Then if KNGk < 1, there exists a unique invariant family of functions ϕλ of class Ck . Theorem 9.4 Let fλ , gλ , with hλ = gλ−1 be as above. We moreover suppose that there exist a non-empty closed subset  ⊂ U and a sequence of open neighborhoods U ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uk ⊃ · · · of , such that for each k: 1. hλ (Uk ) ⊂ Uk . −1 2. For gλ,k = gλ |Uk , putting NGk = supλ∈P {dgλ,k 0 , 1} for a well-chosen system k of coordinates on Uk , we have KNGk < 1. Then there exists a unique family of invariant functions ϕλ that are asymptotically smooth along .

9.2 Ck -Invariant Manifolds for Vector Fields Our principal interest is in slow–fast systems that are families of vector fields. In this section we will derive results for vector fields. We deal with smooth vector fields on ∂ U × [z0 , z1 ] ⊂ Rn × R of the form X(x, z) = G(x) + f (x, z) ∂z , satisfying the conditions (V ): (V1 ) (V2 ) (V3 ) (V4 )

U is a compact n-submanifold with boundary of Rn . G(x) is a smooth vector field defined on a neighborhood of U into Rn . Let Gt (x) be the flow of G. We assume that Gt (U ) ⊂ U for t ∈ ]−∞, 0]. f (x, z) is a smooth function defined on a neighborhood of U × [z0 , z1 ] such that for any x ∈ U we have that f (x, z0 ) > 0 and f (x, z1 ) < 0. ¯ There exists a positive constant K¯ > 0 such that ∂f ∂z (x, z) < −K for all (x, z) ∈ U × [z0 , z1 ].

Remark 9.6 As U is compact, condition (V2 ) implies the existence of a t0 > 0 so that Gt is defined for (x, t) ∈ U × ]−∞, t0 ]. Like in the case of diffeomorphisms, it is easy to extend f on U × R keeping its properties and with bounded partial derivatives:

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Lemma 9.4 We can smoothly extend f on U × R such that: 1. f keeps its value on U × [z0 , z1 ]. ¯ 2. We have ∂f ∂z (x, z) < −K. 3. Each partial derivative of f is bounded.



Let Xt (x, z) = (Gt (x), f t (x, z)) be the flow of X. It follows from (V2 ) that it is defined for any t ∈ ]−∞, t0 ]. Definition 9.2 Let W be a submanifold of U × R. We say that W is invariant by the flow of X if for each (x, z) ∈ W and any t ≤ 0 we have that Xt (x, z) ∈ W . This definition even works for topological submanifold. If W is of class C1 , it is invariant by the flow of X if and only if X is tangent to W . If W is a graph of a function ϕ : U → R, we will say that ϕ is invariant by the flow of X. For a given δ ∈ ]0, t0 ], we consider the diffeomorphism F (x, z) = (g(x), f (x, z)) = (Gδ (x), f δ (x, z)). This diffeomorphism is similar to the one we have considered in the previous section, with g(x) = Gδ (x) and h = g −1 = G−δ . It is clear that conditions (Vk )k=1,...,4 imply (Hk )k=1,...,4 ; in particular we have h(U ) ⊂ U . It is easy to obtain the following estimates for its related constants Kδ , NGδ : Lemma 9.5 1. Kδ = 1 − δ(K¯ + O(δ)). 2. NGδ ≤ 1 + δ(dG0 + O(δ)). Proof Using Lemma 9.4 and the usual estimates for a flow, we have G−δ (x) = x − δG(x) + O(δ 2 ) and f δ (x, z) = z + δf (x, z) + O(δ 2 ). The estimates in the statement follow. 

It follows from Lemma 9.5 that Kδ NGk δ = 1 + δ(kdG0 − K¯ + O(δ)). Then we have Kδ N k δ < 1 if kdG0 − K¯ < 0 and if δ is chosen small enough. Applying G

Theorem 9.1 we obtain the following result: If we suppose that kdG0 − K¯ < 0. Then, for δ small enough, the diffeomorphism Xδ has a unique invariant manifold W , graph of a Ck -function ϕ : U → ]z0 , z1 [.

Now, if t < 0 the image Xt (W ) is also a graph of a function, well defined on U , as Gt (U ) ⊂ U . For t ∼ 0 we have that Xt ◦ Xδ ≡ Xδ ◦ Xt . Then, for any t < 0 small enough, the manifold W is invariant by Xt . It follows that if k ≥ 1, the vector field X is tangent to W (we will say that W is invariant by the field X). We hence have proved the result: ∂ Theorem 9.5 Let X(x, z) = G(x) + f (x, z) ∂z be a vector field as above, and assume G and f satisfy (V1 )–(V4). Then, if kdG0 − K¯ < 0, there exists a unique function ϕ : U → ]z0 , z1 [ of class Ck whose graph is invariant by X, i.e. is tangent to X.

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143

We also obtain sufficient conditions to have an asymptotically smooth invariant manifold: Theorem 9.6 Let X be a vector field as above, satisfying (V1 )–(V4 ). We moreover suppose that there exist a non-empty closed subset  ⊂ U and a sequence of open neighborhoods U ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uk ⊃ · · · of , such that for each k: 1. Gt (Uk ) ⊂ Uk , for all t ∈ ]−∞, 0]. 2. for Gk = G|Uk , and denoting by dGk k0 the norm for a well-chosen system of coordinates on Uk , we have that kdGk k0 − K¯ < 0. Then there exists a unique invariant function ϕ : U → ]z0 , z1 [ that is asymptotically smooth along . Finally, we can also consider parameter families of vector fields Xλ (x, z) = Gλ (x) + fλ (x, z)

∂ , ∂z

with λ = (λ1 , . . . , λk ) ∈ P , some p-submanifold of Rp . We state the parametric versions of Theorems 9.5 and 9.6 and infer that they follow from the similar parametric results on diffeomorphisms, stated in the previous section: Theorem 9.7 Let Xλ be a λ-family of vector fields, verifying the conditions (V ) for ρ any λ ∈ P . Write NX = supλ∈P dGλ 0 . Then, if kNX − K¯ < 0, there exists for Xλ a unique family of invariant functions ϕλ of class Ck . Theorem 9.8 Let Xλ be a λ-family of vector fields, which verifies the conditions (V ) for any λ ∈ P . We moreover suppose that there exist a non-empty closed subset  ⊂ U and a sequence of open neighborhoods U ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uk ⊃ · · · of , such that for each k: 1. Gtλ (Uk ) ⊂ Uk , for all t ∈ ]−∞, 0]. 2. Writing Gλ,k = Gλ |Uk , dGλ,k k0 a norm for a well-chosen system of coordinates on Uk , and NXk = supλ∈P dGλ,k 0 , we have that kNXk − K¯ < 0. Then there exists a unique family of invariant functions ϕλ that are asymptotically smooth along . Remark 9.7 We can replace the condition (V2 ) by the condition: Gt (U ) ⊂ U for t ∈ [0, +∞[. The results in this case are completely similar to the ones proved below by exchanging G and −G and following Remark 9.1. For instance the invariance condition in Definition 9.2 is written: for each (x, z) ∈ W and any t ≥ 0 we have that Xt (x, z) ∈ W . The condition in Theorem 9.5 remains the same. The same remark is also valid for families of vector fields.

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9.3 Smooth Invariant Manifolds in Slow–Fast Systems Let us now apply the general results established in the previous section to the particular case of a family of slow–fast systems X,λ , defined on a surface M and with a small parameter  and an extra parameter λ in some subset of Rp . We will ∂ consider the family of three-dimensional vector fields Xλ = X,λ + 0 ∂ defined + on M × R (in a neighborhood of { = 0}). We want to prove the existence of a smooth family of center manifolds Wλ in the two following cases: along a closed arc of the critical curve, containing no contact points nor singular point of the slow dynamics and near critical semi-hyperbolic points of saddle type (semi-hyperbolic points located on the blow-up locus and isolated on it). To obtain the existence of smooth families of center manifolds we first use Theorem 9.8 giving the existence of a unique asymptotically smooth family of center manifolds. We will have to adapt the dynamics to fulfill the prescribed conditions: we will embed the closed arc into a closed curve diffeomorphic to the circle and we will modify the family at the boundary of a chosen neighborhood of the critical semi-hyperbolic point. The unicity, which will be guaranteed in the above two cases, is the key property that we use to finally obtain smoothness. This unicity is not the case in general for the local situations we consider in this book. In fact it is the lack of unicity that is a crucial fact. To pass from asymptotic smoothness to smoothness, it will be important to use that the center manifold is of codimension 1 and also to use the dynamical properties (for instance the fact that the set  does not contain singular points of the slow dynamics). This step is very specific to the dimension 2.

9.3.1 The Case of a Closed Critical Curve As a preliminary step we will consider a closed critical curve γ , i.e. diffeomorphic to the circle. In this case, for each λ, an invariant manifold Wλ has a topological characterization: for each  small enough its trace is just the persistence of the curve γ , as this curve is normally hyperbolic for  = 0. This persistence could be deduced for instance from the stability of normally hyperbolic submanifolds (see [HPS77]). This topological characterization implies the uniqueness. But we will preferably use Theorem 9.8. In fact it will possible to choose a constant neighborhood U for all the Uk and as it was explained in Remark 9.4, we will obtain directly a smooth family of center manifolds: Theorem 9.9 Consider a family of slow–fast systems X,λ defined on a surface M, with λ ∈ P , a compact p-dimensional submanifold of Rp . Let γ be a connected component of the critical set Sλ , diffeomorphic to S 1 (without loss of generality we can assume, to simplify notations, that γ is independent of λ). We assume that γ contains no contact points nor singular points of the slow dynamics.

9.3 Smooth Invariant Manifolds in Slow–Fast Systems

145

Then there exists a unique smooth family of local center manifolds Wλ along γ ; Wλ is the graph of a smooth λ-family of functions U ≈ S 1 × [0, 0 ] for 0 > 0 small enough. Proof We can assume that γ is an hyperbolically attracting curve for X0,λ (if not, replace X,λ by −X,λ . Let us choose a tubular neighborhood A ⊂ M of γ with a parameterization by (z, θ ) ∈ [−¯z, +¯z] × S 1 , z¯ > 0, such that γ = {z = 0}. On the annulus A the system is written as  z˙ = H (z, θ, , λ) θ˙ = G(z, θ, , λ). As γ is supposed to be an attracting hyperbolic curve without contact points nor zeros of the slow dynamics, we have that G(0, θ, 0, λ) = 0 and ∂H ∂z (0, θ, 0, λ) < 0 for all (θ, λ) ∈ S 1 × P . We can choose an 0 > 0 and z¯ small enough and the orientation of θ such that G(z, θ, , λ) > 0 and ∂H ∂z (z, θ, , λ) < 0 for all 1 (z, θ, , λ) ∈ U = [−¯z, +¯z] × S × [0, 0 ] × P . To look for invariant manifolds, we will replace the family X,λ by a smoothly equivalent one: we divide the system by G, i.e. we can suppose that the system in A × [0, 0 ] is written as 

z˙ = H (z, θ, , λ) θ˙ = ,

(9.18)

z, +¯z] × S 1 × [0, 0 ] × P . (In doing with ∂H ∂z (z, θ, , λ) < 0 for all (z, θ, , λ) ∈ [−¯ so we can make sure to preserve the condition ∂H ∂z < 0 because the division by G ¯ ∂ H ∂ 1 ∂H causes H to be replaced by H¯ = H /G and ∂z = ∂z (1/G)H + G ∂z < 0 since ¯ we may use H (0, 0, 0, λ) = 0.) By compactness, there exists K > 0 such that ∂H ¯ ∂z (z, θ, , λ) < −K. We consider the three-dimensional family of vector fields ⎧ ⎨ z˙ = H (z, θ, , λ) Xλ (x, z) : θ˙ =  ⎩ ˙ = 0,

(9.19)

where x = (θ, ) ∈ S 1 × [0, 0 ]. ∂ ∂ , with Gλ =  ∂θ , verifies the conditions in the The family Xλ = Gλ + Hλ ∂z statement of Theorem 9.8. Conditions (V) are trivially fulfilled. Moreover, for any k ∈ N, we can take Uk = U = S 1 × [0, ]. We first see that Gtλ (U ) = U for any t ∈ R so that the first condition in the statement of Theorem 9.8 is trivially verified. Next, to fulfill the second condition, it suffices to change the coordinates on U . We consider the coordinate change θ = θ,  = δk E. The family of vector ∂ fields Gλ then changed into Gλ,k = δk E ∂θ , for some δk to be chosen. We have

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M

γ γε,λ A

0

ε0 ε

Fig. 9.2 Hyperbolically attracting limit cycle γ,λ inside A

¯

dGλ,k 0 = δk , for any λ, and hence NXk = δk . Then, if we choose δk < Kk , the second condition in the statement of Theorem 9.8 is fulfilled. As the neighborhood Uk = U for any k, the family of functions ϕλ → ]−¯z, z¯ [ given by Theorem 9.8 is smooth (see Remark 9.4). 

Remark 9.8 For any  ∈ ]0, 0 ], λ ∈ P the vector field Xλ is tangent to the annulus A×{} and is entering along the boundary. It is easy to see that, inside A×{}, there is a unique limit cycle γ,λ , which is hyperbolically attracting, see Fig. 9.2. Let Wλ be the family of center manifolds obtained in Theorem 9.9. We necessarily have that Wλ ∩ A × {} = γ,λ . By the Implicit Function Theorem, the family γ,λ depends smoothly on (, λ), when  > 0. It is another way to obtain the smoothness of the family of center manifolds Wg λ, for  > 0. We can reinterpret Theorem 9.9 as a confirmation that the family of limit cycles γ,λ , defined for  > 0, has a smooth continuation by the critical curve γ at  = 0.

9.3.2 The Case of a Closed Critical Interval Theorem 9.10 We consider a λ-family of slow–fast systems X,λ defined on a surface M. Let γ be a segment of the critical set Sλ (without loss of generality we can assume, to simplify notations, that γ is independent of λ). We assume that γ contains no contact points nor singular points of the slow dynamics. Then there exists a smooth family Wλ of local center manifolds along γ , i.e. a smooth family

9.3 Smooth Invariant Manifolds in Slow–Fast Systems

147

of invariant manifolds Wλ in M × R+ , defined for  small enough and such that Wλ ∩ Sλ = γ . Proof The difficulty is that families of center manifolds along an interval are not unique. To overcome this difficulty, the idea is to embed the system near the closed interval γ into a system defined in a neighborhood of a closed curve . We can assume that γ is a hyperbolically attracting segment for X0,λ . Let us choose a rectangular  neighborhood B ⊂ M of γ with a parameterization by (z, u) ∈  [−1, +1]× − 18 , 98 such that γ ⊂ {0}× 14 , 34 . On the rectangle B, we can suppose that the system is written as 

z˙ = H (z, u, , λ) u˙ = ,

(9.20)

with ∂H ∂z (z, u, , λ) < 0 for all (z, u, , λ) ∈ B × [0, 0 ] × P . (To obtain the above form, we have used the liberty to rescale the system like in the proof of the previous theorem, exploiting the absence of singularities in the slow dynamics.) Expression (9.20) represents X,λ up to C∞ -equivalence. It is easy to change the system X,λ in a new smooth one, with an expression as in (9.20) but such that the function H has the following properties:   1. It is preserved for u ∈ 14 , 34 .     2. It is equal to −z for u ∈ − 18 , 18 ∪ 78 , 98 . We can trivially extend H on [−1, +1] × R × [0, 0 ] × P as a u-periodic smooth function verifying H (z, u, , λ) ≡ H (z, u + 1, , λ). The system so defined induces a smooth system Xˆ ,λ on the quotient [−1, +1] × R × [0, 0 ] × P /{u + 1 ∼ u}. This system is defined in a neighborhood of the closed critical curve , given by {z = 0} in the annulus A, quotient of [−1, +1] × R. We have that γ ⊂  and the system Xˆ ,λ coincides with the system X,λ in a neighborhood of γ . We can apply Theorem 9.9 to Xˆ ,λ . This gives a smooth family of center manifolds along γ , which gives by restriction a smooth family of center manifolds along γ . 

9.3.3 The Case of a Critical Semi-Hyperbolic Point During the blow-up of a regular contact point, the contact point is replaced by a blow-up locus, which is a half-sphere. Along the blow-up locus the threedimensional blown-up family of vector fields X¯ λ has semi-hyperbolic points that are end points of an interval of regular zeros of X¯ λ . For instance, for a regular contact point of contact order two we have found two different types of semihyperbolic points. Up to smooth equivalence (i.e. smooth coordinate change with

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a multiplication by a positive smooth function), the first type has the following equation, see (8.26) with n = 2: ⎧ ⎨ z˙ = −4H (z, u, ¯ , λ) − 2X¯ λ : u˙ = ¯ u ⎩˙ ¯ = −3¯ 2 ,

(9.21)

where H (0, 0, 0, λ) = 0 and H (z, u, ¯ , λ) = z(1 + O(z)) + O(u, ) ¯ is a smooth function. The second type has the following equation (also up to smooth equivalence, see (8.25) with n = 2): ⎧ ⎨ z˙ = −4H (z, u, ¯ , λ) 2X¯ λ : u˙ = −¯ u ⎩˙ ¯ = 3¯ 2 ,

(9.22)

where H (0, 0, 0, λ) = 0 and H (z, u, ¯ , λ) = z(1 + O(z)) + O(u, ) ¯ is a smooth function. In the two cases, the vector field X¯ λ has a semi-hyperbolic point q located at X = 0 on the singular set {u = ¯ = 0} of the blow-up locus {u = 0}. This point q has a nonzero eigenvalue −4 in the z-direction and two zero eigenvalues. Moreover, up to the multiplicative factor ¯ , the projection of the vector field X¯ λ on the (u, ¯ )plane is a saddle point. The line γλ = {H (z, u, ¯ , λ) = ¯ = 0} is the local critical manifold but the point q is isolated among the singular points on the local blow-up locus {u = 0}. The difference between the two cases is that the slow vector field along γλ points away from q in case (9.21) and toward q in case (9.22). For more degenerate contact points it might be necessary to iterate the blowup and semi-hyperbolic points that appear on the blow-up locus (located on the singular set of the blow-up locus) that may have a more general form. Sometimes, we also have to adapt the parameters λ prior to blowing up. It implies that the blownup vector field is not a necessarily a λ-family of vector fields but, for instance after blowing up a generic turning point, may be a (a, ¯ μ)-family instead. We will therefore consider η-families of vector fields now, in order to present results that can be applied in all cases, including the blow-up of regular contact points and generic turning points. In all cases there exists a chart (z, w, v), neighborhood of the semi-hyperbolic point Q = (0, 0, 0) such that the X-axis is the local singular set of the local blow-up locii that itself are one of the plane {u = 0}, {v = 0} or the union of these two planes. The differential equation of the blown-up vector field X¯ η is written as following, up to a nonzero factor (positive or negative): ⎧ ⎨ z˙ = −H (z, w, v, η) X¯ η = Xη : w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(9.23)

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149

in some neighborhood U of Q in R3 with H (0, 0, 0, η) = 0 and

∂H (0, 0, 0, η) > 0. ∂z

We also have that p, q, α, β ∈ N with p, q > 0 and that one of the coefficients α or β is not zero (this means that Q belongs to a set of non-isolated zeros). The case (9.21) corresponds to α = 1, β = 0 and the case (9.22) corresponds to ¯η α = 0, β = 1. In these two cases we have that p = q = 1. Moreover the family X is just defined for u ≥ 0 and v ≥ 0, but we can smoothly extend it in a neighborhood of 0 ∈ R3 having the same properties. We suppose it in the sequel. The conditions on Xη imply that Q = (0, 0, 0) is a semi-hyperbolic point with a negative eigenvalue − ∂H ∂z (0, 0, 0, η) and two zero eigenvalues. We also see that Q belongs to one line of zeros (if α or β equal zero) or two lines of zeros (if α = 0 and β = 0). These lines have for equation {H (z, 0, v, η) = w = 0} and {H (z, w, 0, η) = v = 0}. As ∂H ∂z (0, 0, 0, η) = 0, they can be expressed near Q, as graphs z(w) or z(v), respectively in the planes {v = 0} and {w = 0}. We will suppose that it is the case in the whole neighborhood U . Let us call γη this set of zeros. Up to a family of smooth diffeomorphisms (z, w, v, η) → (G(z, w, v, η), w, v, η) we can suppose that H (0, w, 0, η) ≡ H (0, 0, v, η) ≡ 0,

(9.24)

and hence that γη ⊂ {z = 0}. This allows to identify the zeros of Xη with the zeros ∂ ∂ of its projection Gη = G = wα v β (pw ∂w −qv ∂v ), identifying the plane (w, v) with 3 the 2-plane {z = 0} in R (as Gη is independent of η we will use the notation G). We want to prove the existence of a smooth center manifold passing through the singular point Q. In a first step, using Theorem 9.8, we will prove the existence of an asymptotically smooth family of center manifolds. This family will be unique due to the boundary condition we will impose in order to apply Theorem 9.8. In a second step we will prove that this unique family of center manifolds is smooth. Proposition 9.3 We consider a smooth family of vector fields as in (9.23), verifying (9.24) and defined in a neighborhood of Q = (0, 0, 0), with parameter η ∈ P as above. Then, there exists a neighborhood U of (0, 0) ∈ R2 (of coordinates (w, v)) and an asymptotically smooth family of functions ϕη (w, v) : U → R whose graph Wη = {z = ϕη (w, v)} is tangent to Xη (we will also say that Wη is invariant by the flow of Xη ). Remark 9.9 Of course, U must be small enough such that Wη is contained in the domain of definition of Xη . Necessarily we have γη ∩ (U × R) ⊂ Wη . Proof We will suppose that α = 0: then the v-axis is contained in γη and along it the slow dynamics goes toward Q. In the other case α = 0, the w-axis is contained in γη and not the v-axis. Along the w-axis the slow dynamics goes away from the point Q. In this case, we would have to replace the time t ≤ 0 by the time t ≥ 0

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(see Remark 9.7). As the proof is completely similar to the one we give below for the case α = 0, we will just treat this last case. Let us suppose that G is defined on U = [0, w0 ] × [−v0 , v0 ], with w0 , v0 > 0. We can suppose that w0 , v0 and z¯ > 0 are small enough such that: 1. H (¯z, w, v, η) > 0 and H (−¯z, w, v, η) < 0 for (w, v, η) ∈ U × P . 2. inf{ ∂H z, z¯ ] × U × P } = K¯ > 0. ∂z (z, w, v, η) | (z, w, v, η) ∈ [−¯ Then, Xη verify the conditions (V) on U × [−¯z, z¯ ]. To fulfill the first condition of Theorem 9.8, we have to modify the vector field G. We consider a smooth function ψ(v) : R → R, such that ψ(v) ≥ 0, ψ(v) = 0 for |v| ≥ 3v40 and ψ(v) = 1 for |v| ≤ v20 . We will replace the vector field family G by   ˜ = pwα+1 v β ∂ + wα 10ψ(v)(qv β+1 − 1) ∂ . G (9.25) ∂w ∂v   ∂ This new vector field coincides with G on [0, w0 ]× − v20 , + v20 and with walpha ∂v +     3v0 3v0 ∂ α+1 β δ pw v ∂w on [0, w0 ]× −v0 , − 4 ∪ 4 , v0 . If U = [0, δw0 ]×[−v0 , v0 ], ˜ verifies that G ˜ t (U δ ) ⊂ U δ for 0 < δ ≤ 1 ˜ t of G for 0 < δ ≤ 1, we see that the flow G and t ≤ 0. To fulfill the second condition of Theorem 9.8, we have to change the coordinates in Uδ : we put w = δW and keep v. Let U¯ be the rectangle [0, w0 ] × [−v0 , v0 ], in the coordinates (W, v). We have that U δ = δ U¯ (with U 1 = U¯ ≡ U ). In coordinates ¯ with ˜ is equal to δ α G (W, v) on U δ , the vector field G   ¯ = pW α+1 v β ∂ + W α 1 − ψ(v)(qv β+1 − 1) ∂ . G ∂W ∂v As U¯ is compact, we have that ¯ sup{d G(W, v) | (W, v) ∈ U¯ } = M < +∞. For each k, we choose a δk and take Uk = U δk . For the vector field Gk = G|Uk , the bound NXk is equal to δk M and then the second condition of Theorem 9.8 reduces to ¯ kδk M − K¯ < 0, i.e. to δk < K . This finishes the proof. 

kM

We want to prove now that the function ϕη obtained in Proposition 9.3 is smooth. ˜ that This is a consequence of the following dynamical property of the vector field G we have introduced in the proof of Proposition 9.3 (see formula (9.25)). Recall that ˜ the flow G(w, u) is defined for each t ≤ 0, from (w, v) ∈ U = [0, w0 ] × [−v0 , v0 ]. We have the following easy result: ˜ defined by (9.25) on U . Then the wLemma 9.6 Consider the vector field G t ˜ component of the flow G (w, v) converges toward 0 when t → −∞.

9.3 Smooth Invariant Manifolds in Slow–Fast Systems

151

We can now prove our final result: Theorem 9.11 Consider a smooth family of vector fields as in (9.23), verifying (9.24) and defined in a neighborhood of Q = (0, 0, 0), with parameter η ∈ P as above. Then, there exists a neighborhood U of (0, 0) ∈ R2 (of coordinates (w, v)) and a smooth family of functions ϕη (w, v) : U → R whose graph Wη = {z = ϕη (w, v)} is tangent to Xη (we will also say that Wη is invariant by the flow of Xη ). Proof We want to prove that the unique invariant family of functions ϕη given by Proposition 9.3 is in fact smooth. For this, we have to prove that each point (w, ¯ v) ¯ ∈ U has a neighborhood on which ϕη is of class Ck . It follows from lemma 9.6 that ˜ Tk (w, for each k, there exists a time Tk < 0 for which G ¯ v) ¯ ∈ Uk . Let Sk be a ˜ Tk (Sk ) ⊂ Uk . As ϕη is invariant by the flow G ˜ t, neighborhood of (w, ¯ v) ¯ such that G Tk

˜ (Sk ), i.e. on Sk we can write we can express ϕη through its values on G ˜ −Tk ◦ ϕη ◦ G ˜ Tk (w, v). ϕη (w, v) = G ˜ Tk , G ˜ −Tk are smooth and ϕη is of class Ck on GTk (Sk ). Then ϕη is The two maps G of class Ck on Sk . Let us notice that this argument works even when w¯ = 0. 

9.3.4 The Case of Singularities of the Slow Vector Field Extending Theorem 9.10 in case the critical curve γ has a singularity of the slow vector field is in general not possible: one typically does not obtain smoothness but only smoothness to any finite degree. An easy counterexample is given by  x˙ = −x y˙ = −y + f (x). Suppose it would have a smooth invariant manifold y = Y (x, ), then it is easy to n see that the graph of the n-th order derivative y = ∂∂YYn (x, ) is a smooth invariant manifold of  x˙ = −x y˙ = (−1 + n)y + f (n) (x). dy = nf (n) (x), which would imply that It implies that for  = 1/n, it solves −x dx f (n) (0) = 0. If we, however, choose f to have all Taylor coefficients nonzero, there is no smooth solution. This counterexample is inspired by the counterexample by van Strien (see [vS79]). We refer to [DMD08] where detailed results are presented in case of the presence of singularities in the slow vector field.

Chapter 10

Normal Forms

In this chapter we will obtain smooth normal forms, i.e. simplified ways of (locally) expressing a smooth slow–fast family X,λ . As we are considering X,λ as a family of vector fields on a surface M, a normal form for conjugacy is just a choice of a convenient chart. Besides choosing convenient charts, we will sometimes even simplify the expressions further by dividing the vector field by a positive function, hence changing the time parameterization of the orbits. If we do so, we will explicitly mention this and talk about equivalences of slow–fast families of vector fields. The system is supposed to depend on a parameter λ ∈ P , a compact domain (for instance a closed disk) of some Euclidean space. The interest to have a normal form is that it will make the study of transitions along the critical curve much easier. This will be considered in Chap. 12. First, we will consider closed intervals of regular points on the critical curve, not containing singularities of the slow dynamics nor contact points and singular points obtained by blowing up contact points. Next, we will consider points that are contained in the blow-up locus of a contact point. We can have two different types of such points. On one hand semi-hyperbolic points that are end points of intervals of regular points of the critical curve, shifted in the blown-up space, and on the other hand hyperbolic saddles that are isolated from other singular points. Rather paradoxically the best results are obtained for the most degenerate cases, i.e. for the regular and semi-hyperbolic singularities. In these cases it is possible to obtain a result of linearization in the hyperbolic direction. We could use results of Takens [Tak71], as they have been brought to the context of slow–fast systems by Bonckaert [Bon96]. To have a self-contained text we have chosen to present a completely independent proof. Moreover we will give a proof for a smooth normal form, in contrast to the references above that limit to a finite class of differentiabilities. A prerequisite for our proof is the existence of smooth center manifolds as established in Chap. 9. The difficulty to treat hyperbolic saddles appearing after blowing up is that their eigenvalue spectrum is resonant. Moreover these hyperbolic saddles are linear in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_10

153

154

10 Normal Forms

certain directions and we want to find changes of coordinates respecting these directions (it is also the case for the regular and semi-hyperbolic singularities). Even if the result of normal form for this type (or similar one) has already appeared in previous papers, we have chosen again to give here a new proof, probably more precise and complete. With Sect. 10.1, this chapter starts with some preliminaries. We briefly present the path method that will be used in each case. We also indicate how to treat the linear first order equations that we will encounter after reduction of the conjugacy problem, putting emphasis on the solution of hyperbolically attracting linear equations. Section 10.2 is devoted to arcs of regular points on the critical curve. In Sect. 10.3 we consider the semi-hyperbolic points in the blow-up locus (after blowing up a contact point). In Sect. 10.4 we use the normal forms obtained in Sects. 10.2 and 10.3 to construct families of center manifolds starting with given initial conditions. This will be useful to construct “long” center manifolds that will be used to choose well-adapted coordinates on sections in order to study compositions of transition maps in Chap. 12. Finally, Sect. 10.5 is devoted to hyperbolic points in the blow-up locus.

10.1 Preliminaries 10.1.1 The Path Method At several places in the text, we will use the path method introduced by J. Moser in [Mos65]. This method has the advantage of linearizing an a priori non-linear problem. In this method one introduces a 1-parameter path between the two objects one wants to conjugate. Here, we consider (n + 1)-dimensional families of vector fields Vλ in Rn+1 expressed in coordinates (z, x) ∈ R × Rn , and a perturbation of ∂ the form λ (z, x)∂z , with ∂z a short notation for ∂z . We want to conjugate Vλ with the family Vλ + λ ∂z . To this end we introduce a path between Vλ and Vλ + λ ∂z , parameterized by s ∈ [0, 1]: Vλs = Vλ + sλ ∂z . We notice that Vλ0 = Vλ and that Vλ1 = Vλ + ∂z , where the term ∂z is onedirectional. For each s, we look for a λ-family of diffeomorphisms Hλs , with Hλ0 = Id, such that we have the conjugacy (Hλs )∗ Vλ = Vλs ,

∀s ∈ [0, 1] .

If such a path exists, Vλ + λ ∂z and Vλ will be conjugated through Hλ1 .

(10.1)

10.1 Preliminaries

155

Let ξλs be the s-parameter path of λ-dependent vector fields obtained by deriving the family Hλs w.r.t. s: ξλs ◦ Hλs =

∂Hλs . ∂s

Deriving (10.1) w.r.t. s we obtain the following equation in ξλs : [Vλs , ξλs ] = λ ∂z ,

(10.2)

where [·, ·] denotes the usual Lie bracket operator. Conversely, if ξλs is a solution of (10.2), we will obtain the path Hλs by integration. Throughout this chapter the path method will be applied to families Vλ satisfying [∂z , Vλ ] = Sλ ∂z ,

(10.3)

for a smooth function Sλ = Sλ (z, x)). In this case we can look for a λ-family of diffeomorphisms Hλs preserving the variable x. Then, the unknown family of vector fields can be chosen of the form ξλs = Kλs ∂z , and Eq. (10.2) reduces to the one-dimensional equation Vλs · Kλs − (Sλ Kλs + s∂z λ ) = λ .

(10.4)

10.1.2 First Order Differential Equation 10.1.2.1 Generalities Equation (10.4) is a first order differential equation. We want to give some general ideas for solving such a system, ideas that will be useful at several points in this chapter. To this end we consider a general smooth vector field X(x) defined on a neighborhood W ⊂ Rn , for an arbitrary n (we include the possible parameters in the vector field equation). We consider also a smooth function G(x, y) on W × R and we consider the first order differential equation X · K(x) = G(x, K(x)),

(10.5)

where K(x) is the unknown function. We want to construct solutions of (10.5) under the following conditions. We suppose that there exists an open section  ⊂ W , transverse to X. If ϕ(t, x) is the flow of X, we choose an open domain W with the property that for any x ∈ W ,

156

10 Normal Forms

there exists a unique smooth time t (x), such that ϕ(t (x), x) ∈  (for instance, if τ > 0 is small enough ϕ([−τ, τ ] × ) is an example of domain W . This example would not be sufficient for our applications as we will also have to consider an unbounded function t (x)). We consider the vector field Z(x, y) = X(x) + G(x, y)∂y on W × R and we assume that this vector field has a complete flow (i.e. defined for all times t ∈ R). This flow takes the form (ϕ(t, x), ψ(t, x, y)) where ϕ is the flow of X. Then, it is easy to solve (10.5) in W by using the flow of Z. One can in fact easily verify that K(x) is a solution of (10.5) if and only if the graph {y = K(x)} is a surface tangent to the vector field Z, or equivalently if this graph is contained in a union of trajectories of Z.

For instance, using this idea we easily obtain a unique solution K(x) for (10.5) on a given domain W , verifying K| ≡ 0. This solution is given by K(x) = ψ(−t (x), ϕ(t (x), x), 0).

(10.6)

We can understand this formula as follows: starting from x ∈ W , we follow the flow ϕ of X until it arrives at , at the point ϕ(t (x), x); then we follow the flow of Z for a time −t (x) to return above x at the z-value that is equal to K(x) by construction. In a similar way it is easy to justify the following implicit formula for the same solution K(x): ψ(t (x), x, K(x)) = 0.

(10.7)

10.1.2.2 Affine First Order Equations In our applications the function G is affine in y and is given by G(x, y) = L(x)y + (x),

(10.8)

where L,  are smooth. Then (10.5) is an affine differential equation and it is possible to give a more explicit expression for K(x). First, we can obtain by integration the expression of the y-component ψ(t, x, y) for the flow of Z. If we ¯ x) = L(ϕ(t, x)) and (t, ¯ x) = (ϕ(t, x)), where ϕ is the flow of X, we put L(t, have for ψ the following affine differential equation: dψ ¯ x)ψ(t, x, y) + (t, ¯ x), (t, x, y) = L(t, dt which we can integrate with the initial condition ψ(0, x, y) = y. We obtain 

t

ψ(t, x, y) = exp 0

¯ L(τ, x)dτ

    t ¯ (τ, x) exp − y+ 0

0

τ

  ¯ L(σ, x)dσ dτ .

10.1 Preliminaries

157

t ¯ Of course this expression is affine in y. As exp[ 0 L(τ, x)dτ ] > 0 we can solve the implicit Eq. (10.7) to obtain the solution K(x). This gives  K(x) = − 0

t (x)

  (ϕ(τ, x)) exp[−

τ

 L(ϕ(σ, x))dσ ] dτ,

(10.9)

0

where ϕ is the flow of X and t (x) is the time to go from x to the section , along this flow. 10.1.2.3 Solution for Hyperbolically Attracting Affine Equations In this section we continue to assume that G is affine, given by (10.8). Moreover we assume that the vector field X is partially hyperbolically attracting in the following sense that we will encounter several times. We have that n = p + q and we write Rn = Rp × Rq with u the coordinate on Rp and z the coordinate on Rq . We write x = (u, z). If  ·  is the Euclidean norm of Rp or Rq , we suppose that X is given on W = D × ∂ . Here, D is a domain diffeomorphic to a ball in Rp and is a ball in Rq . We choose = ρ0 for some ρ0 > 0, where ρ = {z ≤ ρ | z ∈ Rq }. Let X(x) = U (x) + Z(x) be the corresponding decomposition of X, where U is the component along Rp and Z the component along Rq . We assume that Z = 0 on Rp × {0} (i.e. X is tangent to the space Rp × {0}). We also assume that at each point x = (u, z) the partial differential Dz Z(u, 0) has all its eigenvalues with a strictly negative real part (we can notice that this property is not coordinate-free!). It is easy and classical to verify that under this spectral condition, there exists a smooth map u ∈ D → A(u) ∈ Gl(n, R) such that, if we make the change of coordinates (u, z) → (u, A(u)z) the vector field X will be entering along D × ∂ ρ , for 0 < ρ ≤ ρ0 , if we choose ρ0 small enough. We can notice that this property depends on Z only. Then we have the following result: Proposition 10.1 Assume that the partial differential Dz Z(u, 0) has all its eigenvalues with a strictly negative real part and that ρ0 is small enough as explained above. Let B be any domain diffeomorphic to a closed ball inside the interior of D and assume that the function (x) in (10.8) is smooth and flat along D × {0} (this means that  and all of its partial derivative are of order O(zN ), for any N ∈ N). Then, (10.9) is a smooth solution in B × , which is flat along B × {0}. Proof Let X(u) : Rp → [0, 1] be a smooth function that is equal to 1 on the ball B and equal to 0 in a neighborhood of ∂D. We consider in W the vector field T = Z + XU. This vector field coincides with X on B × . Moreover this vector field is tangent along ∂D × and entering along D × ∂ . Then, if ϕ(t, x) = (ϕu (t, x), ϕz (t, x)) ∈ Rp × Rq

158

10 Normal Forms

is the flow of T , we have that ϕ(t, x) ∈ D × for all x ∈ D × and all t ≥ 0. As a consequence of the conditions on Z, there exists a positive constant E > 0 with the property that ϕz (t, x) ≤ ze−Et ,

(10.10)

for any x = (z, u) ∈ D × and t ∈ [0, +∞[. We intend to apply the formula (10.9) to the flow ϕ of T and with t (x) = +∞. Strictly speaking, this is not the time to arrive to a section but the time to converge to the ω-limit set D × {0}. As the flow keeps D × {0} invariant and  ≡ 0 on this set, integrating to t (x) = +∞ is the same as taking a section  with an initial condition equal to 0 on it (here, by analogy, we can think that  = D × {0} and W = D × ). Then formula (10.9) with the flow ϕ of T and t (x) = +∞, and if the integral converges, will give a solution K(x) to the equation T · K = LK +  on D × and hence to the equation X · K = LK +  on B × , as we want. Here, the formula (10.9) has the following expression: 

+∞

K(x) = −

  (ϕ(τ, x)) exp[−

0

τ

 L(ϕ(σ, x))dσ ] dτ.

(10.11)

0

We will now prove that (10.11) defines a smooth function on D × that is flat along D × {0}. It suffices to prove that K and each partial derivatives exist and are equal to 0 on D × {0}. Let us begin with K itself. As the function L is bounded, there exists M0 > 0 such that  τ   exp[− L ϕ(σ, x) dσ ] ≤ exp(M0 τ ). 0

Take an arbitrary N ∈ N. As  is flat in z, there exists a PN > 0 such that |(u, z)| ≤ PN zN , and hence, as a consequence of (10.10) we have that   | ϕ(τ, x) | ≤ PN zN exp(−NEτ ). Using these majorations we obtain 

+∞

|Kλ (x)| ≤ PN zN 0

exp[(M0 − NE)τ ]dτ.

(10.12)

10.2 Regular Points of the Critical Curve

159

The integral in (10.12) is convergent if N is large enough (strictly greater than ME0 , so that M0 − NE < 0). This proves that the right hand side of (10.11) defines a function K that is continuous and that is equal to 0 on D × {0}. We consider now any partial derivation operator ∂α . Write    H (τ, x) =  ϕ(τ, x) exp[−

τ

  L ϕ(σ, x) dσ ],

0

 +∞ the integrand in (10.11). We have to prove that the integral 0 ∂α H (τ, x)dτ is convergent and that there is a majoration similar to (10.12) for N large enough. We do not want to give all the details and report the reader to [Rou98] for instance for a similar estimation. The idea is that ∂α H (τ, x) is a finite sum of terms such that each of these terms is a product of factors that are partial derivatives in x of the form: 1. (∂α1 )(ϕ(τ, x)). As  is smooth and flat along D × {0}, it is the same for ∂α1 . Then |(∂α1 )(ϕ(τ, x))| can be bounded by PNα1 zN exp[−NEτ ], for constants PNα1 > 0, when N is arbitrarily large. 2. ∂α2 ϕ(τ, x), respectively, ∂α2 ϕ(σ, x) (for τ or σ ; let us notice that 0 ≤ σ ≤ τ ). By the usual variational method along trajectories, there exist constants Eα2 > 0 such that this factor can be bounded in absolute value by exp[Eα2 τ ], respectively, exp[Eα2 σ ]. 3. (∂α3 L)(ϕ(τ, x)). As L is smooth in the compact domain D × , these factors are bounded bya constant  Mα3 . τ 4. exp[− 0 L ϕ(σ, x) ]dσ , which is bounded by exp(M0 τ ). Taking into account that a factor of the first type is present in any term of the sum expanding ∂α H , we conclude as in the case k = 0, by taking N large enough, that +∞ the integral 0 ∂α H (τ, x)dτ is convergent and equal to 0 for x ∈ D × {0}. Then, the partial derivative ∂α K(x) exists, is continuous, and is equal to 0 on D ×{0}. This concludes the proof that the formula (10.11) defines a function K on D × , which is smooth and flat along D × {0}. 

10.2 Regular Points of the Critical Curve To simplify the notations, we can suppose that the critical curve S is independent of the parameter λ. We consider a closed interval γ ⊂ S \ Cλ ∪ λ , i.e. an interval not containing contact points nor singular points of the slow dynamics (for any λ). We have shown in Chap. 9, Theorem 9.10 that the system has smooth families of center manifolds Wλ along γ , i.e. such that γ ⊂ Wλ ∩ S. We want to prove that X,λ has the following normal form along γ : Theorem 10.1 (Smooth Normal Form for Regular Slow Curves) Consider an interval γ and a family of center manifolds Wλ as above. Then, there exist smooth

160

10 Normal Forms

local coordinates (z, u) in a neighborhood W of γ in which X,λ has the following expression, up to smooth equivalence: 

z˙ = ±z u˙ = .

(10.13)

The sign is − if γ is attracting or + if γ is repelling. Moreover we have Wλ ∩ W = {z = 0} ∩ W. The system can be seen as a λ-family of three-dimensional vector fields if we add to (10.13) the trivial equation ˙ = 0. At each point m of the critical curve S this family of vector fields is normally hyperbolic (with a two-dimensional center space). We suppose that the given interval γ is attracting (if not, we change X,λ in −X,λ ). We can choose a neighborhood W of γ in M ×R+ with smooth coordinates (z, u, ) such that W ∩ {z = 0} = W ∩ Wλ . In these coordinates, the system X,λ is written as  z˙ = zF (z, u, , λ) u˙ = G(z, u, , λ). where F, G are smooth functions defined for (z, u) ∈ W, λ ∈ P and  ∼ 0. We have F (0, u, 0, λ) = V (u, λ) < 0 as transverse eigenvalue along W ∩ S. As there is no singular point of the slow dynamics on W ∩ S, we have also that G(0, u, 0, λ) = 0 for any (u, λ). Choosing W small enough we can suppose that F (z, u, , λ).G(z, u, , λ) = 0 everywhere. Choosing the orientation of u we can also assume that G > 0. Then, up to smooth equivalence, we can divide X,λ by G, i.e. we can assume that G ≡ 1:  z˙ = zF (z, u, , λ) (10.14) u˙ = , with F (z, u, , λ) < 0 for all (z, u, , λ). In the next subsections we will prove the following result: Theorem 10.2 There exists a smooth family of diffeomorphisms (z, u, , λ) → (g(z, u, , λ), u, , λ), with g(0, u, , λ) ≡ 0, defined for (z, u) ∈ W , chosen small enough, λ ∈ P and  ∼ 0, which changes (10.14) into the linear family 

z˙ = −L(u, , λ)z u˙ = ,

with L(u, 0, λ) > 0 for all (u, λ) ∈ (S ∩ W ) × P .

(10.15)

10.2 Regular Points of the Critical Curve

161

Theorem 10.2 easily implies Theorem 10.1 as follows. A smooth equivalence changes (10.15) into {˙z = −z, u˙ = /L(u, , λ)}. The one-dimensional family 1 of C∞ differential equations u˙ = L(u,,λ) has no zeros. Then, using the flow-box theorem we can find a smooth family of diffeomorphisms in a neighborhood of γ of 1 the form (u, , λ) → G(u, , λ), , λ that changes u˙ = L(u,,λ) into u˙ = 1. The 1 same family of diffeomorphisms changes u˙ =  L(u,,λ) into u˙ = , which is what we need to obtain (10.13). The remainder of this section is devoted to the proof of Theorem 10.2. We could base it on the work of Takens [Tak71] or more precisely on the work of Bonckaert [Bon96]. In order to have a self-contained text we prefer to give a direct proof. Following [Bon96] the proof splits in three steps. In Sect. 10.2.1 we obtain a formal solution along γ . In Sect. 10.2.2 we reduce the question to a flat perturbation of the normal form along the plane {z = 0}, which we call, following [Bon96], a semi-formal solution. We will finish the proof in Sect. 10.2.3 by applying Proposition 10.1. Our proof differs from [Bon96]. We in fact first prove the existence of a family of smooth center manifolds (see Chap. 9), permitting to make a proof for smooth systems and hence obtaining a smooth linearization. In [Bon96] the author, using the Fenichel theory, just assumes that there exist Ck center manifolds, for any k, and he obtained a linearization in an arbitrary finite class of differentiabilities. We will look at the system (10.14) as a three-dimensional λ-family of vector fields ⎧ ⎨ z˙ = zF (z, u, , λ) Vλ : u˙ =  (10.16) ⎩ ˙ = 0.

We write F (z, u, , λ) = A(u, λ) + O(z) + O(),

(10.17)

recalling that A(u, λ) < 0 for (u, λ) ∈ (S ∩ W ) × P .

10.2.1 The Formal Solution Let  be any interval contained in S ∩ W . We write j∞ ( × P ) the space of formal series α in (z, ) along  × P : α=

!

aij (u, λ)zi  j ,

i,j

where aij are smooth functions on  × P .

162

10 Normal Forms

At each smooth λ-family of function f (z, u, , λ) defined in a neighborhood of {0} ×  × {0} × P , we can associate its Taylor series j∞ f = fˆ along  × P , which is an element of j∞ ( × P ). In the same way we can define the infinite jet of a smooth family of vector fields. Every classical operation on functions or vector fields as derivation, Lie bracket, local conjugacy can be induced on the spaces of infinite jets. The following result gives a formal version of Theorem 10.2: Proposition 10.2 Let γ˜ ⊂ S ∩ W be an interval containing γ in its interior. There exists a family of formal diffeomorphisms (z, u, , λ) → (g(z, ˆ u, , λ), u, , λ), with gˆ ∈ j∞ (γ˜ × P ), which brings the infinite jet of (10.14) to the formal linear family 

ˆ z˙ = −L(u, , λ)z u˙ = ,

(10.18)

ˆ 0, λ) > 0 for all (u, λ) ∈ γ˜ ×P . where Lˆ ∈ j∞ (γ˜ ×P ) is independent of z, with L(u, Proof The proof is based on a recurrence on the  degree. For each k ∈ N we introduce the space Hk of polynomials of degree k: i+j =k aij (u, λ)zi  j . For the recurrence we will expand any series α ∈ j∞ (γ˜ × P ) as α = α1 + · · · + αk + h.o.t, with α ∈ H and the symbol h.o.t is for a series on monomials of degree strictly greater than k. A monomial z p for some p ∈ N is said to be resonant. This terminology comes ∂ from the fact that the monomial vector fields z p ∂z , with p ∈ N, generate the kernel of the Lie-derivation by z∂z . We can rewrite the statement of Proposition 10.2 by saying that the system (10.16) is formally conjugate to a system written on resonant monomials. This can be seen as a version in our case of the Poincaré– Dulac Theorem. It follows from (10.17) that, for any (i, j ) and any smooth function a = a(u, λ), we have that Vˆλ · (azi  j ) = iAazi  j + iazi+1 j O(1) + ∂u azi  j +1 , and then Vˆλ · (azi  j ) = iAazi  j + h.o.t.

(10.19)

This means that the derivation is a diagonal operator, up to higher order terms.

10.2 Regular Points of the Critical Curve

163

To prove Proposition 10.2, it suffices to prove the following recurrence step Rk , for any k ≥ 1: Rk : Consider a λ-family Vλk as in (10.16) with z˙ = zFˆ k = A(u, λ)z +

k =2

α + h.o.t,

where α are resonant. Then, there exists a polynomial !

Gk+1 =

aij zi  j

i+j =k+1

such that the λ-family of diffeomorphisms (z, u, , λ) → (z + Gk+1 , u, , λ) brings Vλk to a λ-family Vλk+1 as in (10.14) with z˙ = zFˆk+1 = A(u, λ)z +

k+1 =2

α + h.o.t,

where αk+1 is also resonant.



To prove Rk we expand zFˆk at order k + 1: zFˆk = A(u, λ)z +

k !

!

α +

hij zi  j + h.o.t.

i+j =k+1

=2

The unknown polynomial Gk+1 must verify the following equation: Vˆλk · (z + Gk+1 ) = Fˆk+1 (z + Gk+1 ), which, writing Gk+1 = Az +

k ! =2

α +



! i+j =k+1

i+j =k+1 gij z

hij zi  j +

i j

and taking into account (10.19), is also

!

iAgij zi  j =

i+j =k+1

Az +

k !

α + αk+1

!

Agij zi  j + h.o.t.

i+j =k+1

=2

Writing αk+1 = ak+1 (u, λ)z k , we obtain for the terms of order k + 1 the following equations for i + j = k + 1: h (u,λ)

ij 1. (1 − i)Agij = hij , solved by gij (u, λ) = (1−i)A(u,λ) , if i = 1. 2. We take g1k = 0 and we have ak+1 (u, λ) = h1k (u, λ).

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Using Borel’s Theorem we obtain that there exists a smooth λ-family of functions g(z, u, , λ) defined in a neighborhood W of {0} × γ × {0} × P , whose infinite jet along γ × P is the formal series gˆ obtained in Proposition 10.2 (we have to restrict γ˜ to γ in order to get a smooth function g on W ). If W is a sufficiently small neighborhood of γ in R3 , the map (z, u, , λ) → (g(z, u, , λ), u, , λ) is a smooth λ-family of diffeomorphisms on W × P . This family changes (10.14) into a system whose first line can be written: z˙ = −L(u, , λ)z + (z, u, , λ), ˆ the formal series obtained in where L,  are smooth functions such that j∞ L = L, Proposition 10.1, and j∞  = 0. We can expand :  = z1 (u, , λ) + z2 2 (z, u, , λ), where 1 , 2 are smooth with infinite jets equal to 0 (the term independent of z is divisible by  as, for  = 0, the critical curve is given by {z = 0}). Replacing L by L + 1 , we obtain finally: Proposition 10.3 There exists a smooth family of diffeomorphisms (z, u, , λ) → (g(z, u, , λ), u, , λ), defined for (z, u) ∈ W , chosen small enough, for λ ∈ P and  ∼ 0, bringing (10.14) into the system: 

z˙ = −L(u, , λ)z + z2 2 (z, u, , λ) u˙ = ,

(10.20)

with j∞ 2 = 0 and L(u, 0, λ) > 0 for all (u, λ) ∈ (S ∩ W ) × P . In the next two subsections we will apply the path method introduced in Sect. 10.1.1 to conjugate Vλ = −Lz∂z + ∂u with Vλ + z2 2 ∂z . The function 2 from Proposition 10.3 is flat along {z =  = 0}. Recall that we consider the s-family Vλ + sz2 2 ∂z . Here the condition (10.3) of Sect. 10.1.1 is fulfilled. Then we have to solve Eq. (10.4) that here is equal to   Vλs · Kλs + L − sz(22λ + z∂z 2λ ) Kλs = z2 2λ .

(10.21)

10.2.2 The Semi-Formal Solution We choose an 0 > 0 small enough and a neighborhood W such that (10.20) is defined on W for  ∈ [0, 0 ]. Let γ˜ be a closed interval in S ∩ W , containing γ in its interior. We write M = γ˜ × [0, 0 ] × P × [0, 1]. Let C∞ (z, M) be the space of germs of smooth functions f (z, u, , λ, s) along M and M(z, M) the ideal of germs equal to 0 along M.

10.2 Regular Points of the Critical Curve

165

In this step we will work in j ∞ (z, M), the space of formal series in z, along M: α=

∞ !

ai (u, , λ, s)zi ,

i=0

where the ai are smooth functions on M. The finite sum at order i corresponds to finite jets whose space is just C∞ (z, M)/Mi+1 (z, M) and the space of infinite jets is j ∞ (z, M) = C∞ (z, M)/M∞ (z, M), where M∞ (z, M) is the space of functions flat on z. Each function f defined in a neighborhood of {0} × M has a Taylor series in z: j ∞ f = fˆ, along M. In the same way we can consider the infinite jet of a smooth family of vector fields and classical operations can be induced on infinite jets along M. In this step we want to solve Eq. (10.21) in the space j ∞ (z, M). Following Bonckaert in [Bon96] we call this step the semi-formal solution, to distinguish it from the formal solution of the previous subsection. In this semi-formal context, Eq. (10.21) takes the form:   ˆ 2 ) Kˆ λs = z2  ˆ 2λ , ˆ 2λ + z∂z  Vˆλs · Kˆ λs + L − sz(2

(10.22)

where Kˆ λs =

∞ !

ˆ 2λ = Ki (u, , λ, s)zi and 

i=2

∞ !

i (u, , λ)zi .

(10.23)

i=2

As 2λ is a function on M, flat on (z, ), the functions i are flat on . Putting m = γ˜ × P × [0, 1], we will write that i ∈ M∞ (, m) ⊂ C∞ (M). We want to prove the following result: ˆ λ as in (10.23) with the i ∈ M∞ (, m). Then Proposition 10.4 Consider  s Eq. (10.22) has a solution Kˆ λ as in (10.23) with Ki ∈ M∞ (, m). Proof Expanding (10.22) we have an inductive system of equations for the functions Ki . For i = 2 we have ∂u K2 − LK2 = 2 , and for any i > 2: ∂u Ki + (1 − i)LKi = i +

i−1 !

j

Ui Kj ,

j =2

where Ui are smooth functions on M: Ui = Ui (u, , λ, s) ∈ C∞ (M). By induction on i we suppose that we have already proved the existence of Kj for j

j

j

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10 Normal Forms

˜ i = i + j < i (this condition being empty if i = 2). We see that  ∞ M (, m) and it remains to prove the inductive step:

i−1

j j =2 Ui Kj

∈

˜ i ∈ M∞ (, m), then the equation If  ˜i ∂u Ki + (1 − i)LKi = 

(10.24)

has a solution in M∞ (, m), for any i ≥ 2.

Equation (10.24) is of the form (10.5) with G of the form (10.8); more specifically one has that x = (u, , λ, s), X = ∂u and G(x, y) = L(x)y + (x), ˜ i . Then we can apply the formula (10.9) affine in y with L = (i − 1)L and  =  to prove the claim. The flow reduces u(t, x) = t + u. To apply (10.9) we choose a section  ⊂ m ∩ {u = u} ¯ ∩ { > 0}. If γ˜ = [u0 , u1 ], we choose u¯ in W ∩ S and such that u¯ > u1 . With this choice it is clear that the domain W contains m ∩ { > 0} and (10.9) defines a smooth function in it. The only question is to prove that the solution given by (10.9) on W extends smoothly by 0 on { = 0}. We now prove this point. ¯ The time t (x) is equal to t (x) = u−u  , and ¯ ¯ ˜ i (τ + u, , λ, s), L(σ, (τ, x) =  x) = (i − 1)L(σ + u, , λ). Then (10.9), with x = (u, , λ, s), takes the form: 

u−u ¯ 

Ki (x) = −

 τ   ˜ i (τ + u, , λ, s) exp[−  (i − 1)L(σ + u, , λ)dσ ] dτ. 0

0

(10.25) Making the changes of variables v = τ + u and w = σ + u, formula (10.25) becomes  Ki (x) = − u

u¯

 v  dv  L(w, , λ) ˜ i (v, , λ, s) exp[− dw] , (i − 1)    u

(10.26)

where we have u ≤ v ≤ u. ¯ We want to prove that the function Ki is well defined on M by the right hand side of (10.26), which is smooth and flat in  (i.e. belongs to M∞ (, m)). We will proceed in a way rather similar to the one used to prove Proposition 10.1, replacing “exponential flatness” by flatness in . ˜ i (v, , λ, s) ∈ M∞ (, m) we can write: As  ˜ i (v, , λ, s)  =  N N i (v, , λ, s), 

10.2 Regular Points of the Critical Curve

167

for any N ∈ N, with a smooth function N i and (10.26) is written 

u¯

Ki (x) = − N u

 v   L(w, , λ) dw] dv. N (v, , λ, s) exp[− (i − 1) i  u (10.27)

v As L(w, , λ) > 0 and u ≤ v, we have that exp[− u (i − 1) L(w,,λ) dw] ≤ 1. Then  N the integrand in (10.27) is continuous in x and Ki (x) = O( ): as this is true for any N, Ki is C0 -flat in . To prove that Ki is C∞ -flat we have to proceed as in the proof of Proposition 10.1: we consider any partial derivative ∂α Ki of the right hand member of (10.26), well defined for  > 0, and using the -flatness of each partial derivative ˜ i is C0 -flat in ; we obtain as in the proof of Proposition 10.1 that ∂α Ki is also of  0 C -flat in . This finishes the proof of Proposition 10.4. 

Using Borel’s theorem we can now choose a smooth function Kλs whose infinite jet along {z = 0} is the solution given by Proposition 10.4. We next apply the path method with the s-parameter family of vector fields Kλs ∂z to obtain the following: Proposition 10.5 There exists a smooth family of diffeomorphisms (z, u, , λ) → (g(z, u, , λ), u, , λ), defined for (z, u) in a neighborhood of γ˜ and for λ ∈ P and  ∼ 0, which brings (10.22) given in Proposition 10.2 on the system: 

z˙ = −L(u, , λ)z + z2 2 (z, u, , λ) u˙ = ,

(10.28)

defined for (z, u, , λ) ∈ W × [0, 0 ] × P , where W is a neighborhood of γ˜ , 0 > 0 is small enough, and 2 is flat along {z = 0} (this means that 2 and all its partial derivatives are 0 along (W × [0, 0 ] × P ) ∩ {z = 0}).

10.2.3 The Final Step To prove Theorem 10.2 it suffices now to use Proposition 10.1: applying the path method to (10.28) we have a linear first order equation X · K = LK +  written in a domain of coordinates (z, u, , λ, s) that is flat along {z = 0} and the component of X along ∂z is zero with a negative eigenvalue precisely along this set. To apply Proposition 10.1 we choose the domains D, to be neighborhoods of the given interval γ . The path method gives a smooth family of diffeomorphims, defined in a neighborhood of γ that brings (10.28) to the z-linear family (10.15), as claimed in Theorem 10.2.

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10 Normal Forms

10.3 Semi-Hyperbolic Points in the Blow-up Locus We consider semi-hyperbolic points in the blow-up locus, obtained during the blowup process. We want to look at a more general form as introduced in (9.23) of Chap. 9. Let us recall it. There exists a chart (z, w, v), neighborhood of the semihyperbolic point Q = (0, 0, 0) such that the z-axis in the local singular set of the local blow-up locus that itself is one of the planes {w = 0}, {u = 0} or the union of these two planes. The differential equation of the blown-up family of vector fields X¯ η is defined in a domain W = [−z0 , z0 ] × [0, w0 ] × [0, v0 ] containing Q at the boundary. It is written as following, up to a nonzero factor (positive or negative): ⎧ ⎨ z˙ = −H (z, w, v, η) X¯ η : w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.29)

with H (0, 0, 0, η) = 0 and ∂H ∂z (0, 0, 0, η) > 0. (Recall Remark 8.3 on why we use a parameter η instead of λ in this section.) We also have p, q, α, β ∈ N, with p, q > 0 and at least one of the coefficients α or β is not zero. These conditions imply that Q = (0, 0, 0) is a semi-hyperbolic point with a nonzero eigenvalue ∂H ∂z (0, 0, 0, η) and two zero eigenvalues. We also see that there arrives at Q one line of zeros (if α or β equal zero) or two lines of zeros (if α = 0 and β = 0). The z-axis is the strong hyperbolic invariant manifold. An important point is that the vector field X¯ η projects on the space (w, v) that is a linear saddle with eigenvalues p, −q, up to the factor wα v β . A consequence is that the vector field X¯ η has the first integral wq v p . We have proved in Theorem 9.11 of Chap. 9 that there exists a smooth family of center manifolds Wη that is a graph of a smooth family of functions z = ϕη (w, v) and we suppose such a family has been chosen in the following result: Theorem 10.3 Consider a smooth family of center manifolds Wη , given by a smooth family of functions: z = ϕη (w, v). There exists a smooth family of diffeomorphisms (z, w, v, η) → (g(z, w, v, η), w, v, η), such that g(z, 0, v, η) = g(z, w, 0, η) = 0, defined for (z, w, v) ∈ W , with z0 , w0 , v0 small enough, η ∈ P , which changes (10.29) into the z-linear family ⎧ ⎨ z˙ = −L(w, v, η)z w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.30)

with L(w, v, η) > 0 for all (u, v, η). Moreover Wη ∩ W = {z = 0} ∩ W . Using the family of center manifolds Wη we can change the coordinates by a smooth η-family of diffeomorphisms in a neighborhood of Q, keeping fixed (w, v),

10.3 Semi-Hyperbolic Points in the Blow-up Locus

169

such that (10.29) takes the form: ⎧ ⎨ z˙ = F (z, u, v, η)z Vη : w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.31)

with F (0, 0, 0, η) < 0. We suppose that W is chosen such that F (z, w, v, η) < 0 for all (z, w, v, η) ∈ W ×P . We see that Vη is tangent to the three planes of coordinates. Moreover Wη ∩ W = {z = 0} ∩ W . The proof of Theorem 10.3 will follow the same lines as the proof given in the previous section for regular points of the critical curve, based on the three same steps. Then, using similar notations, we will just insist on the differences. We will ∂ ∂ write Z = wα v β (pw ∂w − qv ∂v ), the “central projection” of Vη , which we consider ¯ on W = W ∩ {z = 0} ≈ [0, w0 ] × [0, v0 ]. With this notation we write Vη = F (z, u, v, η)z

∂ + Z(w, v). ∂z

10.3.1 The Formal Solution Let  = {wα v β = 0} be the set of zeros of Z in W¯ . We want to prove a formal result analogous to Proposition 10.3: Proposition 10.6 There exists a smooth family of diffeomorphisms (z, w, v, η) → (g(z, w, v, η), w, v, η), with g(0, 0, v, η) = g(0, w, 0, η) = 0, defined on W × P , for W with z0 , w0 , v0 small enough, which changes (10.31) into the system: ⎧ ⎨ z˙ = −L(w, v, η)z + z2 2 (z, w, v, η) w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.32)

where 2 is flat along , i.e. all the partial derivatives of 2 are zero at each point of . A new difficulty is that the set of zeros of Z is not a regular line but the union of two transversal ones, when αβ = 0. We begin by proving that we can solve the problem along a single line of zeros. This gives the result when there exists only one line of zeros, i.e. when αβ = 0. We next consider the case αβ = 0.

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10 Normal Forms

10.3.1.1 Solution Along a Line of Zeros As the slow dynamics will not play a role in the proof, we can suppose that the line of zeros is  = [0, v0 ] ≈ W¯ ∩ {w = 0}. This means that α = 0 ( the other coefficient β may be or not equal to zero). The case of  = [0, w0 ] can be treated in a completely similar way. As in the above, we introduce the space j∞ ( × P ) of formal series α in (z, w) along  × P : α=

!

aij (v, η)zi wj ,

i,j

where aij are smooth functions on  × P . We note j∞ f = fˆ the formal series along  × P of a smooth function f . We have the following result: Proposition 10.7 There exists a family of formal diffeomorphisms (z, w, v, η) → (g(z, ˆ w, v, η), w, v, η), with gˆ ∈ j∞ ( × P ), which brings the infinite jet of (10.30) to the formal linear family ⎧ ˆ v, η)z ⎨ z˙ = −L(w, w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.33)

ˆ where Lˆ ∈ j∞ ( × P ) is independent of z, with L(w, v, η) > 0 for all (w, v, η) ∈  × P. Proof The proof is completely similar to the one given in the previous section. For any (i, j ) and any smooth function a = a(v, η), we have that Vˆη · (azi wj ) = iAazi wj + iazi+1 wj O(1) + Z · (azi wj ), and as Z · (azi wj ) = (jpa − qv

∂a i α+j )z w , ∂v

we have that Vˆη · (azi wj ) = iAazi wj + h.o.t, a formula completely similar to (10.19).

(10.34)

10.3 Semi-Hyperbolic Points in the Blow-up Locus

171

The recurrence step Rk is exactly the same as in the previous section, replacing (u, ) by (w, v), and is proved the same way, using (10.34). In fact, the proof ∂ depends just on the “semi-simple” part Az ∂z and not on Z (whose contribution is contained in the “h.o.t” of (10.34)). 

Next, using Borel’s Theorem we obtain: Proposition 10.8 There exists a smooth family of diffeomorphisms (z, w, v, η) → (g(z, w, v, η), w, v, η), defined on W × P , for W chosen small enough, which brings (10.31) to the system: ⎧ ⎨ z˙ = −L(w, v, η)z + z2 2 (z, w, v, η) w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v,

(10.35)

with j∞ 2 = 0 and L(w, v, η) > 0 for all (w, v, η) ∈ W × P . This finishes the proof of Proposition 10.6 when αβ = 0. 10.3.1.2 The Case αβ = 0 We proceed in two steps. First, applying Proposition 10.8, we obtain a system Vη as in (10.35) with a function 2 that is flat along the v-axis. Next we apply Proposition 10.7 along  = [0, w0 ]. For this, we expand the first component of Vη along . In this expansion we have j∞ 2 =

!

hij (w, η)zi v j ,

which is a series in j∞ ( × P ) whose coefficients hij are smooth functions that are flat along {w = 0} × P . w ( × P ) be the subspace of series that are flat along {w = 0} × P , Let j∞ i.e. whose coefficients are flat functions along {w = 0} × P . We observe that the w ( × P ): as mentioned above we proof of Proposition 10.7 preserves the space j∞ 2 start the recurrence with a series j∞  in this space and in the recurrence step Rk , if the functions hij are flat along {w = 0} × P , it is the same for the solutions gij and ak+1 . Then, Proposition 10.7 gives a formal family of diffeomorphisms (z, w, v, η) → (g(w, ˆ v, η), w, v, η), whose series is flat along {w = 0} × P . Using Borel’s Theorem, we can find a smooth function g(w, v, η) such that j∞ g = gˆ and also such that g is flat along {w = 0} × P . If z0 , w0 , v0 are small enough, the map (z, w, v, η) → (g(w, v, η), w, v, η) is a smooth family of diffeomorphims bringing the system Vη into a system that is flat along  (this family of diffeomorphisms brings Vη on a system whose function 2 is flat along the w-axis and preserves the

172

10 Normal Forms

flatness along the v-axis, already obtained in the first step). This finishes the proof of Proposition 10.6 when αβ = 0. 

In the next two subsections we will apply the path method introduced in Sect. 10.1.1 to conjugate Vη = −Lz∂z + Z with Vη + z2 2η ∂z . Recall that Z = wα v β (pw

∂ ∂ − qv ). ∂w ∂v

We start with a family 2η as in Proposition 10.6, i.e. flat along . Recall that we consider the s-family Vη + sz2 2η ∂z . Here the condition (10.3) of Sect. 10.1.1 is fulfilled. Then we have to solve Eq. (10.4) that in the current case has the form given in (10.21).

10.3.2 The Semi-Formal Solution Let us recall that the set of zeros  = {wα v β = 0} is equal to one of the intervals [0, w0 ] , [0, v0 ] or to the union of these two intervals when αβ = 0. In the first case we will suppose that  = [0, w0 ], i.e. that α = 0 (the other case is completely similar). We write W¯ = [0, w0 ] × [0, v0 ] and M = W¯ × P × [0, 1], with coordinates (w, v, η, s). As in the previous section we denote by j ∞ (z, M), the space of formal series in z, along M, and write j ∞ f = fˆ for a germ of a smooth function f along M. We also consider germs and jets of vector fields or diffeomorphism (in families). In this step we want to solve Eq. (10.21) in the space j ∞ (z, M). In this semiˆ 2η given formal context equation (10.21) takes the form (10.22) with series Kˆ ηs and  by Kˆ ηs =

∞ ! i=2

ˆ 2η = Ki (w, v, η, s)zi and 

∞ !

i (w, v, η)zi .

(10.36)

i=2

As 2η is a function on M, flat on  × P × [0, 1], it is the same for the functions i . Let M(M, ) be the ideal of smooth functions on M that are zero on . We can write that i ∈ M∞ (M, ), the ideal of functions that are flat along . We want to prove the following result: ˆ 2η as in (10.36) with the i ∈ M∞ (M, ). Then Proposition 10.9 Assume that  s Eq. (10.22) has a solution Kˆ η as in (10.36) with Ki ∈ M∞ (M, ). Proof Expanding (10.22) we have an inductive system of equations for the functions Ki . For i = 2 we have Z · K2 − LK2 = 2 ,

10.3 Semi-Hyperbolic Points in the Blow-up Locus

173

and for any i > 2: Z · Ki + (1 − i)LKi = i +

i−1 !

j

Ui Kj ,

j =2

where Ui are smooth functions on M: Ui = Ui (w, v, η, s) ∈ C∞ (M). By induction on i we suppose that we have already proved the existence of Kj for j < i  ˜ i = i + i−1 U j Kj ∈ (this condition being empty if i = 2). Then we have that  j =2 i M∞ (M, ) and it remains to prove the inductive step: j

j

j

˜ i ∈ M∞ (M, ), then the equation If  ˜i Z · Ki + (1 − i)LKi = 

(10.37)

has a solution in M∞ (M, ), for any i ≥ 2.

Equation (10.37) is of the form (10.5) with x = (w, v, η, s), X = Z and the ˜ i. function G(x, y) = L(x)y + (x), affine in y with L = (i − 1)L and  =  Then we can apply the formula (10.9) to prove the claim. To apply (10.9) we choose a section σ = {w = w0 }, parameterized by (v, η, s) ∈ ]0, v0 ] ×P ×[0, 1]. With this choice, it is clear that the domain Wσ = M \{wv = 0} and (10.9) defines a smooth function on it. The only question is to prove that the solution given by (10.9) on Wσ extends in a smooth way on M ∩ {wv = 0}, flat on  (if αβ = 0 we have that  = M ∩ {wv = 0}). We now prove this point. Let x = (w, v) ∈ M \  and ϕ(t, x) = (w(t, x), v(t, x)) be the trajectory of Z through x. We have that dw dv (t, x) = pw(t, x)α+1 v(t, x)β , (t, x) = −qw(t, x)α v(t, x)β+1 . dt dt Let us observe that the monomial wq v p is a first integral of Z, i.e. that w(t, x)q v(t, x)p ≡ wq v p . This allows to eliminate v(t, x) in the equation for w(t, x). This gives the following equation for w(t, x): β dw α+1− qβ p . (t, x) = p(wq v p ) p w(t, x) dt

This differential equation can be integrated to give:  qβ−pα qβ−pα  β = (qβ − pα)(wq v p ) p t, w(t, x) p − w p

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10 Normal Forms

if qβ − pα = 0. We obtain the trajectory by solving it in w(t, x) and next using the first integral to compute v(t, x). Then, when qβ − pα = 0, we have: ⎧  p  ⎪ ⎨ w(t, x) = w 1 + (qβ − pα)wα v β t qβ−pα ϕ(t, x) :  − q ⎪ ⎩ v(t, x) = v 1 + (qβ − pα)wα v β t qβ−pα .

(10.38)

The time t (x) to go from x to σ is given by the condition: w(t (x), x) = w0 and then t (x) =

 qβ−pα qβ−pα  1 −β (wq v p ) p w0 p − w p . qβ − pα

(10.39)

Consider now the case qβ −pα = 0. We have that α = rq, β = rp for an integer r > 0 and then wα v β = (wq v p )r is a first integral. In this case we find  ϕ(t, x) :

α β

w(t, x) = wepw v t α β v(t, x) = ve−qw v t ,

(10.40)

and 1 w0 . ln pwα v β w

t (x) =

(10.41)

As in Sect. 10.1 we write ¯ ˜ i (w(τ, x), v(τ, x), η, s), (τ, x) = 

¯ L(τ, x) = (i − 1)L((w(τ, x), v(τ, x), η).

We now have to study (10.9) in which we introduce the above expressions of ¯ We choose σ = v(t, x) as a new variable of integration. We have ¯ L. t (x), ,  q  α β+1− pα q dt so that σ ∈ v, (w0−1 w) p v when t ∈ [0, t (x)]. dσ = −q(wq v p ) q σ With this new variable of integration, we have α q p −q



q

(w0−1 w) p v

Ki (x) = q(w v )

p

p

˜ i (wv q σ − q , σ, η, s)· 

v

  −α exp − (i − 1)q(wq v p ) q

σ

p

L(wv q ν v

− pq

, ν, η)ν

pα q −1−β

(10.42)  pα −1−β dν σ q dσ,

10.3 Semi-Hyperbolic Points in the Blow-up Locus

175

when qβ − pα = 0 and 

q

(w0−1 w) p v

Ki (x) = qwα v β

p

p

˜ i (wv q σ − q , σ, η, s)· 

v

  −α exp − (i − 1)q(wq v p ) q

σ

p

L(wv q ν

− pq

(10.43)

 , ν, η)νdν σ dσ,

v

when qβ − pα = 0. In (10.42) and (10.43), we have q

(w0−1 w) p v ≤ σ ≤ ν ≤ v ≤ v0 .

(10.44)

The remainder of the proof is very similar to the one given in Sect. 10.2. We have to replace the flatness in  by the flatness along the set . We consider separately the two different cases αβ = 0 and α = 0: 1. The case αβ = 0. This case includes the subcase qβ − pα = 0 where necessary we have the condition αβ = 0. We will simultaneously treat the two formulas (10.42) and (10.43). This case αβ = 0 is more degenerate but also easier than the case αβ = 0. We have that  = {wv = 0}. The integral (10.42) or (10.43) defines a smooth function Ki (x) in M \  × P × [0, 1]. We have to prove that Ki (x) extends into a flat function along  × P × [0, 1], i.e. that Ki and any of its partial derivative extend to a C0 -flat function along  ×P ×[0, 1]. The proof is completely similar to the one given in Sect. 10.2, replacing  by wq v p . ˜ i (w, v, η, s) = O((wq v p )N ) for any N ∈ N and as we In fact, as  p p p −p ˜ i (wv q σ − q , σ, η, s) = have that [wv q σ q ]q σ p ≡ wq v p , we have also that  O((wq v p )N ), uniformly in (σ, η, s). As the function between the brackets of exp[·] is negative, we see that the right hand side of (10.42) or (10.43) extends by 0 to a C0 -flat function along  × P × [0, 1]. ˜ i is also O((wq v p )N ) uniformly in (t, η, s). Next, each partial derivative of  Now, if any partial derivative ∂m Ki of the right hand side of (10.42) or (10.43) is a sum of terms (functions or integrals), each of them containing in factor a function p p ˜ i (wv q σ − q , σ, η, s), for some multi-index θ . Such a factor is O((wq v p )N ), ∂θ  uniformly in (σ, η, s), for arbitrary N. The other factors, which may be equal to σ θ , wθ , v θ (which may be negative exponents!), exponential of negative functions, are all dominated by (wq v p )N for N large enough. It follows from these considerations that ∂m Ki extends as a C0 -flat function along  ×P ×[0, 1].

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10 Normal Forms

2. The case α = 0. We have that qβ − pα = qβ = 0 and we can use the expression (10.42) that reduces to 

q

(w0−1 w) p v

Ki (x) = q

p

p

˜ i (wv q σ − q , σ, η, s)· 

v

  · exp − (i − 1)q

σ

p

L(wv q ν

− pq

(10.45)

 , ν, η)ν −(1+β) dν σ −(1+β) dσ.

v

˜ i is just flat in v. The proof of the flatness As  = [0, w0 ] × {0}, the function  of Ki in v follows the same lines as the above proof for the first case. We just have to replace the monomial wq v p by v. We do not give more details. The new point is that we have to prove now that Ki has a smooth extension for w = 0, i.e. along {0} × ]0, v0 ] × P × [0, 1]. To avoid the fractional expression of q the integration limit (w0−1 w) p v we return to the time t as integration variable and ˜ i as a smooth function, with compact support for (w, v, η, s) ∈ M = extend  + R × [0, v0 ] × P × [0, 1]. Then we can define Ki by the formula: 

+∞

Ki (x) = − 0

˜ i (w(τ, x), v(τ, x), η, s)· 

  exp − (i − 1)q

τ

 L(w(ν, x), v(ν, x), η)dν dτ,

(10.46)

0

where w(t, x), v(t, x) are given by (10.38) with α = 0. The integrand is smooth for (t, x) ∈ R+ × M. We just have to show that the improper integral (10.46) as well as the integrals obtained by derivation converges uniformly for t → +∞. In fact, looking at (10.38), it is clear v(t, x) → 0 for t → +∞, uniformly in −1

x. More precisely we have that v(t, x) = O t β uniformly on x (of course v(t, x) depends just on (t, v)). Then, the uniform convergence of (10.46) follows ˜ i in v, by the same considerations as explained above. for the flatness of 

This finishes the proof of Proposition 10.9.



Using Borel’s theorem we can now choose a smooth function Kηs whose semiformal jet (infinite jet along ) is the solution given by Proposition 10.9. We next apply the path method with the s-parameter family of vector fields Kηs ∂z to obtain the following: Proposition 10.10 There exists a smooth family of diffeomorphisms (z, w, v, η) → (g(z, w, v, η), w, v, η),

10.4 Construction of Center Manifolds

177

defined for (z, w, v) ∈ W , for z0 , w0 , v0 small enough, and for η ∈ P which brings (10.32) given in Proposition 10.6 to the system: ⎧ ⎨ z˙ = −L(w, v, η)z + z2 2 (z, w, v, η) w˙ = pwα v β · w ⎩ v˙ = −qpwα v β · v,

(10.47)

with 2 flat along {z = 0} (this means that 2 and all its partial derivatives are 0 along (W × P ) ∩ {z = 0}).

10.3.3 The Final Step To prove Theorem 10.3 it now suffices to use Proposition 10.1: applying the path method to (10.47) we have a linear first order equation X · K = LK +  written in a domain W × P × [0, 1] of (z, w, v, η, s). The term  is flat along {z = 0}, and the component of X along ∂z is zero with a negative eigenvalue precisely along this set. The solution given by Proposition 10.1 gives by integration a smooth family of diffeomorphisms defined on a smaller W .

10.4 Construction of Center Manifolds In Chap. 9 we have proved the existence of a smooth family of center manifolds along an interval γ of regular points of the critical curve and at semi-hyperbolic points on the blow-up locus. As usual, we assume that γ does not depend on η. These families are not unique. In fact it is easy to see that the smooth center manifolds are formally unique: if W and W are two smooth center manifolds for the same value of the parameter η, defined along a neighborhood of an interval γ , then W and W have a flat contact along a neighborhood of γ (the same is true for center manifolds at a semi-hyperbolic point). In this section, we want to show how to use the normal forms established in the previous sections to construct new smooth families of center manifolds. This will be used to glue local center manifolds in order to construct arbitrarily long ones.

10.4.1 The Case of a Regular Interval As in Sect. 10.2 we consider an interval γ of regular points on the critical curve and the normal form chart W given by Theorem 10.1, with coordinates (z, u, ). Recall

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10 Normal Forms

this chart is a neighborhood of γ . Let us suppose that γ = [u0 , u1 ] × {0} × {0} and that X,η is given by (10.13) with the minus sign (we assume that γ is attracting). In the next proposition we will show how to create different smooth center manifolds starting with smooth initial conditions. Proposition 10.11 Consider a normal form chart as in (10.13). Let the smooth family of curves ση in W be defined by the smooth map (, η) → (z(, η), u(, η)), with u(0, η) < u0 (the ending point for  = 0 has an ω-limit point for the fast dynamics that is located outside γ on the side of the entering end u0 of γ ). Let W η be the closure of the union of positive trajectories of X,η in W , starting at the points of ση . Then, W η is a family of center manifolds that is smooth in a neighborhood of γ , if  is small enough. See Fig. 10.1. Proof The manifold W η is smooth and depends smoothly on η outside the critical set: z =  = 0. This manifold is adherent to the points (u, 0, 0) for u > u(0, η). We want to prove that W η is smooth at these points. For u > u(0, η), the set W η can be seen as a family of graphs of functions z = ϕη (u, ). Let us suppose that  is small enough to have u(, η) < u0 for any (, η). We easily compute that ϕη (u, ) is given by ϕη (u, ) = z(, η)e

u−u(,η) 

.

We claim that the function ϕ(u, , η) = ϕη (u, ) is smooth for u ≥ u0 and flat along { = 0} for these values of u. As this family is smooth for { > 0}, the claim

Wη

ε

W

u u1 γ ση

u0 0

z Fig. 10.1 The family of center manifolds in Proposition 10.11

10.4 Construction of Center Manifolds

179

will follow if we can prove that each of the partial derivatives ∂a ϕ is C0 -flat along  = 0 (where a = (a0 , . . . , ap ) is a multi-index in (u, , η)). We hence need to show that ∂a ϕ is of order O( ) for any ∈ N. Now ∂a ϕ(u, , η) =

Ma (u, , η) u−u(,η) e  ,  |a|+1

where |a| = a1 + · · · + ap and Ma is a smooth function. We choose a compact W˜ = [u0 , u1 ] × P × [0, 0 ] inside the normal form chart. If 0 > 0 is small enough, there exists a K > 0 such that u0 − u(, η) > K on W˜ . Write g the Sup-norm of any continuous function of (u, , η) ∈ W˜ . Then, we have that |∂a ϕ(u, , η)| ≤

Ma  − K e   |a|+1

  on W˜ . The C0 -flatness follows from the fact that the functions ξ → ξ − exp − Kξ are bounded on R+ for any ∈ N and any K > 0.



Remarks 10.1 1. We can replace the family of curves ση by any family given by a smooth map (, η) → m(, η) with value in M such that the ending point m(0, η) has an ωlimit for the fast dynamics, located on the critical curve, outside γ on the side of the entering end of γ , inside the connected component of Sη \ Cη containing γ (Cη is the set of contact points on the critical curve Sη ): using the flow of X,η , we can replace the given family of curves by a family contained inside the normal form chart W . 2. It reveals that the smoothness of the initial conditions ση is not really necessary at  = 0. From the proof it easily follows that it would suffice to require that the functions z(, η) and u(, η) be smooth for  > 0 and that these functions and all their partial derivatives w.r.t. η be -admissible: Definition 10.1 A function f (, x), with (, x) ∈ V , for V a neighborhood of (0, q) ∈ R+ × Rp , is -admissible if f (, x) is C∞ for  > 0, C0 at  = 0, and ∀n ∈ N there exists N(n) so that ∂ nf (, x) = O( −N(n) ), ∂ n as  → 0, uniformly in x.



Typical examples of non-smooth -admissible functions are ln  and  1/r for some r > 1. We will encounter such functions in Chap. 11 where we will study them in great detail. Obviously smooth functions are -admissible with N(n) = 0 (we can even take N(n) < 0). 

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10 Normal Forms

10.4.2 The Case of a Semi-Hyperbolic Point As in Sect. 10.3 we consider a semi-hyperbolic point q and the normal form chart W0 given by Theorem 10.3, with coordinates (w, v, z). The system X¯ η is given by (10.30) and we suppose that α = 0 such that the v-axis {z = w = 0} is contained into the set of zeros (the slow dynamics along this axis goes in the direction of q). In this paragraph we want to study how the different center manifolds that we found in Proposition 10.11 (and the Remark 10.1) extend when they come close to a semi-hyperbolic point q, obtained after blowing up. This will now be expressed in Proposition 10.12. We will rely on the fact that such an extension “enters” any sufficiently small neighborhood of q as a center manifold that is, along {z = w = 0, v > 0}, infinitely tangent to the center manifold given by {z = 0}. The same property clearly also holds on its intersection with any plane {v = v0 }, for v0 > 0 sufficiently small. Recall that we study the extension near q in a normal form (10.30). The required restrictions on v0 , w0 , and 0 will become clear in the proof of the proposition. Proposition 10.12 Suppose that the system X¯ η is given by (10.30) in a normal chart W0 , neighborhood of a semi-hyperbolic point q. We suppose that α = 0. Let (w, η) ∈ [0, w0 ] × P → (v0 , z(w, η)) with v0 > 0 and w0 > 0 small enough be a smooth application that defines a smooth family of curves ση in W0 . We also assume that j∞ (zη )(0) = 0. Let W η be the closure of the union of positive trajectories of X,η in W0 , starting at the points of ση . Then, W η is a family of smooth center manifolds in a small enough neighborhood W ⊂ W0 of q. More explicitly, we can take W = D × [−Z, Z], where D is a small enough neighborhood of (0, 0) in the quadrant Q = {(v, w) ∈ R2 | v ≥ 0, w ≥ 0}, and the family of manifolds W ∩ W η is the graph of a smooth family of functions z = ϕη (v, w), defined on D. Moreover, these functions ϕη have an infinite jet equal to 0 along the set {vw = 0}. Remark 10.2 W η is a surface with boundary and a corner at q. (It is diffeomorphic to D.) In W˜ 0 = W0 ∩ {w > 0, v > 0}, we consider the family of vector fields To simplify the computation we use the coordinate change: w = W p,

v = V q,

1 ¯N X . wα v β η

z = Z.

This change, which is not differentiable on {w = 0}∪{v = 0}, has the good property to change the eigenvalues p, −q by 1, −1. For instance w˙ = pW p−1 W˙ = pw = pW p and then: W˙ = W . The first integral is equal to  = wq v p = (W V )pq .

10.4 Construction of Center Manifolds

181

In the coordinates (Z, W, V ), Eq. (10.5) is replaced by ⎧ p ,V q ,η) ⎪ Z˙ = − L(W ⎨ pα V qβ Z W 1 X˜ η = α β X¯ η : W˙ = W ⎪ w v ⎩ V˙ = −V .

(10.48)

We write G(W, V , η) = L(W p , V q , η). We have that G(0, 0, η) = L(0, 0, η) > 0. The trajectory of wα1v β X¯ η = W pα1V qβ X¯ η is given by W (t) = W et , V (t) = V e−t , and Zη (t), solution of the one-dimensional differential equation: Z˙ η (t) = −

  G W (t), V (t), η W (t)pα V (t)qβ

Zη (t).

A direct integration gives    Zη t, (Z, W, V ) = Z exp −

1 pα W V qβ



t

 G(W eτ , V e−τ , η)e(qβ−pα)τ dτ .

0

(10.49) In the coordinates (W, V , Z), the curves ση are graphs of a smooth family of 1/p functions W → (V0 , Z(W, η)), where V0 = v0 > 0 and where j∞ Zη (0) = 0. It is sufficient to prove that the family of center manifolds W η is given by the graph of a smooth family of functions Z = ϕη (W, V ), flat on {V W = 0} (this flatness will imply that these functions are also flat and smooth in (w, v, η)). We choose a neighborhood M of [0, V0 ] × {0} in W˜ 0 ∩ {Z = 0}, such that the trajectory of X¯ η for each (W, V ) ∈ M reaches the curve σ˜ η = {V = V0 }. We will denote the points on σ˜ η by (V0 , W¯ , Z(W¯ , η)). The time t (W, V , η) to go from σ˜ η to the point (W, V ) is given by t (W, V , η) = ln

V0 . V

(10.50)

The family of functions ϕη (V , W ) we are looking for is given by ϕη (W, V ) = Zη (ln V0 /V , (Z(W¯ , η), W¯ , V0 )), meaning  I (W, V , η)  ϕη (W, V ) = Z¯ exp − , W¯ pα V¯ qβ

(10.51)

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10 Normal Forms

with 

ln

I = I (W, V , η) =

V0 V

G(W¯ eτ , V0 e−τ , η)e(qβ−pα)τ dτ,

(10.52)

0

and with Z¯ = Z(W¯ , η). The function Z is smooth in (W¯ , η), but it can also be considered as a smooth function in (E, η) with E = W V = W¯ V0 . Recall that j∞ (Zη )(0) = 0, so that Z could also be considered as a smooth function in , with E =  1/pq , when appropriate. We can also remark that I does not depend on the precise choice of Z(W¯ , η). There is hence no problem in replacing an a priori chosen V0 > 0 by a smaller one. We want to use the expressions (10.51), (10.52) to prove the smoothness of ϕη and its flatness. The proof is rather long and we have divided it into three steps (a), (b), (c). (a) Computation of the Integral I The first tool is to decompose the function G into G(W, V , η) = G1 (W V , W, η) + G2 (W V , V , η), with G1 (0, 0, η) = L(0, 0, η) > 0 and G2 (W V , 0, η) ≡ 0 (see Lemma 12.3). Then, we compute the integral I for the functions G1 and G2 , respectively, and next summing the results. We will write I = I1 or I2 , depending on the case. The simplification will come from the fact that W¯ eτ V0 e−τ ≡ W¯ V0 = W V = E is independent of τ . We will just describe the essential steps. More details can be found in a somewhat similar treatment in Chap. 12. We separately treat the resonant case: {qβ − pα = 0} and the non-resonant case: {qβ − pα = 0}: 1. The resonant case {qβ − pα = 0} for G1 (W V , W, η). We have 

ln

I1 =

V0 V

G1 (V W, W¯ eτ , η)dτ.

0

We make the change of variable s = W¯ eτ to obtain  I1 =

W W¯

G1 (V W, s, η)

ds . s

We get I1 = G1 (E, 0, η) ln where H11 is a smooth function.

V0 + H11 (V , W, η), V

(10.53)

10.4 Construction of Center Manifolds

183

2. The resonant case {qβ − pα = 0} for G2 (W V , V , η). We have that 

ln

I2 =

V0 V

G2 (W V , V¯0 e−t , η)dτ.

0

We make the change of variable s = V0 e−t to obtain that 

V

I =−

G2 (E, s, η) V0

ds . s

˜ 2 (E, s, η) for a smooth As G2 (E, 0, η) = 0, we can write G2 (E, s, η) = s G ˜ function G2 , leading to I2 = H12(V , W, η),

(10.54)

where H12 is a smooth function. 3. The non-resonant case {qβ − pα = 0} for G1 (V , W, η). We have that 

ln

I1 =

V0 V

G1 (W V , W¯ eτ , η)e(qβ−pα)τ dτ.

0

We make the change of variable s = W¯ eτ to obtain  I1 =

V0 E

qβ−pα 

E/V

G1 (E, s, η)s qβ−pα−1 ds.

E/V0

(3.1) If qβ − pα > 0, we introduce the function 

σ

K(σ, V , W, η) =

G1 (W V , s, η)s qβ−pα−1 ds.

0

Recalling that W¯ = E/V0 and W = E/V , we get  I1 =

V0 E

qβ−pα (K(E/V , V , W, η) − K(E/V0 , V , W, η)).

If V is sufficiently small w.r.t. V0 , we can write  K(E/V , V , W, η) − K(E/V0 , V , W, η)) =

E V

qβ−pα

˜ where K˜ is a smooth function such that K(0, 0, η) =

˜ , W, η), K(V

L(0,0,η) qβ−pα

> 0.

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10 Normal Forms

As a consequence, we have I1 =

 V qβ−pα 0

V

(10.55)

L(V , W, η),

where L is a smooth function such that L(0, 0, η) = (3.2) If qβ − pα < 0, we expand G1 in powers of s:

L(0,0,η) qβ−pα

> 0.

˜ 1 (E, s, η), G1 (E, s, η) = a0 + a1 s + · · · + apα−qβ s pα−qβ + s pα−qβ+1 G ˜ 1 is also where ai are smooth functions of (E, η), a0 = G1 (E, 0, η), and G smooth. Putting this expansion in the integral I1 , we obtain that I1 can be written: I1 = a(E, η)(E/V )pα−qβ ln W + H12 (W, V , η),

(10.56)

where a = apα−qβ and H12 are smooth functions. Moreover, as a0 (0, η) = L(0, 0, η) we have that L(0, 0, η) > 0. pα − qβ

H12 (0, 0, η) =

4. The non-resonant case {qβ − pα = 0} for G2 (E, V , η). We have that 

ln(V0 /V )

I2 =

G2 (E, V0 e−τ , η)e(qβ−pα)τ dτ.

0

In the integral, we make a change of variable s = V0 e−τ . Taking into ˜ 2 (E, s, η), with G ˜ 2 smooth, we obtain account that G2 (E, s, η) = s G qβ−pα

I2 = V0



V0

˜ 2 (E, s, η)s pα−qβ ds. G

V

(4.1) If qβ − pα > 0, we have that qβ−pα

I2 = V0



V0 V

˜ 2 (E, s, η) G ds. s qβ−pα

We proceed as in the case (3.2) to obtain I2 =

1 V qβ−pα−1

  a(E, η)V qβ−pα−1 ln V + H¯ 22 (V , W, η) ,

(10.57)

and hence I2 = a(E, η) ln V + H2 2(V , W, η), where a, H¯ 22, H22 are smooth functions.

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185

(4.2) If qβ − pα < 0, we introduce 

σ

K(σ, V , W, η) =

G2 (V W, s, η)s pα−qβ−1 ds.

0

K is a smooth function. Then I2 (E, V , η) = K(V0 , V , W, η) − K(V, V , W, η) = H22(V , W, η), (10.58) where H22 is a smooth function, with the property that H0,0,η = pα−qβ O(V0 ). (b) Expression for ϕη (V , W ) Summary: we can sum up the integrals I1 and I2 in the different cases and get a final form for (10.51): 1. The resonant case qβ − pα = 0. Using (10.53) and (10.54) we have   J1 ¯ ϕη (W, V ) = Z(E, η). exp − pα , E with J1 = −(L(0, 0, η) + O(E)) ln V + H (V , W, η), where O(E) and H are smooth functions. 2. The non-resonant case qβ − pα > 0. Using (10.55) and (10.57) we have  ¯ ϕη (W, V ) = Z(E, η). exp −

J2 E pα V qβ−pα

 ,

with  J2 =

 L(0, 0, η) + O(V , W ) + a(E, ¯ η)V qβ−pα ln V + H˜ 22(V , W, η), qβ − pα

where O(V , W ), a, ¯ and H˜ 22 are smooth functions. 3. The non-resonant case qβ − pα < 0. Using (10.56) and (10.58) we have   pα−qβ J3 ¯ ϕη (W, V ) = Z(E, η). exp −V0 , E pα

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10 Normal Forms

with J3 =

L(0, 0, η) pα−qβ + O(V , W ) + O(V0 ) + a(E, η)W pα−qβ ln W, pα − qβ pα−qβ

where O(V , W ), O(V0

), and a are smooth functions.

(c) Proof of Proposition 10.12 If we now take V0 , V , W sufficiently small, then we see that J1 , J2 , and J3 are strictly positive, due to the fact that L(0, 0, η) > 0 (and 0 < V0 < 1 in case 1). It is not hard to check that the functions of the form exp [·] in either of the three above expressions for ϕη are smooth and infinitely flat along E = 0, inducing the same property for ϕη (W, V ). There is an explicit proof of this result in Lemma 11.6, and ¯ this in a much larger context. We also recall that Z(E, η) has this property, so that it can also be expressed as a smooth function in (, η), using that E =  pq . In any case, since ϕη is smooth and infinitely flat in terms of (W, V ), it remains having this property if we replace (W, V ) by (w, v) with W = w1/p and V = v 1/q . Let us now summarize the obtained result in the next section.

10.4.3 Intervals Ending at a Semi-Hyperbolic Point If we consider an arc of singular points as in Proposition 10.11, ending at a contact point, the blow-up of the contact point produces an arc of singular points ending at a semi-hyperbolic point located on the blow-up locus. We can combine the two previous sections to obtain a smooth family of center manifolds all along the arc γ : Proposition 10.13 Let γ be a regular arc of the critical curve (i.e. without zeros of the slow dynamics), ending (after blowing up) at a semi-hyperbolic point q of the blow-up locus. We suppose that a neighborhood of γ on the critical curve is parameterized by u: the arc γ is equal to [u0 , 0], with u0 < 0; it starts at u0 and ends at q = {u = 0}. We consider a smooth family of curves ση , graph of a family of applications  → mη () ∈ M. We assume that the ω-limit point of mη (0) for the fast dynamics on the critical curve has a coordinate u 0 < u0 (this point is outside γ ) on the side of u0 . Let W η be the union of positive trajectories of X,η starting at the points of ση . Then, W η is a family of center manifolds that is smooth in a neighborhood of γ , if  is small enough (neighborhood including the point q). Proof We choose a normal form chart W around the semi-hyperbolic point q. Next we take u1 ∈ ]u0 , 0[ such that u1 corresponds to a point of γ inside W . Let  be a section through u1 , transverse to γ . We can apply Proposition 10.11, in fact Remark 10.1(1) to obtain that the family W η is smooth in a neighborhood of [u0 , u1 ]. This family cuts  along a smooth family of curves ση . It suffices now to apply Proposition 10.13 at this family of curves ση to obtain that W η is also smooth along [u1 , 0], and then in a whole neighborhood of γ . 

10.5 Hyperbolic Saddle Points in the Blow-up Locus

187

Remark 10.3 As observed in the second remark after Proposition 10.11, the smoothness of the initial condition ση is not necessary at  = 0. It suffices to require that  → mη () and all partial derivatives w.r.t. η be -admissible (see Definition 10.1).

10.5 Hyperbolic Saddle Points in the Blow-up Locus In this section we want to establish normal forms for the singular saddle points obtained by blowing up in Chap. 8. These points can be seen as a η-family of saddle points in dimension 3. They have special characteristics: in the given coordinates y, u, ¯ , up to a translation in y, the family of vector fields is linear in the directions u, ¯ . Moreover, the eigenvalues are fixed (η-invariant) and resonant, and the family of vector fields is linear for ¯ = 0. We will consider a more general setting: we consider (germs of) families of (k + 2)-dimensional vector fields Vη at (0) ∈ Rk+2 , with coordinates (u, v1 , . . . , vk , y), which are quasi-linear smooth saddle points of the form: ⎧ ⎨ u˙ = u (10.59) Vη : v˙i = −qi vi , i = 1, . . . , k ⎩ y˙ = −ry + Fη (u, v, y), where v = (v1 , . . . , vk ) ∈ Rk and p, q1 , . . . , qk , r ∈ N. The smooth family of functions Fη is of order 2 at the origin. At the saddle points s+ and s− of Chap. 8 the family X¯ η or the family −X¯ η in Sect. 10.1 is a family as Vη , with k = 1, q = 2n − 1, r = n. We want to prove the following normal form result: Theorem 10.4 We consider a family Vη . There exists smooth η-families of coordinate changes (u, v1 , . . . , vk , y) → (u, v1 , . . . , vk , Y = Gη (u, v1 , . . . , vk , y)), which brings Vη into the normal form VηN :

VηN

⎧ ⎨ u˙ = u : v˙i = −qi vi , i = 1, . . . , k ⎩ ˙ Y = −rY + Nη (u, v, Y ),

(10.60)

with Nη a function of order 2 of the form: Nη (u, v, Y ) = η (uq1 v1 , . . . , uqk vk , ur Y )Y +

!

j

j

uα v β ηj (uq1 v1 , . . . , uqk vk ),

j =1

(10.61)

188

10 Normal Forms j

j

j

where the functions η , η are smooth and α j ∈ N, β j = (β1 , . . . , βk ) ∈ Nk , for j = 1, . . . , , satisfy α j = #q, β j $ − r ≥ 0 for j = 1, . . . , . The β j are minimal in the sense of Definition 10.2. As Nη is of order 2, we have that j

β

j

β

j

η (0, . . . , 0) = 0. We use the notation v β for the product v1 1 · · · vk k . Remark 10.4 The coordinates (u, v1 , . . . , vk , Y ) will be called normal form coordinates for Vη . Theorem 10.4 says that Vη can be put in the normal form (10.60) by change y → Y = Gη (u, v1 , . . . , vk , y) of the coordinate y into the normal coordinate Y . In other words we can simply say that VηN is the given family Vη written in the smooth normal coordinates (u, v1 , . . . , vk , Y ). We proceed in successive steps.

10.5.1 Resonant Monomial Vectors uα v β y γ ∂y As we only want to work in the last dimension (in y-direction) we are just interested β β in resonant monomial vectors uα v β y γ ∂y , where v β = v1 1 · · · vk k . Of course, we will just write the monomial components uα v β y γ of these resonant monomial vectors that we will simply call resonant monomials. We have α ∈ N, β ∈ Nk , γ ∈ N. We use |β| = β1 + · · · + βk and assume that α + |β| + γ ≥ 2.

(10.62)

The resonance conditions are α − #q, β$ − rγ = −r, for β ∈ Nk , γ ∈ N and (10.50), where #q, β$ = q1 β1 + · · · + qk βk . We consider two cases: (a) The case γ = 0. In this case α = #q, β$ − r, with the condition α ≥ 0. There are a finite number of β : β 1 , . . . , β that are the minimal elements (for the lexicographic order) among the set R of β satisfying #q, β$ − r ≥ 0. Definition 10.2 A minimal element in R is an element β0 such that if β ∈ R then β is greater than β0 or β and β0 are not comparable in the lexicographic order. Remark 10.5 If k = 1 the lexicographic order coincides with the order in N  is  and r r r 1 1 the total. Then in this case we have = 1 and β = q if q ∈ N and β = q + 1 ([·] means the integer part) if qr ∈ N.

10.5 Hyperbolic Saddle Points in the Blow-up Locus

189

If k > 1 it is clear that it would be possible to compute and the vectors β 1 , . . . , β in terms of the eigenvalues q1 , . . . , qk and r . We will not make this computation here. Each other admissible β ∈ R is equal to β j + β¯ for some β¯ ∈ Nk and some j = 1, . . . , (for a given β the values of j and β¯ are not unique). Then, if we put ¯ for some j ∈ 1, . . . , and α j = #q, β j $ − r, any possible α is equal to α j + #q, β$ β¯ ∈ Nk and the resonant monomials are given by uα

j +#q,β$ ¯

vβ

j +β¯

¯

¯

¯

¯

= uα v β u#q,β$ v β = uα v β (uq1 v1 )β1 · · · (uqk vk )βk , j

j

j

j

(10.63)

with β¯ = (β¯1 , . . . , β¯k ) ∈ Nk and j = 1, . . . , . (b) The case γ ≥ 1.

We put γ = 1 + c with c ≥ 0. The resonant conditions are α = #q, β$ + rc.

The corresponding resonant monomials are u#q,β$+rc v β y 1+c = u#q,β$ v β (ur y)y = (uq1 v1 )β1 · · · (uqk vk )βk (ur y)c y, (10.64) with β = (β1 , . . . , βk ) ∈ Nk and c ∈ N.

10.5.2 Formal Normal Form Using the Theorem of Dulac–Poincaré, it is possible to formally reduce the family Vη on the resonant vector fields, with smooth functions of η as coefficients. Here, as the k + 1 first lines are already linearized we can obtain a reduction on the resonant monomial vectors uα v β y γ ∂y , where the monomials uα v β y γ are given by (10.63), (10.64). More precisely one can prove: ˆ η (u, v, y) such that the family of Proposition 10.14 There exists a formal series G formal diffeomorphisms ˆ η (u, v, y)) (u, v, y) → (u, v, Y = G is a formal conjugacy of the formal Taylor series of Vη with a formal system u˙ = u, v˙i = −qi vi , Y˙ = −rY + Nˆ η (u, v, Y ), where Nˆ η is a formal series (of order 2) on the monomials in (10.63) and (10.64) ˆ η and Nˆ η are smooth (with the coordinate y replaced by Y ). The coefficients of G functions of η defined on a same small neighborhood of 0 in the parameter space.

190

10 Normal Forms

Proof We use the proof given by Takens in [Tak71] (see also [Dum93] for instance) ˆ and Fˆ by induction on the degree. to construct the series G Recall that at the step of order s ∈ N, s ≥ 2, we have to consider the Lie operator Z → [u∂u −

k

i=1 qi vi ∂vi

− ry∂y , Z]

on the homogeneous polynomial vector fields Z of degree s. The aim of the step of order s is to replace the term Xs of order s in the partial normal form of order s − 1, to a sum of resonant terms of order s. Xs is a homogeneous polynomial vector field of degree s. We can decompose it into Xs = Fs + Us , where Fs is a sum of resonant terms and Us is in the image of the Lie operator. Then the term of Fˆ of order s is ˆ of order s is the counter-image gs of Us by the operator. Fs , and the term of G Here in our case we remark that the Lie operator leaves invariant the space K∂y of the homogeneous vector fields proportional to ∂y . Then, at each step we can work in this space and obtain Fs and gs in it. This gives the desired result. 

Let us consider the expression of Nˆ η in the monomials of type (10.63) or (10.64). The terms on monomials of type (10.64) can be grouped together in a series in uq1 v1 , . . . , uqk vk , ur Y . This means that there exists a formal series η in k + 1 variables (with smooth coefficients in η) such that the sum of these terms is equal to η (uq1 v1 , . . . , uqk vk , ur Y ). In the same manner, we can partition the terms on 1 1 monomials of type (10.63) in sets, the first set grouping terms factor of uα v β , and so on, the -th set grouping terms factor of uα v β . There exists a formal series 1 η in k variables (with smooth coefficients in η) such that the sum of the terms in 1

1

the first set is equal to uα v β η1 (uq1 v1 , . . . , uqk vk ). We have the same thing for the sums of terms in the other sets. Then, we have proved the following result: ˆ η in k + 1 variables,  ˆ η1 , . . . ,  ˆ η in k Lemma 10.1 There exist formal series  variables such that Nˆ η (u, v, Y ) is equal to  ! j j ˆ η (uq1 v1 , . . . , uqk vk , ur Y ) Y + ˆ ηj (uq1 v1 , . . . , uqk vk ).  uα v β 

(10.65)

j =1

ˆ η and  ˆ ηj , for j = 1, . . . , , are represented by The coefficients of the series  smooth functions of η defined on a same small neighborhood of 0 in the parameter space. As a consequence of Borel’s Theorem, there exist smooth functions Gη , η , ˆ η,  ˆ η,  ˆ η1 , . . . ,  ˆ η . This induces the η1 , . . . , η whose Taylor series are equal to G following result of normal form, “up to a flat term”:

10.5 Hyperbolic Saddle Points in the Blow-up Locus

191

Proposition 10.15 There exists a smooth parameter depending on the change of coordinates (u, v1 , . . . , vk , y) → (u, v1 , . . . , vk , Y = Gη (u, v1 , . . . , vk , y)) in which Vη is written: Vη = VηN + η ∂Y .

(10.66)

Here VηN is the smooth normal form given in (10.60) and (10.61), and j ∞ η (0, 0, 0) = 0, for all η. (We say that η is flat at (u, v, Y ) = (0, 0, 0).)

10.5.3 Reducing to a Differential Equation on Flat Functions Let M∞ η (u, v, Y ) be the ideal of smooth flat germs of families Kη (u, v, Y ), i.e. such that j ∞ Kη (0, 0, 0) = 0 for all η. To deduce Theorem 10.4 from Proposition 10.15 we have to find a smooth ηdepending change of coordinates removing the flat term η ∈ M∞ η (u, v, Y ) in (10.66). It is a general result if X is a smooth hyperbolic germ of vector field at 0 ∈ Rn and P is any flat germ of vector fields at 0, then X + P is smoothly conjugated to X. Here we have a special case of a family of vector fields that is already linear in all variables except the last one and we want to find a conjugacy that only changes this last variable (hence preserving the partial linearization). As this result is not explicit in the literature we will now give a proof of it. Precisely we have the following result: Proposition 10.16 Let VηN be the family of normal forms (10.49) and let η ∈ M∞ η (u, v, Y ). Then there exist a smooth η-family of diffeomorphisms (u, v, Y ) → (u, v, Y + Pη (u, v, Y )) N N with Pη ∈ M∞ η (u, v, Y ), reducing Vη + η ∂Y to Vη .

We will give a proof based on the path method of Moser [Mos65], recalled in Sect. 10.1. Recall that this method reduces the question to solve the Eq. (10.4) that here is written as VηNs · Kηs −

∂ (Nη + sη )Kηs = η . ∂y

(10.67)

It is clear for experts that this hyperbolic partial differential equation of order one has a flat solution when the right hand term is flat. For instance a similar equation was studied in [Rou98], but without term of order zero and without parameter. To

192

10 Normal Forms

be self-contained we present a proof here, based on Proposition 10.1, which is a particular case.

10.5.4 Solving the Differential Equation on Flat Functions There is no new difficulty if we replace VηNs by a more general family of hyperbolic saddles. Then, let M be a compact parameter space in some Euclidean space with coordinate η (in our case M is a compact neighborhood of {0} × [0, 1] ⊂ Rp × R and η¯ = (η, s) with η ∈ (Rp , 0)). We consider a η-family ¯ of smooth germs of hyperbolic singular points Xη¯ at 0 ∈ Rn . We assume that Rn = Rd × Rn−d with coordinates (y, z) = (z1 , . . . , zd , u1 , . . . , un−d ) and that Rd × {0} and {0} × Rn−d are the stable and unstable invariant vector spaces of DXη (0) for all η (in our case n = k + 2, d = k + 1, (v, y) = z and u stands for (u1 , . . . , un−d )). It is not a restriction to suppose that the singular point and the invariant spaces do not depend on η. ¯ This can easily be obtained using the Implicit Function Theorem. We write C∞ ¯ (η¯ ∈ M) and Mη∞ ¯ (z, u) η¯ (z, u) for the space of germs of η-families for the ideal of germs of η-families ¯ fη¯ of smooth functions at 0 ∈ Rn , such that j ∞ fη¯ (0) = 0. We have the following result: ∞ Proposition 10.17 We consider Gη¯ (z, u) ∈ C∞ η¯ (z, u) and η¯ ∈ Mη¯ (z, u). Then the equation

Xη¯ · Kη¯ + Gη¯ Kη¯ = η¯

(10.68)

has a solution Kη¯ ∈ M∞ η¯ (z, u). ∞ Proof Let M∞ ¯ fη¯ of smooth η¯ (z) and Mη¯ (u) be the spaces of germs of η-families n functions at 0 ∈ R , flat along {z = 0} and {u = 0}, respectively. Using a blow-up of 0 ∈ Rn it is easy to prove that ∞ ∞ M∞ η¯ (z, u) = Mη¯ (z) + Mη¯ (u).

It follows from this formula that it is sufficient to solve (10.68) in the spaces Mη∞ ¯ (z) and M∞ η¯ (u), respectively. Of course the two cases are equivalent, up to the change of Xη¯ by −Xη¯ and Gη¯ by −Gη¯ . In the two cases the family of vector fields Xη¯ or −Xη¯ satisfies the condition of Proposition 10.1: it is hyperbolically attracting to a linear subspace through the origin. Then, applying Proposition 10.1 we have that for η¯ in M∞ η¯ (z) or ∞ ∞ in M∞ η¯ (u), we can solve (10.68) with Kη¯ and η¯ in Mη¯ (z) or in Mη¯ (z), respectively. This finishes the proof. 

Chapter 11

Smooth Functions on Admissible Monomials and More

11.1 Admissible Monomials and Functions in Admissible Monomials Definition 11.1 An Admissible monomial in coordinates z1 , . . . , zn is an expression ω = z1α1 · · · znαn lnβ1 z1 · · · lnβn zn

(11.1)

with αj ∈ R+ and βj ∈ Z, verifying the condition that αj > 0 if βj > 0, for j = 1, . . . , n (for some indices j we may have αj = 1 and βj = 0) (we write lnβ z for (ln z)β ). Using the usual Einstein convention, we will sometimes contract the expressions (11.1) in ω = zα lnβ z with α = (α1 , . . . , αn ) ∈ Rn+ and β = (β1 , . . . , βn ) ∈ Zn . Let us consider the positive cone Rn+ = {z1 ≥ 0, . . . , zn ≥ 0}. Each such monomial is smooth when the zi > 0 and can be extended by continuity to 0 on the boundary ∂Rn+ = {ni=1 zi = 0}. Based on admissible monomials, we consider functions f (y, z) defined on a neighborhood W of (0, 0) ∈ Rp × Rn+ in the following way: Definition 11.2 Let  = {ω1 , . . . , ω } be a finite set of admissible monomials. n p C∞ W (y, ) will be the set of functions defined on a neighborhood W of 0 in R ×R+ that can be written as f (y, z) = f˜(y, ω1 , . . . , ω ), with f˜ a smooth function defined on a neighborhood of the image of W by the map (y, z) → (y, ω1 , . . . , ω ) ∈ Rp+ . A function as f will be called a smooth function in (y, ω1 , . . . , ω ) or, in short, a smooth function in (y, ) on W . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_11

193

194

11 Smooth Functions on Admissible Monomials and More

In other words a function f that is a smooth function in (y, ) on W is the composition of the map (y, z) → (y, ω1 , . . . , ω ) with the smooth map f˜. Sometimes, we will omit the notation f and simply say that f˜ is smooth in y, ω1 , . . . , ω . If we do not want to specify the set  of admissible monomials, then we simply say that f is a Smooth function in admissible monomials (SFAM) type germ, or that f is of Smooth function in admissible monomials (SFAM) type germ (smooth function in admissible monomials). Remarks 11.1 1. In the definition of admissible monomial we admit negative exponents of logarithmic terms. For instance ln−1 z1 is an admissible monomial. 2. The expression of an admissible monomial is unique, but in general it is not the case that functions of SFAM-type have a unique expression in terms of admissible monomials. For instance, considering  = {ω1 = z1/3 , ω2 = z1/2 ln z, ω3 = z1/3 ln z}, then ω1 ω32 and ω22 are both expressions of the same function f (z) = z ln2 z of SFAM-type. We also remark that f = o(z2/3) but it is not true that z−2/3 f is smooth in . It shows that working with functions of SFAM-type will require some precaution. 

Algebras of Functions of SFAM-Type Clearly C∞ W (y, ) is an algebra of functions for the usual definition of sum and product. It is a subalgebra of C0R (W ). If we introduce the map  : (y, z) ∈ (Rp × Rn+ , 0) → (y, ξ ) ∈ (Rp+ , 0) defined by (y, z) = (y, ω1 , . . . , ω ), then this algebra is the algebra induced from p+ by the map : the algebra C∞ (W ) (y, ξ ) of smooth functions on (W ) ⊂ R ∞ C∞ W (y, ) = ∗ C(W ) (y, ξ ).

If we consider two sets of admissible monomials  = {ω1 , . . . , ω } and ∞  = {ω1 , . . . , ω }, the minimal algebra containing C∞ W (y, ) and CW (y,  ) is ∞ the algebra CW (y,  ∪  ), which is the sup (least upper bound) of the two algebras for the inclusion order. Also the union of all these algebras is an algebra. We call it ∞ ∞ the algebra C∞ W (y,  ) of SFAM-type functions on W , where  is the infinite set of all admissible monomials. ∞ ∞ ∞ Then C∞ W (y,  ) = ∪ CW (y, ) is the sup of the algebras CW (y, ). By ∞ ∞ definition f ∈ CW (y,  ) if and only if there exists a finite set f of admissible monomials such that f ∈ C∞ W (y, f ). Germs of SFAM-Type Functions We consider the algebra of germs of SFAM-type functions at 0 = (0, 0) ∈ Rp ×Rn+ : Definition 11.3 C∞ 0 (y, ), with  = {ω1 , . . . , ω }, is the algebra of germs of functions (f, 0) with a representative f defined in a neighborhood of 0 ∈ Rp × Rn+ ,

11.1 Admissible Monomials and Functions in Admissible Monomials

195

which can be written as f (y, z) = f˜(y, ω1 , . . . , ω ) for a smooth function f˜, defined on some neighborhood of 0 ∈ Rp+ , representative of a germ (f˜, 0) at 0 ∈ Rp+ . A germ (f, 0) will be called an Smooth function in admissible monomials (SFAM) type germ in (y, ). ∞ We denote by C∞ 0 (y,  ) the algebra of germs of SFAM-type functions. Of course these algebras of germs inherit the properties that we have mentioned above for the algebras defined on a neighborhood W . Moreover it is easy to consider inverses in these algebras of germs: Lemma 11.1 An element (f, 0) ∈ C∞ 0 (y, ) is invertible if and only if f (0) = 0. ∞ ). The same result holds in C∞ (y,  0 Proof If f (0) = 0, then f (y, z) = f˜(y, ω1 , . . . , ω ) with f˜(0) = 0. Then f˜ has an inverse g, ˜ which is a smooth germ at 0 ∈ Rp × R + and we have that 1 ˜ ω1 , . . . , ω ) locally at 0. 

f (y, z) = g(y, The above proof contains an idea that is trivial but which we will often use to study and work with SFAM-type functions and germs: we deduce the properties of f (y, z) = f˜(y, ω1 , . . . , ω ) from the properties of the smooth function f˜, followed by the substitution of the admissible monomials. An application of this idea easily leads to two propositions, for which we do not provide the evident proof. Proposition 11.1 Let us consider f ∈ C∞ 0 (y, ) with  = {ω1 , . . . , ω }. If f (0) = 0, then there exist h1 , . . . , hp , g1 , . . . , g ∈ C∞ 0 (y, ), such that f =

p !

yi hi +

!

ωj gj .

j =1

i=1

The second proposition is a consequence of Malgrange Preparation Theorem [Mal64], which is stated in the one-dimensional setting p = 1: Proposition 11.2 (Division Theorem) Let C∞ 0 (y, ) be the algebra of SFAM-type germs at (0, 0) ∈ R × Rn+ . Let f ∈ C∞ (y, ) have order k ≥ 1 in y at 0: 0 f (0, 0) =

∂f ∂ k−1 f (0, 0) = · · · = (0, 0) = 0 ∂y ∂y k−1

and

∂kf (0, 0) = 0. ∂y k

Then, there exists U ∈ C∞ 0 (y, ), with U (0) = 0 and smooth germs g˜ 0 , . . . , g˜ k−1 at 0 ∈ R such that # f (y, z) = U (y, z) y + k

k−1 ! i=0

$ g˜ i (ω1 , . . . , ω )y

i

.

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11 Smooth Functions on Admissible Monomials and More

11.2 Derivation We will consider the Lie-derivation LX by a linear vector field X=

n ! i=1

λi zi

∂ . ∂zi

We write λ = (λ1 , . . . , λn ). We furthermore introduce the following notation for vectors u = (u1 , . . . , un ), v = (v1 , . . . , vn ), w = (w1 , . . . , wn ) ∈ Rn :  1. #u, v$ = ni=1  ui vi . 2. #u, v, w$ = ni=1 ui vi wi . A direct computation gives: Lemma 11.2 Let us consider an admissible monomial ω = zα lnβ z. We have (using Einstein convention) LX ω = (#λ, α$ + #λ, β, ln−1 z$)ω.

(11.2)

  Explicitly #λ, α$ = ni=1 λi αi and #λ, β, ln−1 z$ = ni=1 λi βi ln−1 zi . Recall that, in an admissible monomial, we admit negative powers of ln zi . Then we have that ∞ LX ω is a finite sum of admissible monomials and belongs to C∞ 0 (y,  ). As a consequence we have ∞ ∞ ∞ Lemma 11.3 If f ∈ C∞ 0 (y,  ) then LX f ∈ C0 (y,  ).

Proof By Definition 11.2, we write f (y, z) = f˜(y, ω1 , . . . , ω ), where f˜(y, ξ1 , . . . , ξ ) is a smooth germ and the ωi are admissible monomials. We have LX f (y, z) =

! ∂ f˜ (y, ω1 , . . . , ω )LX ωi . ∂ξi i=1

˜

∂f ∞ (y, ω1 , . . . , ω ) and LX ωi belong to C∞ As the functions ∂ξ 0 (y,  ), the result i ∞ ∞ follows from the algebra property of C0 (y,  ). 

Remark 11.2 It is easy to extend (11.2) to any derivation. Of course the derivation LX does not act on the variable y that can be considered as a parameter. But the derivations in the zi , including the simplest ones ∂z∂ i , do not in general produce SFAM-type functions. It is the reason why we prefer to use a linear operator as LX . Let us also notice that the derivation LX may reduce the exponent in the power of

11.2 Derivation

197

logarithmic terms. It is the reason to allow negative power in logarithmic terms in the ∞ definition of admissible monomials and to consider the whole algebra C∞ W (y,  ). ∞ As a consequence of Lemma 11.3, C∞ 0 (y,  ) is a X-derivation algebra. It is ∞ in general not the case for the algebras C0 (y, ) depending on a finite set  of admissible monomials. A noticeable exception is when the elements of  do not have logarithmic factors. In this case the X-derivation acts as a diagonal linear operator on monomials: Eq. (11.2) simplifies to LX zα = #λ, α$zα . We will say that the monomial zα is invariant by X or is first integral of X if #λ, α$ = 0 (the flow of X leaves invariant the function zα ).

Elementary Properties of X-Derivation ∞ We list now some properties of this X-derivation on C∞ 0 (y,  ). We will use the following Landau order: a function g is said to be oz (1) if it is zero on {ni=1 zi = 0} or o(1), if it is zero at 0 ∈ Rp × Rn+ . As such: 1. LX f = oz (1). Indeed, from (11.2) we have that LX ω = oz (1) for any admissible monomial ω. Following the proof of Lemma 11.3, we can write for any SFAM ∂ f˜ type function f : LX f (y, z) = i=1 ∂ξi (y, ω1 , . . . , ω )LX ωi , hence LX f = oz (1). 2. Let ω = zα lnβ z be an admissible monomial for which #λ, α$ = 0 (zα is not ∞ X-invariant). Then we have the following implication for f ∈ C∞ 0 (y,  ):   f = zα lnβ z 1 + o(1)

⇒

  LX f = #λ, α$zα lnβ z 1 + o(1) .

(11.3)

Proof We write f = ω(1 + g) with g = o(1), and we recall that by supposition ∞ g ∈ C∞ 0 (y,  ). Then:  LX g  #λ, β, ln−1 z$  (1 + g) + 1+ #λ, α$ #λ, α$  LX g  #λ, β, ln−1 z$ (1 + g) + . = #λ, α$ω 1 + g + #λ, α$ #λ, α$

LX [ω(1 + g)] = #λ, α$ω

(11.4)

Between the brackets, the function g is o(1) and the two others terms are oz (1) (hence also o(1)). The result follows. 

In (11.3) we can replace the order o(1) by the order oz (1). z-Regularly Smooth Functions In [DR07a] the following notion has been introduced: Definition 11.4 A function f (y, z) for (y, z) ∈ W , with W a neighborhood of 0 ∈ Rp × R+ , is said to be z-Regularly smooth in y if f and any of its partial derivatives w.r.t. y are defined and continuous in (y, z) including for z = 0. The interest of this notion is that properties of f (y, 0) that are stable for the C∞ ∞ topology remain true for z > 0 small enough. Of course a function f ∈ C∞ W (y,  )

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11 Smooth Functions on Admissible Monomials and More

is also z-regularly smooth in y but the opposite is false. It is in general not possible to relate the properties of functions that are z-regularly smooth in y to properties ∞ of globally smooth functions as it is the case for f ∈ C∞ W (y,  ). For instance we cannot derive simple properties like the ones stated in Propositions 11.1 and 11.2 or the derivation rules given above.

11.3 Counting the Number of Roots First we generalize the notion of admissible monomial: Definition 11.5 A Generalized admissible monomial in the variables z1 , . . . , zn is an expression ω = zα lnβ z = z1α1 · · · znαn lnβ1 z1 · · · lnβn zn with αj ∈ R and βj ∈ Z for j = 1, . . . , n ((α, β) ∈ Rn × Zn ). The generalization consists in taking the powers αj , βj without any restriction on sign. For instance we admit negative powers αj for the zj . An important consequence is that if ω is a generalized admissible monomial, then it is also the case for ω−1 . In fact, generalized admissible monomials could be defined as quotients of admissible monomials. A generalized admissible monomial does not necessarily define a continuous function on the closed quadrant Rn+ but just a smooth function on the open cone Int Rn+ = {z1 > 0, . . . , zn > 0}.  We consider now a linear vector field X = ni=1 λi zi ∂z∂ i on Rn . Definition 11.6 We say that a generalized admissible monomial ω = zα lnβ z is a n X-Resonant monomial if #λ, α$ = i=1 λi αi = 0. The derivation formula (11.2) remains valid for generalized admissible monomials. It implies that a monomial ω = zα lnβ z is X-resonant if and only if zα is a first integral of X on Int Rn+ , i.e. if and only if zα has a constant value on each orbit of X. It is also possible to generalize (11.4) to obtain the following formula that will play a key role in the proof of Theorem 11.1: Lemma 11.4 Let zα lnβ z be a generalized admissible monomial that is not X∞ resonant (#λ, α$ = 0) and g ∈ C∞ 0 (y,  ) of order o(1). Then   LX zα lnβ z(1 + g(y, z)) = #λ, α$zα lnβ z(1 + g(y, ˜ z)),

11.3 Counting the Number of Roots

199

∞ for another function g˜ ∈ C∞ 0 (y,  ) of order o(1). Roughly speaking, if the symbol o(1) indicates SFAM-type functions, we have

  LX zα lnβ z(1 + o(1)) = #λ, α$zα lnβ z(1 + o(1)).

(11.5)

 Proof Putting ω = zα lnβ z and we refer to (11.4) for an expression for LX ω(1 +  ∞ g) . As LX h = oz (1) for any h ∈ C∞ 0 (y,  ) we have that the function g˜ = g +

LX g #λ, β, ln−1 z$(1 + g) + #λ, α$ #λ, α$

∞ appearing in (11.4) belongs to C∞ 0 (y,  ) and is of order o(1).



Remark 11.3 In Lemma 11.4 we can replace the order o(1) by the order oz (1), but the result is less interesting. We want to use the algorithm of division-derivation introduced in [Rou86] (see also [Rou98]) in order to prove the following result: Theorem 11.1 Let F (y, z) be a function on Rp × Int Rn+ that in some neighborhood W0 of 0 may be written as F (y, z) =

k !

  Ai (y)ωi (z) 1 + gi (y, z) ,

i=1

where: 1. 2. 3. 4.

i

i

The ωi = zα lnβ z are generalized admissible monomials, for i = 1, . . . , k. ∞ The functions gi belong to C∞ W0 (y,  ) and are of order o(1). The functions Ai (y) are continuous and Ak (0) = 0. The monomials ωj ωi−1 , i = j , are non-resonant. #λ, α i − α j $ = 0,

∀i = j.

Then, there exists a neighborhood W of 0 in W0 , such that the function F has at most k − 1 roots counted with their multiplicity, in restriction to each orbit of X in W. Proof The division-derivation algorithm consists of the production of a sequence of functions: F0 = F, F1 , . . . , Fk−1 . Each Fj is expressed as a sum similar to F , but only with k − j terms and defined on a smaller neighborhood Wj ⊂ W0 of 0.

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11 Smooth Functions on Admissible Monomials and More

To define F1 we first divide F by ω1 (1 + g1 ). This is done on a neighborhood W1 ⊂ W0 chosen such that 1 + g1 (y, z) = 0 for each (y, z) ∈ W1 . We obtain on W1 the function:   ! F = A1 (y) + Ai (y)ωi ω1−1 (z) 1 + g˜ i (y, z) . ω1 (1 + g1 ) k

i=2

Using the properties of SFAM-type functions we have that the function g˜i , given by 1+gi (y,z) g˜i (y, z) = 1+g − 1, is of SFAM-type and o(1) (the division step). 1 (y,z) Next we apply the operator LX , using Lemma 11.4 (the derivation step). We obtain the function F1 on W1 : F1 (y, z) = LX



   ! F #λ, α i − α 1 $Ai (y)ωi ω1−1 (z) 1 + gi1 (y, z) . = ω1 (1 + g1 ) k

i=2

The effect of the derivation is to eliminate the first term A1 (y) and then to reduce by one unity the number of terms in the sum. Apart from this fact, the terms of the summation are completely similar to the ones in F but with the functions Ai replaced by #λ, α i −α 1 $Ai and the monomials ωi replaced by the monomials ωi ω1−1 . For the recurrence step of order j + 1 = 1, . . . , k − 1, we assume that we have a function: Fj (y, z) =

j k  % ! i=j +1

   j #λ, α i − α m $ Ai (y)ωi ωj−1 (z) 1 + gi (y, z)

m=1 j

defined on some neighborhood Wj ⊂ W0 , with functions gi of SFAM-type and of order o(1). It is now easy to verify the recurrence step. By the computations made to pass  j −1 from F to F1 we obtain that the division of Fj by ωi ωj (z) 1 + gi (y, z) , which is possible on some neighborhood Wj +1 ⊂ Wj , followed by the derivation by X, produces a function Fj +1 (y, z) =

+1 k  j% ! i=j +2

   j +1 #λ, α i − α m $ Ai (y)ωi ωj−1 (z) 1 + g (y, z) , i +1

m=1

j +1

are admissible and of order o(1). where gi Performing the k − 1 steps of the recurrence we finish with a function   −1 Fk−1 (y, z) = #λ, α k − α 1 $ · · · #λ, α k − α k−1 $Ak (y)ωk ωk−1 (z) 1 + gkk (y, z) , where gkk (y, z) is of SFAM-type and of order o(1).

11.3 Counting the Number of Roots

201

As Ak is a continuous function such that Ak (0) = 0, we can choose a final neighborhood W = Wk ⊂ Wk−1 on which the function Ak (y)(1 + gkk (y, z)) is nowhere zero. Then, the function Fk−1 itself is nowhere zero on W . Consider now any orbit γ of X on W . Recall that the derivation of a function G by the vector field X corresponds to the derivation of G along the flow. Then, as Fk−1 is equal to the derivation of Fk−2 up to a division by a nonzero (smooth in z) function, Rolle’s Theorem applied to Fk−2 implies that this function has at most one zero counted with its multiplicity, in restriction to γ (recall that γ is connected!). The same argument based on Rolle’s Theorem can be applied by recurrence for each j ≤ k, to obtain that the function Fk−j has at most j − 1 zeros, counted with their multiplicity, in restriction to γ . We finally obtain the desired result: there exists a neighborhood W ⊂ W0 , such that on each orbit of X contained in W, the function F has at most k − 1 zeros counted with their multiplicity. 

Remarks 11.4 1. Even if F is a sum of admissible monomials, it is clear that in general, the division step produces admissible monomials that may be of generalized type and we have to consider them in the recurrence. In Theorem 11.1 there is hence no reason to begin with admissible monomials, even if the functions we have in this text are sums of admissible monomials. ∂ and the unique resonant 2. If z is one-dimensional, it suffices to consider X = z ∂z monomials are the constant ones. The orbits of X are intervals parallel to the z-axis and we can consider to deal with a y-family of functions in one variable z. 3. In the one-dimensional case, there is a natural total order of flatness at the origin, among the generalized admissible monomials. A one-dimensional monomial is just an expression zα lnβ z with α ∈ R, β ∈ Z. The order of flatness ω % ω (for flatter than) is given by



{zα lnβ z % zα lnβ z} ⇐⇒ {α > α } or {α = α and β < β }. A better result can then be obtained by ordering the terms in the sum in this %order for the monomials ωi : if we suppose that k0 ≤ k is the first index such that Ak0 (0) = 0, then the number of roots is bounded by k0 − 1. In fact it is possible to group together all the terms from k0 to k in a single term, as, for j > k0 , in some small neighborhood (where Ak0 (y) is nowhere zero) we have Aj ωj (1 + o(1)) = Ak0 ωk0

Aj ωj ωk−1 (1 + o(1)) = Ak0 ωk0 o(1). 0 Ak0

4. As an example of application we see that the function z → y 3 + y exp y ln−1 z + y 2 z1/3 ln z + z has at most 3 roots on the interval ]0, z0 ] for a z0 > 0 small enough and for y near 0.

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11 Smooth Functions on Admissible Monomials and More

5. In dimension one, a finite sequence of ordered generalized admissible monomials is an example of a Chebyshev system. Linear combinations of these systems of functions are studied in [Mar98]. In [Rou98] and also in [Mar98] one can find, for dimension one, a more general proof than the one given here, including functions on any interval in R+ and for monomials that can include compensator functions. Also, under generic assumptions on the functions Aj it is proved in [Mar98] that the bifurcation diagram for the roots is topologically equivalent to the one encountered in catastrophe theory. 

11.4 Asymptotically Smooth Functions in Admissible Monomials In this section we will replace smooth functions by Ck -functions in the definition of functions of SFAM-type: Definition 11.7 Let  = {ω1 , . . . , ω } be a finite set of admissible monomials. For any k ∈ N, CkW (y, ) will be the set of functions defined on a neighborhood W ⊂ Rp × Rn+ , which can be written as f (y, z) = f˜(y, ω1 , . . . , ω ) with f˜ a Ck -function defined on a neighborhood of the image of W by the map (y, z) → (y, ω1 , . . . , ω ) ∈ Rp+ . A function as f will be called a Ck -function in (y, ) on W . Clearly CkW (y, ) is an algebra of functions. If we consider two sets of monomials  = {ω1 , . . . , ω } and  = {ω1 , . . . , ω }, the minimal algebra containing CkW (y, ) and CkW (y,  ) is the algebra CkW (y,  ∪  ), which is the Sup-bound of the two algebras for the inclusion order. Also the union of all these algebras is an algebra. We call it the algebra CkW (y, ∞ ) of Ck -functions in admissible monomials on W (or of CkSFAM-type) where ∞ is the infinite set of all admissible monomials. By definition f ∈ CkW (y, ∞ ) if and only if there exists a finite set f of admissible monomials such k ∞ ∞ that f ∈ CkW (y, f ). It is also clear that Ck+1 W (y,  ) ⊂ CW (y,  ), for any k. k ∞ The algebra CW (y,  ) is not invariant by derivation: if f ∈ CkW (y, ∞ ) for ∞ k ≥ 1, then LX f ∈ Ck−1 W (y,  ). To have an algebra closed by LX -derivation we will consider the following: Definition 11.8 We say that a function f is an Asymptotically smooth function in admissible monomials (or of ASFAM-type) if for each k it belongs to CkWk (y, ∞ ), where Wk is a neighborhood of 0 ∈ Rp × Rn+ . The set of all asymptotically smooth admissible functions is the algebra C→∞ (y, ∞ ) = ∩k,Wk CkWk (y, ∞ ). 0

11.5 Functions of Exponentially Flat Type

203

The functions f of C→∞ (y, ∞ ) are the functions that for any k can be 0 k written as f = f˜k (y, ω1 , . . . , ω kk ) for a Ck -function f˜k defined near 0 and a set of admissible monomials {ω1k , . . . , ω kk } that also may depend on k. Of course →∞ (y, ∞ ) but it is dubious that these two algebras are equal. ∞ C∞ W (y,  ) ⊂ C0 The interest for this notion is that it is technically much easier to prove that a given function f is of ASFAM-type rather than of SFAM-type. For instance, in this text, the construction of SFAM-type functions is made through theorems (local normal form of vector fields, existence of center manifold) for which the smooth version (if it exists) is much more involved than the version “Ck for any k.” On the other side almost all the results we want to prove do not need smoothness but just k-differentiability, for k large enough, depending for instance on the codimension or the number of steps in the proof. If we start with an initial class of differentiability k0 , this differentiability can decrease at each step (for instance if we have to use the Division Theorem in a finite differential class). It will be sufficient to start with k0 large enough such that at the end, the remaining differentiability is sufficient to prove the desired result. The reason using in this text smooth functions in admissible monomials is that the theorems we will use have a smooth version and of course the results are simpler to state (although more difficult to prove) and also stronger.

11.5 Functions of Exponentially Flat Type In this section, in view of the notations that we will use in the next chapters, we replace y by (y, μ) and z by , a 1-dimensional variable. The admissible monomials we encounter will hence be of the form  α lnβ . If we use a variable z, it will serve to describe a similar type of variable as the variable y. In Chap. 12 we will see that transition maps Tμ, along the critical curve, including possible passage near jump points or turning points, have the following particular structure: there exist two sets 0 ⊂ 1 of admissible monomials in  such that Tμ, : y → z = α(μ, ) ± e

I˜(y,μ,) 

,

(11.6)

where α is smooth in (μ, 0 ), I˜ is smooth in (y, μ, 1 ), I˜(y, μ, ) < 0. This will justify Definition 11.9 below, which focuses on this kind of expressions. The variables (y, μ, ) are defined for (y, μ) ∈ W = σ × P , where σ is a compact real interval and P is some connected compact domain in Rp . The parameter  ∈ [0, 0 ], where 0 > 0 can be chosen arbitrarily small; equivalently, we will also write  ∼ 0. This means that we want to consider germs of functions on W × R+ along the compact set W × {0}. Then, taking into account the compactness of W it will be sufficient to express open conditions for the function I˜(y, μ, ) on its restriction to { = 0}.

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11 Smooth Functions on Admissible Monomials and More

For the remainder of this chapter, we will use the following: Notation 1 If J˜(z, ) is defined for (z, ) ∈ W × [0, 0 ], we write J˜(z, 0) = J (z), for z ∈ W .

 Remark 11.5 Consider any set  of admissible monomials in . In the sequel we will always assume that the monomial  belongs to the monoid generated by the elements of . (It will be always the case in Chap. 12 and in the sequel of the book). This condition is equivalent to say that  and then any function smooth in  are also smooth in . We can strictly order the monomials of . Let ωmin be the monomial of minimal 1 1 order of flatness (for instance, if  = { 2 ,  ln } we have that ωmin =  2 ). As the function J (z) is smooth, it will be equivalent to say that J˜(z, ) is smooth in (z, ) or that J˜(z, ) = J (z)+O(ωmin) with the remainder O(ωmin ) smooth in (z, ). We will often prefer the second formulation, which has the advantage to put emphasis on the function J (z) and give a more manageable expression. Moreover it will often happen to have a remainder of an order O(ω) greater than O(ωmin ), and in this case the expression J˜(z, ) = J (z) + O(ω) will give more precise information. Definition 11.9 Let 0 ⊂ 1 be sets of admissible monomials in  and W = σ ×P as above. An expression Tμ, (y) defined by (11.6), involving the affine term α and exponent I˜, will be called a family of Functions of exponentially flat type (on the sets 0 , 1 ), if: 1. I (y, μ) := I˜(y, μ, 0) < 0 for any (y, μ) ∈ W . 2. α is smooth in (μ, 0 ) and I˜ smooth in (y, μ, 1 ). Under these conditions, (11.6) defines such a family for (y, μ) ∈ W and  ∼ 0. If moreover ∂I ∂y = 0 on W we will say, because it is the case for  > 0, that Tμ, (y) is a family of Diffeomorphisms of exponentially flat type (on the sets 0 , 1 ). We want to give some general properties of this type of functions, properties that will be intensively used in the sequel. Notation 2 In the text, if a ∈ R \ {0}, we will write sign(a) to denote the number ±1 that has the same sign as a. 

11.5.1 Some Basic Properties of the Exponential Term I˜(y,μ,)

Lemma 11.5 Let I˜(y, μ, ) be defined as in Definition 11.9. Then, e  is smoothly flat along  = 0, i.e. all partial derivatives, which are well defined for  > 0, tend to 0 when  tends to 0. More precisely, if 0 < δ < inf{|I (y, μ)| |(y, μ) ∈ W },

11.5 Functions of Exponentially Flat Type

205

then we have that each partial derivative is of order O(e−  ). As a consequence, the δ

function e

I˜(y,μ,) 

is smooth on W × [0, 0 ].

Proof Any partial derivative of e

I˜(y,μ,) 

is a finite sum of terms of the form

 a | ln |b g(y, μ, )e

I˜(y,μ,) 

δ

with a, b ∈ R and g smooth in (y, μ, 1 ). Each term is clearly of order O(e−  ), if δ is chosen as in the statement of the Lemma. The same order estimate is valid for each partial derivative, so smoothness and flatness follow. 

Lemma 11.6 Let ω = z1a1 · · · znan lnb1 z1 · · · lnbn zn be an admissible monomial in z = (z1 , . . . , zn ). We assume that a1 > 0, . . . , an > 0. Let D be a compact neighborhood of the origin in the quadrant Q = {z1 ≥ 0, . . . , zn ≥ 0}. We also consider a compact neighborhood P of 0 ∈ R (with coordinate y). Let ∞ N(z, y), M(z, y) be admissible functions, i.e. N, M ∈ C∞ that D (y, ). We assume  is smooth N > 0 on D × P . Then the function F (z, y) = M(z, y) exp − N(z,y) ω on D × P and flat along ∂D × P , where ∂D = ∂Q ∩ D is the boundary of D, i.e. ∂D = D ∩ (∪i {zi = 0}).

Proof Of course, the function F is smooth on D ∩ {z1 > 0, . . . , zn > 0}. Then, to prove the claim, it suffices to prove that each partial derivative ∂a F is C0 -flat along ∂D × P , i.e. is of order O((i zi ) ) for any ∈ N, where i zi = z1 · · · zn . ∞ As it is seen in Chap. 11, the space of admissible functions C∞ D (y,  ), which is not invariant by the usual partial derivations, is invariant under derivations by the linear operators ∇i = zi ∂z∂ i and also by the partial derivations in the coordinates

of y. As these basic derivations commute, we can replace each derivation ∂z∂ i by the derivation ∇i , to associate with each multi-index a in the coordinates of (z, y), a derivation operator ∇a . It is easy to see that the flatness of F along ∂D × P is then given by the C0 -flatness along ∂D × P of each function ∇a F, for any multi-index a. A particular case of a general formula established in Chap. 11 is that ∇i ω = (αi + βi lnβi −1 zi )ω. Using iteratively this formula and the invariance of ∞ C∞ D (y,  ) that we have recalled above, we obtain that ∇a F =

 N(z, y)  Ma exp − , ω ω

(11.7)

where Ma is an admissible function. Write A = sup{a1 , . . . , an }) > 0 and B = inf{a1 , . . . , an }) > 0. Consider any δ > 0 small enough, chosen such that B −δ > 0. There exist positive constants C− = C− (ω, δ) > 0, C+ = C+ (ω, δ) > 0 such that C− (i zi )A+δ ≤ ω(z) ≤ C+ (i zi )B−δ ,

206

11 Smooth Functions on Admissible Monomials and More

for z ∈ D (we need constant δ > 0 if ω contains some logarithmic terms). Let us write G the Sup-norm of a continuous function G on D × P . As D × P is compact, we have that 1/N > 0. Using these different estimates in (11.7), we have  −1 ∇a F  ≤ Ma C− (i zi )−(A+δ) exp −

 1/N . C+ (i zi )B−δ

(11.8)

As the functions ξ → ξ − exp[− Kξ ] are bounded on R+ for any ∈ N and any K > 0, the inequality (11.8) implies that ∇a F  = O((i zi )s ) for any s ∈ N. This means that ∇a F is C0 -flat along ∂D. 

Lemma 11.7 Let I˜(y, μ, ) be defined as in Definition 11.9 and let g(y, ˜ μ, ) also be smooth in (y, μ, 1 ), with the same domain of definition. Moreover we assume that g(y, ˜ μ, 0) > 0 for any (y, μ) ∈ W . Then, for 0 > 0 small enough: g(y, ˜ μ, )e

I˜(y,μ,) 

=e

I˜(y,μ,)+ H˜ (y,μ,1 ) 

,

where H˜ is smooth in (y, μ, 1 ). If ωmin is the monomial of minimal order in 1 , we can write g(y, ˜ μ, )e

I (y,μ)+O(ωmin ) 

=e

I (y,μ)+O(ωmin ) 

,

for remainders O(ωmin ) smooth in (y, μ, 1 ), i.e. that g˜ is absorbed by the exponential term. Proof As g(y, ˜ μ, 0) > 0 for any (y, μ) ∈ W , the function ln g˜ is well defined in the same domain as g˜ and is smooth in (y, μ, 1 )). Then, we have the above formula with H˜ = ln g. ˜ The second formulation is a direct consequence of Remark 11.5.  Lemma 11.8 Let I˜1 (y, μ, ), . . . , I˜k (y, μ, ) be functions defined as in Definition 11.9, on the same domain W × [0, 0 ]. Assume that I1 (y, μ) > Ij (y, μ) for j = 2, . . . , k and all (y, μ) ∈ W . Then, for 0 > 0 small enough: I˜1

I˜k

I˜2

e  ± e  ± ··· ± e  = e

I˜1 +∞ 

,

where ∞ (y, μ, ) is smooth and -flat of order O(e−  ) for some δ > 0. δ

Proof We can write I˜1

e  ±e

I˜2 

I˜k

± ··· ± e  = e

I˜1 



1±e

I˜2 −I˜1 

± ··· ± e

I˜k −I˜1 

 .

If 0 > 0 is small enough, there exists a δ > 0 such that I˜j (y, μ, ) − I˜1 (y, μ, ) < −δ, for any (y, μ, ) and j = 2, . . . , k. As a consequence the function H˜ =

11.5 Functions of Exponentially Flat Type

207

I˜k −I˜1

I˜2 −I˜1

±e  ± · · · ± e  in the bracket is smooth and -flat of order O(e−  ), by Lemma 11.5. Now, if 0 > 0 is small enough, we have that 1 + H˜ (y, μ, ) > 0, for any (y, μ, ), and we can rewrite the expression: δ

I˜1

I˜2

e  ± e  ± ···± e

I˜k 

=e

I˜1 + ln(1+H˜ ) 

.

δ As the function ∞ =  ln(1 + H˜ ) is smooth and -flat of order O(e−  ), we have finished the proof. 

Remark 11.6 The result and the proof of Lemma 11.8 are rather trivial because one of the functions Ij is strictly larger than the other. The situation may be much more delicate if it is not the case. In Part III we will see for instance how to manage (at I˜1

I˜2

least partially!) the situation for a difference e  −e  when for some value (y0 , μ0 ) we have the equality I1 (y0 , μ0 ) = I2 (y0 , μ0 ). For the next property we have to restrict to families of diffeomorphisms: Lemma 11.9 Let I˜(y, μ, ) be defined as in Definition 11.9 and so that 0 for any (y, μ) ∈ W . Then, for 0 > 0 small enough:

∂I ∂y (y, μ)

=

 ∂I  I˜(y,μ,)− ln + H˜ (y,μ,) ∂  I˜/   (y, μ, ) = sign e e , ∂y ∂y where H˜ is smooth in (y, μ, 1 ). 1 Let ωmin be the monomial of minimal order in 1 and ωmin the monomial of minimal order in 1 ∪ { ln } (in the order of flatness). We can say in short that 1 )  ∂I  I +O(ωmin ∂  I +O(ωmin )    = sign e e , ∂y ∂y

1 ) smooth in for a remainder O(ωmin) smooth in (y, μ, 1 ) and a remainder O(ωmin (y, μ, 1 ∪ { ln }).  I˜  ˜ ˜ I˜ ∂ I˜ I ∂ e . We can apply Lemma 11.7 to | ∂∂yI |e  . This e  = 1 ∂y Proof We have that ∂y

gives the result, taking into account that

1 

=e

− ln  



.

Remarks 11.7 1. In the proof of Lemma 11.9 we have used the possibility to “absorb” the term I˜ 

1 

inside the exponential. We can easily generalize this idea to  k e for any k ∈ Z: I˜

I +O(ω1 ) min

1 ,  , with a remainder smooth in (y, μ, 1 ∪ { ln }) and ωmin k e  = e the monomial of minimal order in 1 ∪ { ln }.

208

11 Smooth Functions on Admissible Monomials and More

2. We can iterate Lemma 11.9. After one step of derivation, the set of monomials is stabilized at the set 1 ∪ { ln } and we have for any k that 1 )  ∂I k I +O(ωmin ∂ k  I +O(ωmin )     e = sign e . ∂y k ∂y

This means that, for any  > 0, any diffeomorphism y → I˜(y, μ, ) < 0 u composed with the map u → e  becomes infinitely monotone, in the sense that the derivatives at any order have a definite sign (like the exponential map!). 

11.5.2 Coherence of Definition 11.9 A natural question is to ask if the expression (11.6) for a given family Tμ, is unique or not. The answer is no, even for a family of diffeomorphisms. Let us consider for I˜ instance any expression α + e  . As I , with I (y, μ) = Y˜ (y, μ, 0), is continuous and negative on the compact set σ × P , there exist reals K1 , K2 , such that K1 < I˜

I (y, μ) < K2 < 0, for any (y, μ). Take now any K < K1 . We can write α + e  = I˜ K K α − e  + e  + e  . If 0 is small enough, we have that K < I˜(y, μ, ) for any K

I˜

(y, μ, ), and then, by Lemma 11.8 we know that e  + e  = e smooth function of order O(e I˜

I˜1

−

α + e  = α1 + e  ,

|K1 −K| 

I˜+∞ 

, with ∞ a

). We have obtained a second expression

with α1 = α − e  and I˜1 = I˜ + ∞ . K

As I˜1 (y, μ, 0) = I (y, μ), this second expression is permitted. It is also interesting to take a K such that K2 < K < 0. In this case we have a third expression with K again α2 = α − e  but with I˜2 = K + ∞ , where ∞ a smooth function of |K2 −K| order O(e−  ). As we now have I˜2 (y, μ, 0) ≡ K < 0, this third expression is again permitted, and it shows that in general even the function I is not intrinsically defined. Nevertheless, the following lemma shows that the function I is uniquely defined in an expression (11.6) for a given family of diffeomorphisms: I˜1 (y,μ,)

for a family Lemma 11.10 Consider any expression Tμ, (y) = α(μ, ) ± e  of diffeomorphisms of exponentially flat type, as in Definition 11.9. Then

∂

 

α(μ, 0) = Tμ,0 (y) and I (y, μ) = lim  ln Tμ, . →0 ∂y As a consequence α(μ, 0) and I (y, μ) do not depend on the choice of the expression (11.6).

11.5 Functions of Exponentially Flat Type

209

J˜

∂

Proof The first claim is trivial. Next, Lemma 11.9 says that ∂y Tμ, = e  , for a J˜ ˜ that verifies  J (y, μ, 0) = I (y, μ). It follows from this that we can write: I (y, μ) =

∂



lim→0  ln ∂y Tμ, (y) . Changing the coordinate y or z is the same as to compose on the right or on the left by a smooth (μ, )-family of diffeomorphisms. It is this last point of view that we choose in the following lemma, in order to prove that the notion of a “family of functions of exponentially flat type” does not depend on the choice of coordinates: I˜

Lemma 11.11 Let z = Tμ, (y) = α ± e  be a family of functions of exponentially flat type as in Definition 11.9, with α smooth in (μ, 0 ) and I˜ smooth in (y, μ, 1 ), with 0 ⊂ 1 . Let y = μ, (Y ) and Z = μ, (z) be two smooth families of diffeomorphisms, preserving the orientation and defined on compact domains, such that I (μ,0 (Y ), μ) is well defined for all (Y, μ) and μ,0 (α(μ, 0)) is well defined for all μ. Then the compositions Tμ, ◦ μ, and μ, ◦ Tμ, are well defined for 0 small enough and are a family of functions of exponentially flat type on the monomial sets 0 , 1 . More precisely, with ωmin the monomial of minimal order of 1 : I (

(Y ),μ)+O(ω

)

μ,0 min with O(ωmin ) smooth in 1. Tμ, ◦ μ, (Y ) = α(μ, ) ± exp  (Y, μ, 1 ). )+O() 2. μ, ◦ Tμ, (y) = μ, (α(μ, )) ± exp I +O(ωmin , with O(ωmin ) + O()  smooth in (y, μ, 1 ); the term μ, (α(μ, )) is smooth in (μ, 0 ).

Proof Let us compute the two compositions. For 0 > 0 small enough, we can write: (1) I˜(μ, , μ, ) = I (μ,0 (Y ), μ) + O(ωmin ) and then Tμ, ◦ μ, (Y ) = α ± e

I (μ,0 (Y ),μ)+O(ωmin ) 

with O(ωmin ) smooth in (y, μ, 1 ). (2) There exists a smooth family of functions in 2 variables 1μ, (u, v) such that we can write: μ, (u + v) = μ, (u) + 1μ, (u, v)v, with 1μ, (u, v) > 0 if (u, μ) ∈ W and v is small enough (by the Mean Value Theorem, the d function 1μ, is equal to dy μ, (θ ) for some θ ∈ [u, u + v] and then is strictly positive). Then, for 0 small enough, we can write: μ, (Tμ, (y)) = μ, (α)± I˜

I˜

I˜

1μ, (α, ±e  )e  . Recall that e  is a smooth -flat function by Lemma 11.5. As the monomial  is supposed to be smooth in 0 (see Remark 11.5), I˜

1μ, (α, ±e  ) is a smooth and strictly positive function in (y, μ, 0 ). Then, I˜

the function H˜ (y, μ, ) = ln |1μ, (α, ±e  )| is well defined and also smooth in (y, μ, 0 ), and as such in (y, μ, 1 ). We obtain μ, ◦ Tμ, (y) = μ, (α(μ, )) ± e

I˜+ H˜ 

210

11 Smooth Functions on Admissible Monomials and More

and hence μ, ◦ Tμ, (y) = μ, (α(μ, )) ± e

I +O(ωmin )+O() 

,

where the term O(ωmin ) + O() is smooth in (y, μ, 1 ).



11.5.3 Composition of Families of Diffeomorphisms of Exponentially Flat Type We consider two families of diffeomorphisms of exponentially flat type, defined, respectively, on W1 = σ1 × P and W2 = σ2 × P for  ∈ [0, 0 ], 0 > 0 small enough: I˜1 (y,μ,) 

,

I˜2 (w,μ,) 

,

1 Tμ, : w = α1 (μ, ) + θ1 e

and 2 Tμ, : z = α2 (μ, ) + θ2 e

with θ1 , θ2 = ±1. We assume that: 1 is defined on the monomial sets 1 , 1 and that T 2 is defined on the 1. Tμ, μ, 0 1 monomial sets 20 , 21 . 2. I˜1 (y, μ, ) = I1 (y, μ) + O(ω1 ) and I˜1 (y, μ, ) = I1 (y, μ) + O(ω2 ) with ω1 smooth in 11 and ω2 smooth in 21 (i.e. in the monoid generated, respectively, by 11 and 21 ). 2 ◦T1 3. α1 (μ, 0) ∈ σ2 for all μ ∈ P , so that we can consider the composition Tμ, μ, for 0 small enough.

We want to prove that this composition is also a family of diffeomorphisms of exponentially flat type: 1 (y) and T 2 (w) be as above, i.e. verifying conditions Proposition 11.3 Let Tμ, μ, (1),(2), and (3). Then, if ωmin = min{ω1 , ω2 ,  ln }): 2 1 Tμ, ◦ Tμ, : z = α(μ, ) + θ1 θ2 sign

 ∂I  I1 (y,μ)+I2 (α1 (μ,0),μ)+O(ωmin ) 2  e , ∂w

I˜2 (α1 (μ,),μ,)

 with the term α(μ, ) := α2 (μ, ) + θ2 e 1 2 O(ωmin ) smooth in (y, μ, 1 ∪ 1 ∪ { ln }).

smooth in (μ, 20 ) and the term

11.5 Functions of Exponentially Flat Type

211

Proof We have that 2 1 ◦ Tμ, : z = α2 (μ, ) + θ2 e Tμ,

I˜2 (T 1 ,μ,) 

.

Like in the proof of Lemma 11.11, there exists a family of functions I˜21 (u, v, μ, ), now smooth in (u, v, μ, 12 ), such that we can write I˜2 (u + v, μ, ) = I˜2 (u, μ, ) + I˜21 (u, v, μ, )v, 2 with I˜21 (u, v, μ, ) = 0 (having the sign of ∂I ∂w ) if (u, μ) ∈ W and (v, ) ∼ (0, 0). 1 The function I˜2 (Tμ, , μ, ) can be written as

    I˜1 (y,μ,) I˜1 (y,μ,) 1 I˜2 (Tμ, (y), μ, ) = I˜2 α1 , μ,  + θ1 I˜21 α1 , θ1 e  , μ,  e  .

(11.9)

Let us consider the function  I˜1 (y,μ,) I˜1 (y,μ,) 1  K(y, μ, ) = θ1 I˜21 α1 (μ, ), θ1 e  , μ,  e  .  By Lemmas 11.5 and 11.7, this function is smooth and flat at  = 0. Introducing ζ the strictly positive analytic function ξ(ζ ) = e ζ−1 , we have that eK = 1 + Kξ(K). This gives    I˜1 (y,μ,) I˜1 (y,μ,) 1  eK(y,μ,) = 1 + θ1 I˜21 α1 (μ, ), θ1 e  , μ,  ξ K(y, μ, ) e  . 

   

I˜1 (y,μ,) Now, the function H (y, μ, ) =

I˜21 α1 (μ, ), θ1 e  , μ, 

ξ K(y, μ, ) , is strictly positive and smooth in (y, μ, 21 ). Then, we can rewrite eK(y,μ,) = 1 + θ1 sign

 ∂I  2

∂w

e

J˜(y,μ,) 

,

(11.10)

with J˜ = I˜1 −  ln  +  ln H,

(11.11)

with ln H smooth in (y, μ, 21 ). From (11.9), (11.10) we obtain that e

1 ,μ,) I˜2 (Tμ, 

=e

I˜2 (α1 (μ,),μ,) 

eK = e

I˜2 (α1 (μ,),μ,) 

  ∂I  J˜  2 1 + θ1 sign e ∂w

212

11 Smooth Functions on Admissible Monomials and More

and then, using (11.11): e

1 ,μ,) I˜2 (Tμ, 

=e

I˜2 (α1 (μ,),μ,) 

+ θ1 sign

 ∂I  I˜1 (y,μ,)+I˜2 (α1 (μ,),μ,)− ln + ln H 2  e . ∂w

As I˜2 (α1 (μ, ), μ, ) is smooth in (μ, 10 ∪ 21 ) and as I˜1 (y, μ, ) + I˜2 (α1 (μ, ), μ, ) −  ln  +  ln H = I1 (y, μ) + I2 (α1 (μ, 0), μ) + O(ω1 ) + O(ω2 ) + O( ln ), we finally obtain the desired expression 2 1 Tμ, ◦ Tμ, : z = α(μ, ) + θ1 θ2 sign I˜2 (α1 (μ,),μ,)

 ∂I  I1 (y,μ)+I2 (α1 (μ,0),μ)+O(ωmin ) 2  , e ∂w

 where α(μ, ) = α2 (μ, ) + θ2 e , as an -flat perturbation of α2 (μ, ), is smooth in (μ, 20 ) and the term O(ωmin ) is smooth in (y, μ, 11 ∪21 ∪{ ln }). 

Part III

Results and Open Problems

Chapter 12

Local Transition Maps

In this chapter we present results about transition maps along the flow of a slow–fast system X,λ on a surface M, and also for families of vector fields X¯ λ obtained from X,λ by blowing up the system at contact points. The blown-up family of vector fields X¯ λ is defined on a three-dimensional space, the blown-up space. By including the trivial equation {˙ = 0}, we can also consider X,λ as a λ-family defined on the three-dimensional space M × R+ (with  ∈ R+ ). We will hence always consider the phase space to be three-dimensional. We will consider transition near regular points of the slow dynamics and near singular points located on the singular part of the blow-up locus. These singular points may be hyperbolic or semi-hyperbolic. Even if it seems paradoxal, the computation near hyperbolic points is more involved than computations near semihyperbolic points. In fact, due to possible resonances, normal forms obtained in Chap. 10 are more complicated for hyperbolic points than for semi-hyperbolic ones. Based on the normal forms, we will present simple expressions for transitions. The advantage of using normal form coordinates is to simplify considerably the computation of transition maps in comparison to a direct study in a general coordinate system. After computing the transition maps in normal form coordinates, it will be easy to return in initial coordinates. We will study transitions near the singular points introduced in Chap. 10, which are more general than the points obtained here by blowing up and which are used in Part III (for instance, for the case of hyperbolic points, we do not restrict to the dimension 3). As transitions in case of points obtained by blowing up have a simplified expression, which will be useful in Part III, we will also deduce explicit expressions for transitions near these particular points. To obtain a more precise control on the properties of differentiability, we will not just consider transitions between two sections transverse to the flow, but we will also compute the map induced by the trajectories starting at points m in a well-chosen open set W in phase space and ending on a section D, transverse to the flow (at least

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_12

215

216

12 Local Transition Maps

when  = 0). From it, it will be easy to obtain expression for transitions between section  chosen in W and the section D. All the transitions are local but we will also consider global transitions, i.e. along arbitrarily long arcs of regular points in the critical curve. This is explained in the first section. In Sect. 12.4 we will consider transitions at a jump point. To study it we will have to compose transitions at different singular points with different normal forms, obtained in the blow-up of the jump point. In Sect. 12.5 we will compose several transitions at jump points to obtain a formula for the transition along an attracting sequence passing through several jump points, as defined in Chap. 4. In the last section we will extend this formula to a Hopf attracting sequence where the last jump point is replaced by a turning point and where the exit section is chosen after blowing up the turning point. Transitions along (Hopf) attracting sequences are used all along Part III. In general, we will have to adopt different notations for coordinates in each particular case. Of course, these transitions degenerate for  = 0 and for this limit value the section may be non-transverse to the vector field. Moreover we will only consider transitions in a direction where the flow is attracting: in applications we will replace, if necessary, X,λ by −X,λ . The advantage in considering the flow in an attracting direction is that the transition can be defined on a whole open set W , neighborhood of a point on the critical curve, once continuously extended by a constant value for  = 0. The image of a section  ⊂ W in the exit section D reduces to a narrow tongue of exponential size: exp − K , for some K > 0. We will encounter two types of transition, rather similar but with a slightly different expression, depending on whether that the starting section is exterior to the critical curve or is cutting it. The first type is more common for applications in Part III. An example is given by (12.3). This type of transition is a family of diffeomorphisms of exponentially flat type as introduced in Chap. 11, Sect. 11.5, and we will make extensive use of such families in this chapter. The second type corresponds to a quasi-linear map. An example is given by (12.2). Notation 3 In the text, if a ∈ R \ {0}, we will write sign(a) to denote the number ±1 which has the same sign as a. 

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics We can suppose that the critical curve S is independent of the parameter λ, and we write S instead of Sλ . We consider a segment γ ⊂ S \ Cλ ∪ λ of regular points, i.e. a compact interval not containing contact points or singular points of the slow dynamics (for any λ). We suppose that γ is attracting. In this whole section, by transverse section we will mean a section for the flow of X,λ , transverse for  > 0 but possibly tangent to X0,λ .

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics

217

12.1.1 Transition in a Normal Form Chart It follows from Theorem 10.1 that near γ there exist smooth local coordinates (z, u) defined on a neighborhood T = ]u¯ 0 , u¯ 1 [ × [−¯z, +¯z] of γ ⊂ ]u¯ 0 , u¯ 1 [ × {0}, in which X,λ has the normal form up to a smooth time rescaling; 

z˙ = −z u˙ = ,

where  ∈ [0, 0 ], with 0 > 0 small enough. We assume that γ = [u0 , u1 ] × {0} (with [u0 , u1 ] ⊂ ]u¯ 0 , u¯ 1 [). Choose a neighborhood W of (u0 , 0) in T , such that W ∩ {u1 } × [−¯z, +¯z] = ∅. We take τ0 = {u1 } × [−¯z, +¯z]. It is a transverse section for  > 0. We parameterize τ0 by z. Integrating the above equation, we obtain a transition map (see Fig. 12.1) Tλ (u, z, ) = (Zλ (u, z, ), ) along the flow of X,λ , from W × ]0, 0 ] to T0 = τ0 × ]0, 0 ]. It is given by   u1 − u , Zλ (u, z, ) = z exp −  where u, z are normal form coordinates.

12.1.1.1 Changing the Exit Section For the composition of transitions, it is convenient to change the section T0 by any section, even if we continue working in the normal form coordinates (u, z). We therefore consider an arbitrary section T through the point (u1 , 0, 0). We assume that it is a graph u = u1 + 1 (z, , λ) for some smooth function 1 . This means that the section is tangent to the limit vector field X0,λ and then is cutting { = 0} T0 ε W

(u, z, ε)

u z

u ¯0

u0

Tλ γ

τ0 (Zλ , ε) S u1 u ¯1 T

Fig. 12.1 Normally hyperbolic passage in normal form coordinates

218

12 Local Transition Maps

T ε

τ0 (Zλ , ε)) S u1 u ¯1

W u

u ¯0

Tλ

(u, z, ε)

u0

γ

z T Fig. 12.2 Normally hyperbolic passage in normal form coordinates, mapping toward a curved exit section

along the line {u = u1 }; the section T may depend on λ. If W is a small enough neighborhood of (u0 , 0, 0), a transition map is defined from any point (z, u) ∈ W to the section T, see Fig. 12.2. Let Zλ (u, z, ) be the corresponding z-component of the transition map. To compute it we first look to the transition time t = t (u, z, , λ) to arrive at the section T; this time is given by the following equation: u + t = u1 + 1 (ze−t , , λ). If 1 ≡ 0 this time is equal to perturbation term

u1 −u  .

Then it is natural to write the equation for the

τ (u, z, , λ) = t (u, z, , λ) −

u1 − u . 

This term is solution of the equation τ = 1 (ze−

u1 −u 

e−τ , , λ).

u1 −u

As the function e−  is smooth in (u, ) and flat in  it can be solved, using the Implicit function theorem, to give a smooth function  u1 −u  . τ (u, z, , λ) = 1 (0, , λ) + O ze−  The transition function Zλ is equal to ze−t (u,z,,λ). This gives   u1 −u  ⎞ u1 − u +  1 (0, , λ) + O ze−  ⎠. Zλ (u, z, ) = z exp ⎝−  ⎛

(12.1)

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics

219

We have proved the following result: Proposition 12.1 Let z, u be normal form coordinates. Let W and T be a neighborhood and a section as above. Then, the z-component of the transition map from W to T is given by   u1 − u + O() , Zλ (u, z, ) = z exp −  where the O() is a smooth function in (u, z, , λ) as described in (12.1).

12.1.1.2 Restricting to a Starting Section Transverse to the Critical Curve We can restrict the above transition to a section  ⊂ W through the point (u0 , 0, 0), given as a graph of a function u = u0 + 0 (z, , λ) (so that the section is tangent to X0,λ along { = 0}, see Fig. 12.3). The transition time t (z, , λ) in going from  to T can be estimated as above. One obtains that  u1 −u0  u1 − u0 . + (1 − 0 )(0, , λ) + O ze−  t (z, , λ) =  This gives the following result for the transition function Zλ (z, λ) from  to T: Proposition 12.2 Let z, u be normal form coordinates. Let  and T be two sections through (u0 , 0, 0) and (u1 , 0, 0), given as graph of the functions u0 + 0 (z, , λ) and u1 + 1 (z, , λ), respectively. Then, the z-component of the transition map from  to T is given by   u1 − u0 + O() Zλ (z, ) = z exp − , 

Σ

T

ε

τ0 u

u0

γ

u1

z Fig. 12.3 Normally hyperbolic passage in normal form coordinates, between two curved sections

220

12 Local Transition Maps

where O() is a smooth function in (z, , λ), of the form   u1 −u0  .  (1 − 0 )(0, , λ) + O ze−  Remark 12.1 The term −(u1 −u) in Proposition 12.1 is the slow divergence integral I ([u, u1 ] , λ) from the point (u, 0) to the point (u1 , 0) on the critical curve, and the term −(u1 − u0 ) in Proposition 12.2 is the slow divergence integral I (γ , λ) along the arc γ , a notion that is invariant for C∞ -equivalence (see Lemma 5.2). ˜ Then, in Proposition 12.1, we can write Zλ (u, z, ) = zeI (u,z,,λ)/ , where I˜ is smooth and I˜(u, z, 0, λ) = I ([u, u1 ] , λ) and in Proposition 12.2, we can write ˜ Zλ (z, ) = zeI (z,,λ)/ , where I˜ is smooth and I˜(z, 0, λ) = I (γ , λ).

12.1.2 General Expressions for Regular Transitions In the last section, we have used normal form coordinates for smooth equivalence in order to parameterize the domains and sections. We want to show now that it is possible to use any coordinate system in order to obtain similar expressions for transitions. We consider as above an attracting arc γ of regular points on the critical curve S, starting at m0 and finishing at m1 . Using Chap. 9, we choose a smooth family of center manifolds Wλ intersecting S along a neighborhood of γ in S. We now consider any smooth local (u, z, ) defined along the attracting arc γ . We may assume that: 1. z is a smooth coordinate transverse to Wλ such that Wλ is contained in {z = 0}, 2. γ is parameterized by u and the direction of the slow dynamic corresponds to increasing values of u. We write u0 = u(m0 ) and u1 = u(m1 ). We will first consider, as in Proposition 12.2 a transition between two sections transverse to the critical curve. Next we will give the expression for a transition starting at an exterior section, i.e. a section transverse to the fast vector fields.

12.1.2.1 Transition Between Two Sections Transverse to the Critical Curve Let  and T be sections transverse to S, through the points m0 and m1 , respectively, and tangent to the fast vector fields for  = 0. We assume that a transition map is defined from  to T, when  ∈ ]0, 0 ], see Fig. 12.4. The two sections are parameterized by (z, ). Let Zλ (z, ) be the z-component of the transition. Then Zλ has an expression similar to the one given in Remark 12.1:

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics

221

Σ T m0 γ

m1

Fig. 12.4 Normally hyperbolic passage in arbitrary coordinates, between two curved sections

Theorem 12.1 We consider a smooth family of center manifolds Wλ along a neighborhood of γ , any smooth coordinates (z, u) such that the center manifolds are contained in {z = 0}, and two sections , T through the extremities of γ as above. Then Zλ (z, ) = z exp

I˜(z, , λ) , 

(12.2)

where I˜ is smooth and I˜(z, 0, λ) = I (γ , λ), the slow divergence integral along γ . Proof If Z is a normal form coordinate along Wλ (given in Chap. 10) it follows from Proposition 12.2 that the transition function Zλ has precisely the desired I˜0 (Z,,λ) expression: Z → Ze  , where I˜0 is smooth and such that I˜0 (z, 0, λ) = I (γ , λ) (see Remark 12.1). Now, on each section, we can pass from the coordinate Z to the coordinate z by a smooth family of diffeomorphisms Z = H,λ (z) such that H,λ (0) = 0 (we use the same name H,λ on each section). We can write that Z = H,λ (z) = a(, λ)z(1 + O(z)) with a(, λ) > 0. For the inverse map, we have 1 that z = G,λ (Z) = a(,λ) Z(1 + O(Z)). In the variable z the transition function is   I˜0 (H,λ (z),,λ)  z → F,λ (z) = G,λ H,λ (z)e . Clearly we have that I˜0 (H,λ (z)) is smooth and equal to I (γ , λ) + O(). Also, as a(, λ) is strictly positive and smooth, ln[a(, λ)z(1 + O(z))] is a well-defined smooth function if |z| is small enough and we can write H,λ (z) = zeln[a(,λ)z(1+O(z))]. Using this, we obtain H,λ (z) exp

I (γ , λ) + O() I˜0 (H,λ (z), , λ) = z exp ,  

222

12 Local Transition Maps

where the term O() is a smooth function of (z, , λ). Next, we have that  I (γ ,λ)+O()   = F,λ (z) = G,λ ze

  I (γ ,λ)+O()  I (γ ,λ)+O() 1   1 + O ze , ze a(, λ)

I (γ ,λ)+O()

 where the terms O(), O(ze ) are smooth.   I (γ ,λ)+O() I (γ ,λ)+O()  1 ) , we finally obtain  ) = e− ln a(,λ)+O(ze As a(,λ) 1 + O(ze

F,λ (z) = ze

I (γ ,λ)+O() 

, 

where the term O() is a smooth function of (z, , λ).

12.1.2.2 Transition from An Exterior Section As above, T is an arbitrary section transverse to the critical curve, through the point m1 , and tangent to the fast vector fields for  = 0. We consider now an exterior section  transverse to X,λ , through a point n0 ∈ S, contained in an orbit of the fast dynamics, with ω-limit the point m0 (there are two such fast orbits). If  is small enough, there is a transition map along the flow of X,λ , defined from  to T, for  > 0. We assume that  is parameterized by (v, ) and that T is parameterized by (z, ), where v and z are arbitrary smooth coordinates. If  is small enough, the fast orbit through (v, 0) ∈  has an ω-limit point u = (v, λ) ∈ S. Of course, we have that (v(n0 ), λ) = u0 = u(m0 ) and  is a smooth map. The section  is located on the side {z > 0} or on the side {z < 0} of the critical curve. See Fig. 12.5.

Σ (v, ε) v

v(n0 )

T

u = π(v, λ) u(m0 )

(Zλ , ε)

γ

m1

Fig. 12.5 Normally hyperbolic passage in arbitrary coordinates, starting from a section transverse to the fast flow

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics

223

Let Zλ (v, ) be the z-component of the transition map from  to T, defined for  > 0. This (, λ)-family is a smooth family of exponentially flat type as defined in Chap. 11 (Sect. 11.5): Theorem 12.2 We consider a smooth family of center manifolds Wλ along a neighborhood of γ and a coordinate z such that the center manifolds are contained in {z = 0}. We also consider an exterior section  and a section T transverse to S as above. We assume that  is a sufficiently small section through the point n0 , such that the transition map is well defined. The section  is parameterized by (v, ) and T by (z, ). Then Zλ (v, ) = θ exp

I˜(v, , λ) , 

(12.3)

where I˜ is smooth and I˜(v, 0, λ) = I ([(v, λ), u1 ] , λ), the slow divergence integral along the arc from (v, λ) to u1 (u1 = u(m1 )). The sign of the Index θ is equal to +1 if  is located on the side {z > 0} and equal to −1 if  is located on the other side. Proof We consider a section 0 contained in the neighborhood W given by Proposition 12.1 with normal form coordinates (u, Z) associated to the family of center manifolds Wλ . We assume that 0 is chosen in {Z = Z0 } and such that a smooth transition u = h(v, , λ) is defined from  to 0 along the flow of X,λ . If we parameterize S by the normal coordinate u, we have clearly that (v, λ) = h(v, 0, λ).

(12.4) I (u,u1 )+(Z,,λ)

 If the transition from W to T is given by (u, Z) → Ze , for a smooth function , then the transition family from  to T is given by (v, ) → (Z˜ λ (v, ), ) with

I (h(v, , λ), u1 ) + (Z0, , λ) , Z˜ λ (v, ) = sign(Z0 )|Z0 | exp  where sign(Z0 ) = θ , the coefficient introduced in the statement. Using (12.4) we get I (h(v, , λ), u1 ) = I ([(v, λ), m1 ] , λ) + O(), where the O()-term is I˜0 (v,,λ) smooth. Writing |Z0 | = eln |Z0 | we obtain that Z˜ λ (v, ) = θ e  where I˜0 is ˜ smooth and I0 (v, 0, λ) = I ([(v, λ), m1 ] , λ). It remains to compose Z˜ λ with a smooth family of diffeomorphisms Z → z = h,λ (Z) = a(, λ)Z(1 + O(Z)).

224

12 Local Transition Maps

Recall that h,λ (0) = 0 because the family Wλ is contained in {z = 0} as well as in {Z = 0}. The transition is then given by the family    ˜ ˜ Zλ (v, ) = aeI/ 1 + O aeI / . ˜

˜

As in the proof of Theorem 12.1 the expression aeI0 / (1 + O(aeI0 / )) is easily ˜ transformed into eI (v,,λ)/ , where I˜ is a smooth function such that I˜(v, 0, λ) = I ([(v, λ), m1 ] , λ). 

This finishes the proof.

12.1.2.3 Remark Concerning the Choice of the Coordinate z The single constraint on the choice of the coordinate z is that the given smooth family of center manifolds Wλ is contained in {z = 0}. In particular, the orientation of z is not specified. It is interesting to verify that the statements of Theorems 12.1 and 12.2 are coherent with this remark: in Theorem 12.1, if we change the sign of z, we change at the same time the sign of Zλ in (12.2); in Theorem 12.2, if we change the sign of z, we change at the same time the sign of θ in (12.3).

12.1.3 Properties of Transitions Along Regular Arcs An important property of the formulas (12.2) and (12.3) obtained for the transition maps is that they behave well w.r.t. standard operations as derivation and composition. First, let us recall that if u is any smooth parameterization of S in a neighborhood of an attracting arc γ = [u0 , u1 ] of regular points, putting ∂I (u, λ) > 0 for u < u1 . I (u, λ) = I ([u, u1 ] , λ), we have that I (u, λ) < 0 and ∂u Lemma 12.1 The following holds: (1) Let Z,λ (z) be the transition function given in (12.2). If the section  is small enough, we can write ∂Z,λ I˜1 (z, , λ) = exp , ∂z  where I˜1 is smooth and I˜1 (z, 0, λ) = I (γ , λ). Z,λ is hence a smooth family of diffeomorphisms of exponentially flat type.

12.1 Transition Along an Arc of Regular Points of the Slow Dynamics

225

(2) If Z,λ (v) is the transition function given in (12.3), then   ∂Z,λ ∂ I˜2 (v, , λ) +  ln  = θ sign , exp ∂v ∂v  where I˜2 is smooth and I˜2 (v, 0, λ) = I ([(v, λ), u1 ] , λ). Also the derivative may hence be considered as a smooth family of diffeomorphisms of exponentially flat type. Proof (1) Let Z,λ (z) be given by (12.2). We get O() it implies

∂Z,λ ∂z (z, )

∂Z,λ ∂z (z, )

= (1 +

I˜ 

˜

z ∂ I˜ I  ∂z )e .

As

∂ I˜ ∂z

=

= (1 + O(z))e . If |z| is small enough, ln(1 + O(z)) ∂Z

is a well-defined smooth function and we can write ∂z,λ as in the statement, with I˜1 = I˜ +  ln(1 + O(z)). (2) As the map Z,λ (v) given by (12.3) is a smooth family of diffeomorphisms of exponentially flat type, the result follows from Lemma 11.9 of Sect. 11.5:   ∂Z,λ I˜2 (v, , λ) +  ln  ∂ = θ sign . [I ([(v, λ), u1 ] , λ)] exp ∂v ∂v  As the slow dynamics in the sense of increasing u, we have that then as

∂I ∂u

∂ ∂I ∂ ((v, λ), , λ)) (v, λ), [I ([(v, λ), u1 ] , λ)] = ∂v ∂u ∂v



∂ we have that sign ∂v [I ([(v, λ), u1 ] , λ)] = sign ∂ ∂v .

> 0 and



Remark 12.2 Let us consider the transition Zλ (z, ) given by (12.2), along the arc λ γ . It follows from Lemma 12.1 that:  ln ∂Z ∂z (z, ) → I (γ , λ) for  → 0. This proves an important property of the slow divergence integral given in Chap. 5. In Theorem 12.1, we can choose the segment γ arbitrarily large. This is in agreement with the fact that the type of maps as in (12.2) is invariant by composition: Lemma 12.2 Consider two functions as in (12.2): I1 (λ) + 1 (z1 , , λ) ;  I2 (λ) + 2 (z2 , , λ) 2 z3 = Z,λ (z2 ) = z2 exp ,  1 z2 = Z,λ (z1 ) = z1 exp

226

12 Local Transition Maps

where I1 , I2 , 1 , 2 are smooth and I1 < 0, I2 < 0. Assume that we can compose these functions. Then 2 1 Z,λ ◦ Z,λ (z1 ) = z1 exp

I1 (λ) + I2 (λ) + (z1 , , λ) , 

where  is smooth. Proof It is direct to see that 2 1 Z,λ ◦ Z,λ (z1 ) = z1 exp

I1 (λ) + I2 (λ) + 1 (Z 1 , , λ) . 

1 (z ) is Putting (z1 , , λ) = 1 (Z 1 , , λ), the result follows from the fact that Z,λ 1 smooth in (z1 , , λ). 

Remark 12.3 We can iterate the formula in Lemma 12.2 to a composition formed by an arbitrarily large number of functions of similar types. In a similar way, it is easy to prove that the composition of a function of type as in (12.3) followed by a finite number of functions of type as in (12.2) is a function of type as in (12.3).

12.2 Transition Near Semi-Hyperbolic Points We will work in the normal coordinates given in Theorem 10.3 in which the family of vector fields has the normal form X¯ ηN given by ⎧ ⎨ z˙ = −L(w, v, η)z w˙ = pwα v β · w ⎩ v˙ = −qwα v β · v.

(12.5)

Moreover, we can choose the normal form such that a given family of center manifolds is contained in {z = 0}. Let ν = wq v p be the first integral of (12.5). Put η = L(0, 0, η). Recall that −L < 0 is the z-eigenvalue at (0, 0, 0) of (12.5). We suppose that this normal form is defined in a neighborhood W0 of Q = (0, 0, 0) in the quadrant {(Z, w, v) | w ≥ 0, v ≥ 0}, of the coordinate space (Z, w, v). We choose in W0 a section T ⊂ {w = w1 > 0} and parameterize it by (Z˜ = Z, ν = q w1 v p ). If W is a small enough neighborhood of Q in W0 , the trajectory, of the vector field wα1v β X¯ ηN , through an arbitrary point (Z, w, v) ∈ W ∩ {w > 0} cuts the section T in a positive time equal to tw = − p1 ln ww1 . See Fig. 12.6. This defines a transition   Tη (Z, w, v) = ν = wq v p , Z˜ η (Z, w, v) .

12.2 Transition Near Semi-Hyperbolic Points

227

v

T W Tη

Q

w1

w

z Fig. 12.6 Transition map near semi-hyperbolic points

12.2.1 Equation for the Transition Component Z˜ We will now compute the component Z˜ η (Z, w, v). In W˜ 0 = W0 ∩ {w > 0, v > 0}, we consider the family of vector fields wα1v β X¯ ηN . To simplify the computation, we use the coordinate change: w = W p,

v = V q.

This change, which is not differentiable on {w = 0} ∪ {v = 0}, has the property to change the eigenvalues p, −q by 1, −1. For instance, w˙ = pW p−1 W˙ = pw = pW p and then W˙ = W . The first integral ν is equal to ν = wq v p = (W V )pq . In the coordinates (Z, W, V ), the Eq. (12.5) is replaced by ⎧ ,η) ⎪ Z˙ = −Lη G(W,V Z W pα V qβ 1 ¯N ⎨ ˙ X : W =W wα vβ η ⎪ ⎩ V˙ = −V ,

(12.6)

with G such that G(0, 0, η) = 1 and L(w, v, η) = Lη · G(W, V , η) (hence Lη := L(0, 0, η) > 0). The trajectory of wα1v β X¯ ηN = W pα1V qβ X¯ ηN , starting at an initial point (Z, W, V ) is given by W (t) = W et , V (t) = V e−t and Zη (t), solution of the one-dimensional linear differential equation: G(W (t), V (t), η) Z˙ η (t) = −Lη Zη (t). W (t)pα V (t)qβ

228

12 Local Transition Maps

A direct integration gives   Zη (t, (Z, W, V )) = Z exp −Lη

t 0

 G(W (τ ), V (τ ), η) dτ . W (τ )pα V (τ )qβ

p

We introduce W1 > 0 such that w1 = W1 . The time tw for w = W p is now given W in function of W by tW = − ln W . Then, the Z-component of the transition Tη is 1 given by   W ˜ , (Z, W, V ) . Zη (Z, W, V ) = Zη − ln W1 This equation can be written as  Z˜ η (Z, W, V ) = Z exp −

 Lη I (W, V , ν, η) , W pα V qβ

(12.7)

G(W (t), V (t), η)e(qβ−pα)t dt.

(12.8)

with  I (W, V , ν, η) =

− ln

W W1

0

Recall that G(0, 0, η) = 1. For simplicity in notation, we do not explicitly mention that I also depends on the choice of W1 .

12.2.2 A Simple Case To have an idea of the expression of Z˜ η , let us suppose for a moment that G is constant and equal to 1. In this simple case, we have that  ˜ Zη (Z, W, V ) = Z exp −

Lη pα W V qβ



− ln

W W1

 e

(qβ−pα)t

dt .

0

We see that there are two cases: The resonant case {qβ − pα = 0}.

β

α

In this case, as (W V )qβ = ν p = ν q , we have

  Lη W1 ˜ . Zη (Z, W, V ) = Z exp − β ln W νp

12.2 Transition Near Semi-Hyperbolic Points

The non-resonant case {qβ − pα = 0}. 

− ln

W W1

e

(qβ−pα)t

0

229

In this case,

1 dt = qβ − pα

#

W1 W

$

qβ−pα

−1 .

Depending on the sign of qβ − pα, we have that: (a) If qβ − pα < 0: 

Lη 1 Z˜ η (Z, W, V ) = Z exp − pα − qβ W pα V qβ

#

 1−

W W1

pα−qβ $

which can be written as 

#  pα−qβ $ 1 W L η pα−qβ Z˜ η (Z, W, V ) = Z exp − 1− . α V pα − qβ ν q W1 (b) If qβ − pα > 0: 

  Lη  qβ−pα 1 qβ−pα W1 −W . Z˜ η (Z, W, V ) = Z exp − qβ − pα ν βp

12.2.3 Preparing the Function G We now consider the general case of a function G which may be non-constant. The fact that G(X, Y, η) may depend on X and on Y may cause serious difficulties. We show how to use the first integral XY to overturn this problem. Lemma 12.3 Let G(X, Y, η) be a family of smooth functions defined in a neighborhood U of (X, Y, η0 ) = (0, 0, η0 ) with G(0, 0, η) = A(η). Then, we can write G(X, Y, η) = G1 (XY, X, η) + G2 (XY, Y, η), for two smooth families G1 , G2 defined on U , such that G1 (0, 0, η) = A(η) and G2 (XY, 0, η) ≡ 0. Proof To make the reading more agreeable, we exchange the roles of (X, Y ) and (x, y).

230

12 Local Transition Maps

(1) Study on formal series of function families. We first consider the formal series of G at (x, y) = (0, 0). !

ˆ G(x, y, η) =

aij (η)x i v j ,

i≥0,j ≥0

where the aij (η) are smooth functions defined on a same neighborhood M of η0 in the parameter space. We can partition the set of monomials Xi Y j into two sets, 1 = {x i v j | i ≥ j } and 2 = {x i v j | i < j }. The monomials in 1 are equal to (xy)j wi−j with i − j ≥ 0, and the monomials in 2 are equal to (xy)i v j −i with j − i > 0. Then, the sum of terms in 1 is a formal series ˆ 1 (xy, x, η) and the sum of terms in 2 is a formal series G ˆ 2 (xy, y, η). We G ˆ ˆ have that G1 (0, 0, η) = a00 (η) = A(η), G2 (xy, 0, η) ≡ 0 and ˆ 2 (xy, y, η). ˆ ˆ 1 (xy, x, η) + G G(x, y, η) = G Using Borel’s Theorem, we can find smooth functions G1 (xy, x, η) and ˆ 1 (xy, x, η) and G ˆ 2 (xy, y, η), G2 (xy, y, η) whose Taylor series are equal to G respectively, with the conditions G1 (0, 0, η) = A(η) and G2 (xy, 0, η) ≡ 0. We have that G(x, y, η) = G1 (xy, x, η) + G2 (xy, y, η) + Hη (x, y), where Hη is flat in (x, y) at any point (0, 0, η). (2) Study on flat function families. It remains to prove the result for such smooth family of flat functions Hη , supposed to be defined on W = {(x, y) | |x| ≤ A, |y| ≤ B} for some A, B > 0 to be chosen later. See Fig. 12.7. Given a δ > 0 small enough, a family of flat functions Hη can be split on X into a sum Hη = Hηx + Hηy , y W

¯ W

B

{y = −δx}

Y B

{y = δx}

A x

Ψ

Fig. 12.7 The set W in the proof of Lemma 12.3

X

12.2 Transition Near Semi-Hyperbolic Points

231

y where Hηx , Hη are smooth, Hηx (x, y) ≡ 0 for {(x, y) ∈ W | yx ≤ δ} and



y Hη (x, y) ≡ 0 for {(x, y) ∈ W | xy ≤ δ}. This claim can easily be proved as follows. We consider the polar coordinate map (r, θ ) = (r cos θ, r sin θ ) for (r, θ ) ∈ R+ × S 1 and the induced function H˜ η = Hη ◦ . Recall that the map Hη → H˜ η is an isomorphism between the space of families of flat functions on (x, y) at (0, 0) ∈ R2 and the space of flat families of functions on (r, θ ) along the circle {r = 0}. Let us suppose that δ < 1. Taking a partition of unity in a neighborhood of {r = 0}, we can split H˜ η in a sum H˜ η = H˜ ηx + H˜ ηy , y sin θ ˜y where H˜ ηx , H˜ η are smooth, H˜ ηx (x, y) ≡ 0 for | cos θ | ≤ δ and Hη (x, y) ≡ 0 for y θ x | cos sin θ | ≤ δ. The families of functions Hη , Hη are the corresponding ones through the map . We will now prove that there exist smooth function families Hη1 (X, Y ) and 2 Hη (X, Y ), defined in a neighborhood of (0, 0) in the (X, Y )-plane and flat at this point, such that on X:

Hηy (x, y) = Hη1 (xy, x) and Hηx (x, y) = Hη2 (xy, y). We will just consider the case of Hηx , as the treatment of the other case is completely similar. Let us notice that the map  : X = xy, Y = y is the projective blow-up map sending the line {x = a} of the (x, y)-plane on the line {X = aV } of slope a of the (X, Y )-plane. The image (W) is the conic sector ¯ = {|X| ≤ A|Y |, |Y | ≤ B}. W If Hη1 (X, Y ) is the family of functions we are looking for, we see that the corresponding blown-up family of functions Hη1 ◦  is the family of functions Hηx . ¯ by H 1 (X, Y ) = Hηy ( X , Y ). Then, the function Hη1 is defined on the conic sector W η Y ¯ and flat at (0, 0). In fact its support is contained This function is smooth on W

¯ |

Y 2

≥ δ}. Let us assume that ¯ δ = {(X, Y ) ∈ W inside the quadratic sector W X ¯ ¯ B < δA. Under this condition

X the sector Wδ is in interior of W, in the sense

¯ that if (X, Y ) ∈ Wδ then Y < A. As a consequence the family of functions ¯ on a whole neighborhood of Hη1 (X, Y ) can be extended smoothly by 0 outside W (X, Y ) = (0, 0). In conclusion, we have found a family of functions Hη1 smooth on y a neighborhood of (0, 0), flat at (0, 0), such that Hη (x, y) = Hη1 (xy, x). 

232

12 Local Transition Maps

12.2.4 Estimates for the Integral I in (12.8) Thanks to Lemma 12.3 and to the linearity of (12.8), it suffices to consider the functions 1

1

G1 (W V , W, η) = G1 (ν pq , W, η) and G2 (W V , V , η) = G2 (ν pq , V , η). We also treat the resonant case and the non-resonant case separately, studying the integral I defined by (12.8) in each case: 1

1. The resonant case {qβ − pα = 0} for G1 (ν pq , W, η). We have 

− ln

I = I (W, ν, η) =

W W1

1

G1 (ν pq , W et , η)dt.

0

We make the change of variable s = W et to obtain 

W1

I=

W

1

G1 (ν pq , s, η) ds. s

1

As G1 (ν pq , 0, η) = 1, we find 

W1 I (W, ν, η) = ln W



1

+ H11 (W, ν pq , η),

(12.9)

for a smooth function H11 (depending on W1 ). 1

2. The resonant case {qβ − pα = 0} for G2 (ν pq , V , η). We have that 

− ln

I =

W W1

1

G2 (ν pq , V e−t , η)dt.

0

We make the change of variable s = V e−t to obtain  I =− V 1

ν 1/pq W1

1

G2 (ν pq , s, η) ds. s 1

1

˜ 2 (ν pq , s, η) for a As G2 (ν pq , 0, η) = 0, we can write G2 (ν pq , s, η) = s G ˜ smooth function G2 . Then 1

I = I (V , ν, η) = H12 (V , ν pq , η), where H12 is a smooth function (depending on W1 ).

(12.10)

12.2 Transition Near Semi-Hyperbolic Points

233 1

3. The non-resonant case {qβ − pα = 0} for G1 (ν pq , W, η). We have 

− ln

I (W, ν, η) =

W W1

1

G1 (ν pq , W et , η)e(qβ−pα)t dt.

0

We make the change of variable s = W et to obtain  I (W, ν, η) = W pα−qβ

W1

1

G1 (ν pq , s, η)s qβ−pα−1 ds.

W

(a) If qβ − pα > 0, we introduce  K(W, ν, η) =

W

1

G1 (ν pq , s, η)s qβ−pα−1 ds.

0

We have that I (W, ν, η) = W pα−qβ (K(W1 , ν, η) − K(W, ν, η)). The 1

function K is smooth in (W, ν pq , η) and moreover K(W, ν, η) ∼ CW qβ−pα for some C > 0. Then I (W, ν, η) =

K(W1 , ν, η) + O(W qβ−pα ) , W qβ−pα

(12.11) 1

where K(W1 , ν, η) + O(W qβ−pα ) is a smooth function of (W, W1 , ν pq , η) and K(W1 , ν, η) > 0 (if W1 is small enough). More precisely, if we write G1 (ν 1/pq , s, η) = 1 + α(ν 1/pq , η) + O(s), with α(ν 1/pq , η) = O(ν 1/q ), then we get K(W, ν, η) = =

1 + α(ν 1/pq , η) qβ−pα .W + O(W qβ−pα+1 ) qβ − pα 1 + α(ν 1/pq , η) qβ−pα W (1 + O(W )). qβ − pα

(b) If qβ − pα < 0, we expand G1 in powers of s: 1

1

˜ 1 (ν pq , s, η), G1 (ν pq , s, η) = 1 + a1 s + · · · + apα−qβ s pα−qβ + s pα−qβ+1 G 1

˜ 1 is smooth. Putting this where the ai are smooth functions of (ν pq , η) and G expansion in the integral I , we obtain that I can be written as ⎞ 1 pq , η) 1 P (W, ν ⎝ + apα−qβ ln W + H˜ 12 (W, ν pq , η)⎠ , W pα−qβ ⎛ I (W, ν, η) = W

pα−qβ

234

12 Local Transition Maps

where H˜ 12 is a smooth function and P a polynomial in W . It follows that 1

1

I (W, ν, η) = a(ν pq , η)W pα−qβ ln W + H12(W, ν pq , η),

(12.12)

where a and H12 are smooth functions. Moreover, 1

1

H12 (0, ν pq , η) = P (0, ν pq , η) =

1 pα−qβ−1 W > 0. pα − qβ 1

4. The non-resonant case {qβ − pα = 0} for G2 (ν, V , η). We have 

− ln

I=

W W1

1

G2 (ν pq , V e−t , η)e(qβ−pα)t dt.

0

We make the change of variable s = V e−t to obtain  I =V Let us notice that V =

V

qβ−pα

ν 1/pq W1

ν 1/pq W

>

G2 (ν 1/pq , s, η)s pα−qβ−1 ds.

ν 1/pq W1 .

(a) If qβ − pα > 0, we have  I = I (V , ν, η) = V

qβ−pα

V ν 1/pq W1

1

G2 (ν pq , s, η) ds. s qβ−pα+1

We proceed as in the case 3(b) to obtain I (V , ν, η) = a(ν 1/pq , η)V qβ−pα ln V + H22 (V , ν 1/pq , η),

(12.13)

1

where a and H22 are smooth functions. Because G2 (ν pq , 0, η) = 0, we have H22 (0, ν 1/pq , η) = 0. (b) If qβ − pα < 0, we introduce 

V

K(V , ν, η) =

G2 (ν, s, η)s pα−qβ−1 ds.

0

K is a smooth function of (V , ν 1/pq , η) and is O(V pα−qβ ). Then I (V , ν, η) = V pα−qβ K(V , ν, η) = H22 (V , ν 1/pq , η), where H22 is a smooth function of order O(V pα−qβ ).

(12.14)

12.2 Transition Near Semi-Hyperbolic Points

235

12.2.5 Theorems for the Transition Map We consider a family of smooth vector fields X¯ η in normal coordinates (Z, w, v), given by (12.5) at a semi-hyperbolic point q = (0, 0, 0). Recall that the transition map Tη is defined from a neighborhood W ⊂ {(Z, w, v) | w ≥ 0, v ≥ 0} of (0, 0, 0) to a section T ⊂ {w = w1 }, parameterized by (Z, ν). We have   Tη (Z, w, v) = ν = wq v p , Z˜ η (Z, w, v) .

(12.15)

In the following theorems, we give the expression of the transition component Z˜ η in the three different cases qβ − pα = 0, qβ − pα > 0 and qβ − pα < 0. To 1

1

1

simplify the expression we remark that, as ν pq = w p v q , a smooth function of 1

1

1

1

1

(w p , v q , ν pq , η) is also a smooth function of only (w p , v q , η). Theorem 12.3 (Resonant Case qβ − pα = 0)  ˜ Z(Z, w, v) = Z exp −

Lη β

νp

  1 1 1 − ln w + H˜ (w p , v q , η) , p

(12.16)

where H˜ is a smooth function. If Wη is a given smooth family of center manifolds at q, we can choose the coordinate Z such that Wη is contained in {Z = 0}. Proof We collect (12.9) and (12.10) to obtain the expression of the integral I in 1

1

function of W, V and we substitute W = w p and V = v q . This gives (12.16).



Theorem 12.4 (The Case qβ − pα > 0) ⎡ Z˜ η (Z, w, v) = Z exp ⎣−Lη

K(W1 ,ν,η)−K(w,ν,η)+w

qβ−pα p

⎤  qβ−pα av q ln v+H

⎦,

β

(qβ−pα)ν p

(12.17) 1

1

1

1

1

˜ p , ν pq , η), a = a(ν where K(w, ν, η) = K(w ˜ pq , η), and H = H˜ (v p , ν pq , η), for 1 ˜ ˜ smooth functions K, a, ˜ and H . Moreover, we have H (0, ν pq , η) = 0 and ˜ K(W, E, η) = (1 + ϕ(E, η))W qβ−pα (1 + O(W )), for ϕ smooth in (E, η), ϕ(0, η) = 0 and the remainder O(W ) smooth in (W, E, η). qβ−pα

Roughly speaking K(w, ν, η) = w p (1 + O(w1/p )). If Wη is a given smooth family of center manifolds at q, we can choose the coordinate Z such that Wη is contained in {Z = 0}.

236

12 Local Transition Maps

Proof We collect (12.11) and (12.13) to obtain the expression of the integral I in 1

1

function of W, V and we substitute W = w p and V = v q . This gives (12.17).



Remark 12.4 Taking (ν, η) in some fixed compact set, we have to choose W1 > 0 small enough in order to have K(W1 , ν, η) > 0. Then we have to choose w  W1 even smaller in order that the numerator in (12.17) was > 0. Theorem 12.5 (The Case qβ − pα < 0)    pα−qβ Lη pα−qβ q p ˜ aw ln w + H , Z(Z, w, v) = Z exp − α v νq 1

1

(12.18)

1

where a = a(ν ˜ pq , η) and H = H˜ (w p , v q , η), for smooth functions a˜ and H˜ , and with the property that H˜ (0, 0, η) > 0. If Wη is a given smooth family of center manifolds at q, we can choose the coordinate Z such that Wη is contained in {Z = 0}. Proof We collect (12.12) and (12.14) to obtain the expression of the integral I in 1 1 function of W, V , and we substitute W = w p and V = v q . Moreover, H˜ (0, 0, η) = H12 (0, 0, η) + H22(0, 0, η) > 0. This gives (12.18). 

12.2.6 Transitions Near Particular Semi-Hyperbolic Points In this section, we specify the above theorems to the semi-hyperbolic points which are obtained when we blow-up the more generic contact points: the jump points and the turning points which are the only contact points which are used in Part III. The blown-up vector field near these points is described in Chap. 8. The two possible semi-hyperbolic points q1 , q2 obtained by blowing up a jump point are given by Eqs. (8.25) for q1 and (8.26) for q2 , see Fig. 12.8. To obtain the corresponding normal form, we have to change the first lines of (8.25) and (8.26) by

u u

¯λ X

T

¯λ −X

T q1 Z

Tλ ε¯

q2

Tλ ε¯

Z

Fig. 12.8 Transitions near the semi-hyperbolic points in the blow-up of a jump point

12.2 Transition Near Semi-Hyperbolic Points

237

Z˙ = ±4F (u, ¯ , λ)Z, where F is a smooth function with F (0, 0, λ) = 1. This gives the two following particular normal forms: ⎧ ⎨ Z˙ = −4F (u, ¯ , λ)Z ¯ Xλ at q1 : ˙¯ = 3¯ 2 ⎩ u˙ = −¯ u,

(12.19)

⎧ ⎨ Z˙ = 4F (u, ¯ , λ)Z − X¯ λ at q2 : ˙¯ = 3¯ 2 ⎩ u˙ = −¯ u.

(12.20)

and

Recall that  = u3 . ¯ The two possible semi-hyperbolic points s2 , s3 obtained by blowing up a turning points are given by Eqs. (8.39) for s2 and (8.40) for s3 . As the shape of the blown-up vector field near s2 is the time-reversed version of the shape of the vector field near s3 , where the nonzero eigenvalue is negative, we will just consider this last point s3 . Again, to obtain the corresponding normal form at s3 , we have to change the first lines of (8.40) by Z˙ = −4F (u, ¯ , a, ¯ μ)Z, where F is a smooth function such that F (0, 0, 0, λ) = 1. This gives the two following particular normal forms: ⎧ ¯ a, ¯ μ)Z ⎨ Z˙ = −4F (u, v, 3 ˙ X¯ a,μ at s : ν ¯ = ν ¯ ¯ 3 ⎩ u˙ = −¯ν 2 u,

(12.21)

and its time-reversed opposite for s2 . Here ν = u¯ν is a parameter. We recall that in all of the above cases the coordinates Z may be chosen such that {Z = 0} contains a given smooth family of center manifolds Wλ respectively Wa,μ ¯ . 12.2.6.1 Transition Near the Semi-Hyperbolic Point q1 (for X¯ λ ) In (12.19), the coordinate ¯ corresponds to w and the coordinate u corresponds to v. We have that T ⊂ {¯ = ¯1 } for some ¯1 > 0 small enough and that ν =  = u3 ¯ . Moreover, in this case we have p = 3,

q = 1,

α = 1,

β = 0,

ν = .

Then qβ−pα = −3 < 0 and we can apply Theorem 12.5. The transition component Z˜ λ is given by   u3 ˜ Z(Z, u, ) ¯ = Z exp −4 (H + ρ ¯ ln ¯ ) , 

(12.22)

238

12 Local Transition Maps

where ρ = ρ( ˜ 1/3, λ) and H = H˜ (¯ 1/3 , u, λ), for some smooth functions ρ˜ and H˜ . Moreover, we have that H˜ (0, 0, λ) > 0. ¯ λ) 12.2.6.2 Transition Near the Semi-Hyperbolic Point q2 (for −X In (12.20), the coordinate u corresponds to w and the coordinate ¯ corresponds to v. We have that T ⊂ {u = u1 } for some u1 > 0 small enough and that ν =  = u3 ¯ . Moreover, in this case we have p = 1,

q = 3,

α = 0,

β = 1,

ν = .

Then qβ − pα = 3 > 0 and we can apply the Theorem 12.4. For a well-chosen coordinate Z, the transition component Z˜ λ is given by   4  3 ˜ A + u (a ¯ ln ¯ + H ) , Z(Z, u, ) ¯ = Z exp − 3

(12.23)

˜ a, ˜ and where a = a( ˜ 1/3, λ) and H = H˜ (u, ¯ 1/3 , λ), for smooth functions A, ˜ ˜ H , with H (0, 0, λ) = 0. Moreover A = K(u1 , , λ) − K(u, , λ), with K = ˜ K(u,  1/3 , λ), for some ˜ K(u, E, λ) = (1 + α(E, λ))u3 (1 + O(u)), with α smooth in (E, λ), O(u) smooth in (u, E, λ) and α(0, λ) = 0.

12.2.6.3 Transition Near the Semi-Hyperbolic Point s3 (for Xa,μ ¯ ) In (12.21), the coordinate ν¯ corresponds to w and the coordinate u corresponds to ν. We have that T ⊂ {¯ν = ν¯ 1 } for some ν¯ 1 > 0 small enough and that  = ν 2 = (u¯ν )2 . Moreover, in this case we have p = 1,

q = 1,

α = 2,

β = 0,

 = ν2.

Then qβ − pα = −2 < 0 and we can apply the Theorem 12.5. The transition is given by component Z˜ a,μ ¯   4 2 2 H + a ν ¯ u (Z, u, ν) ¯ = Z exp − ln ν ¯ , Z˜ a,μ ¯ 

(12.24)

where a = a( ˜ 1/2 , a, ¯ μ) for a smooth function a˜ and H (¯ν , u, a, ¯ μ) is smooth. Moreover, we have that H (0, 0, a, ¯ μ) > 0.

12.3 Transition Near Hyperbolic Saddle Points

239

12.3 Transition Near Hyperbolic Saddle Points In this section, we want to study transition near the hyperbolic saddle points obtained by blowing up in Chap. 8. These points can be seen as λ-family of saddle points in dimension 3. These points have special characteristics: In the given coordinates y, u, ¯ , up to a translation in y, the family of vector fields is linear in the directions u, ; ¯ moreover, the eigenvalues are fixed (λ-invariant) and resonant, and the family of vector fields is linear for ¯ = 0. We will consider here a more general setting: germs of families of (k + 2)dimensional vector fields Vλ at (0) ∈ Rk+2 , with coordinates (u, v1 , . . . , vk , y), which are quasi-linear smooth saddle points of the form: ⎧ ⎨ u˙ = pu Vλ : v˙i = −qi vi , i = 1, . . . , k ⎩ y˙ = −ry + Fλ (u, v, y),

(12.25)

where v = (v1 , . . . , vk ) ∈ Rk and p, q1 , . . . , qk , r ∈ N. The smooth family of functions Fλ is of order 2 at the origin. At the saddle points s+ and s− , of Chap. 8 the family X¯ λ or the family −X¯ λ in Sect. 8.3, is a family as Vλ , with k = 1, p = 1, q = 2n − 1, r = n. We want to study transition maps for (12.25), in the region Q = {u ≥ 0, v1 ≥ 0, . . . , vk ≥ 0, } ⊂ Rk+2 near the origin. More precisely, let W be a neighborhood at the origin and a section D ⊂ {u = U0 }, for U0 > 0. We will consider the transition Tλ from the points in W ∩ {u > 0} to the section D, which can be disjoint from W . The result for Tλ is given in Theorem 12.6 below. This theorem is proved, using a smooth normal form established in Chap. 10 for the case p = 1. Just recall that in this case p = 1, so the normal form is given, up to a smooth parameter-depending change of coordinates, as

VλN

⎧ ⎨ u˙ = u : v˙i = −qi vi , i = 1, . . . , k ⎩ ˙ Y = −rY + Nλ (u, v, Y ),

(12.26)

with Nλ a smooth family of functions of order 2 of the form Nλ (u, v, Y ) = λ (uq1 v1 , . . . , uqk vk , ur Y )Y +

!

j

j

j

uα v β λ (uq1 v1 , . . . , uqk vk ),

j =1

(12.27) j

with λ and λ smooth, α j − #q, β j $ = −r and λ (0, . . . , 0, 0) = 0. One can find more details in Chap. 10.

240

12 Local Transition Maps

12.3.1 The Transition Map in the Case p = 1 We consider first a family Vλ as in (12.25) with p = 1. We restrict this family to a neighborhood V of the origin of Rk+2 in the domain {u ≥ 0, v1 ≥ 0, . . . , vk ≥ 0}. We assume that we can use coordinates (u, v1 , . . . , vk , Y ) in V, parameterizing V such that in these coordinates, the family Vλ coincides with its normal form VλN given by (12.26) and (12.27). We choose U0 > 0, such that (U0 , 0, . . . , 0) ∈ V and consider a section D ⊂ V, through this point and contained in {u = U0 }. We notice that VλN has k first integrals uqi vi = i for i = 1, . . . , k. We parameterize D by (1 , . . . , k , Y˜ ) where Y˜ is just another name for the coordinate Y . If W is a small enough neighborhood of the origin in V, the trajectory of VλN through any point (u, v, Y ) ∈ W ∩ {u > 0} reaches D in the finite positive time − ln Uu0 . This defines a λ-family of transition maps   Tλ : (u, v1 , . . . , vk , Y ) → uq1 v1 , . . . , uqk vk , Y˜λ (u, v1 , . . . , vk , Y ) from W ∩ {u > 0} to D. Let (u(t), v1 (t), . . . , vk (t), Y (t)) be the trajectory of VλN through a point (u, v1 , . . . , vk , Y ) ∈ W ∩ {u > 0}. We have that u(t) = uet ,

v1 (t) = v1 e−q1 t ,

vk (t) = vk e−qk t .

...,

We obtain a differential equation for Y (t) by substituting these functions in the last line of (12.26). This gives Y˙ = −rY + Nλ (u(t), v1 (t), . . . , vk (t), Y ). j

j

Taking into account that ui (t)qi vi (t) = i , for i = 1, . . . , k, and that u(t)α v(t)β = j j e−rt uα v β for j = 1, . . . , , as α j − #q, β j $ = −r, this equation is equal to Y˙ = −rY +λ (1 , . . . , k , ert ur Y )Y +e−rt

!

j

j

j

uα v β λ (1 , . . . , k ).

(12.28)

j =1

In order to eliminate the linear term −rY in (12.28), we look for Y (t) in the form ˙ Y (t) = e−rt Z(t). As Y˙ (t) = e−rt Z(t)−rY (t), we obtain the following autonomous differential equation for Z(t): Z˙ = λ (1 , . . . , k , ur Z)Z +

! j =1

j

j

j

uα v β λ (1 , . . . , k ).

(12.29)

12.3 Transition Near Hyperbolic Saddle Points

241

Then Z(t) is the trajectory of a smooth λ-family of one-dimensional vector fields ⎛ Zλ (Z, u, v) = ⎝λ Z +

!

⎞ αj

u v

βj

j λ ⎠

j =1

∂ ∂Z

 r with initial condition Z(0) = Y . The function Y˜λ is equal to Uu0 Z(− ln Uu0 ). An important difficulty is that we have to integrate the Eq. (12.29) for a time going to +∞ when u → 0. To avoid this difficulty, it is sufficient that Zλ (Z, u, v) is divisible by u. But it is not necessarily the case if for some j = 1, . . . , we have α j = #q, β j $ − r = 0. To express this condition, we introduce the monoid  N{q1 , . . . , qk } =

k !

. ai qi | a1 , . . . , ak ∈ N .

i=1

N{q1 , . . . , qk } is the smallest subset of N that contains {q1 , . . . , qk } and is closed for summation. Then, in order to have the property that α j = #q, β j $ − r > 0, for j = 1, . . . , , it is sufficient that the following condition is fulfilled: Condition D :r ∈ N{q1 , . . . , qk }.

Remark 12.5 When k = 1 (i.e. if Vλ is a family of vector fields in dimension 3), the condition D amounts to the condition that q is not a factor of r. It is the case when q > r, like for the saddle points in Chap. 8 where q = 2n − 1 and r = n with n ≥ 2.  j j j Under this condition D, we obtain that j =1 uα v β λ (1 , . . . , k ) is divisible r by u. Moreover, as λ (0) = 0, the term λ (1 , . . . , k , u Z) is also divisible by u. Finally Zλ is divisible by u, under the condition D. From now on we will assume that this condition is satisfied. ˜λ = If ϕλ (t, u, v, Z) represents the flow of the smooth family of vector fields Z 1 u Zλ , we get Z(t) = ϕλ (ut, u, v, Y ) and hence Y˜λ (u, v, Y ) =



u U0

r

  ϕλ −u ln Uu0 , Y, u, v .

The function ϕλ (t, u, v, Z) is smooth. As the integration time −u ln Uu0 is bounded (this integration time even tends to zero when u → 0), the function   ϕλ −u ln Uu0 , u, v, Y

242

12 Local Transition Maps

˜ λ (u, v, Y ) + is smooth in (u, u ln u, v, Y, λ). We have that ϕλ (t, u, v, Y ) = Y + t Z O(t 2 ). This gives   ϕλ −u ln Uu0 , u, v, Y = Y + O(u ln u). We hence obtain the following result: Proposition 12.3 Under condition D (r ∈ N{q1, . . . , qk }), the Y -component of the λ-family of transition maps Tλ is given by Y˜λ (u, v, Y ) =



u U0

r (Y + λ (u, v, Y )) .

The function λ (u, v, Y ) is of order O(u ln u) and smooth in (u, u ln u, v, Y, λ) in the sense of Chap. 11.

12.3.2 Transition in the General Case (for p ∈ N) We consider now a family Vλ as in (12.25) for any p. This general case easily reduces to the particular case p = 1. It is sufficient to notice that the change (U, v, y) → (u = U p , v, y) brings (12.1) into ⎧ ⎨ U˙ = U (12.30) V¯λ : v˙i = −qi vi , i = 1, . . . , k ⎩ p y˙ = −ry + Fλ (U , v, y). This family is smooth (even if the coordinate change fails to be a diffeomorphism along {U = 0}) and is a family of type (12.1) with p = 1, the coordinate u named U and the function Fλ replaced by Gλ (U, v, y) = Fλ (U p , v, y). We can apply Proposition 12.3 to (12.30) and next substitute u1/p to U . Putting now i = uqi v p for i = 1, . . . , k, we can parameterize the section D ⊂ {u = U0 } by (1 , . . . , k , Y˜ ). We obtain that the transition Tλ from W ∩ {u > 0} to D ⊂ {u = U0 } is given by   p p Tλ : (u, v1 , . . . , vk , Y ) → uq1 v1 , . . . , uqk vk , Y˜λ (u, v1 , . . . , vk , Y ) , where the Y˜ -component is given by the following result: Theorem 12.6 Under condition D (r ∈ / N{q1 , . . . , qk }), the Y -component of the λ-family of transition maps Tλ is given by Y˜λ (u, v, Y ) =



u U0

r/p (Y + λ (u, v, Y )) .

12.3 Transition Near Hyperbolic Saddle Points

243

The function λ is of order O(u1/p ln u) and is smooth in (u1/p , u1/p ln u, v, Y, λ) in the sense of Chap. 11.

12.3.3 The Saddle Points of Chap. 8 In this section, we want to apply Theorem 12.6 to the three-dimensional families X¯ λ of saddle points obtained in Chap. 8 by blowing up. As in Chap. 8, we will use the coordinates u, ¯ for v and y. Changing X¯ λ by −X¯ λ when the eigenvalue in the y-direction is positive, we have a family Vλ with coefficients p, q, r equal to p = 1, q = 2n − 1, and r = n ≥ 2 (with n = 2 in the generic case). The germ of the family Vλ at (0, 0, 0) ∈ R3 has the form: ⎧ ⎨ u˙ = u Vλ : ˙¯ = −(2n − 1)¯ ⎩ y˙ = −ny + ¯ F¯λ (u, ¯ , y),

(12.31)

with F¯ (0, 0, 0) = 0. We first consider the normal form VλN recalled above. As k = 1 we also have = 1 and we have a unique pair (α, β) ∈ N × N satisfying α = (2n − 1)β − n with β minimal in N. This pair is β = 1 and α = n − 1. Then the normal form Nλ is equal to ¯ Y ) = λ (u2n−1 ¯ , un Y )Y + un−1 ¯ λ (u2n−1 ¯ ), Nλ (u, ,

(12.32)

with λ (0, 0) = 0. Clearly Nλ is divisible by u (in fact by un−1 ). This is in accordance to the fact that the condition D is fulfilled for any n ≥ 2, as it was remarked above (as r = n < q = 2n − 1). A particular property, that we have not used up to now, is that Vλ is linear along {¯ = 0}. As a consequence, the function Nλ in the normal form (see (12.27)) is divisible by . ¯ A simple proof of this divisibility now follows. As we have seen in Sect. 12.1, it suffices to verify formally this divisibility. This reduces to prove that, in this special case, we can find a formal normal form Nˆ λ on resonant monomials, each divisible by . ¯ If we return to the proof of Theorem 10.4, we see that it is sufficient to prove that the Lie bracket: Z → [u∂u − (2n − 1)¯ ∂¯ − ny∂y , Z] leaves invariant the spaces {¯ K∂y | K}, where K belongs to a space of homogeneous polynomials of some fixed degree ≥ 2. But this invariance is clear as ˜ y, [u∂u − (2n − 1)¯ ∂¯ − ny∂y , K∂ ¯ y ] = ¯ K∂

244

12 Local Transition Maps

with K˜ = u∂u K − (2n − 1)¯ ∂¯ K − ny∂y K − nK. This finishes the proof of the divisibility. To say that Nλ is divisible by ¯ is equivalent to say that λ (u2n−1 ¯ , u2 Y ) is divisible by , ¯ and so by u2n−1 . ¯ Then we find ˜ λ (u2n−1 ¯ , un Y )Y + un−1 ¯ λ (u2n−1 ¯ ), Nλ (u, ¯ , Y ) = u2n−1 ¯  ˜ λ . We can summarize the above considerations in the for a smooth function  following version of a theorem of normal form for family (12.31): Theorem 12.7 We consider a family Vλ in dimension 3, given by (12.31). There exists a smooth parameter-dependent change of coordinates (u, , ¯ y) → (u, ¯ , Y = Gλ (u, , ¯ y)) bringing Vλ in the normal form VλN : VλN

⎧ ⎨ u˙ = u : ˙¯ = −(2n − 1)¯ , ⎩˙ Y = −nY + Nλ (u, ¯ , Y ),

(12.33)

with Nλ a function of the form: ˜ λ (u2n−1 , Nλ (u, , ¯ Y ) = u2n−1 ¯  ¯ un Y )Y + un−1 ¯ λ (u2n−1 ¯ ),

(12.34)

˜ λ , λ are smooth (we see that such Nλ is of order at least 2). where the functions  We now consider the transition map Tλ (u, , ¯ Y ) = (u2n−1 , ¯ Y˜λ (u, , ¯ Y )) defined, as in the general case, from W ∩ {u > 0} to a section D ⊂ {u = U0 }, parameterized by (, Y˜ = Y ). Recall that to compute Y˜λ , we have to integrate the last line of Eq. (12.28), which, using u2n−1 ¯ = , is now written as ˜ λ (, ent un Y )Y + e−nt un−1 ¯ λ (). Y˙ = −nY +  

(12.35)

Putting Y (t) = e−nt Z(t), we have the following autonomous differential equation for Z(t): ˜ λ (, un Z)Z + un−1 ¯ λ () Z˙ =  

(12.36)

to be integrated from the initial condition Z(0) = Y and from t = 0 to t = − ln Uu0 .  n The function Y˜λ is equal to Uu0 Z(− ln Uu0 ).

12.3 Transition Near Hyperbolic Saddle Points

245

As  = u2n−1 ¯ , the family of vector fields ˜ λ (, un Z)Z + un−1 ¯ λ ()) Zλ = ( 

∂ ∂Z

is divisible by un−1 ¯ . The divided family of vector fields is ˜λ= Z

  ∂ n˜ n Z . = u (, u Z)Z +  ()  λ λ λ un−1 ¯ ∂Z 1

(12.37)

˜ λ . We have Let ϕλ (t, u, ¯ , Z) be flow of the smooth family of vector fields Z ¯ Y) Z(t) = ϕλ (un−1 ¯ t, u, , and as a consequence Y˜λ (u, ¯ , Y ) =



u U0

n

  ϕλ −un−1 ¯ ln Uu0 , u, , ¯ Y .

As the integration time −un−1 ¯ ln Uu0 is bounded (even this integration time tends ¯ is smooth in (u, un−1 ¯ ln u, Y, λ). to zero when u → 0), ϕλ (−un−1 ¯ ln Uu0 , Y, u, ) The flow ϕλ can be expanded in t:   ˜ λ (, un Y )Y + λ () + O(t 2 ). ϕλ (t, u, , ¯ Y ) = Y + t un 

(12.38)

This gives   ϕλ −un−1 ¯ ln Uu0 , u, , ¯ Y = Y + O(un−1 ¯ ln u). We can make this estimate more precise. To this end, we use (12.36) to obtain ∂ ˜ λ (u, , . Z ¯ Y ) = (λ (0) + O(un )) ∂Z Bringing this estimate in (12.35), we find ϕλ (−un−1 ¯ ln Uu0 , u, ¯ , Y ) is equal to Y − un−1 ¯ ln

u λ (0) + O(u2n−1 ¯ ln u) + O(u2n−2 ¯ 2 ln2 u), U0

expression that we can summarize in Y − un−1 ¯ ln

u λ (0) + O(u2n−2 ¯ ln2 u). U0

246

12 Local Transition Maps

We hence obtain the following result: Theorem 12.8 Let Tλ be the transition map for the family of three-dimensional saddle points (12.33). The Y -component of the λ-family of Tλ is given by Y˜λ (u, , ¯ Y) =



u U0

n ¯ Y )) . (Y + λ (u, ,

(12.39)

The function λ is of order O(un−1 ¯ ln u) and smooth in (u, un−1 ¯ ln u, , ¯ Y, λ) in the sense of Chap. 11. More precisely, we have λ (u, ¯ , Y ) = −un−1 ¯ ln

u λ (0) + O(u2n−2 ¯ ln2 u), U0

(12.40)

where λ is the smooth function introduced for the normal form in Theorem 12.7. Remark 12.6 To use the formula (12.40) in practice, one has to compute the term λ (0) which is the coefficient of the monomial vector field un−1 ¯ ∂Y in the normal form VλN .

12.4 Transition at a Jump Point In this section, we consider a transition which includes passage near a jump point J . We will obtain the form of the transition by composing several local transitions that we have studied in the previous sections. Recall that at a jump point we have local coordinates (x, y) where the system X,λ can locally be written as 

x˙ = F (x, y, , λ) = y − x 2 + O(x 3 ) y˙ = G(x, y, , λ) = (1 + o(1)).

The jump point J in located at (0, 0) for any λ. We consider an attracting arc γs arriving at J (we assume that γs includes J as end point). It is contained in the right branch of the critical curve, locally given by {F (x, y, 0, λ) = 0, x ≥ 0} (but there is no need for γs to be visible in the chart of the local normal form coordinates). We will assume γs to be chosen so that all the points in γs \ J are regular (neither a contact point nor a zero of the slow dynamics). The arc γs is oriented by the slow dynamics which is directed toward J . We can also define a natural transverse orientation along γs . Locally, in the coordinates (x, y) we define the right side of γs as the side where F > 0 and the left side of γs to be the side defined locally by F < 0. This local orientation is extended by continuity along γs and we will speak of the right and left side all along γs . We consider an exterior section  for γs , i.e. a two-dimensional section  = σ × [0, 0 ], where σ is an interval transverse to the orbits of the fast dynamics and such that each point in σ has an ω-limit in γs \ J (see σ and σr in Fig. 12.9

12.4 Transition at a Jump Point

247

T

τ

J

J

Σ

τ

σl

σr

γf

γf σ

σm γs

γs

Fig. 12.9 Left and right sides with appropriate sections σ and σr

for example). Let v be a smooth parameterization of σ . The section σ may be taken outside the domain of the local coordinates (x, y) as long as the fast orbits through it in forward time enter the domain of the local coordinates. By continuity, it permits to define whether σ is located on the right side or on the left side of γs and both possibilities are permitted. Occasionally, we will also work with a transverse section  = σm × [0, 0 ], transversally cutting γs . In that case we take σm tangent to the fast vector field for  = 0, like in Theorem 12.1. On σm we take a regular variable z in a way that {z = 0} agrees with σm ∩ γs , while {z < 0} is situated to the right of γs and {z > 0} to the left. There is a fast orbit γf whose α-limit is the jump point J . We take a section T = τ × [0, 0 ] at a point m of γf , transverse to γf . Let z be smooth parameterization of τ , with z(m) = 0. We will assume that z > 0 corresponds to fast trajectories passing below the jump point (direction corresponding to y < 0 in the local coordinates). We call (v, λ) the smooth family of maps which associates to each  v ∈ σ its ω-limit point on γs \ J . For two points m, m ∈ γs , we write I ( m, m , λ) for the slow divergence integral on the arc from m to m . As the point J is a jump point, we know that this integral is also defined for m = J and that it is a smooth function of (m, m , λ) ∈ γs × γs × P (see Chap. 5 and [DMDR11]). In the following theorem, we want to prove that the transition near the jump point J exists and give an expression of the transition map: Theorem 12.9 Let  = σ × [0, 0 ] and T = τ × [0, 0 ] be sections as above: we choose any smooth coordinates v and z on σ and τ . We assume that z is oriented as above and is such that z = 0 at the point m = γf ∩ τ . If 0 > 0 is small enough, any trajectory starting at a point of \{ = 0} reaches the section T and defines a transition map (z, ) → (Z,λ (z), ) from  \ { = 0} to T \ { = 0}. The z-component Z,λ has the following expression:  I ([(v, λ), J ], λ) + O( 1/3 ) , Z,λ (v) =  (, λ) + θ exp  2 3



(12.41)

248

12 Local Transition Maps

where I ([(v, λ), J ], λ) is the slow divergence integral from the point (v, λ) to the jump point J . The function  is smooth in ( 1/3 ,  1/3 ln , λ), with (0, λ) = 0 and the term O( 1/3) is smooth in (v,  1/3 ,  1/3 ln , λ). The coefficient θ is equal to +1 if σ is located on the left of γs (with “left” defined as above, see σ in Fig. 12.9) and is equal to −1 in the other case (see σr in Fig. 12.9). In the sense of Sect. 11.5, Z,λ is a family of diffeomorphisms of exponentially flat type on the sets of admissible monomials 0 = 1 = { 1/3,  1/3 ln }. Remarks 12.7 (1) From (12.41) it is clear that Z,λ extends continuously by 0 at  = 0. In (12.41) we write that the remainder is of order O( 1/3 ) and say that it is smooth in (v,  1/3 ,  1/3 ln , λ). This provides more precise information than just writing that the function I˜ in the exponential is smooth in (v,  1/3 ,  1/3 ln , λ) and reduces to I ([(v, λ), J ], λ) for  = 0. It is for a similar reason that we write the “translation” term as  2/3 (, λ). A similar remark can be made about Theorems 12.10 and 12.11 below. (2) Since (, λ) is smooth in ( 1/3 ,  1/3 ln , λ) and starts with a nonzero term, it implies that the function  2/3 (, λ) has at  = 0 a formal development in the variables  1/3 and  1/3 ln , with coefficients depending smoothly on λ and starting with a leading term c(λ) 2/3 for some c(λ) = 0. Formula (12.41) is known in asymptotic analysis for individual systems and has also been obtained in [vGKS05] using blow-up. In the same paper is proven that the transition map is a contraction with rate O(e−d/ ) for some d > 0. This result can easily be deduced from (12.41), an expression in which we give detailed information on the exponent of the exponentially flat function. We show that it is a smooth function in (v,  1/3 ,  1/3 ln , λ) having as leading term I ((v, λ), J ), the slow divergence integral calculated along γs between (v, λ) and J . Observe that the  used in Theorem 2.1 of [vGKS05] does not have the same meaning as the map  that we use here, but stands for Z,λ . (3) If  is not exterior to γs but cuts γs transversally, i.e.  = σm × [0, 0 ], as described before, then we get following result: ¯ λ) + z. exp Z,λ (v) =  2/3 (,



 I (γ , λ) + O( 1/3) , 

(12.42)

where I (γ , λ) stands for the slow divergence integral along γs from γs ∩ σm to ¯ p, while (, λ) and O( 1/q ) have the same properties as in (12.41). 

The remainder of this section is devoted to the proof of Theorem 12.9. We begin by some preliminary results. As explained in Chap. 8, in order to prove the existence of this transition and to study it, we blow-up the system X,λ at the point J . Recall that the blow-up formula is x = ux, ¯ y = u2 y, ¯  = u3 . ¯ The system X,λ is blown up into a family of blown-up vector fields X¯ λ defined on a three-dimensional singular space E. The 2 = {x¯ 2 + y¯ 2 +  2 = 1, ¯ ≥ 0}, the point J is blown-up into a half-sphere S+

12.4 Transition at a Jump Point

249

blow-up locus. The lift of γs on E, that we will identify with γs , arrives at a semi2 whose form is given in (8.27). On the blow-up locus, hyperbolic point q1 ∈ ∂S+ 2, a separatrix starts at q1 and tends toward a hyperbolic saddle point s− ∈ ∂S+ whose form is given in (8.14). The lift of γf on E, that we will identify with γf , has s− as α-limit. The lift of trajectories starting on  \ { = 0} first follow γs , next and next γf until eventually arriving at T. Then, for any  > 0, we can see the map Z,λ (v) : v → z = Zλ (v, ) as a composition of several transition maps as it is explained below (on , we do not have to look at the coordinate  which is preserved in the transition). Let (Z, u, ) ¯ be the normal coordinates near q1 , recalled in Sect. 12.2.6. In these coordinates, γs is given by {¯ = Z = 0} and the separatrix by u = Z = 0, see Fig. 12.10. We take a section 1 = σ1 × [0, 0 ], transverse to γs , in {u = u1 } for a u1 > 0. We take also a section 2 = σ2 × [0, 0 ], transverse to , through a point {¯ = ¯0 } for an ¯0 > 0. We consider also a normal form chart, neighborhood of the saddle point s− , with normal coordinates (Y, u, ), ¯ as defined in Sect. 12.3. We choose two sections in this chart: an entry section 3 = σ3 × [0, 0 ], in {¯ = ¯1 } for an ¯1 > 0; this section is chosen large enough such that the separatrix cuts it in its interior; an exit section 4 = σ4 × [0, 0 ], in {u = u0 } for a u0 > 0. For any  > 0, we can see the map Z,λ as the composition: 5 4 3 2 1 ◦ Z,λ ◦ Z,λ ◦ Z,λ ◦ Z,λ , Z,λ = Z,λ 1 : σ → σ , Z 2 : σ → σ , Z 3 : σ → σ , Z 4 : σ → σ , and where Z,λ 1 1 2 2 3 3 4 ,λ ,λ ,λ 5 Z,λ : σ4 → τ . We can choose any section  and T as in the statement. Then, the value 0 > 0 has to be chosen small enough, and the other sections also can be chosen such that the different maps are well defined for any λ and  ∈]0, 0 ] and can be composed.

2 ∂S+

2 S+

T

l s− Σ

q1

l

τ

q2

q1

σ γf γs

σ2

σ3

σ4 s−

q2

σ1 σ

γf

v π(v, λ)

γs

Fig. 12.10 The sections σ and τ and intermediate sections σ1 , σ2 , σ3 , σ4

τ

250

12 Local Transition Maps

This proves the first part of the statement: a transition map from  toward T is well defined and hence also its z-component Z,λ by the above composition. It follows from Proposition 10.12 that we can choose a smooth family of center manifolds Wλ along the arc γs (including the semi-hyperbolic point q1 ). We assume that σ1 and σ2 are taken in a normal form chart W , of coordinates (Z, u, ¯ ), associated to the family Wλ . We parameterize these sections by the corresponding coordinate Z and then the family of center manifolds Wλ cuts W along {Z = 0}. This coordinate Z is z1 on σ1 and z2 on σ2 . On the sections σ3 , σ4 we will use the coordinate Y : we denoted Y1 on σ3 and Y2 on σ4 . i obtained in the We recall now the expressions of the different transitions Z,λ previous sections. 1 (v). We apply Theorem 12.2 with a coordinate z1 on σ1 oriented The map Z,λ toward the left. Then, with the coefficient θ defined in the statement of Theorem 12.9, we have that   I˜(v, , λ) 1 z1 = Z,λ (v) = θ exp , (12.43) 

where I˜ is smooth and I˜(v, 0, λ) = I ([(v, λ), u1 ], λ). 2 (z ). The map Z,λ It is given by (12.22). On the section σ2 , we have that  = u31 ¯ . 1 Then, for any fixed u1 > 0, we can use  instead of ¯ as variable on σ2 . Mind that, as ¯ is bounded, we have to assume that  ∈ [0, 0 (u1 )], with 0 (u1 ) of order O(u31 ). On this interval in , this gives 

z2 =

2 Z,λ (z1 )

u3 = z1 exp −4 1 

#

#

 H + ρ ln u31

$$ ,

(12.44)

1/3 , u , λ), for smooth functions ρ˜ where ρ = u−3 ˜ 1/3, λ) and H = H˜ (u−1 1 1 ρ( 1  and H˜ . Moreover, we have that H˜ (0, 0, λ) > 0. 3 (z ). This map is smooth. We will just need an expansion of order The map Z,λ 2 one in z2 : 3 Y1 = Z,λ (z2 ) = a(, λ) + b(, λ)z2 (1 + O(z2 )),

(12.45)

where a, b and the term O(z2 ) are smooth. Moreover we have that b(, λ) > 0, as 3 is a smooth family of diffeomorphisms in z . We have proved in Sect. 8.5.3 Z,λ 2 that a(0, λ) = 0. This means that the separatrix does not coincide with the axis {Y1 = 0} in the normal coordinates at the saddle point s− .

12.4 Transition at a Jump Point

251

4 (Y ). The map Z,λ The normal form at s− is given by (12.33) with n = 2 and 1 the transition near s− by (12.39) and (12.40). In formula (12.39) we have to put  1/3 , U0 = u0 , ¯ = ¯1 . We obtain u = ¯1

# Y2 =

4 Z,λ (Y1 )

=



$2/3

¯1 u30



 Y1 + O( 1/3) ,

(12.46)

where the term O( 1/3) is smooth in (Y1 ,  1/3 ,  1/3 ln , λ) in the sense of Chap. 11. 5 (Y ). 5 is smooth. As the fast orbit γ corresponds The map Z,λ The transition Z,λ 2 f to {Y2 = 0} on σ4 and to z = 0 on T, we have that 5 z = Z,λ (Y2 ) =

5 ∂Z,λ

∂Y2

(0)Y2 (1 + O(Y2 )),

(12.47)

∂Z 5

with ∂Y,λ (0) > 0. This claim follows from the assumption that z = 0 at γf ∩ τ 2 5 (Y ) is a smooth family of diffeomorphisms preserving orientation. and that Z,λ 2 1 and Z 2 : Let us first compute the composition of Z,λ ,λ

Lemma 12.4 2 1 ◦ Z,λ (v) = θ exp z2 = Z,λ

I ([(v, λ), J ], λ) + O( 1/3) , 

(12.48)

where the term O( 1/3) is smooth in (v,  1/3 ,  ln , λ). Proof We first take a fixed u1 > 0. By composition of (12.43) and (12.44), we get   J˜(v, , λ) 2 1 Z,λ ◦ Z,λ(v) = θ exp ,  where

#

#

 1/3 , u1 , λ) + a( ˜ 1/3 , λ) ln J˜(v, , λ) = I˜(v, , λ) − 4 u31 H˜ (u−1 1  u31

$$ . (12.49)

We notice that the function J˜ is independent of u1 , contrary to what might be suggested by the above formula (12.49). This formula proves that the function J˜ is smooth in (v,  1/3 ,  ln , λ). We also want to use (12.49) to compute J˜(v, 0, λ). To this end, we take its value for  = 0. We obtain for u1 > 0, small enough: J˜(v, 0, λ) = I ([(v, λ), u1 ], λ) − 4u31 H˜ (0, u1 , λ).

252

12 Local Transition Maps

Here, H˜ (0, u1 , λ) is smooth in (u1 , λ) as well as I ([(v, λ), u1 ], λ) in (v, u1 , λ). Taking the limit u1 → 0 in this formula, we see that J˜(v, 0, λ) = I ([(v, λ), J ], λ). As J˜ is smooth on (v,  1/3 ,  ln , λ), we can write J˜(v, , λ) = I ([(v, λ), J ], λ) + O( 1/3 ), with a term O( 1/3), smooth in (v,  1/3 ,  ln , λ). 

3 , Z 4 , and Z 5 : Next we consider the composition of Z,λ ,λ ,λ

Lemma 12.5 We have the following expansion: 5 4 3 z = Z,λ ◦ Z,λ ◦ Z,λ (z2 ) (12.50) $2/3 # 3 5   ∂Z0,λ ∂Z0,λ 2  1/3 (0) (0)z 1 + O(z ) + O( ) , =  3 (, λ) + 2 2 ∂z2 ∂Y2 ¯1 u30

where the function  is smooth in ( 1/3 ,  1/3 ln , λ) such that # (0, λ) =

$2/3

1 ¯1 u30

5 ∂Z0,λ

∂Y2

(0)a(0, λ) = 0.

The terms O(z2 ), O( 1/3) are smooth in (z2 ,  1/3 ,  1/3 ln , λ). Proof Using (12.45) and (12.46) we obtain that # Y2 =

4 Z,λ

3 ◦ Z,λ (z2 )

=

 ¯1 u30

$2/3

  a(, λ) + b(, λ)z2 (1 + O(z2 )) + O( 1/3 ) ,

where a, b are as above and where the O(z2 ) and O( 1/3 ) terms are smooth in (z2 ,  1/3 ,  1/3 ln , λ). As b(, λ) > 0, we can decompose the term O( 1/3) into a sum  1/3 c(, λ) + b(, λ)z2 O( 1/3 ), and then rewrite  Y2 =

 ¯1 u0

2/3 

  a(, λ) +  1/3 c(, λ) + b(, λ)z2 1 + O(z2 ) + O( 1/3) , (12.51)

with c smooth in ( 1/3 ,  1/3 ln , λ) and where the O( 1/3 ) term is smooth in 5 (Y ). Putting (z2 ,  1/3 ,  1/3 ln , λ). We compute now z = Z,λ 2 # α=

 ¯1 u30

$2/3

  a(, λ) +  1/3 c(, λ)

12.4 Transition at a Jump Point

253

and # β=



$2/3

¯1 u30

  bz2 1 + O(z2 ) + O( 1/3 ) ,

5 (α + β) at α, in first order in β: we expand Z,λ 5 ∂Z,λ

5 5 (α + β) = Z,λ (α) + Z,λ

∂Y2

(α)β(1 + O(β)),

with a term O(β) smooth function of (α, β). 5 (α) = As Z,λ

5 ∂Z,λ ∂Y2 (0)α(1 +

# (, λ) =

1 ¯1 u30

2

5 O(α)), we have that Z,λ (α) =  3 (, λ), with

$2/3

5 ∂Z,λ

∂Y2

  (0)a(0, λ) 1 + O( 2/3)

which is a smooth function in ( 1/3 ,  1/3 ln , λ). Moreover, we have that # (0, λ) =

$2/3

1

5 ∂Z0,λ

¯1 u30

∂Y2

(0)a(0, λ) = 0.

We obtain (12.50) by substituting the expression of β and taking into account that b(, λ) =

3 ∂Z0,λ ∂z2 (0) +

O() and

5 ∂Z,λ ∂Y2 (α)

=

5 2 ∂Z0,λ 3 ∂Y2 (0) + O( ).



Proof (Proof of Theorem 12.9) It remains to put together the two partial transitions 2

∂Z 3

computed in Lemmas 12.4 and 12.5. Putting C = (¯1 u30 )− 3 ∂z0,λ (0) 2 in (12.50) and I = I ([(v, λ), J ], λ) in (12.48) to simplify, we have z = Z,λ (v) =  1/3 (, λ) + θ C 2/3 e

I +O( 1/3 ) 

5 ∂Z0,λ ∂Y2 (0)

> 0

  I +O( 1/3 ) 1 + O(e  ) + O( 1/3) , I +O( 1/3 )

where terms O( 1/3) are smooth in (v,  1/3 ,  1/3 ln , λ). The function e  is flat in  and then smooth and of order O( 1/3 ): We can incorporate the term I +O( 1/3 ) 

O( 1/3 ) in the above expression. Then, we have that  ) into the term 1/3 I +O( ) ) + O( 1/3) is smooth in (v,  1/3 ,  1/3 ln , λ) and we α = ln 1 + O(e  O(e

can write C

2/3

e

I +O( 1/3 ) 



 1+O e

I +O( 1/3 ) 



 + O(

1/3

) = exp

I +O( 1/3 )+ 32  ln + ln C+α . 

254

12 Local Transition Maps

The function O( 1/3 ) + 23  ln  +  ln C + α is smooth in (v,  1/3 ,  1/3 ln , λ) and 

of order O( 1/3). This finishes the Proof of Theorem 12.9. Remarks 12.8 1. A completely similar proof leads to expression (12.42). We only have to rely 1 . on Theorem 12.1 instead of Theorem 12.2 in describing the structure of Z,λ During the construction we also have to make sure that the smooth family of center manifolds Wλ along γs (including the semi-hyperbolic point q1 ) is given by {z = 0} in the chosen coordinates (z, ) on . i 1 has an exponentially flat type. 2. Among the five maps Z,λ , just the first one Z,λ Its behavior dominates the one of the other maps. The only effect of these maps is to replace the smoothness by a smooth dependence on the monomials  1/3 ,  1/3 ln . 

12.5 Transition Along an Attracting Sequence Let us now consider an attracting sequence A, as defined in Definition 4.8 for a system X,λ . Such a sequence contains k ≥ 1 regular jump points J1 , . . . , Jk . Let γs1 , . . . , γsk be the attracting arcs of the critical curve ending respectively at the contact points J1 , . . . , Jk . We assume that all the points in ∪i {γsi \ Ji } are regular for the slow dynamics, pointing toward Ji at points of γsi \ Ji . For each i < k, a fast orbit γfi “jumps” from the point Ji to a regular point mi+1 on γsi+1 . This orbit arrives on the right side or on the left side of the arc γsi+1 (right and left side, as defined in Sect. 12.4). As this fact will change the expression for the jump transition at the point Ji+1 , we define an index θJi+1 at each jump point Ji+1 , for i = 1, . . . , k − 1: It is equal to +1 if, starting from Ji , we arrive on the left side and equal to −1 if we arrive on the right side, “left” and “right” as defined in Sect. 12.4, see Fig. 12.11.

Ji+1 σi

mi+1

Ji

γsi+1 (a)

+1

= +1

σi+1

σi+1

Ji+1 mi+1

σi

γsi+1 (b)

+1

Ji

= −1

Fig. 12.11 Passage from Ji to Ji+1 in an attracting sequence. (a) Ji connects to γsi+1 via the left. (b) Ji connects to γsi+1 via the right. The sign θJi+1 in both cases does not depend on the orientation of σi , whereas the orientation of σi+1 is chosen upward w.r.t. the normal form coordinates at Ji+1

12.5 Transition Along an Attracting Sequence

255

A last fast orbit γfk “starts” at the point Jk . Of course, all these data depend smoothly on λ, but, in order to simplify the notations, we do not explicitly indicate this dependence. The number k will be called length of the sequence A. We can define a transition map associated to the attracting sequence A. To this end, as in Sect. 12.4, we choose an exterior section  = σ × [0, 0 ] for γs1 and a section T = τ × [0, ] at a point m of γfk . If 0 is small enough, the flow of X,λ defines a transition map from  \ { = 0} to T \ { = 0}. This a direct consequence of the exponential contraction property of transitions at jump points, which has been made precise in Theorem 12.9. We introduce (exterior) sections i = σi × [0, 0 ] at points mi on the fast orbits γfi , for i = 1, . . . , k − 1. We notice that, using the notation of Sect. 12.4, we have θi = θJi+1 , for i = 1, . . . , k − 1. Aside the projection  on γs1 introduced in Sect. 12.4, we introduce in a similar way, projections i (vi , λ) from σi to γsi+1 . In order to apply Theorem 12.9 at each jump, we parameterize each σi by a coordinate vi which is zero at the point γfi ∩ σi and such that vi > 0 corresponds to fast trajectories passing “above” the jump point Ji (see Fig. 12.12). Now, relative to the fast orbit γfi , the next jump point Ji+1 may be “located on the side vi > 0” (when the slow vector field on γsi+1 at i (vi , λ), with vi ∼ 0, points in the direction of increasing vi when parameterizing γsi+1 by vi → i (vi , λ) ∈ γsi+1 ), or may be “located on the side vi < 0” (in the opposite sense as in the former definition). For i = 1, . . . , k − 1, we denote by the Index θfi this relative position: +1 in the first case and −1 in the second case, see Fig. 12.12. i If ui is any smoothparameterization  of γs , oriented as the slow dynamics, we can see that θfi = sign ∂v∂ i [ui+1 ◦ i ] . Let us summarize the list of different coefficients that we have associated to the configuration of the sequence A (the precise definitions are given above) and let us define a global coefficient for A. 1. At each jump point Ji , for i = 2, . . . , k is associated the coefficient θJi , equal to +1 if the fast trajectory γfi−1 arrives on γsi “on the left side of the jump point at Ji ” and equal to −1 if not.

Ji

σi

Ji+1 mi+1 γsi+1

γsi (a)

= +1

Ji

σi

mi+1 γsi+1

γsi

Ji+1 (b)

= −1

Fig. 12.12 Passage from Ji to Ji+1 in an attracting sequence. (a) Ji+1 lies “above” Ji . (b) Ji+1 lies “below” Ji

256

12 Local Transition Maps

2. At each fast orbit γfi , for i = 1, . . . , k − 1 is associated the coefficient θfi , equal to +1 if the point Ji+1 is “above γfi ” and equal to −1 if not. 3. We define a coefficient θA for the whole sequence A by θA =

k %

θ Ji

k−1 %

j

θf .

(12.52)

j =1

i=2

Moreover, independently of the previous coefficients associated to A itself, we will use the index θ1 associated to the position of the starting section  in relation to γs1 .   For two points m, m on γsi , we write Ii ( m, m , λ) for the slow divergence integral along the arc from m to m on γsi (recall that this integral is well defined even when m or m coincides with the jump point Ji ). If v ∈ σ , the slow divergence integral of the sequence, starting at the point (v, λ), is the sum: I (v, λ) = I1 ([(v, λ), J1 ], λ)

(12.53)

+I2 ([1 (0, λ), J2 ], λ) + · · · + Ik ([k−1 (0, λ), Jk ], λ). We will call I (v, λ): slow divergence integral of the sequence A. It is a smooth function of (v, λ). We notice that the dependence on the variable v comes just from the first term, other terms depending only on the parameter λ. The transition studied in Sect. 12.4 is the particular case of a sequence of length 1, i.e. a sequence with a unique jump point. It is easy to extend Theorem 12.9 to a general sequence A. In particular, this result will be useful to study the jump breaking mechanism. Theorem 12.10 Consider an attracting sequence of arbitrary length. We consider as above, an exterior section  = σ × [0, 0 ] associated to the first slow arc γs1 and a section T = τ × [0, 0 ], transverse to the last fast orbit γfk . We take smooth coordinates v on σ and z on τ such that z is oriented as above and such that 0 is the coordinate of the point τ ∩ γfk . If 0 > 0 is small enough, any trajectory starting at a point of  \ { = 0} reaches the section T and defines a transition map (z, ) → (Z,λ (z), ). The z-component Z,λ (v) of the transition along this sequence, seen as a map from σ to τ , has the following expression:  Z,λ (v) =  2/3 (, λ) + θ1 θA exp

 I (v, λ) + O( 1/3 ) , 

(12.54)

where I (v, λ) is the slow divergence integral defined in (12.53). The function  is smooth in ( 1/3 ,  1/3 ln , λ) with (0, λ) = 0 and the term O( 1/3) is smooth in (v,  1/3 ,  1/3 ln , λ). Furthermore, Z,λ is, in the sense of Sect. 11.5, a family

12.5 Transition Along an Attracting Sequence

257

of diffeomorphisms of exponentially flat type on the sets of admissible monomials 0 = 1 = { 1/3 ,  1/3 ln }. Proof From Theorem 12.9 the first statement in the theorem is clear. We will now prove (12.54) by recurrence on the length k of the sequence. For k = 1, Theorem 12.10 reduces to Theorem 12.9 (the term θA is equal to 1 in this case). Consider now a sequence A of length k ≥ 2. We can write the z-component Z,λ of 2 ◦ Z 1 where Z 1 is the component its transition map as a composition Z,λ = Z,λ ,λ ,λ 2 is the component for a sequence for a sequence of length 1 (the jump at J1 ) and Z,λ of length k − 1, associated to the jump points J2 , . . . , Jk . By the hypothesis of recurrence we can write that 2

1 (v) : v1 =  3 1 (, λ) + θ1 exp Z,λ

˜ M(v, , λ) , 

and, using the recurrence hypothesis: 2

2 Z,λ (v1 ) : z =  3 2 (, λ) + θ2 θA exp

N˜ (v1 , , λ) . 

These two maps are families of diffeomorphisms of exponentially flat type, on the sets of admissible monomials 0 = 1 = { 1/3,  1/3 ln }. Moreover, we have that ˜ ˜ M(v, , λ) = I1 ([(v, λ), J1 ], λ)+O( 1/3) and N(w, , λ) = IA (w, λ)+O( 1/3 ), where IA is the slow divergence integral of the sequence A . 2 ◦ We can apply Proposition 11.3 of Sect. 11.5 to the composition Z,λ = Z,λ 1 Z,λ: Z,λ (v) =  2/3 (, λ)



+θ1 θ2 θA sign

∂I2 ∂v1

 exp

I1 ([(v, λ), J1 ], λ) + IA (0, λ) + O( 1/3 ) , 

where we have used the fact that the monomial  ln  is flatter than the monomial  1/3 . The slow divergence integral I (v, λ) of the sequence A is equal to I1 ([(v, λ), J1 ], λ) + IA (0, λ). Moreover, we have that 

∂I2 sign ∂v1



 ∂ [u2 ◦ 1 ] = θf1 . = sign ∂v1 

This gives 

∂I2 θ2 θ sign ∂v1 A

 = θ2 θf1 θA = θA .

The result follows with a function (, λ) that is smooth in ( 1/3 ,  1/3 ln , λ) and such that (0, λ) = k (0, λ) = 0. 

258

12 Local Transition Maps

Remarks 12.9 2

2

1. As (0, λ) = 0, the “translation” term  3  is equivalent to  3 (0, λ). By an argument of recurrence, we see that this term, equal to k (0, λ), is equal to the translation term of the last jump point Jk . This term is given in Lemma 12.5. The fact that it is nonzero follows from the geometric properties of the blownup vector field (explained in Sect. 8.5.3) on the blow-up locus of the last jump point Jk . Precisely, from the fact that the separatrix of the semi-hyperbolic point s3 does not coincide with the unique smooth invariant curve of the linear node given by the restriction of the blown-up vector field on the blow-up locus in the neighborhood of the saddle point s− , in normal coordinates (this curve is {Y = 0} in the normal coordinates). This corresponds to the fact that the coefficient a(0, λ) = 0 (see Sect. 12.4). It is easy to see that the fact that the equivalence of the translation term with 2  3 K, for a K = 0, does not depend on the choice of the smooth coordinate on τ . 2. If in the statement of Theorem 12.10 we would not work with an exterior section , but a section cutting the first slow arc γs1 transversally, with  = σm ×[0, 0 ], where σm is as in Remark 12.7(3) w.r.t. γs1 , then (12.54) changes into 

Z,λ (z) = 

2/3

 I (γs1 , λ) + O( 1/3) (, λ) + θA z exp , 

where I (γs1 , λ) stands for the slow divergence integral along γs1 from γs1 ∩ σm to the end point. The expressions  and O( 1/3) have the same properties as in (12.54). 

12.6 Transition Along a Hopf Attracting Sequence For the Hopf breaking mechanism, we have to consider a slightly different type of attracting sequence A than the one considered in Sect. 12.5. In such a sequence, that we will call a Hopf attracting sequence, every contact point associated to it is a jump point, except for the last one which is a turning point. Then the sequence contains k − 1 jump points J1 , . . . , Jk−1 and a last contact point which is a turning point T . In this section, we will use the rescaled parameter a¯ =  −1/2 a. In view of proving the results in this book, Proposition 6.1 can be applied to restrict a¯ to a compact set, in other words to justify the use of the rescaled parameter. Let again γs1 , . . . , γsk be the attracting arcs of the critical curve composing the sequence A (the last one γsk ends at the turning point T ) and let γf1 , . . . , γfk−1 denote the fast orbits present in A. Again, all these data depend smoothly on a parameter λ which is not explicitly mentioned and there is by assumption no singularity of the slow dynamics along the sequence. The formula (12.53) for I (v, λ) = I (v, a, μ) unfortunately does not remain well defined when Jk = T is a turning point, for values a different from 0 since the slow–fast Hopf point will bifurcate into a jump

12.6 Transition Along a Hopf Attracting Sequence

259

point together with a nearby slow singularity of saddle type on the attracting or repelling branch. We can define, for a = 0, the slow divergence integral of the Hopf attracting sequence A: it is a μ-family of integrals, denoted I (v, μ). Distinguishing  as a singular parameter from the other parameters (a, ¯ μ), one could also say that the integrals depend on (a, ¯ μ) and write I (v, a, ¯ μ) (the dependence on a¯ being trivial though), in order to be consistent with the previous section, where also the slow divergence integrals were dependent on the nonsingular parameters. We define a transition map associated to the Hopf sequence A, after blowing up the turning point T. This blowing up is described in Chap. 8. It produces a (a, ¯ μ)2 family of three-dimensional vector fields X¯ a,μ ¯ . If S+ is the blow-up locus, the lift of γsk that we identify with γsk arrives at the semi-hyperbolic point s3 (see Chap. 8). 2 , the point s has a unique separatrix . We choose an interior In restriction to S+ 3 section T = τ × [0, 0 ], in the blown-up space, transverse to . See Fig. 12.13. 2 The section τ is a one-dimensional section, transverse to X¯ a,μ ¯ , on S+ . We define an orientation of the section τ , i.e. a transverse orientation to the separatrix . This separatrix is part of the boundary of a disk D on the blow-up locus, in which X¯ a,μ ¯ is a center (see Fig. 8.6 in Chap. 8). We choose the orientation toward the exterior of D: i.e. the coordinate z on τ increases from the interior to the exterior of D. As in the previous section, we choose also an exterior section  = σ × [0, 0 ] associated to γs1 . Again, it is easy to see that a transition map is defined from \{ = 0} to T \ { = 0}. Let v and z be coordinates on σ and τ , respectively. As  is preserved by the transition, we are just interested to its z-component z = Z,a,μ ¯ (v). The coefficients θJi and θfi , for i = 1, . . . , k − 1 are defined as in Sect. 12.5. We define θT in a similar way as θJk in Sect. 12.5: this coefficient is equal to +1 if the fast orbit γfk−1 arrives to the left of γsk and equal to −1 if it arrives to the right, where left and right are defined in the same way as for a jump. We define θA as in Sect. 12.5, replacing θJk by θT in the formula for θA . For k = 1 the sequence is reduced to a single slow arc which ends at the turning point T . We will call the associate transition: interior jump at T. For k ≥ 2 the transition is the composition of the transition, from  to k−1 (an exterior section γsf γsf s3

s2

γsf

Blow up τ T

Fig. 12.13 Blow-up of the turning point and choice of section T

T

s3

260

12 Local Transition Maps

transverse to the last fast orbit γfk−1 ), along the sequence A associated to the jump points J1 , . . . , Jk−1 , with the interior jump at T from k−1 to the interior section T. We first consider the interior jump: Proposition 12.4 Let T be a turning point at which an attracting arc γs arrives (we suppose that γs \ {T } does not contain zeros of the slow dynamics). Choose an exterior section k−1 = σk−1 × [0, 0 ]. Let w be a smooth parameterization of the interval σk−1 and  be the projection from σk−1 to γs along the trajectories of X0,a,μ ¯ . As above, we blow-up the point T and take a section T = τ × [0, 0 ] transverse to the blow-up locus. Let z be a smooth parameterization of τ such that z = 0 is the coordinate of the point ∩ τ . If 0 > 0 is small enough, any trajectory starting at a point of k−1 \ { = 0} reaches the section T and defines a transition map (w, ) → (Z,a,μ ¯ (w), ). T The z-component Z, (w) of the interior jump, seen as a map from σk−1 to τ , a,μ ¯ has the following expression: 

T Z, a,μ ¯ (w)

=

1/2

 I ([(w), T ], a, ¯ μ) + O( 1/2 )  (, a, ¯ μ) + θk−1 exp ,  (12.55) T

¯ μ) and the O( 1/2) is smooth in (w,  1/2 ,  ln , a, ¯ where  T is smooth in ( 1/2 , a, T μ). The coefficient θk−1 is defined as in Sect. 12.5. In the sense of Sect. 11.5, Z, a,μ ¯ is a family of diffeomorphisms of exponentially flat type on the sets of admissible monomials 0 = {} and 1 = { 1/2,  ln }. Proof To simplify notations, we denote k−1 by , and the notations i below should not be confused with . The proof is rather similar and even simpler than the proof given for the transition at a jump point in Theorem 12.9. T We can see Z, a,μ ¯ as a composition of three maps: T 3 2 1 Z, = Z, a,μ ¯ ¯ ◦ Z,a,μ ¯ . a,μ ¯ ◦ Z,a,μ i we introduce two sections in a normal chart of the To define the transitions Z, a,μ ¯ semi-hyperbolic point s3 where the equation of the blown-up vector field is given by (12.21) in the coordinates (Z, v, ¯ u): 1 = σ1 × [0, 0 ] with σ1 ⊂ {u = u1 } and 2 = σ2 × [0, 0 ] with σ2 ⊂ {v¯ = v¯1 }, for some u1 > 0, v¯1 > 0, small 1 enough. The variable Z is called z1 on σ1 and z2 on σ2 . We define Z, a,μ ¯ (w) as the 3 2 transition from σ to σ1 , Z,a,μ ¯ (z2 ) as ¯ (z1 ) as the transition from σ1 to σ2 , and Z,a,μ the transition from σ2 to τ . We choose the orientation of the normal coordinate Z 3 directed outside the disk D, i.e. such that ∂z∂ 2 Z, > 0. a,μ ¯ 2 1 1 ◦ Z, is the To begin, we consider the composition Z,a,μ ¯ a,μ ¯ . The map Z,a,μ ¯ 2 same as the one in Sect. 12.4, and the map Z,a,μ is quite similar to the one in ¯ Sect. 12.4: Comparing Eqs. (12.24) and (12.22), we see that ¯ is changed into v¯ and that now, on the section σ1 we have that v¯ 2 = 2 . Using a quite similar proof, we u1

12.6 Transition Along a Hopf Attracting Sequence

261

obtain for the present composition, up to the change of notations, the same result as the one in Lemma 12.4. We give this result without a new proof (the coefficient θ1 in (12.56) comes from the choice of the orientation of Z): Lemma 12.6 

z2 =

2 Z, a,μ ¯

1 ◦ Z, a,μ ¯ (w)

 I ([(w, a, ¯ μ), J ], a, ¯ μ) + O( 1/2) = θ1 exp ,  (12.56)

¯ μ). where the term O( 1/2) is smooth in (w,  1/2 ,  ln , a, We can now finish the proof of Proposition 12.4. The section 2 is smoothly 3 parameterized by (u, z2 ). As u =  1/2/v, ¯ on 2 , the map Z, a,μ ¯ (z2 ) can be expanded as 3 1/2 Z, α(, a, ¯ μ) + β(, a, ¯ μ)z2 (1 + O(z2 )), a,μ ¯ (z2 ) = 

where α, β are smooth in ( 1/2 , a, ¯ μ) and the remainder is smooth in (z2 ,  1/2 , a, ¯ μ) (as z = 0 is the coordinate of ∩ τ , the term of order 0 is divisible by ). Moreover 3 T β(, a, ¯ μ) > 0. We obtain Z, ¯ : a,μ ¯ (w) by substituting (12.56) in Z,a,μ T Z, a,μ ¯ (w)

=

1/2

# $$  # I˜ I˜ 1 + O exp , α(, a, ¯ μ) + θ1 β(, a, ¯ μ) exp  

where we write I˜ for I ([(w, a, ¯ μ), J ], a, ¯ μ) + O( 1/2) to simplify the notation. ˜ The function f (w, , a, ¯ μ) = exp I / is smooth and flat in  at  = 0. It is hence the same for the function ln(1 + f ). Using this idea and the fact that β > 0, we can T rewrite the expression for Z, a,μ ¯ :  ˜ I + (ln β + ln(1 + f )) T 1/2 Z, . α(, a, ¯ μ) + θ1 exp a,μ ¯ (w) =   

As the function ln β + ln(1 + f ) is smooth in ( 1/2 , a, ¯ μ), this finishes the proof of Proposition 12.4 with  T = α, which is smooth in ( 1/2 , a, ¯ μ). 

We consider now the case of a general Hopf attracting sequence A of length k ≥ 2. We have the following result, similar to Theorem 12.10: Theorem 12.11 Consider a Hopf attracting sequence A of arbitrary length, ending at a turning point T . We consider as above, an exterior section  = σ × [0, 0 ] associated to the first slow arc γs1 and a section T = τ × [0, 0 ] transverse to the blow-up locus of T . Let also z be a smooth parameterization of τ oriented as above and such that z = 0 is the coordinate of the point ∩ τ .

262

12 Local Transition Maps

If 0 > 0 is small enough, any trajectory starting at a point of  \ { = 0} reaches the section T and defines a transition map (v, ) → (Z,a,μ ¯ (v), ). The z-component Z,a,μ ¯ (v) of the transition along this sequence, seen as a map from σ to τ , has the following expression:  1/2 Z,a,μ (, a, ¯ μ) + θ1 θA exp ¯ (v) = 

 I (v, a, ¯ μ) + O( 1/3 ) , 

(12.57)

where I (v, a, ¯ μ) is the slow divergence integral defined in (12.53). The function  is ¯ μ); the term O( 1/3) is smooth in (v,  1/2 ,  1/3 ,  1/3 ln , a, ¯ μ). smooth in ( 1/2 , a, Furthermore, Z,a,μ is, in the sense of Sect. 11.5, a family of diffeomorphisms of ¯ exponentially flat type on the sets of admissible monomials 0 = {} and 1 = { 1/2,  1/3 ,  1/3 ln }. Proof In the way that  was chosen in Proposition 12.4, let 1 = σ1 × [0, 0 ] be an exterior section associated to the last slow arc γsk , with σ1 parameterized by w. We can see the map Z,a,μ ¯ (v) associated to A as the composition of the map T Z, (v) associated to the sequence A with the map Z, a,μ ¯ a,μ ¯ (w) given by (12.55). T The study of the composition Z,a,μ ◦ Z,a,μ is quite similar to the study of the ¯ ¯ 2 1 composition Z,a,μ ◦ Z,a,μ made in the proof of Theorem 12.10. The map Z, ¯ ¯ a,μ ¯ T is given by (12.54) and the map Z,a,μ (w) given by (12.55). For the map Z , we ¯ ,a,μ ¯ write   (v, , a, ˜ 2 ¯ μ) I 3 Z, , ¯ μ) + θ1 θA exp a,μ ¯ (v) : w =   (, a,  where I˜ (v, , a, ¯ μ) = I (v, a, ¯ μ) + O( 1/3): I is the slow divergence integral of the sequence A and the term O( 1/3) is smooth in (v,  1/3 ,  1/3 ln , a, ¯ μ). The function  is smooth in ( 1/3 ,  1/3 ln , a, ¯ μ). T The map Z, a,μ ¯ is given by (12.55), with θ = θk = θT . Then, we can write 

T Z, a,μ ¯ (w)

:

z=

1/2

 ¯ μ) + O( 1/2 ) I˜T (w, , a,  (, a, ¯ μ) + θT exp ,  T

where I˜T (w, , a, ¯ μ) = I ([(w), T ], a, ¯ μ)+O( 1/2). The term O( 1/2) is smooth 1/2 in (v,  ,  ln , a, ¯ μ). The function  T is smooth in ( 1/2 , a, ¯ μ). We have T = Z, : z Z,a,μ ¯ a,μ ¯ ◦ Z,a,μ ¯

¯ μ) + θfk−1 θ1 θA θT exp =  1/2  T (, a,

I˜T (Z, ¯ μ) a,μ ¯ , , a,



.

12.6 Transition Along a Hopf Attracting Sequence

263

We see that θfk−1 θA θT = θA . The term I˜T (Z, ¯ μ) can be written as a,μ ¯ , , a,

  ¯ μ) = I˜T  1/3  (, a, ¯ μ), , a, ¯ μ I˜T (Z, a,μ ¯ , , a, $ # ˜ (v, , a, I˜ (v, , a, ¯ μ) ¯ μ) I T , , a, ¯ μ exp , +I˜1 exp   where I˜1T (w, , a, ¯ μ), as well as I˜T (w, , a, ¯ μ), is smooth in (w,  1/2 ,  ln , a, ¯ μ) T ˜ ¯ μ) < 0. and I1 (w, , a, Moreover, taking into account the smoothness properties of  , we see that the function I˜T ( 1/3 (, a, ¯ μ), , a, ¯ μ) is smooth in ( 1/2 ,  1/3 ,  1/3 ln , a, ¯ μ). The remainder of the proof is completely similar to the proof of Theorem 12.10. We will not repeat it. We finally obtain 1/2 (, a, ¯ μ) + θ1 θA exp Z,a,μ ¯ (v) = 

I˜(v, , a, ¯ μ) , 

with (, a, ¯ μ) =  T (, a, ¯ μ) +  −1/2 exp

¯ μ), , a, ¯ μ) I˜T ( 1/3  (, a, . 

Then  is smooth in ( 1/2 , a, ¯ μ). We also have ¯ μ), , a, ¯ μ) + I˜ (v, , a, ¯ μ) −  ln  +  ln H, I˜(v, , a, ¯ μ) = I˜T ( 1/3  (, a, where H > 0 is smooth. We see that I˜(v, 0, a, ¯ μ) = I ([(w), T ], a, ¯ μ) + I (v, 0, a, ¯ μ) is the slow divergence integral I (v, a, ¯ μ) of A. This gives the desired expression: : Z,a,μ ¯

z =  1/2 (, a, ¯ μ) + θ1 θA exp

I (v, a, ¯ μ) + O( 1/3 ) , 

where  is smooth in ( 1/2 , a, ¯ μ) and O( 1/3) is smooth in (v,  1/2 ,  1/3 ,  1/3 ln ). 

If, instead of z, we would like to use the coordinate y, ¯ as encountered in the family chart, defined by (x, y, v) = (ux, ¯ u2 y, ¯ u2 ), then expression (12.57) would change (since not only the orientation of z and y¯ are reversed, but one also has to take into account that z = 0 corresponds to a a-dependent ¯ value y-coordinate): ¯ : y¯,a,μ ¯

y¯ = (, a, ¯ μ) − θ1 θA exp

I (v, a, ¯ μ) + O( 1/3) , 

(12.58)

264

12 Local Transition Maps

where the properties of I (v, a, ¯ μ) and O( 1/3) are unchanged while  is still 1/2 smooth in ( , a, ¯ μ), but (0, a, ¯ μ) < 0 since cuts {x¯ = 0} at a negative value of y¯ (see Sect. 8.5.4). This observation introduces a possibility to express Theorem 12.11 in a way for which a blow-up is not needed in the statement of the theorem. We however do not know the existence of a proof that does not rely on blow-up. Theorem 12.12 Consider a Hopf attracting sequence A of arbitrary length, ending at a turning point T . Consider the (, a, μ)-family of vector fields in normal 1/2 a. form (6.4) near T as in Chap. 6, and consider the family X,a,μ ¯ ¯ by setting a =  Let, as above,  = σ × [0, 0 ] be an exterior section associated to the first slow arc γs1 and let T = τ × [0, 0 ], with τ = {x = 0} in the normal form coordinates (x, y). On τ we use z = y/ as a parameter, that is regular for  > 0. If 0 > 0 is small enough, any trajectory starting at a point of \{ = 0} reaches the section T and defines a transition map (v, ) → (Z,a,μ ¯ (v)). The z-component Z,a,μ ¯ (v) of the transition has an expression as in (12.58). We can also parameterize τ simply by y and as a result get Y,a,μ ¯ μ) − θ1 θA exp ¯ (v) = Z,a,μ ¯ (v) = (, a,

I (v, a, ¯ μ) + O( 1/3) ,  (12.59)

where (, a, ¯ μ) is a smooth function in ( 1/2 , a, ¯ μ) with (0, a, ¯ μ) < 0 and the 1/3 O( ) term in (12.59) has similar properties as O( 1/3) in (12.58). Proof The theorem is a direct consequence of Theorem 12.11 together with (12.58) if we recall that in the blow-up procedure we successively used (, a) = (ν 2 , ν a) ¯ and (x, y, ν) = (ux, ¯ u2 y, ¯ u). As such y = u2 y¯ = ν 2 y¯ =  y. ¯ We have that Y,a,μ ¯ μ) − θ1 θA exp ¯ (v) = Z,a,μ ¯ (v) = (, a,

I (v, a, ¯ μ) + O( 1/3) , 

and the -factor in front of the exponential can be added to the remainder term inside  O( 1/3 ) the exponential by writing  = exp(  log ). 

 ) = exp(  If in Theorems 12.11 and 12.12 we would not work with an exterior section , but with a section cutting the first slow arc γs1 transversally, with  = σm × [0, 0 ] as in Remark 12.7(3), then (12.57), (12.58), and (12.59) change in, respectively, 

I (γs1 , a, ¯ μ) + O( 1/3) (, a, ¯ μ) + θA z exp Z,a,μ ¯ (z) =     ¯ μ) + O( 1/3) I (γs1 , a, ¯ μ) − θA z exp y¯,a,μ ¯ (z) = (, a,  1/2

 (12.60) (12.61)

12.6 Transition Along a Hopf Attracting Sequence

265



 ¯ μ) + O( 1/3 ) I (γs1 , a, Y,a,μ ¯ μ) − θA z exp , ¯ (z) = (, a, 

(12.62)

¯ μ) stands for the slow divergence integral along γs1 from γs1 ∩ σm to where I (γs1 , a, the end point of γs1 , while the O( 1/3) terms have the same properties as in (12.57) and the functions  and  are smooth in ( 1/2, a, ¯ μ).

Chapter 13

Ordinary Canard Cycles

13.1 Introduction We consider a smooth slow–fast family X,λ defined on some surface M with a parameter λ in some compact domain P and  ∼ 0 and positive, as usual. In this chapter we are interested in canard cycles that are the simplest possible ones, meaning the ordinary one-layer canard cycles or equivalently where only one canard mechanism is part of the canard cycle, see Chap. 4. Because our results are local around a given canard cycle, it will be sufficient to know X,λ in a neighborhood of this canard cycle in M (nevertheless, some results will concern a whole family of canard cycles; in this case we have to know X,λ in a neighborhood of the support of this family of canard cycles). We will use the notions and definitions introduced in Part I (particularly in Chap. 4). As explained in Part I, we will just consider in this chapter and also in the following chapters of Part III, a canard cycle  with one layer. This means that the canard cycle contains just one attracting sequence and one repelling one. The passage from the attracting sequence A to the repelling sequence R is made through a canard connection of jump type or of Hopf type (in this last case we speak of Hopf attracting or repelling sequences, as defined in Chap. 12, Sect. 12.6). For canard connections of jump type, we will work directly with the family of vector fields X,λ , where λ = (a, μ). For canard connections of Hopf type, we  will work with the induced, restricted, family of vector fields X√ , as explained ,a,μ ¯ in Chap. 8. The results that we obtain for this induced family of vector fields are trivially extended to results for the initial family of vector fields X,λ , in view of Proposition 6.1. For the sake of simplicity, we will introduce a uniform notation for the parameters. In the jump case, it consists simply of a new notation for the breaking parameter: we will write b instead of a. In the Hopf case, we will define b = a. ¯ This way we can work with (, b, μ)-families of vector fields, and we will occasionally © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_13

267

268

13 Ordinary Canard Cycles

write η = (b, μ) for the nonsingular parameter. We will keep (b, η) in a compact set [−b0 , b0 ] × Q. The passage from the repelling sequence to the attracting one is through a fast orbit γ . We suppose the canard cycle to be regular in the sense that the slow curves belonging to γ consist of regular points of the slow dynamic outside the contact points. The α-limit of γ belongs to the last arc of R and its ω-limit belongs to the first arc of A. If we replace γ by a nearby fast orbit, we obtain a similar regular canard cycle. This means that γ belongs to a layer of fast orbits. In order to parameterize this layer we choose a one-dimensional open section σ transverse to the fast orbits and parameterized by a smooth coordinate v. We obtain a family of fast orbits γv (μ) (at b = 0, through the v ∈ σ ). Each of this orbit γv (μ) belongs to a regular canard cycle v (μ). We will also assume that there is no zero of the slow vector field all along the canard cycles v (μ). The results in this chapter extend the known results explained in Chap. 7.

13.2 Basic Settings We also consider a section  = σ × [0, 0 ], where we recall that σ is a onedimensional section transverse to the layer of fast orbits, defined above. At ω-limits of the orbits in the layer, attracting sequences start toward a canard breaking mechanism p of jump type or of Hopf type, and similarly, at α-limits of the orbits in the layer we assume repelling sequences start toward p. In case of a jump breaking mechanism we take a section T = τ × [0, 0 ] associated to the breaking mechanism, with z a smooth coordinate on the onedimensional section τ . The section τ is transverse to the fast orbit γf that “jumps” for a = 0 from the last jump point J A of the attracting sequence to the first jump point J R of the repelling sequence. See Fig. 13.1. We refer to Chap. 6 for more information. In case of a Hopf mechanism, the section τ is taken on the blow-up locus of the turning point T , transverse to the separatrix joining the two semi-hyperbolic points situated in the boundary of the blow-up locus (see Sect. 8.5.5). Recall that in this case we replace the parameter a by a rescaled parameter a¯ with a =  1/2 a, ¯ as in Chap. 8. See Fig. 13.2.

13.2.1 Difference Functions In order to study the bifurcation of limit cycles from the canard cycles v (μ), we could consider the return map P,λ , on the section  for instance. The map P,λ is a composition of several transitions and among them the transition along the repelling sequence R. But, it is preferable to restrict to transitions along attracting sequences, whose structure was established in Chap. 12. The reason not to consider transitions

13.2 Basic Settings

269

τ JA

JR

attracting sequence

repelling sequence

layer

σ Fig. 13.1 Ordinary 1-layer canard cycle with a canard jump breaking mechanism

repelling sequence

Hopf point

τ

Blow-up

σ

attracting sequence Fig. 13.2 Ordinary 1-layer canard cycle with a canard Hopf breaking mechanism

along repelling sequences (which are of course just inverses of transitions along attracting sequences!) is that such a transition “explodes” when  → 0, with the consequence that we have to restrict its domain to an interval depending on , of K size of order O(e−  ) for some K > 0. As such, in order to study the bifurcation of limit cycles, we prefer to work with the difference function ,λ on . This function is defined as follows. If F,λ is the forward transition from  to T along the flow of X,λ and if B,λ is the backward transition from  to T along the flow of −X,λ , we define ,λ = F,λ − B,λ .

(13.1)

270

13 Ordinary Canard Cycles

−1 The return map on T is equal to Q,λ = F,λ ◦ B,λ and we see that −1 (z). Q,λ (z) − z ≡ ,λ ◦ B,λ

(13.2)

−1 As B,λ is a diffeomorphism (for  > 0), a value z is a fixed point of Q,λ if and −1 (z) is a root of ,λ . Moreover, Eq. (13.2) says that the two families only if v = B,λ of functions Q,λ (z) − z and ,λ (v) are smoothly right equivalent through the −1 −1 family of diffeomorphisms B,λ , when  > 0. This implies that B,λ sends the roots of ,λ to the roots of Q,λ − Id (which are the fixed points of Q,λ ) and preserves their multiplicities (for instance a simple root of ,λ (v) is sent to a hyperbolic fixed −1 point of Q,λ ). But, as we have the conjugacy P,λ = B,λ ◦Q,λ ◦B,λ , the transition B,λ sends the fixed points of Q,λ bijectively to fixed points of P,λ , preserving their multiplicities. This means that the equations P,λ (v) − v = 0 and ,λ (v) = 0 have the same roots (as said above), but also with the same multiplicities. In conclusion, the bifurcations of the fixed points of the return map on  coincide with the bifurcations of the difference function ,λ , function that we will consider from now on.

13.2.1.1 The Case of a Jump Mechanism In Sect. 12.5, we have proved the existence of a forward transition map from  to T, A , ). The component Z A , seen as a map from along the flow of X,λ , equal to (Z,λ ,λ σ to τ , is given by the formula (12.54) with properties given in Theorem 12.10. To apply this formula we have to take on τ a coordinate that is equal to 0 at the point zA (λ). We take z − zA (λ) and moreover we assume that z > zA (λ) corresponds to the fast orbits passing above the jump at J A . Then, returning to the coordinate z on J (v) = zA (λ) + Z A (v). This gives σ , the transition map is equal to F,λ ,λ J (v) F,λ



 I A (v, λ) + O( 1/3 ) = z (λ) +   (λ, ) + θA θA exp ,  A

2 3

A

(13.3)

where I A is the slow divergence integral of the attracting sequence A as defined in Sect. 12.5. The function  A is smooth in ( 1/3 ,  1/3 ln , λ); the term O( 1/3) is smooth in (v,  1/3 ,  1/3 ln , λ) and we write θA instead of θ1 , taking care of the position of  w.r.t. A. R¯ , ) from  to T, along the flow of We also consider the backward transition (Z,λ −X,λ , where R¯ denotes the attracting sequence, inverse of the repelling sequence R¯ as in (12.54), we have to consider two different cases for the R. To express Z,λ choice of the right coordinate on τ : the bumps at J A and J R may be on the same side of γf or in opposite sides, see Fig. 13.3. We introduce a coefficient θC equal to

13.2 Basic Settings

271

Fig. 13.3 Different cases: the bumps of J are on the same side of γf or in opposite sides.

+1 in the first case and to −1 in the second case. Then, with the coordinate z, the J R¯ (v). This gives transition map is given by B,λ (v) = zR (λ) + θC Z,λ 2

¯

J (v) = zR (λ) +  3  R (λ, ) + θC θR¯ θR¯ e B,λ

¯ I R (v,λ)+O( 1/3 ) 

,

(13.4)

¯ where I R is the slow divergence integral of the attracting sequence R¯ as defined ¯ in Sect. 12.5. The function  R is smooth in ( 1/3 ,  1/3 ln , λ),the term O( 1/3) is 1/3 1/3 smooth in (v,  ,  ln , λ), and we again write θR¯ for θ1 , taking care of the ¯ position of  w.r.t. R. The intersection points of limit cycles with σ are the zeros of the difference J − B J given by function J,λ = F,λ ,λ 2

J,λ (v) = a +  3  J (λ, ) + θA θA e

I A (v,λ)+O( 1/3 ) 

− θC θR¯ θR¯ e

¯ I R (v,λ)+O( 1/3 ) 

. (13.5) The function  J is smooth in ( 1/3 ,  1/3 ln , λ) and terms O( 1/3) are smooth in (v,  1/3 ,  1/3 ln , λ).

13.2.1.2 The Case of a Hopf Mechanism In this case the two sequences are of Hopf type, and the corresponding forward and backward transitions are given by Theorem 12.11. A simplification in comparison with the previous case is that we can choose the same coordinate z on τ for the two transitions, in order to apply Theorem 12.11 (a coordinate that is zero at the point τ ∩ and is oriented in the direction pointing outside the center disk; see Sect. 12.6). The separatrix is broken when a¯ = 0 and we have two different separatrices, related to A and R, respectively, cutting τ at the points of coordinate zA (λ) and zR (λ), respectively. In this case we prefer to take λ = (a, ¯ μ). The forward and

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13 Ordinary Canard Cycles

backward transitions have the following expression: A F,Ha,μ ¯ μ) +  A (a, ¯ μ, ) + θA θA e ¯ (v) = z (a,

1/3 ) I A (v,a,μ)+O( ¯ 

,

(13.6)

¯ 1/3 ) I R (v,a,μ)+O( ¯ 

,

(13.7)

and ¯

H R B, ¯ μ) +  R (a, ¯ μ, ) + θR¯ θR¯ e a,μ ¯ (v) = z (a, ¯

where I A and I R are the slow divergence integrals of the attracting sequences A and ¯ as defined in Chap. 12. The functions  A ,  R¯ are smooth and terms O( 1/3) are R, smooth in (v,  1/2 ,  1/3 ,  1/3 ln , a, ¯ μ). (Here, H stands for Hopf.) In the Hopf case, we see that zA (a, ¯ μ) − zR (a, ¯ μ) is a smooth function equal ∂β to β(a, ¯ μ) such that ∂ a¯ (a, ¯ μ) = 0, for any (a, ¯ μ) (one can find some more precisions on β in [DR01b]). It follows that the difference function has the following expression: H ¯ μ) +  H (a, ¯ μ, ) + θA θA e ,a,μ ¯ (v) = β(a,

1/3 ) I A (v,a,μ)+O( ¯ 

− θR¯ θR¯ e

¯ 1/3 ) I R (v,a,μ)+O( ¯ 

.

(13.8) In this equation, the function  H is smooth in (a, ¯ μ,  1/2 ) and the remainder term 1/3 1/2 1/3 1/3 O( ) is smooth in (v,  ,  ,  ln , a, ¯ μ).

13.2.1.3 A Common Expression for the Two Difference Functions The two different canard mechanisms can be treated in a similar way. We denote by ,η (v) the difference map in both cases, continuing writing F,η (v) and B,η (v) for the respective forward and backward transitions. In the formulas (13.5) and (13.8) for the difference functions, we have indicated a difference of smoothness in , between the translation terms and the other terms entering in the exponentials. These precisions will not be very useful in the remainder of the text. In order to unify the expression we introduce the following definition. Definition 13.1 The Index θC is defined as above for the jump case and is taken conventionally equal to +1 for the Hopf mechanism. The set  of -monomials is equal to { 1/3,  1/3 ln } in the jump case and is equal to { 1/2,  1/3 ,  1/3 ln } in the Hopf case. With these conventions, we replace the two formulas for the difference functions by a unique one: ,η (v) = α(η, ) + θA θA e

I A (v,η)+o (1) 

− θC θR¯ θR¯ e

¯ I R (v,η)+o (1) 

,

(13.9)

13.2 Basic Settings

273

where α and the remainder terms o (1) are smooth in the appropriate set  of monomials and in the other variables (v, η). The function α verifies α((b, μ), 0) = b.

13.2.2 Tubular Neighborhood of the Canard Cycle Let us consider a canard cycle  = v0 (μ0 ). It is a simple curve homeomorphic to the circle S 1 . As it is piecewise smooth, it is easy to see that it has tubular neighborhoods diffeomorphic to the annulus S 1 × [0, 1] or to the Möbius band. We will say that  is an orienting curve in the first case and a disorienting curve in the second case. The second case may only occur when the surface M is non-orientable. First, we introduce the following topological index associated to : Definition 13.2 Let θC be defined in Definition 13.1 above. For the two cases of jump or Hopf mechanism, the Topological index of canard cycle  is defined by θ = θC θA θA θR¯ θR¯ sign

 ∂I A  ∂v

sign

 ∂I R  , ∂v

(13.10)

where the coefficients are θA , θA , θR¯ , θ R¯ are defined in Chap. 12. The topology of tubular neighborhood of the canard cycle  can be related with its topological index θ : Proposition 13.1  is an orienting curve if and only if θ = +1 and a disorienting curve if and only if θ = −1. Proof Let us first consider the case of a jump mechanism. We consider the family of difference functions given by (13.5) and just retain that it is smooth in (, v, b, μ). For  = 0 this function reduces to the parameter b. It follows that there exist functions b(, μ) and v(, μ), defined for  > 0 small enough, such that for η = (b(, μ), μ) we have that J,η (v(, μ)) = 0. Moreover, we can assume that v(, μ) → v0 and b(, μ) → 0, when  → 0, uniformly in μ. The orbit of X,η through v(, μ) is closed, i.e. it is a limit cycle γ (, μ). The return map on the J )−1 ◦ F J . Clearly, the sign section σ , defined near v(, μ), is the composition (B,η ,η of its derivative is precisely θ . Then, the limit cycle γ (, μ) is an orienting curve if and only if θ = +1 and disorienting curve if and only if θ = −1. Now, as γ (, μ) →  in the Hausdorff sense when  → 0, we have that γ (, μ) and  have diffeomorphic tubular neighborhoods for  small enough and the result follows for the case of a jump mechanism. Let us now consider the case of a Hopf mechanism. As we know, the study is based on the blow-up of the turning point T . For b = 0, there exists a blown-up canard cycle ˆ that is the union of the strict lift of  in the blown-up space with the separatrix . This curve ˆ belongs to the singular boundary ∂E of the blown-up

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13 Ordinary Canard Cycles

space E (see Chap. 8). A similar argument as above proves the result for this blownup canard cycle, seen as a curve on ∂E, i.e. that ˆ has oriented or non-oriented tubular neighborhood, depending on θ . Now, it is easy to see that any sufficiently thin tubular neighborhood of ˆ is isotopic in ∂E to an open set that is blown down into a tubular neighborhood of  (a thin tubular neighborhood of ˆ blows down into a “pinched” tubular neighborhood of , and of course we can just consider ˆ The results follow for the case of a Hopf “rather thin” tubular neighborhood of ). mechanism. 

Remark 13.1 A consequence of Proposition 13.1 is that configurations of canard cycle corresponding to θ = −1 cannot happen if M is orientable. In other words, if M is orientable we have necessarily that θ = +1 for any canard cycle . On a non-orientable surface M, we may have θ = ±1 depending on the embedding of  into M, as an orienting or a disorienting curve. Let us also notice that we can choose a tubular neighborhood W of  containing  in its interior and containing the strip of all the canard cycles cutting . These canard cycles are simultaneously pinched by the canard mechanism. They are isotopic between themselves in W and we can replace  by any other canard cycle of the strip.

13.3 Results of Bifurcation We want to look for the limit cycles, cutting the interior Int() of the section , for  > 0 small enough. We can choose a tubular neighborhood W of  such that ∂ ⊂ ∂W and such that any such limit cycle is contained in W . When  is an orienting curve, any limit cycle contained into W is isotopic to . In this case and for  > 0 small enough, each limit cycle cutting Int() has a unique intersection with  (this means that the periodic points of a return map are all fixed points). As said above, these intersection points are in 1–1 correspondence with the roots of the difference map. Then, when  is an orienting curve, the limit cycles bifurcating from the canard cycles cutting  are given by the following equation: ,η (v) = 0,

(13.11)

where ,η is given in (13.1). When  is a disorienting curve, there exists only one limit cycle contained in W and that is isotopic to , when  > 0 is small enough. Again the corresponding intersection point with Int() is a root of Eq. (13.11), root that is necessarily simple, as ,η is strictly monotone in this case (see below). But there may also be other limit cycles contained in W , which are isotopic to a double covering of  (and 2 , are cutting  twice). To study them, we could consider the fixed points of P,η where P,η is a return map of the flow of the vector field on . Unfortunately, as explained above, our methods do not allow to consider this return map and surely

13.3 Results of Bifurcation

275

not its square! So, we do not try to study these limit cycles that may just occur on a non-orientable surface.

13.3.1 A Mild Preparation for Eq. (13.11) In this section we improve the structure of the equation ,η (v) = 0. Notice that the layer variable only appears in the exponential terms and not in the affine term α(η, ) = α(b, μ, ). We will now derive an equivalent equation where the breaking parameter b only appears in the affine term and not in the exponential terms, this way separating the layer variable from the breaking parameter in the equation. Deriving the function (v, η, ) = ,η (v) given by (13.1) w.r.t. b, we obtain ∂α − K ), for some positive K, so that ∂ that ∂ ∂b = ∂b + O(e ∂b = 0. Hence, using the Implicit Function Theorem, we can solve Eq. (13.11) to obtain a function b(v, μ, ) that is smooth in (, v, μ) and that is of order o (1). ¯ Recall that the slow divergence integrals I A and I R are functions of the parameter η. With a mild abuse of notations we write ¯

¯

I A (v, μ) = I A (v, (0, μ)) and I R (v, μ) = I R (v, (0, μ)). ˜ ,η by replacing the parameter b with the We construct now a new function function b(v, μ, ) in the exponential terms of (13.1). For instance, the function I˜A (v, η, ) = I A (v, η) + o (1) is changed into a function independent of b: I˜A (v, μ, ) = I˜A (v, (b(v, μ, ), μ)) = I A (v, μ) + o (1). ¯

¯

The same is true for the function I˜R (v, η, ) = I R (v, η)+o (1). With new functions ¯ I˜A and I˜R , we obtain the following expression: ˜ ,η (v) = α(η, ) + θA θA e

I˜A (v,μ,) 

− θC θR¯ θR¯ e

¯ I˜R (v,μ,) 

,

(13.12)

¯ ¯ with functions I˜A (v, μ, ) = I A (v, μ) + o (1) and I˜R (v, μ, ) = I R (v, μ) + o (1) independent of b. One could think that substituting a solution of back into the equation would render a new equation 0 = 0, but it is not the case because we have not substituted the solution for b in the term α. The expression (13.12) is completely similar to (13.1). The remainders o (1) are now independent of b and are smooth in (, v, μ) and α is smooth in (, η). ˜ ,η (v) = 0} define the same set, so Now, the two equations { ,η (v) = 0} and { that we can replace (13.11) by an equivalent equation:

˜ ,η (v) = 0.

(13.13)

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13 Ordinary Canard Cycles

13.3.2 The Canard Phenomenon Let us fix any v0 ∈ σ . Then the system has a limit cycle passing through v0 if ˜ ,η (v0 ) = 0. By the same argument as and only if the parameter (η, ) verifies ˜ ,η (v0 ) = 0 to find a function b = b(μ, ). This above we can solve the equation 2

function is smooth in (μ,  1/3 ,  1/3 ln ) and of order O( 3 ) in the jump case; it is smooth in (μ,  1/2 ) and of order O( 1/2) in the Hopf case. (We will actually see in the next subsection that the smoothness in the Hopf case is better.) The function so defined depends on the choice of v0 , but this dependence only appears in an -flat term. Then its formal series (in ( 1/3 ,  1/3 ln ) or  1/2 , depending on the case) is well defined (independent of the choice of v0 ). Definition 13.3 The surface {b = b(μ, )} is called a Canard surface; it is well defined up to -flat terms. Take a constant K > 0 such that    ¯ K < min min |I A (v, μ)|, |I R (v, μ)| | (v, μ) ∈ σ × Q .

(13.14)

K

The -flat term associated to the change of v0 is bounded by e−  if  ∈ [0, 0 ], for 0 small enough. Then, we can state the so-called canard phenomenon: Proposition 13.2 Let {b = b(μ, )} be a canard surface. There is 0 > 0 small enough such that the vector field has no limit cycle intersecting σ if (μ, ) ∈ Q × K [0, 0 ] and |b − b(μ, )| > e−  . Remark 13.2 The above proposition deals essentially with the induced, (, b, μ)family of vector fields, which in case of a jump breaking mechanism coincides with  X,λ and in case of a Hopf breaking mechanism is given by X, a,μ ¯ . In the latter case, the results from Proposition 13.2 are trivially extended to the original family X,λ , taken into account Proposition 6.1 (see also Remark 8.3). A rough way to state the canard phenomenon is to say that limit cycles can bifurcate from the canard layer associated with the section σ , only when the breaking parameter b belongs to the exponentially flat tongue defined by |b − K b(μ, )| ≤ e−  .

13.3.3 Formal Power Series Expansion of the Canard Surface for Generic Hopf Breaking Mechanisms In case of a generic Hopf breaking mechanism, we √ prove that the function  1/2 b(μ, ) is actually smooth in (μ, ) (instead of in (μ, )).

13.3 Results of Bifurcation

277

The affine term zA (a, ¯ μ) +  A (a, ¯ μ, ) appearing in (13.6) is where a single choice of center manifolds at the point s3 on the blow-up locus of the slow–fast Hopf point is mapped to under positive flow of the vector field. Along a canard surface, we have an (, μ)-family of orbits that blow up to an invariant manifold in blow-up space; it joins together a μ-family of center manifolds at s2 with a μ-family of center manifolds at s3 . It could be noticed that the mentioned center manifold is not of the same type as the ones introduced in Chap. 9. In Chap. 9, the (b, μ)-families of center manifolds are defined in 3-space (x, y, ). Here, the mentioned center manifold is defined for b = b(μ, ): the center manifold is restricted to a codimension 1-manifold of R3+p , where p is the dimension of the parameter μ. In the rest of this subsection b = b(μ, ) will be the implicit solution ˜ ,η = 0, where we disregard the exponential terms in the of the equation equation. The difference between this implicit solution and the function b(μ, ) in Proposition 13.2 is clearly exponentially small, so we can safely proceed with this implicit solution in view of proving our claim concerning the smoothness. Consider again the family chart in the blow-up of the Hopf point. The blown-up invariant manifold is a graph y¯ = Wμ (x, ¯ u) = x¯ 2 −

1 + O(u). 2

Let us now relate the power series expansion of both the canard surface a¯ = b(μ, ) and this invariant manifold to the so-called outer expansions given in Proposition 6.2 but now written in terms of (a, ¯ ν) instead of (a, ): y = ϕ(x, ˆ ν 2 , μ) = x 2 +

∞ !

ϕk (x, μ)ν 2k ,

ˆ ν 2) = a¯ = ν −1 A(μ,

k=1

∞ !

Ak (μ)ν 2k−1 .

k=1

ˆ Proposition 13.3 The Taylor series of b(μ, u2 ) is equal to the series u−1 A(μ, u2 ) −2 2 w.r.t. u. The Taylor series of Wμ (x, ¯ u) is equal to the series u ϕ(u ˆ x, ¯ u , μ) w.r.t. u. Proof We start from the matching chart {x¯ = 1}, where the semi-hyperbolic ¯ u, ) ¯ = (1, 0, 0). There, any smooth center singularity s3 is found at the point (y, manifold is of the form y¯ = Za,μ ¯ Such an expression blows down to ¯ (u, ν). y = x 2 Za,μ ¯ (x, ν/x). Writing the same function again in the family chart, we find y¯ = x¯ 2 Za,μ ¯ 1/x). ¯ It is defined for x¯ sufficiently large. If we restrict to ¯ (ux, a¯ = b(μ, u2 ), then we find y¯ = Wμ (x, ¯ u) = x¯ 2 Zb(μ,u2 ),μ (ux, ¯ 1/x). ¯

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13 Ordinary Canard Cycles

It has a power series in u with coefficient functions in terms of (x, ¯ a, ¯ μ). By construction, the uk -coefficient of this series is bounded in growth by Ck |x|k+2 . It means that the coefficient functions all have at most polynomial growth toward +∞ and, by a similar argument near s2 , also toward −∞. By Proposition 8.1 this polynomial growth property shows that the formal series of b(μ, u2 ) and Wμ (x, ¯ u) necessarily coincide with the blown-up versions of the expansions formulated in front of this proposition. 

Corollary 13.1 The function  1/2 b(μ, ) is C∞ in (μ, ). Proof Since the series expansion of b(μ, u2 ) coincides with the blown-up series −1 ˆ 2 u ) (that is in terms of √ A(u , μ), the original asymptotic expansion of b(μ, ˆ μ). The former is, ) also coincides with the asymptotic expansion  −1/2 A(, however, an expansion in , if we disregard the leading coefficient  −1/2 ; hence also  1/2 b(μ, ) admits an expansion in terms of  and is a smooth function. 

Remark 13.3 By similar arguments, one could prove that the (, μ)-family of limit cycles crossing σ at v0 is, locally in a smooth normal form coordinates around the slow–fast Hopf point, expressed as C∞ graphs of the form y = ϕ(x, , μ). See [DMD06] for details.

13.3.4 Canard Explosion, Flying Canard, and Sitting Canards In the context of Sect. 13.3.2, one can also encounter the notion “Canard explosion,” a name that first appeared in [BBE91] and that was later popularized by M. Krupa and P. Szmolyan in [KS01b]. It describes well that, for  sufficiently small, a limit cycle grows (or shrinks) in a very fast way under changes of the parameter b. Let us explain this remarkable phenomenon on an open layer of unbalanced canard cycles. Let σ be such a layer with v1 ∈ σ and suppose that ¯

I A (v1 , μ) < I R (v1 , μ) < 0. ¯

(13.15)

(The case I R (v1 , μ) < I A (v1 , μ) < 0 can be treated by reversing time.) Limit cycles tending to v for  → 0 occur for b belonging to some (v, μ)-depending control curve that we will now write as b = b(v, μ, ). The curve is implicitly defined by ˜ ,(b(v,μ,),μ)(v) = 0, which we can write as ˜ (v, b(v, μ, ), μ, ) = 0.

13.3 Results of Bifurcation

279

We hence get the following (13.12): $ # ˜ ˜ ∂ ∂b ∂ (v1 , μ, ) = − / (v1 , b, μ, ) ∂v ∂v ∂b $  −1 # ¯ ∂α ∂ I˜A I˜A / ∂ I˜R I˜R¯ / . e e =− · θA θA − θC θR¯ θR¯ ∂b ∂v ∂v Because of (13.15) we see that  ∂b 1  R¯ (v1 , μ, ) = ± exp I (v1 , μ) + o (1) , ∂v 

(13.16)

where o (1) is smooth in (, v, μ). Reversing (13.16) and writing b1 = b(v1 , , μ), we get  ∂v 1  R¯ (b1 , μ, ) = ± exp −I (v1 , μ) + o (1) , ∂b  showing that the “position” (i.e. the v-value) of the limit cycles near an unbalanced canard cycle changes exponentially fast with respect to b for  → 0. In [Ben81] such a limit cycle was called a “Flying canard,” but we prefer to keep the word “canard” for the limiting canard cycle that we find for  = 0. Benoit claims that there can be at most one “flying canard,” a statement that we will make precise now, while proving it in the case the layer of canards under consideration is a terminal layer (see Definition 4.6 for the distinction between terminal layer and dodging layer, which is relevant in the sequel) on an orientable manifold. Afterward, we will discuss the possibility of two flying canards in case of a dodging layer (still on an orientable manifold). We therefore restrict the parameter b to b1 (μ, ) = b(v1 , μ, ) as above and try to find canard cycles v0 that can generate closed orbits keeping b = b(v1 , μ, ). We have to solve ˜ (v, b1 (μ, ), μ, ) = 0. Based on (13.12), this reduces to exp

¯ ¯ I˜A (v1 , μ, ) I˜R (v, μ, ) I˜R (v1 , μ, ) I˜A (v, μ, ) − exp = exp − exp ,     (13.17)

where we have used that θA θA θC θR¯ θR¯ = 1. It is here that we use the fact that we  A have a terminal layer and we work in an orientable setting: θ = 1 and sign ∂I∂v =  R sign ∂I∂v , see also (13.10).

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13 Ordinary Canard Cycles

It immediately follows that no limit cycles can bifurcate from v0 when ¯ ¯ I A (v0 , μ) > I A (v1 , μ) and I R (v0 , μ) > I R (v1 , μ). We can now look for limit ¯ cycles bifurcating from v0 , when I A (v0 , μ) < I A (v1 , μ) and I R (v0 , μ) < ¯ I R (v1 , μ). In that case, (13.17) at v0 can be written as exp

¯ I˜A (v0 , μ, ) I˜R (v0 , μ, ) (1 + o (1)) = exp (1 + o (1)).  

In this equation, the 1 + o (1)-factors can be entered in the exponentials after which ¯ the equation simplifies to I˜A (v0 , μ, ) = I˜R (v0 , μ, ) + o (1). We see that, for ¯  sufficiently small, solutions are only possible if I˜A (v0 , μ, 0) = I˜R (v0 , μ, 0), hence if I A (v0 , μ) = I R (v0 , μ), meaning that we have a balanced canard cycle. Limit cycles near such a balanced canard cycle have been called “sitting canards” in [Ben81]. Both “flying canards” and “sitting canards” have a v-location that is extremely sensitive to changes in the parameter b. The terminology “sitting canards” comes from the fact that ∂v0 = o (1), ∂v1 ∂v0 ∂v1 ∂v1 0 and it implies ∂v ∂b = ∂v1 ∂b = ∂b .o (1). In other words, the ratio of the way how v0 changes w.r.t. b versus how v1 changes w.r.t. b is asymptotically negligible (even exponentially small if one would do the effort to quantify the o (1)-factor). As ∂α ∂b = 0, we can find, by the Implicit Function Theorem, a function b(μ, ), smooth in (μ, ), and solution of α((b(μ, ), μ), ) = 0 with b(μ, 0) = 0. The equation b = b(μ, ) defines a hypersurface H in the parameter space (a curve for ˜ ,η = 0 reduces to each μ) such that in restriction to H , the equation

exp

¯ I˜R (v, μ, ) I˜A (v, μ, ) = exp ,  

and hence also as ¯ I˜(v, μ, ) := I˜A (v, μ, ) − I˜R (v, μ, ) = 0,

with I˜(v, μ, ) = I (v, μ) + o (1), where o (1) is smooth in (v, μ, ) and I (v, μ) is the slow divergence integral of μ (v). The limit cycles for (v, η, ) → (v, (0, μ0 ), 0) hence behave like the roots of the equation I (v, μ) = 0 as long as we consider stable behavior (like number and multiplicity or stable elementary catastrophes of zeros). For example a single

13.3 Results of Bifurcation

281

μ-family of hyperbolic limit cycles will bifurcate from v0 when v0 is a simple zero of I (v, μ0 ). On a dodging layer however, instead of (13.17) we have to consider exp

¯ ¯ I˜A (v, μ, ) I˜A (v1 , μ, ) I˜R (v, μ, ) I˜R (v1 , μ, ) − exp = − exp + exp .     (13.18)

¯ Based on (13.15) the term involving I˜R dominates the one with I˜A , so the equation reduces to

exp

¯ ¯ I˜R (v, μ, ) I˜R (v1 , μ, ) + o(1) I˜A (v, μ, ) = − exp + exp .    R¯

Suppose that I∂v (v, μ) < 0. Then for v < v1 , by keeping again the dominant term in the right hand side of the equation, one can see that the right hand side is negative, so there are no solutions. When v > v1 on the other hand, the equation reduces, for sufficiently small  > 0, to exp

¯ I˜R (v1 , μ, ) + o(1) I˜A (v, μ, ) = exp ,   ¯

which simplifies to the equation I A (v, μ) = I R (v1 , μ) + o(1). Observe that ¯

(v, μ) → I A (v, μ) − I R (v1 , μ) is strictly negative at v = v1 , and since we have assumed that IA ∂v (v, μ)

¯

IR ∂v (v, μ)

< 0 (which

> 0), the above map is monotonically implies in a dodging layer that increasing w.r.t. v, making it feasible that a zero in {v > v1 } is possible in sufficiently large layers. Observe that at a point v = v0 where the above equation is ¯ ¯ ¯ solved, one automatically has I R (v0 , μ) < I R (v1 , μ), so I A (v0 , μ) − I R (v0 , μ) < 0. In other words, a second flying canard is found. Observe also ¯

∂I R ∂I A ∂v0 (v1 , μ)/ (v0 , μ) + o (1) = ∂v1 ∂v ∂v ∂v0 ∂v1 0 and by ∂v ∂b = ∂v1 . ∂b = a comparable speed).

∂v1 ∂b (c

+ o (1)), it shows that indeed both canards “fly” (at

Remark 13.4 In case a second flying canard is found at v0 , there is always a zero of the slow divergence integral between v0 and v1 , but unlike in the case of a terminal layer, there is no extra cycle located near the zero of the slow divergence integral.

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13 Ordinary Canard Cycles

13.3.5 Counting the Limit Cycles over a Whole Layer Strip Because of the presence of exponentially flat terms, it is difficult to study the bifurcations of limit cycles directly based on Eq. (13.13). In this subsection we will simply count the number of bifurcating limit cycles. It can be done at once for limit cycles bifurcating from the whole layer strip of canard cycles cutting σ . ˜ ∂

We therefore consider the equation ∂v,η = 0, which reveals to be equivalent to a simple regular equation, and next to apply Rolle’s Theorem to obtain a bound on the number of bifurcating zeros of (13.13). Using the topological index θ introduced in Definition 13.2 and Lemma 11.9, we get θA θA sign

 ∂I R¯  ∂ ˜ ∂v

,η

∂v

(v) = e

I A (v,μ)+o (1) 

− θ e

¯ I R (v,μ)+o (1) 

,

(13.19)

where the remainders o (1) are smooth in (, v, μ). Depending on the sign of θ , there are two very different cases, as the  may be an orienting or a disorienting curve. See Fig. 13.4.

13.3.5.1 The Non-orientable Case We first assume that θ = −1. Recall that this may occur only when  is a disorienting curve. This implies that the surface M itself is non-orientable. In this ˜ ∂

case, using (13.19), we have that ∂v,η (v) has a constant sign for any (v, η) and  > 0 small enough. Then ,η has at most one root v(η, ) ∈ σ , which must be simple. The return map P,η reverses the orientation of σ and v(η, ) is a hyperbolic fixed point with a negative eigenvalue. As seen above, v(η, ) exists for any μ ∈ Q,  > 0 small enough and b ∈ K [b− (μ, ), b+ (μ, )], where |b− (μ, ) − b+ (μ, )| ≤ e−  , for some K > 0. Then

θΓ = 1 Fig. 13.4 Orienting and disorienting curves.

θΓ = −1

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283

the bifurcation of fixed points of P,η resumes to the passage of a single hyperbolic point, crossing the interval σ when b crosses [b− (μ, ), b+ (μ, )]. Of course, the K  “speed” | ∂v(η,) ∂b | tends to +∞ when  tends to 0, with the order e (the so-called flying canard phenomenon). As explained above, there may exist other bifurcating limit cycles, not isotopic to .

13.3.5.2 The Orientable Case We assume now that θ = +1. In this case, any limit cycle bifurcating from the layer strip is isotopic to , and as said above, this means that all these limit cycles ˜ ,η . Equation (13.19) writes coincide with roots of the difference function  ∂I R¯  ∂ ˜ ,η 1

θA θA sign

∂v

∂v

(v) = e

I˜1A (v,μ,) 

−e

¯ I˜1R (v,μ,) 

,

¯ ¯ where I˜1A (v, μ, ) = I A (v, μ) + o (1) and I˜1R (v, μ, ) = I R (v, μ) + o (1) are smooth functions in (, v, μ). η −eξ We define the analytic function E(η, ξ ) = eη−ξ for η = ξ and E(ξ, ξ ) = eξ . 2 We notice that E(η, ξ ) > 0 for any (η, ξ ) ∈ R . Using this function we can write

θA θA sign

 ∂I R¯  ∂ ˜ ,η 1 ∂v

∂v

¯ ¯ ¯ (v) = E(I˜1A , I˜1R )(I˜1A − I˜1R ) = E(I˜1A , I˜1R )I˜1 (v, μ, ),

(13.20) ¯ ¯ where I˜1 = I˜1A − I˜1R . If I (v, μ) = I A (v, μ) − I R (v, μ) is the slow divergence integral of the canard cycle v (μ), we get

I˜1 (v, μ, ) = I (v, μ) + o (1),

(13.21)

where I˜1 and o (1) are smooth in (, v, μ). ¯ As E(I˜1A , I˜1R ) > 0, Eq. (13.20) shows that the two families of functions  R¯  ˜ ∂I ∂ ,η ˜ θA θA sign ∂v1 ∂v (v) and I1 (v, μ, ) are smoothly equivalent for  > 0, so that these two families have the same roots with the same multiplicity. Remark 13.5 When  → 0, the two sides of (13.20) tend to 0 in a flat way in : in ˜ ∂ some sense, replacing { ∂v,η (v) = 0} by {I˜1 (v, μ, ) = 0} is a way to desingularize the first equation. Recall from Sect. 13.1 that bounds on the number of limit cycles of the (, b, μ)family of vector fields lead to similar bounds for the original family of vector fields X,λ . As a consequence, under non-degeneracy conditions, the limit cycles will be controlled by the roots of the slow divergence integral:

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13 Ordinary Canard Cycles

Theorem 13.1 Let us assume that the slow divergence integral I (v, μ) has just roots with finite multiplicity on σ . Then, the number of these roots, including their multiplicity, is bounded by some finite constant N. Moreover, if 0 > 0 and a0 > 0 are small enough, at most N + 1 limit cycles of X,λ = X,a,μ cut σ , for (, a, μ) ∈ ]0, 0 ] × [−a0 , a0 ] × Q. Proof Let N(μ) be the finite number of roots of I (v, μ), counted with their multiplicity. This function is upper semi-continuous (N(μ0 ) ≥ lim supμ→μ0 N(μ); this property is not true for the number of roots if we do not take into account their multiplicity). Then, as σ and Q are compact, N(μ) has a finite upper bound N. Now, as I˜1 is smooth in (, v, η), there are 0 > 0 and b0 small enough such that the roots in σ of the function I˜1 have a finite multiplicity for (b, μ, ) ∈ [−b0 , b0 ] × Q × [0, 0 ]. If 0 is small enough, the number of these roots remains bounded by N. Now, ˜ ∂ using the equivalence (13.20) between ∂v,η and I˜1 (v, μ, ) and Rolle’s Theorem, we have that at most N +1 limit cycles cut σ , for (b, μ, ) ∈ [−b0 , b0 ]×Q×[0, 0 ]. 

Remark 13.6 The simplest case of Theorem 13.1 is when I has no root at all. In this case, we have that there exists a unique limit cycle γ,η , corresponding to the value v(η, ). This limit cycle is hyperbolic, for each  > 0 small enough. As we have seen in Part I, γ,η is attracting if I < 0 and repelling if I > 0 (this may also ˜ ∂

be checked, in a very painful way, using the sign of ∂v,η given by (13.20)). As we have noticed in the non-orientable case, γ,η “flies” along the section σ when K the breaking parameter b crosses an interval [b− (μ, ), b+ (μ, )] of order e−  , for some K > 0. In any case and as seen above, all possible roots are confined in a K canard tongue |b − b(μ, )| ≤ e−  . If the function v → I (v, μ0 ) has a root v0 of finite multiplicity, we can apply Theorem 13.1 to a small interval in σ around v0 and in a neighborhood of μ0 in Q. Recall from Sect. 13.1 that bounds on the number of limit cycles of the (, b, μ)family of vector fields lead to similar bounds for the original family of vector fields X,λ . In this way we have a result of  finite cyclicity  (see Definition 4.3 in Chap. 4 for a definition of the cyclicity Cycl X,λ , v0 (μ0 ) of the canard cycle v0 (μ0 ) in the family X,λ ): Theorem 13.2 Let us assume that the slow divergence integral I (v, μ0 ) has a root v0 of multiplicity N. Then:   Cycl X,λ , v0 (μ0 ) ≤ N + 1. Remark 13.7 The non-degeneracy condition of Theorem 13.1 is fulfilled if I is an analytic family of functions, as soon as I (v, μ) is not identical to 0 for any μ. We have that I (v, μ0 ) ≡ 0 if the system has a so-called center type for μ = μ0 . It is possible to extend the result of finite cyclicity in this case, for μ ∼ μ0 , by using the ideal of Bautin associated to the germ of I˜ at μ0 . A difficult problem is to extend

13.3 Results of Bifurcation

285

the division of I in the ideal of Bautin into a division of I˜ in the same ideal. We do not treat this center case in this book.

13.3.6 Limit Cycles and Bifurcations in a Rescaled Layer Theorem 13.2 says that at most N + 1 limit cycles bifurcate from a canard cycle  = v0 (μ0 ) corresponding to a root v0 of multiplicity (or order) N of the slow divergence integral I (v, μ0 ). It would of course be interesting to know whether degenerate limit cycles of order N + 1 will be created nearby  and, if so, whether they will be unfolded in an elementary catastrophe of limit cycles of codimension N + 1. This will reveal to be the case if we suppose that I has a zero of multiplicity N at v0 , unfolded in an elementary catastrophe of codimension N. Unfortunately it is not easy to provide such results that hold in a genuine neighborhood of . We will have to restrict to a rescaled layer around  and also in a rescaled neighborhood in the parameter space. The reason for it will be explained in Sect. 13.3.7. In order to permit describing a precise result we will not only make an elaborated presentation of the construction, but we also recall in a detailed way the notions that we will use. As said above, we have no way to study the bifurcations of orbits not isotopic to . Such orbits may occur when  is a disorienting curve, i.e. when θ = −1. We will hence again restrict to the orientable case: θ = +1. In this case, we can simplify the coefficients in expression (13.12). First we write sA = sign

 ∂I A  ∂v

and sR = sign

 ∂I R¯  . ∂v

We have that θ = (θA θA )(θC θR¯ θR¯ )sA sR = 1. Then, θA θA = MsA and θC θR¯ θR¯ = MsR , with M = θA θA sA = θC θR¯ θR¯ sR = ±1 (we recall that all the coefficients are equal to ±1). We can change (13.12) to ˜ ,η (v) = Mα(η, ) + sA e M

I˜1A (v,μ,) 

− sR e

¯ I˜R (v,μ,) 

.

(13.22)

We recall that the slow divergence integral is equal to I (v, μ) = I A (v, μ) − In Remark 13.6 we have already explained what happens in the simple case I (v0 , μ0 ) = 0. Then, from now on we will assume that the canard cycle v0 (μ0 ) is balanced, in the sense that ¯ I R (v, μ).

¯

I (v0 , μ0 ) = I A (v0 , μ0 ) − I R (v0 , μ0 ) = 0.

(13.23)

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13 Ordinary Canard Cycles

We introduce now a non-degeneracy condition of I at (v0 , μ0 ): Definition 13.4 We say that the slow divergence integral I has a root of order (or multiplicity, or codimension) r ≥ 1 at (v0 , μ0 ) if I (v0 , μ0 ) =

∂I ∂ r−1 I ∂rI (v0 , μ0 ) = · · · = r−1 (v0 , μ0 ) = 0 and r (v0 , μ0 ) = 0. ∂v ∂v ∂v (13.24)

In the generic situation I (v0 , μ0 ) = 0, we can say that I is of order 0 at (v0 , μ0 ). We make the variable change w → v = v0 + w and expand I at order r in w: I (v0 + w, μ) = c0 (μ) + c1 (μ)w + · · · + cr−1 (μ)wr−1 + cr (μ)wr + O(wr+1 ), where ci and the remainder are smooth. The slow divergence integral I has a root of order r at (v0 , μ0 ) if and only if c0 (μ0 ) = · · · = cr−1 (μ0 ) = 0 and cr (μ0 ) = 0. Definition 13.5 We say that the μ-unfolding defined by I (v, μ) is generic at (v0 , μ0 ) if the map μ → (c0 (μ), · · · , cr−1 (μ)) is of rank r at μ0 (i.e. is a local submersion at μ0 ). For a generic unfolding we necessarily need p ≥ r. Up to a diffeomorphism on Q, we can then assume that μ = (c, ξ ) with c = (c0 , . . . , cr−1 ) ∈ Q1 and ξ ∈ Q2 , where Q1 is an arbitrarily small connected neighborhood of 0 ∈ Rr and Q2 is an arbitrarily small connected neighborhood of 0 ∈ Rp−r . In this new parameter μ we have that μ0 = 0 ∈ Rp and the above Taylor expansion writes: I (v0 + w, μ) = c0 + c1 w + · · · + cr−1 wr−1 + cr (μ)wr + O(wr+1 ),

(13.25)

again with the condition cr (0) = 0 (taking Q1 , Q2 small enough, we assume that ¯ cr (μ) = 0 for any μ). We also expand I˜R in w: ¯ I˜R (v0 + w, μ, ) = −k0 (μ, ) + k1 (μ, )w + O(w2 ).

(13.26)

The functions k0 , k1 are smooth in (μ, ), with k0 (μ, ) > 0 and k1 (μ, ) = 0, for any (μ, ) (assuming Q1 , Q2 and  small enough). We see that sR = sign(k1 ). ¯ Putting I˜ = I˜1A − I˜R and using the above expansion, we have ¯ ,η (w) = Me

k0 (μ,) 

Me

˜ ,η (v0 + w) =

k0 (μ,) 

α(η, ) + e

k1 (μ,)w+O(w2 ) 

 sA e

I˜(v0 +w,μ,) 

 − sR .

(13.27)

13.3 Results of Bifurcation

287

¯ ,η (w) can be taken as a new difference map. Its translation term This function k0 (μ,) Me  α(η, ) is unbounded when  → 0. The function I˜ expands also in w: I˜(v0 + w, μ, ) = c˜0 + c˜1 w + · · · + c˜r−1 wr−1 + c˜r wr + O(wr+1 ),

(13.28)

where the c˜i = c˜i (μ, ) are smooth in (, μ). As I˜(v, μ, 0) = I (v, μ), we have that c˜i = ci + o (1) and that c˜r (μ, ) = 0 if (c, ξ, ) belongs to a neighborhood Q1 × Q2 × [0, 0 ] small enough. The map (c, ξ, b, ) → (c, ˜ ξ, b, ), with c˜ = (c˜0 , . . . , c˜r−1 ), defines a change of parameters that are smooth in (, c, ξ ). This allows to choose (c, ˜ ξ ) as a new parameter μ in (13.27) and (13.28). (Recall that b is the breaking parameter for both the jump and Hopf breaking mechanisms as explained at the end of Sect. 13.2.1.) Due to the exponential flatness in , it is difficult to study the equation of limit cycles in a whole neighborhood of (v, b, μ, ) = (v0 , 0, μ0 , 0). To get an interesting result, we will restrict to a rescaled layer in the variable w and to a rescaled parameter defined in the following way: Definition 13.6 Choose any w¯ 0 > 0. A Rescaled layer in the rescaled variable w¯ ∈ [−w¯ 0 , w¯ 0 ] is defined by the change of variable w =  w¯ (i.e. v = v0 +  w). ¯ Rescaled parameters c¯i , for i = 0, . . . , r − 1, are defined by c˜i =  r−i c¯i , with c¯ = (c¯0 , . . . , c¯r−1 ) ∈ Q¯ 1 , some compact connected domain in Rr . They are related to the parameters ci by formulas  r−i c¯i = ci + ϕi (μ, ),

(13.29)

where the functions ϕi are smooth in (μ, ) and of order o (1). Moreover, w¯ 0 and Q¯ 1 can be chosen arbitrarily large. Let us write μ¯ = (c, ¯ ξ ) and η¯ = (b, μ). ¯ The translation term is α( ¯ η, ¯ ) = Me

k0 (μ,) 

α(η, ).

(13.30)

¯ ) as a new parameter instead of b, at least for As ∂∂bα¯ = 0, we can choose α¯ = α(η,  > 0. ¯ ,η given by (13.27), in terms of the We can express the difference function rescaled variable and rescaled parameters. Moreover, we introduce c¯r (ξ ) and k¯1 (ξ ) so that c˜r = c¯r (ξ ) + o (1) and k1 = k¯1 (ξ ) + o (1), with terms o (1) that are smooth in (, η). ¯ We have k¯1 (ξ ) = 0 for all ξ , by definition, and c¯r (ξ ) = 0 for all ξ ∈ Q2 , if the neighborhood Q2 is small enough: we simply write c¯r = 0 and k¯1 = 0. As Q2 is connected, it is equivalent to say that these functions have a constant sign. We can write   r−1 ¯ ¯ c,ξ ¯ ¯  (1) ¯ )+o (1)) ¯ ,η¯ (w) sA e (δ(w, (13.31) ¯ = α¯ + ek1 (ξ )w+o − sR ,

288

13 Ordinary Canard Cycles

where ¯ w, δ( ¯ c, ¯ ξ ) = c¯0 + c¯1 w¯ + · · · + c¯r−1 w¯ r−1 + c¯r (ξ )w¯ r , ¯ μ). ¯ with terms o (1) that are smooth in (, w, ¯ ,η¯ (w) Remark 13.8 We want to search for the roots w¯ of the equation ¯ = 0 in an arbitrarily large interval [−w¯ 0 , w¯ 0 ], and for a parameter c¯ in an arbitrarily large compact domain Q¯ 1 (the neighborhood Q2 for the parameter ξ and the interval [0, 0 ] may be chosen small enough). Considering its definition, the parameter α¯ should be taken arbitrarily large. But, we can see that the value of α¯ remains ¯ ,η¯ (w) ¯ = 0}, when (c, ¯ ξ, ) ∈ Q¯ 1 × Q2 × [0, 0 ]. As such bounded along the set { we can assume that α¯ belongs to an arbitrarily large interval [−α0 , α0 ].

13.3.6.1 Equivalence of Families of Functions In order to state the promised results on the bifurcation patterns of limit cycles, we have to recall some definitions for families of functions. We consider smooth families of functions f = f (y, c) = fc (y) defined on a closed interval I ⊂ R and parameterized by a compact Euclidean domain Q (for instance, a closed ball). We suppose that the functions that we consider always have a finite number of roots on I , counted with their multiplicity. As Q is compact this number is uniformly bounded on Q. This parameter space can be stratified, each stratum corresponding to a fixed number of roots, given with their order and multiplicity (for instance, functions with simple roots correspond to open strata). Let f ⊂ Q be the bifurcation set of the families f . Definition 13.7 Let fc (y) and gη (y) be two smooth families of functions on a closed interval I , parameterized by c ∈ Q and η ∈ Q , respectively. These two families are Weakly C∞ -equivalent if there exists a smooth diffeomorphism η = ϕ(c), from Q to Q , such that, for each c ∈ Q, there exists a smooth diffeomorphism hc of I , sending the roots of the function fc on the roots of gϕ(c) , preserving their multiplicities (some of these roots may belong to the boundary ∂I ). A direct consequence of Definition 13.7 and of the definition of bifurcation set is the following: Lemma 13.1 Let f and g be two families as in Definition 13.7. Then a smooth diffeomorphism ϕ defines a weak C∞ -equivalence from f to g if and only if ϕ sends the bifurcation set of f onto the bifurcation set of g, in preserving the type of each stratum (i.e. labeled in terms of roots, given with their order and multiplicity). Remark 13.9 There are stronger forms of equivalence. For instance, one says that the two families are left C∞ -equivalent if they differ by an everywhere nonzero factor; they are said to be C∞ -equivalent if they are weakly C∞ -equivalent through a family hc that is a smooth family of I -diffeomorphisms. Clearly, the left C∞ -

13.3 Results of Bifurcation

289

equivalence implies the C∞ -equivalence with ϕ ≡ Id and hc = Id for all c. These two equivalences imply the weak C∞ -equivalence. Notice that, in the definition of weak equivalence, we do not even ask hc to depend continuously on c. To each notion of equivalence is associated as usual a notion of structural stability. We will just use the structural stability associated with the weak C∞ equivalence. Definition 13.8 A smooth family f (y, c), defined for (y, c) ∈ I × Q, is said to be structurally stable for the weak C∞ -equivalence if there exists a neighborhood U of f in the C∞ topology, such that each g ∈ U is weakly C∞ -equivalent to f with a diffeomorphism ϕg depending smoothly on g. The family g is defined on a parameter domain Qg containing ϕg (Q), which may depend on g. The smoothness of the dependence means that if f (y, c, s) is any smooth function defined on I × Q × B, where B is an Euclidean ball, such that f (y, c, 0) ≡ f (y, c), then there exists a smooth family of diffeomorphisms ϕs (c) inducing a weak C∞ -equivalence between f (y, c) and fs (y, c) = f (y, c, 0). In this chapter we will be interested in families f (y, c, ξ, ) that are smooth in (y, c, ξ, ), where  in a finite set of -monomials. Since the perturbation is not genuinely smooth, we cannot be sure, directly from the above definition, that the stratification for the function f,ξ (y, c) is similar to that of f0,0 (y, c), when we suppose f0,0 to be structurally stable. Nevertheless it is the case as we see in the following proposition: Proposition 13.4 Let f,ξ (y, c) = f (y, c, ξ, ) be defined for (y, c) ∈ I × Q1 , and for  ≥ 0, with  ∼ 0 and ξ ∼ 0 ∈ R , for some (I is an interval and Q1 a compact domain of Rp ). We assume that f (y, c, ξ, ) is smooth in (y, c, ξ, ), where  is a finite set of -monomials (see Chap. 11) (meaning that f (y, c, ξ, ) = f˜(y, c, ξ, ω1 , . . . , ω ) for some function f˜ that is smooth in all its variables and with ω1 , . . . , ω ∈ ) and that f0,0 (y, c) is structurally stable for the weak C∞ -equivalence. Let f,ξ be the bifurcation set of the family f,ξ (y, c). Then, with 0 > 0 small enough and ξ ∈ Q2 small enough, there exists an (, ξ )-family of diffeomorphisms ϕ,ξ (c) : Q1 → Rp , smooth in (, ξ, c) and defined for (, ξ ) ∈ [0, 0 ] × Q2 , such that ϕ,ξ (f0,0 ) = f,ξ . This implies that for each (c, , ξ ) ∈ Q × [0, 0 ] × Q2 , the function y → f (y, ϕ,ξ (c), ξ, ) has the same number of roots counted with their multiplicity as the function y → f (y, c, 0, 0). We can also say that the bifurcation set of the (c, ξ, )-family f (y, c, ξ, ) is the image of the trivial product f0,0 × [0, 0 ] × Q2 by the map (c, ξ, ) → (ϕ,ξ (c), ξ, ) that is smooth in (c, ξ, ). Proof If  = {ω1 , . . . , ω }, there exists a smooth family f˜(y, c, ξ1 , . . . , ξ ) such that f (y, c, ) = f˜(y, c, ω1 , . . . , ω ). To prove the statement, it suffices to use Definition 13.8 for f˜ and then to substitute the monomials ωi to the parameters ξi . 

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13 Ordinary Canard Cycles

The simplest example of a structurally stable family is given by the universal polynomial family Pr (y, c) = y r +

r−1 !

ci y i .

(13.32)

i=0

We could also consider the normalized polynomial with cr−1 = 0. The bifurcation set of this last polynomial is the classical catastrophe bifurcation set of codimension r: fold, cusp, swallowtail, and so on. Here we prefer to keep the parameter cr−1 , because it is in this form that we will use it. For this form (13.32) we have an (extended) catastrophe bifurcation set that is diffeomorphic to the previous catastrophe bifurcation set times R: in particular, instead of one point, we now have a line r consisting of the most degenerate singularity (fold, cusp points, and so on); it can be located at any point y in function of the value of the parameter. It reveals to be this bifurcation set, restricted to an interval I , that we have to use in the statements below. As we restrict to an interval I , we also have to restrict this diagram to bifurcations of roots belonging to I and to add bifurcation strata that correspond to the passage of roots through the boundary ∂I . The domain in the parameter space corresponding to r simple roots in the interior of I is a r-simplex T of Rr whose boundary is contained in the union of strata of the extended catastrophe set and the strata corresponding to the passage by the boundary ∂I . See Fig. 13.5 for the cases r = 2 and r = 3. We now indicate briefly why the family Pr is structurally stable. First, if one considers any m0 = (y0 , ξ 0 ) ∈ r and any germ of family f (y, c) nearby the germ of Pr at m0 , this germ has a root of order r at a point m1 = (y1 , ξ 1 ) ∼ m0 . We can use the preparation theorem to see that the germ (f, m1 ) is left C∞ -equivalent to

T T

Bifurcation diagram of P2

Bifurcation diagram of P3

Fig. 13.5 Bifurcation diagrams of Pr and r-simplex T , for r = 2 and r = 3.

13.3 Results of Bifurcation

291

the germ Pr (y, h(c)) where h(c) is a local diffeomorphism of parameter space. So, the bifurcation set of Pr is locally structural stable at the points of r . A similar argument can be applied at each point of the bifurcation set. This implies that this bifurcation set is locally structural stable. Second, in order to look at global structural stability, we have to choose an arbitrarily large ball Q in Rr , with a boundary ∂Q transverse to the bifurcation set of Pr . We also assume that Q contains T in its interior. We will say that Q is well adapted to Pr . It is now easy to glue together the result on local stability to obtain a result of global stability: if a family f is sufficiently near Pr on Q, there exists a diffeomorphism ϕf : Q → Q that sends each stratum of the bifurcation set of Pr on Q on a corresponding stratum of the bifurcation set of f on Q. The compactness and the transversality of the bifurcation set of Pr to ∂Q are of course essential for this stability property. As the stratification of the bifurcation set corresponds to the stratification in terms of the set of roots, in number and order, it is easy to deduce that Pr is weakly C∞ - equivalent to f . The smooth dependence of ϕf on f is just an easy technical point. It is possible to obtain a stronger result of stability by using the theory of differentiable functions of Thom–Mather: the family Pr is indeed structurally stable in the sense of C∞ equivalence. The polynomial family Pr is a particular case of a more general type of family constructed on complete Chebyshev systems: Definition 13.9 A system H = (h0 (y), . . . , hr (y)) of smooth functions on some closed interval I ⊂ R is called a Complete Chebyshev system of order r on I if the Wronskian determinants Wn (y) = W (h0 , . . . , hn )(y), for n = 0, . . . , r defined by Wn (y) = det

 ∂ih

j ∂y i

 (y)

i,j =0,...,n

,

are nowhere zero on I . In [Mar98] such a system is called a “full differentiable ECT system.” As trivially we have that Wn (1, y, . . . , y n ) = 1!2! · · · n!, so that the systems of monomials (1, y, · · · , y r ) are complete Chebyshev systems for any r ∈ N. Obviously, being a complete Chebyshev system is a local property: a system is a complete Chebyshev system on I if and only if for any p ∈ I there is an interval Ip neighborhood of p such that the system is a complete Chebyshev system on Ip . For future use, we give two other basic properties in the following lemma: Lemma 13.2 Let H = (h0 , . . . , hr ) be a complete Chebyshev system on I . Then, if f is a nowhere zero smooth function, the system (f h0 , . . . , f hr ) is also a complete Chebyshev system on I . Let gi = rj ≤i /mij hj , for i = 0, . . . , r, where (mij ) is an invertible lower triangular matrix ( i mii = 0); then (g0 , . . . , gr ) is also a complete Chebyshev system on I . Proof Let f be a nowhere zero smooth function. Then, by a direct computation we have that W (f h0 , . . . , f hn )(y) = f n W (h0 , . . . , hn )(y), for any n = 0, . . . , r.

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13 Ordinary Canard Cycles

Then the system (f h0 , . . . , f hr ) is a complete Chebyshev system on I , if and only if it is the case for the system (h0 , . . . , hr ). r Consider a linear system of equations gi = j ≤i mij hj as in the statement.  / /n n Then W (g0 , . . . , gn )(y) = i=0 mii W (h0 , . . . , hn )(y). As i=0 mii = 0 for any n = 0, . . . , r, the system (g0 , . . . , gr ) is a complete Chebyshev system on I , if and only if it is the case for the system (h0 , . . . , hr ). 

Definition 13.10 Let {h0 , h1 , . . . , hr } be a complete Chebyshev system on I . The associated affine Chebyshev family (of order r) is the family on I defined by f (y, ξ ) =

r−1 !

ci hi (y) + hr (y),

(13.33)

i=0

with the parameter c = (c0 , . . . , cr−1 ) ∈ Rr . We will say that the affine Chebyshev family is of order r if it is the case for the corresponding Chebyshev system. It is not hard to prove that an affine Chebyshev family of orders r has at most r roots counted with their multiplicity. But there is more: as for the monomial systems, it is rather easy to prove that an affine Chebyshev family f on I is weakly C∞ equivalent to the polynomial family Pr restricted to the interval I . Let us just state this result without proof: Theorem 13.3 All affine Chebyshev families of the same order r are weakly C∞ equivalent. In particular they are weakly C∞ -equivalent to the polynomial family Pr and their bifurcation diagrams are diffeomorphic to the bifurcation diagram of Pr , called the codimension r catastrophe diagram. Moreover, any such Chebyshev combination f is structurally stable for the weak C∞ -equivalence, when the parameter is restricted to compact domain Q image by the equivalence between Pr and f , of a ball that is well adapted to Pr . Remark 13.10 A compact domain Q as in the statement will be said well adapted to f . Using Thom–Mather theory of differentiable functions, one can replace the weak C∞ -equivalence by the C∞ -equivalence in the statement of Theorem 13.3. ¯ ,η¯ (w) 13.3.6.2 Asymptotic Expression for the Difference Function ¯ We have to distinguish two different cases, depending on the value of the codimension r: 1. The case r = 1. We have that   ¯ ¯  (1) ¯  (1) ¯ ,η¯ (w) sA ec¯0 +c¯1 (ξ )w+o ¯ = α¯ + ek1 (ξ )w+o − sR .

13.3 Results of Bifurcation

293

Using that eo (1) = 1 + o (1), with preservation of the smoothness property, we can write ¯ ¯ ¯ ,η¯ (w) ¯ = α¯ + sA ec¯0 e(k1 (ξ )+c¯1 (ξ ))w¯ − sR ek1 (ξ )w¯ + o (1),

(13.34)

with a perturbation term o (1) that is smooth in (, μ, ¯ w). ¯ The function k¯1 + c¯1 (as well as the function k¯1 itself) has a constant sign on a neighborhood Q2 small enough. We have that sA = sign(k¯1 + c¯1 ) and sR = sign(k¯1 ). Recall that c¯1 also has a constant sign. 2. The case r > 1. In this case, sA = sR = sign(k¯1 ). We obtain that   ¯ ¯ w, ¯ ,η¯ (w) ¯ = α( ¯ η, ¯ ) + sA  r−1 ek1 (ξ )w¯ δ( ¯ c, ¯ ξ ) + o (1) . This gives ¯ ,η¯ (w)  −(r−1) sA ¯ = α˜ +

r−1 !

¯

¯

c¯i ek1 (ξ )w¯ w¯ i + c¯r (ξ )ek1 (ξ )w¯ w¯ r + o (1),

(13.35)

i=0

with ¯ η, ¯ ) α˜ =  −(r−1) sA α(

(13.36)

that we can take as a new parameter instead of b and a perturbation term o (1) that is smooth in (, μ, ¯ w). ¯ Recall that c¯r (ξ ) has a constant sign.

13.3.6.3 Bifurcation Results in a Rescaled Layer We now use the expansions (13.34) and (13.35) to study the bifurcations of limit cycles in a rescaled layer. To this end, we will prove the structural stability (for ¯ of these expansions. There are weak C∞ -equivalence) of the principal part ηr¯ (w) some differences between the two cases r = 1 and r > 1. The principal difference is that ηr¯ (w) ¯ is linear in the parameters (α, ˜ c¯0 , . . . , c¯r−1 ) when r > 1, and it is not the case when r = 1. So, we will consider separately the two cases. Recall that we fix an arbitrarily interval I = [−w¯ 0 , w¯ 0 ]. To simplify the notation, we no longer indicate the smooth dependence on ξ of the different constants k¯1 (ξ ), . . . The Case r = 1 Let ¯

¯

1 α, ¯ =  1 (w, ¯ α, ¯ c¯0 ) = α¯ + sA ec¯0 e(k1 +c¯1 )w¯ − sR ek1 w¯ ¯ c¯0 (w)

(13.37)

294

13 Ordinary Canard Cycles

1 (w) be the principal part of (13.34). We consider α, ¯ c0 )-family ¯ c¯0 ¯ as a smooth (α, of functions of w. ¯ ¯

¯

Although the triple {1, e(k1 +c¯1 )w¯ , ek1 w¯ } is a complete Chebyshev system (as a consequence of k¯1 = 0 and c¯1 = 0), we cannot apply Proposition 13.4 and Theorem 13.3 to  1 because the dependence on the parameter c¯0 is non-linear. We have to make a direct (but easy) study. We have that

Then, α1¯ c¯0

∂ 1 ¯ ¯ (w, ¯ α, ¯ c¯0 ) = |k¯1 + c¯1 |ec¯0 +(k1 +c¯1 )w¯ − |k¯1 |ek1 w¯ . ∂ w¯   ¯   k1 | − c ¯ . It is has a unique critical point: w¯ 1 (c¯0 ) = c¯11 ln |k¯ ||+| 0 c¯ | 1

1

∂ 21 (w¯ 1 , α, ¯ c¯0 ) = 0, and then w¯ 1 is a Morse singularity whose ∂ w¯ 2 d w¯ 1 1 position, as d c¯0 = − c¯1 = 0, is a regular function of the parameter c¯0 . This 1 , restricted to the interval I , is weakly C∞ -equivalent implies that the family α, ¯ c¯0 to the polynomial family P2 : its bifurcation set is diffeomorphic to the extended

easy to verify that

catastrophe bifurcation set of codimension 2. This bifurcation set is easy to compute in the whole parameter space R2 of the parameter (α, ¯ c¯0 ). It contains the two curves d− , d+ , corresponding to the passage of a root through the end points −w¯ 0 and +w¯ 0 , respectively. Precisely, we have that ¯

¯

d− : α¯ = sR e−k1 w¯ 0 − sA e−(k1 +c¯1 )w¯ 0 ec¯0 and ¯

¯

d+ : α¯ = sR ek1 w¯ 0 − sA e(k1 +c¯1 )w¯ 0 ec¯0 . We have the following equation in c¯0 for intersection of d− with d+ :   sA sinh (k¯1 + c¯1 )w¯ 0 ec¯0 = sR sinh(k¯1 ).   As sA sinh (k¯1 + c¯1 )w¯ 0 > 0 and sR sinh(k¯1 ) > 0, this equation has a unique solution c¯0 = ln



 sR sinh(k¯1 )  .  sA sinh (k¯1 + c¯1 )w¯ 0

13.3 Results of Bifurcation

295

α ¯ sn

T

¯0 λ d− d+ Fig. 13.6 The bifurcation set with triangle T , and bifurcation curves sn, d− , and d+ . ¯

¯

Moreover, as sA e−(k1 +c¯1 )w¯ 0 = sA e(k1 +c¯1 )w¯ 0 , d− and d+ intersect transversally. The bifurcation diagram is completed by an arc of saddle-node bifurcations sn that is easy to parameterize by the position of the double root w¯ 1 : ⎧ ⎨ c¯0 = −c¯1 w¯ 1 + ln |k¯1 | |k¯1 +c¯1 | sn : ⎩ α¯ = sR c¯1 ek¯1 w¯ 1 .

(13.38)

k¯1 +c¯1

This curve is parameterized by w¯ 1 ∈ [−w¯ 0 , w¯ 0 ]. The end points of the arc sn belong to d− and d+ , respectively, where sn is tangent at these curves. We have two simples roots inside the triangle T bounded by sn, d− , and d+ . As a consequence of the previous study, we have the following result on the bifurcations, see Fig. 13.6: Theorem 13.4 (1) We consider an arbitrarily large interval [−w¯ 0 , w¯ 0 ] ⊂ R and denote by  1 , with the triangle T and the curves the bifurcation diagram of the family α, ¯ c¯0 d− , d+ as defined above. Let us also consider an arbitrarily large domain Q1 in R2 , diffeomorphic to a disk, containing the triangle T in its interior and with a boundary transversely cutting d− and d+ , each in two points. Then, the 1 family α, ¯ 0 , w¯ 0 ] and restricted to Q1 is weakly C∞ -equivalent to the ¯ c¯0 on [−w polynomial family P2 restricted to a well-adapted ball. (2) Let v0 (μ0 ) be a canard cycle such that v0 is a root of order 1 of the slow divergence integral I (v, μ0 ). We assume that μ = (c0 , ξ ) and that the unfolding I (v, c0 , ξ ) is generic in c0 . We consider the rescaled parameters (α, ¯ c¯0 ) given by (13.29) and (13.30). Then, with 0 > 0 small enough and ξ ∈ Q2 small enough, there exists an (, ξ )-family of diffeomorphisms ϕ,ξ (c) : Q1 → R2 ,

296

13 Ordinary Canard Cycles

smooth in (, ξ, c) and defined for (, ξ ) ∈ [0, 0 ] × Q2 , such that ϕ,ξ () = ,ξ , where ,ξ ⊂ Q1 is the bifurcation set of limit cycles, for (, ξ ) ∈]0, 0 ]× Q2 . Proof 1 (1) As the family α, ¯ c¯0 restricted to a domain Q1 , chosen like in the statement, has a bifurcation diagram diffeomorphic to the bifurcation diagram of the polynomial P2 restricted to a well-adapted disk, we can apply Lemma 13.1 . The weak C∞ -equivalence follows. (2) Recall that the intersection points of limit cycles with the interval [−w¯ 0 , w¯ 0 ] are in a one-to-one correspondence with the roots of the difference function (13.34), so that the bifurcation set ,ξ in Q1 is the bifurcation set of the roots of (13.34). ∞ close If α, ¯ c¯0 is a smooth family defined on a neighborhood of Q1 and C 1  enough to α, ¯ c¯0 , it has a bifurcation diagram  diffeomorphic to the bifurcation 1 . Then, there exists a diffeomorphism ϕ from Q onto Q , diagram  of α,  1 1 ¯ c¯0 which sends  to   . As it is clear that ϕ depends smoothly on , the structural 1 stability of α, ¯ c¯0 follows also from Lemma 13.1. Using this structural stability, the second claim follows from Proposition 13.4. 

The Case r > 1 Let r α, ¯ =  r (w, ¯ α, ˜ c) ¯ = α˜ + ˜ c¯ (w)

r−1 !

¯

¯

c¯i ek1 w¯ w¯ i + c¯r ek1 w¯ w¯ r

(13.39)

i=0

be the principal part of (13.35). Recall that c¯ = (c¯0 , . . . , c¯r−1 ) and that α˜ is related r (w) to α¯ by (13.36). We consider α, ˜ c)-family ¯ of functions of w. ¯ ˜ c¯ ¯ as a smooth (α, ¯ As in the above, we will not indicate the smooth dependence of k1 and c¯r on ξ that we consider as constants. Let us recall that k¯1 = 0 and that c¯r = 0. We have: Proposition 13.5 Let r be any integer. If k¯1 = 0, the system ¯

¯

¯

{1, ek1 w¯ , ek1 w¯ w, ¯ . . . , c¯r ek1 w¯ w¯ r } is a complete Chebyshev system of order r + 1 on any interval [−w¯ 0 , w¯ 0 ] ⊂ R. Proof Using both statements in Lemma 13.2 we can replace the given system by the system ¯

{e−k1 w¯ , 1, w, ¯ . . . , w¯ r }.

13.3 Results of Bifurcation ¯

297

¯

As W0 (e−k1 w¯ ) = e−k1 w¯ , it is sufficient to consider the Wronskian determinants ¯ Wn+1 (e−k1 w¯ , 1, w, ¯ . . . , w¯ n ), for n = 0, . . . , r. We have that ¯

¯

Wn+1 (e−k1 w¯ , 1, w, ¯ . . . , w¯ n ) = (−1)n k¯1n e−k1 w¯ Wn (1, w, ¯ . . . , w¯ n ), and then that ¯

¯

Wn+1 (e−k1 w¯ , 1, w, ¯ . . . , w¯ n ) = (−1)n 1! · · · n!k¯1n e−k1 w¯ . 

This implies the Chebyshev property.

r As a consequence, α, ¯ c¯0 is an affine Chebyshev family. We have the following result on the bifurcations:

Theorem 13.5 We choose an arbitrarily large interval [−w¯ 0 , w¯ 0 ] for the rescaled variable w¯ to define a rescaled layer as in Definition 13.6. r . Let us (1) Let  be the bifurcation diagram of the affine Chebyshev family α, ¯ c¯0 r also consider an arbitrarily large compact domain Q1 in R , diffeomorphic to r (see Remark 13.10 for the definition of wella ball, and well adapted to α, ¯ c¯ adapted domain). Then, the bifurcation diagram  on Q1 is diffeomorphic to the bifurcation diagram of the polynomial family Pr+1 , restricted to a welladapted ball. (2) Let v0 (μ0 ) be a canard cycle such that v0 is a root of order r of the slow divergence integral I (v, μ0 ). We assume that μ = (c, ξ ) with c = (c0 , . . . , cr−1 ) and that the unfolding I (v, c, ξ ) is generic in c.

Then, in the rescaled parameters α˜ and c¯ = (c¯0 , . . . , c¯r−1 ) given by (13.29), we have the following result. If 0 > 0 is small enough and if ξ ∈ Q2 is a small enough neighborhood, there exists an (, ξ )-family of diffeomorphisms ϕ,ξ (c) : Q1 → Rr , smooth in (, ξ, c) and defined for (, ξ ) ∈ [0, 0 ] × Q2 , such that ϕ,ξ () = ,ξ , where ,ξ ⊂ Q1 is the bifurcation set of limit cycles, for (, ξ ) ∈ ]0, 0 ] × Q2 . A precise expression of α˜ is α˜ = sA K  −(r−1) ek0 (μ,)/ α(b, μ, ), where α((b, μ), 0) = b and α is smooth in (b, μ, );  is an appropriate set of A monomials. K is positive constant, and sA = ±1 is the sign of ∂I∂v . k0 is a positive constant defined in (13.26). Proof ¯

¯

¯

¯ . . . , c¯r ek1 w¯ w¯ r } is a complete Chebyshev system (1) As the system {1, ek1 w¯ , ek1 w¯ w, of order r + 1 as proved in Proposition 13.5, we can apply Theorem 13.3 to r : if we choose an arbitrarily large compact the affine Chebyshev family α, ¯ c¯0 r r , its bifurcation diagram  in Q domain Q1 in R , well adapted to α, 1 ¯ c¯ is diffeomorphic to the bifurcation diagram of the polynomial family Pr+1 ,

298

13 Ordinary Canard Cycles

restricted to a well-adapted ball. This is point (1) of the statement. We obtain the precise expression of α˜ by combining (13.9), (13.30), and (13.36). r restricted (2) Moreover, using again Theorem 13.3, we see that the family α, ¯ c¯ ∞ to Q1 is structurally stable for weak C -equivalence. Then, we can apply Proposition 13.4 to obtain the point (2) of the statement. 

Remark 13.11 In [DR01b] and with the present notations, one introduces the sys w¯ ¯ tem of functions (d0 , d1 , . . . , dr+1 ) where d0 (w) ¯ = 1 and di (w) ¯ = 0 ek1 z zi−1 dz, ¯ for i = 1, . . . , r + 1. For all i ≥ 1 we see that ek w¯ w¯ i = k¯1 di+1 + idi , while ¯k1 w¯ e = −k¯1 + k¯1 d1 . This means that one passes from the system (d0 , d1 , . . . , dr+1 ) to the system (1, ew¯ , . . . , ew¯ w¯ r ) by an invertible inferior triangular system of linear equations. Then, it follows from Lemmas 13.5 and 13.2 that (d0 , d1 , . . . , dr+1 ) is also a complete Chebyshev system. (This point was proved by a direct computation r in terms in [DR01b].) Of course, we can expand the affine Chebyshev family α, ¯ c¯ of the functions di . This gives ! 1 r α, = η¯ i (c)d ¯ i (w) ¯ + c¯r dr+1 (w), ¯ ¯ c ¯ k¯1 i=0 r

(13.40)

with η¯ 0 = k¯α˜ − c¯0 and η¯ i = c¯i−1 + k¯i c¯i , for i = 1, . . . , r. The advantage of (13.40) 1 1 is that the most degenerate solution occurs at w¯ = 0 for the parameter values η¯ i = 0, as in the case of the standard polynomial family Pr+1 .

13.3.7 Limit Cycles Outside the Rescaled Layer In Theorem 13.5 we have proved that close to a canard cycle v0 with slow divergence integral having a zero of multiplicity k ≥ 1, it is generically possible to find a catastrophe of limit cycles of codimension k +1. The result was not studied on a full neighborhood of the canard cycle, but in a domain that was shrinking to v0 for  → 0. In the proof we had not only to rescale the variables but also the parameters. One could wonder whether it might be possible to have such a catastrophe of codimension k + 1 on a uniform domain. But this is definitely not the case, not even for k = 1. The reason lays in the unavoidable presence of a flying canard (see Sect. 13.3.4) that has to escape from whatever a priori chosen neighborhood and this with increasing speed when  → 0. This induces the unavoidable presence of boundary bifurcations, like it was in the study of the unfoldings of a nilpotent singularity of codimension 3 with an elliptic sector [DRS87]. The result is hence not in contradiction with the one found in Sect. 13.3.4, namely that in the presence of a flying canard, near some unbalanced canard cycle v1 , one can encounter at most k limit cycles near a balanced canard cycle v0 of which the slow divergence integral has a zero of multiplicity k. To find an extra limit cycle near v1 , needed in the catastrophe of codimension k + 1, we definitely have to use the breaking

13.3 Results of Bifurcation

299

parameter. We will work it out in detail for a slow divergence integral only having simple zeros, permitting to exhibit the typical shape of a so-called Position curve (see [Dum13]). For zeros of higher order we will present a structure theorem that holds in an -uniform way, both in the phase space and the parameter space, but which is less precise than Theorem 13.5.

13.3.7.1 Uniform Result for a Simple Zero of the Slow Divergence Integral To study what happens in a uniform way (in phase space and parameter space) near a simple zero of I (v, μ) at v = v0 , we start again with expression (13.22) that we recall here: ˜ ,η (v) = Mα(η, ) + sA e M

I˜A (v,μ,) 

A

− sR e

¯ I˜R (v,μ,) 

(13.41)

R¯

with M = ±1, sA = sign( ∂I∂v ), sR = sign ∂I∂v , α is smooth in (, η), and the ¯ functions I˜A and I˜R are smooth in (, v, μ). Let b = b(v, μ, ) represent the canard manifold (or manifold of closed orbits), ˜ ,η (v) = 0. Seen that ˜ ,η (v, b(v, μ, ), μ, ) = 0 and implicitly defined by ˜ ,η ∂ ∂b

= 0, it follows that imposing ˜ ,η =

˜ ,η ˜ ,η ∂k ∂ = ··· = =0 ∂v ∂v k

is equal to require that b = b(v, μ, ) and ∂kb ∂b = · · · = k = 0. ∂v ∂v We see that M ˜A

¯ ˜ ,η ∂ ∂ I˜A I˜A (v,μ,) ∂ I˜R I˜R¯ (v,μ,) (v) = sA e  e  . − sR ∂v ∂v ∂v ˜R¯

I I and sR ∂∂v are both strictly positive, we can change the expression into Since sA ∂∂v I˜A (v,μ,) I˜R (v,μ,) ˜ ,η ∂ 1 1 (v) = e  −e  , ∂v ¯

M

(13.42)

300

13 Ordinary Canard Cycles

where

∂ I˜A

= I (v, μ, ) +  log

,

∂v

∂ I˜R¯ ¯ ¯

I˜1R (v, μ, ) = I˜R (v, μ, ) +  log

∂v I˜1A (v, μ, )

˜A

and

are both smooth in (, v, μ). Observe that ¯ I˜1 (v, μ, ) := (I˜1A − I˜1R )(v, μ, ) = I (v, μ) + o (1),

where o (1) is smooth in (, v, μ). Since I (v0 , μ0 ) = 0 and also have I˜1 (v0 , μ0 , 0) = 0,

and

∂I ∂v (v0 , μ0 )

= 0, we

∂ I˜1 (v0 , μ0 , 0) = 0, ∂v

hence also ∂ I˜1 (v, μ, ) = 0 ∂v

(13.43)

near (v0 , μ0 , 0). Let 1 = {(v, μ, )|I˜1 (v, μ, ) = 0}. Because of property (13.43) we know that inside sufficiently small neighborhoods of (v0 , μ0 , 0) in (v, μ, )-space, 1 is a manifold of codimension 1. Let us now consider .  ˜ ,η ∂ ˜ C1 = (v, η, )| ,η = (v) = 0 ∂v   ∂b = (v, b, μ, )|b = b(v, μ, ), (v, μ, ) = 0 . ∂v It is given by the two equations: {b = b(v, μ, ),

I˜1 (v, μ, ) = 0},

so that C1 = {(v, (b, μ), )|b = b(v, μ, ), (v, μ, ) ∈ 1 }.

13.3 Results of Bifurcation

301

It is clearly a manifold of the same dimension as 1 , and hence of codimension 2 in (v, η, )-space. The result holds on sufficiently small neighborhoods of (v0 , μ0 , 0). (In the absence of extra parameters μ, it is a curve in (v, b, )-space, smooth in (, v).) Knowing that ∂α > 0, ∂b we get from (13.41) that  sign

∂b ∂v



= −M sign I˜1 ,

at the points of C \ C1 . Position Curves ˜ ,η = 0 as a (v, μ)-family of “control” Besides representing the canard manifold curves in an (, b)-plane it can also be interesting to represent it as an (, μ)-family of curves in a (b, v)-plane. We call these curves position curves. In Sect. 13.3.4 we already saw that the v-value of a flying canard (close to a layer of unbalanced canard cycles) can be written as a function of b with a slope that tends to ±∞ for  → 0. For  = 0, the position curve is given by the v-axis but for  > 0 it shows the number of limit cycles that we can encounter for a given value of the breaking parameter b and how the number and positions of the limit cycles evolve under a changing b. This representation is (, μ)-dependent. Let us consider the situation in which the slow divergence integral only has simple zeros. We orient v in a way that higher values represent canard cycles v that are farther away from the canard connection. This of course only makes sense for a terminal layer (see Definition 4.6). We already saw that through a dodging layer there can pass at most 2 limit cycles. Seen all the requirements on such a position curve, it has a typical shape that we sketch in Fig. 13.7. In this picture, shown for a fixed value of the parameter , we ¯ ¯ ¯ suppose that 0 > I A (v0 ) > I A (v0 ) > I A (v1 ) and 0 > I R (v0 ) > I R (v0 ) > I R (v1 ), ¯ ¯ ¯ I A (v0 ) = I R (v0 ), I A (v0 ) = I R (v0 ) and I A (v1 ) < I R (v1 ), implying that the “flying canard” is attracting. We also suppose that the v-values increase for increasing b. We recall that for every value of b there can be at most one “flying canard” while at the same b-value, there has to be a sitting canard near v0 and near v0 . Moreover, near such a zero we have to encounter a saddle-node bifurcation of limit cycles at some b-value. A typical position curve will have to look as in Fig. 13.7, tending to the line {b = 0} when  → 0. In the figure, v0 and v0 mark zeros of the slow divergence integral; their location is o(1)-close to fold points of the position curve. The same vvalues are o(1)-close to intersections with the line b = b(v1 ) (the vertical dotted line in the figure). Given that b is more or less the difference between two exponentials, its graph lies between the graphs of the two individual exponentials (the dotted

302

13 Ordinary Canard Cycles

Fig. 13.7 Position curve for a relatively small value of : a possible situation in a terminal layer. The curve is pinched between two exponentially small (dashed) curves centered around a central canard value. As  → 0, the pinching becomes stronger and the curve tends toward a straight line.

b v

v1 v0

v0

curves in the figure), alternately lying close to one or to the other one, depending on which of the two exponentials is dominant. Uniform Structure Theorem The uniform result that we proved in this section for canard cycles v0 whose slow divergence integral has a simple zero can be generalized to a case where the multiplicity is ≥ 2, but in the presence of a sufficient number of parameters inducing a stable unfolding of the zero. We suppose that at some (v0 , μ0 ) the slow divergence integral I represents an elementary catastrophe of codimension k, in the sense that I (v0 , μ0 ) =

∂I ∂I k (v0 , μ0 ) = · · · = k (v0 , μ0 ) = 0, ∂v ∂v

∂ k+1 I (v0 , μ0 ) = 0, ∂v k+1

k−1

∂I ∂(I, ∂I ∂v , . . . , ∂v k−1 )

∂(μ1 , . . . , μk )

(v0 , μ0 ) = 0, (13.44)

where μ are the first k components of μ. Generalizing the definition of 1 , we define the sets i in (v, μ, )-space as 

∂ i−1 I˜1 ∂ I˜1 i = (v, μ, ) : I˜1 (v, μ, ) = (v, μ, ) = · · · = (v, μ, ) = 0 ∂v ∂v i−1

.

for 2 ≤ i ≤ k + 1. Because of the conditions imposed in (13.44), one can prove (see e.g. [GG73]) that the sets i are smooth manifolds of respective codimension i, if we restrict to sufficiently small neighborhoods of (v0 , μ0 , 0) in (v, μ, )-space.

13.3 Results of Bifurcation

303

We define the sets Ci in (v, b, μ, )-space as  ˜ ,η = Ci = (v, η, )|

˜ ,η ˜ ,η ∂i ∂ (v) = · · · = (v) = 0 ∂v ∂v i

.



 ∂ib ∂b = (v, b, μ, )|b = b(v, μ, ), (v, μ, ) = · · · = i (v, μ, ) = 0 . ∂v ∂v Extending in a straightforward way the calculations started in the case of a slow divergence integral with simple zeros one can get the following uniform result, first stated and proved in [Dum13]: ˜ ,η (v) as in (13.12), attached to a jump or Theorem 13.6 Consider the function Hopf mechanism, with all the conditions imposed in this chapter. Let v0 be a value at which the canard cycle v0 has a slow divergence integral satisfying (13.44). Then there exists a neighborhood V of (v0 , μ0 , 0) in (v, μ, )-space, and some b0 > 0 such that inside V × [−b0 , b0 ] ∩ { > 0} the sets Ci for 1 ≤ i ≤ k + 1 are given by Ci = {(v, (b, μ), ) : b = b(v, μ, ), (v, μ, ) ∈ i }, and are hence smooth manifolds of the same dimension as i , having i + 1 as respective codimension. For (v, η, ) ∈ Ck+1 we also have k

∂ b ∂( ∂b ∂v , . . . , ∂v k )

∂(μ1 , . . . , μk )

(v, η, ) = 0.

This theorem provides a uniform result near (v0 , μ0 , 0) but surely does not make Theorem 13.5 superfluous. The result is comparable to the results on the so-called Thom–Boardman singularities for zeros of functions. It describes the degeneracy of the limit cycles that we will encounter, but it does not describe their bifurcations with the precision as given in Theorem 13.5. It permits to prove the occurrence of catastrophes of codimension k + 1, without however providing information on the domain, as we did in Theorem 13.5.

Chapter 14

Transitory Canard Cycles With Slow–fast Passage Through a Jump Point

In this chapter we will study a first case of transitory canard cycles, on an orientable surface M. Transitory canard cycles were introduced in Chap. 4, for slow–fast families X,λ = X,a,μ where a is a breaking parameter. The parameter μ is supposed to belong to a compact neighborhood Q ⊂ Rp of μ0 = 0. Recall that such transitory canard cycles appear, for a particular value of the parameter μ0 and for a = 0, at the boundary of the set of one layer canard cycles that have been studied in the previous chapter. A one layer canard cycle contains just one layer of fast orbits connecting a repelling branch of the critical curve (which is the onset of a repelling sequence) to an attracting branch (which is the onset of an attracting sequence), see Chap. 4. The connecting fast orbits in this layer link two normally hyperbolic points of the critical curve. We parameterize the layer by a layer variable v ∈ σ , where σ is a transverse (maximal) open section. Keeping (a, μ) = (0, μ0 ) for the moment, let γv be the fast orbit passing through the point v ∈ σ . Ordinary one-layer canard cycles associated to the section σ form a 1parameter family v , where v is the canard cycle containing the fast orbit γv . A transitory canard cycle, at the boundary of such a family of one layer canard cycles, corresponds to a value v0 = 0 at the boundary of σ . As σ is supposed to be maximal, the fast orbit γ0 must contain a contact point in its closure. See Fig. 14.1 In this chapter and in the following, we are interested in the simplest transitory canard cycles, considering only generic ones in the set of transitory canard cycles. In some sense these generic transitory canard cycles form strata of codimension 1 in this set. In this chapter, the closure of the orbit γ0 contains one and only one contact point q, which must be a quadratic jump point (i.e. of contact order 2, also called generic jump point, see Definition 2.3). We will assume that γ0 links the regular contact point q to a normally attracting point n on a slow arc. As this situation is stable under perturbation of μ, there will be a μ-family of transitory canards linking a regular contact point q(μ) to a normally attracting point n(μ). We may assume without loss of generality that q(μ), n(μ), γ0 (μ) and the critical curve around n and q are independent of μ ∈ Q, where Q is some compact neighborhood of μ0 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_14

305

306

14 Slow–fast Passage through a Jump Point

q

n

q n

σ

σ

p

p

Fig. 14.1 Transitory canard cycles of types I (left) and II (right). The dots represent an attracting or repelling sequence; p is a generic Hopf breaking mechanism or a generic jump breaking mechanism

We will also assume that the orientation of σ is chosen such that σ ∼ ]α, 0[, with α < 0. Then for each (v, μ) ∈ ]α, 0] × Q there exists a canard cycle v (μ). For v = 0 it is an ordinary canard cycle, for v = 0 it is a transitory canard cycle. On the critical curve, the contact point q is the end point of a repelling slow arc which is the onset of a repelling sequence R0 (μ) of 0 (μ) toward the canard connection p. The point n is a regular point on an attracting slow arc of the repelling sequence A0 (μ) of 0 (μ). We distinguish two types of transitory canard cycles. The attracting part A0 (μ) may either start at n in the direction v < 0 (slowly moving along the layer) or in the direction v > 0 (slowly moving away from the layer). In the first case, we will say that the canard cycle is of type I and in the second case that it is of type II. These two situations are illustrated in Fig. 14.1.

14.1 Statement of the Results For any μ, there exists a family of canard cycles v (μ) for v ∈ [−α, 0] finishing with a transitory canard cycle at v = 0. We might call v (μ) a canard cycle without head, thinking about the traditional Van der Pol shape. If α is small enough, then for each value of v ∈ ]−α, 0[ we can also associate to the same value of the layer variable v ∈ σ a canard with head, which contains a first orbit on the same leaf of the fast foliation (0,0,μ)(v), see Fig. 14.2. We will denote this canard with head |v| (μ). Now for each v ∈ ]−α, α[ we have a canard cycle, one without head for v < 0, with head for v > 0, and of transitory kind for v = 0. The slow divergence integral I (v, μ) of the canard cycle v (μ) is defined as the sum of the slow divergence integrals along all slow curves contained in v (μ). This function is smooth for v = 0. We have given the definition in Chap. 5 and will obtain precise expressions in Sect. 14.2 below. In this sum, all the integrals depend on the parameter μ, but only the two integrals directly related to the connecting fast orbit depend on the variable v. (In the statement of the results we will only need the slow divergence integrals defined for a = 0, as they may or may not be defined for a = 0,

14.1 Statement of the Results

0 v

σ

−α canard mechanism Canard without head Γv (μ)

307

0 v

σ

−α canard mechanism Canard with head Γ|v| (μ)

Fig. 14.2 Slow–fast trajectories near a transitory canard. The situation shown in the figure is a Type-I slow–fast transitory canard

but in the proof of the results the a-dependence on these integrals, or pieces of these integrals will be taken into account.) Perturbations from the ordinary canard cycles v (μ), for v = 0, have been studied in the previous chapter, where they were related to the slow divergence integral I (v, μ). In this chapter we want to study the transitory canard cycles 0 (μ0 ), at an interior point μ0 = 0 of the parameter space Q. If the integral I (0, μ0 ) = 0, we already know from Chap. 7 that at most one limit cycle can bifurcate from 0 (μ0 ). We will recall the proof below. We now want to extend the study to the degenerate case of a canard cycle without head 0 (μ0 ), for a parameter value μ0 , such that I (0, μ0 ) = 0: Theorem 14.1 Let 0 (μ0 ) be a slow–fast transitory canard cycle as above. If I (0, μ0 ) = 0, there bifurcates at most one limit cycle from 0 (μ0 ), which is a repelling or attracting hyperbolic limit cycle, depending on the sign of I (0, μ0 ). Suppose now that I (0, μ0 ) = 0. The cyclicity of 0 (μ0 ) in the system X,a,μ is bounded by 2 in case I and by 3 in case II. This means that the number of limit cycles of X,a,μ , near 0 (μ0 ) and for  ∼ 0, a ∼ 0, μ ∼ μ0 is at most 2 in case I and 3 in case II. In case I, we prove the presence of a semi-hyperbolic limit cycle, unfolding in a saddle-node bifurcation. Theorem 14.2 Under the assumptions of Theorem 14.1, case I, and under the presence of a generic canard breaking mechanism with breaking parameter a, there exists a neighborhood U ⊂ [0, 0 ] × Q of (0, μ0 ) and a Hausdorff neighborhood V of the transitory cycle 0 (μ0 ) with the following properties: (a) There exists a continuous function a = a0 (, μ) defined on U such that along this manifold in parameter space, the family of vector fields, for  > 0, has a semi-hyperbolic limit cycle γ (, μ), unique in V , and γ (, μ) → 0 (μ0 ) in the Hausdorff sense. (b) The limit cycles in V are described in a bifurcation diagram of generic saddlenode type in function of the parameter a − a0 (, μ).

308

14 Slow–fast Passage through a Jump Point

When the parameter μ is scalar and the divergence integral I (0, μ) depends ∂ regularly on μ, i.e. ∂μ I (0, μ) = 0, then μ can be seen as another bifurcation parameter together with the canard breaking parameter a. In case I, the parameter μ can then be used to control the location of the non-hyperbolic cycle. More information will be provided in Sect. 14.6. In case II, we prove the presence of a degenerate limit cycle of order 3, and under this extra condition concerning μ, we show in Sect. 14.6 that this limit cycle unfolds in an elementary catastrophe of codimension 2 using the bifurcation parameters (a, μ). The location of the limit cycle of order 3 is fixed in case II.

14.2 Behavior of the Slow Divergence Integral 14.2.1 The Slow Divergence Integrals J , K, and L In a local coordinate system, in which the jump point q is situated at the origin, we can and will assume that the family of vector fields has the form  X,λ :

x˙ = y + x 2

y˙ =  g(x, , λ) + (y + x 2 )h(x, y, , λ) ,

(14.1)

with q = (0, 0) and g(0, , λ) = 1. In Chap. 2, it is shown that a regular jump point can always be put in this normal form in a smooth way. Recalling that λ = (a, μ), we write g(x, 0, λ) = g0 (x, μ) + O(a). Let us calculate the slow divergence integral along segments of the critical curve, expressed in the normal form (14.1). For a rigorous definition of the slow divergence integral in general, we refer to Chap. 5. In view of the calculations that we will perform in this chapter, it is important to recall that this notion is coordinate-free, and hence does not depend on the local coordinate system. It also does not change under time reparameterizations. Let us hence focus on the canard segments near the jump point, in normal form coordinates. First notice that div X,λ = 2x + O(). The slow dynamics, along y = −x 2 , is given by −2x dx dt = g(x, 0, λ). We find that a slow divergence integral along the part of the critical curve between x = x1 and x = x2 is given by 

t2 t1

 2x dt =

x2 x1

−(2x)2 dx = g(x, 0, λ)



x2 x1

−(2x)2 dx + O(a). g0 (x, μ)

Recall that in this chapter, we distinguish two (global) situations, both shown in Fig. 14.3.

14.2 Behavior of the Slow Divergence Integral

309

Fig. 14.3 Slow divergence integrals J , K, and L for canards without head (left) and canards with head (right). The situation shown in the figure is near a type I slow–fast transitory canard

In the local normal form coordinates near q, the two branches of the singular points are given by √ √ x = − −y, x = + −y,

y ≤ 0.

We consider a point (x0 , y0 ) on the left branch, with y0 < 0, and define  K(y0 , a) =

√ − −y0

0

4 (2x)2 dx = − |y0 |3/2 + o(|y0|3/2 ) < 0 g0 (x, μ) 3

√ as the slow divergence integral on the left branch x = − −y, from (x0 , y0 ) to q = (0, 0), see Fig. 14.3(I). Let us now choose the layer section in these normal form coordinates as {x = x1 > 0, y < 0}, parameterized by v = y. The fast orbit through v has √ an α-limit on the repelling slow arc near the jump point q at the point (x0 , y0 ) = ( −v, v). We let L(v, μ), for v ≤ 0,√ be the slow divergence integral along the repelling sequence between (x0 , y0 ) = ( −v, v) and the breaking mechanism p (for a = 0); it is strictly negative because it is computed in negative time. The integral between q and p corresponds to L(0, μ): Definition 14.1 The integral L(0, μ), which is the slow divergence integral between q and p, is called the left slow divergence integral of 0 (μ); it is strictly negative. We will denote it as L0 (μ). Clearly, we have, still for v ≤ 0, √

 L(v, μ) = L0 (μ) + 0

−v

(2x)2 4 dx = L0 (μ) + |v|3/2 + o(|v|3/2 ). g0 (x, μ) 3

Next, we assume, for v < 0, that the fast leaf to the right of (x0 , y0 ) intersects an attracting arc (not visible in local normal form coordinates) at a point nv (μ). From that point on, an attracting sequence Av (μ) connects to the breaking mechanism p. Let us denote the slow divergence integral from nv (μ) to p as J (v, μ).

310

14 Slow–fast Passage through a Jump Point

Definition 14.2 The integral J (0, μ), which is the slow divergence integral between the point n and p, is J0 (μ) := J (0, μ). Clearly, we have J (v, μ) = J0 (μ) + J1 (μ)v + O(v 2 ),

v ≤ 0.

(14.2)

In case I we have J1 (μ) < 0; in case II we have J1 (μ) > 0. Remark 14.1 The slow divergence integral J computed along the attracting sequence and L computed along the repelling sequence is defined for a = 0 and μ ∼ μ0 . Depending on the type of breaking mechanism, it is possible to extend their definition to either nonzero values of a or nonzero values of a, ¯ see Sects. 12.5 and 12.6.

14.2.2 Slow Divergence Integrals of Slow–fast Cycles As in Sect. 14.1, we use v (μ) as a notation for the canard cycles near 0 (μ), recalling that we take v = |y0 | > 0 for a canard with head and v = y0 < 0 for a “round” canard. We hence define  J (v, μ) − L(v, μ), v < 0, I (v, μ) = K(−v, μ) + J (0, μ) − L(−v, μ), v > 0. Both definitions agree at v = 0 and evaluate to J0 (μ) − L0 (μ). We describe the local behavior of I near v = 0. Using the asymptotic expressions for the integrals L, J , and K, we find I (v, μ) =

 J0 (μ) − L0 (μ) + J1 (μ)v + o(v) J0 (μ) − L0 (μ) −

8 3/2 3v

+

o(v 3/2 )

v < 0, v > 0.

Essentially, the behavior is captured in Fig. 14.4, and it depends on the sign of J1 (μ). The slow divergence integral of v (μ) is only piecewise smooth w.r.t. the parameter v controlling the size of the canard.

14.3 Local Study Near the Jump Point q

311

v

J0 (μ) > L0 (μ)

v

v

J0 (μ) = L0 (μ) v

J0 (μ) > L0 (μ)

1(

)0

J0 (μ) < L0 (μ)

v

J0 (μ) = L0 (μ)

Case I:

v

J0 (μ) < L0 (μ)

Fig. 14.4 Piecewise smooth behavior of the slow divergence integral I (v, μ) of the slow–fast cycle v (μ) near v = 0

Proposition 14.1 Consider case I. Suppose that I (0, μ0 ) = 0. Then there exists a neighborhood U of μ0 and a unique continuous function v : U → R with v(μ0 ) = 0 and such that I (v(μ), μ) ≡ 0,

∀μ ∈ U.

Proof Let f (t) = I (t, μ), when t < 0 and f (t) = I (c(μ)t 2/3 , μ) when t > 0. The coefficient c(μ) > 0 is chosen conveniently so that f is continuously differentiable at the origin (in fact c = 14 (−3J1 )2/3 ). We can apply the Implicit Function Theorem 1 since f (0, μ0 ) = 0, ∂f ∂t (0, μ0 ) = J1 (μ) = 0. There exists a C function t (μ) so that f (t (μ), μ) = 0. We now define v(μ) = t (μ) when t (μ) < 0 and v(μ) = c(μ)t (μ)2/3 when t (μ) > 0. Then v is a continuous solution of I (v(μ), μ) = 0. 

14.3 Local Study Near the Jump Point q We recall the local expression (14.1) for the family of vector fields near q, introduced in the previous section:  X,λ :

x˙ = y + x 2

y˙ =  g(x, , λ) + (y + x 2 )h(x, y, , λ) ,

with g(0, , λ) = 1. The interface from round canards to canards with head occurs precisely at the point q, and therefore more information is needed about this point. A detailed study of the point q = (0, 0) requires a desingularization procedure (blow-up). This

312

14 Slow–fast Passage through a Jump Point

procedure is presented in Chap. 8 and the case of a generic jump point like q is treated in Sect. 8.2. We briefly recall this desingularization procedure in Sect. 14.3.1 emphasizing its effect on the canard cycle 0 (μ).

14.3.1 Blowing Up the Jump Point The point q = (0, 0) is a jump point. We consider the blow-up of the system at the point q, given by the formula: ¯  = u3 ¯ , x = ux, ¯ y = u2 y, 2 , u ∈ (R+ , 0). Here S 2 is the half-sphere {x¯ 2 + y¯ 2 + ¯ 2 = with (x, ¯ y, ¯ ¯ ) ∈ S+ + 1,  ≥ 0}, diffeomorphic to a 2-disk, and the blow-up formula defines a map  2 × [0, u ], for any small u > 0 into a neighborhood of from the blown-up space S+ 0 0 3 (q, 0) ∈ R+ = {(x, y, ) |  ≥ 0}. The vector field u1 −1 ∗ (X,λ ) defined for u > 0 extends to the blown-up space ¯ into a vector field X called the blown-up vector field (it is actually a λ-family of vector fields, but we will not denote the dependence explicitly when we refer to the blown-up vector field). In Fig. 14.5 we present the blown-up vector field

s

S

q2

−∞ q1

σ +∞

Γ−∞ 0

Γ−h 0

Γ+∞ 0

Fig. 14.5 Phase portrait on the blow-up locus, after blow-up of the jump point q, and a detail of the way the headless canard 0 unfolds in different limit periodic sets on the blow-up locus

14.3 Local Study Near the Jump Point q

313

2 × {0}. Its phase portrait is independent of λ. The in restriction to the disk S+ ¯ two boundary of this disk (given by u = ¯ = 0) contains four singular points of X: , resonant hyperbolic saddles s, S and two semi-hyperbolic singular points q1 q2 . In 2 × {0}, there is an open set W filled by orbits of X ¯ with q1 as the interior of S+ α-limit and s as ω-limit. Also the circular arcs (q1 , s) and (q1 , q2 ), are orbits of X¯ 2 × {0}. which are contained in the boundary of S+ ¯ and parameterized by h ∈ R (we We choose in W a section σ , transverse to X, will identify σ with R and write h ∈ σ ). We assume that σ cuts each orbit of X¯ in W and denote by γh the orbit through h ∈ σ . The end points of σ¯ belong to the ¯ respectively. They correspond to h = ±∞ and we orbits (q1 , s) and (q1 , q2 ) of X, choose the orientation of σ such that −∞ belongs to (q1 , s) and +∞ belongs to (q1 , q2 ). The canard cycle 0 shown in Fig. 14.6 unfolds in the blow-up space into a full ¯ that we call secondary slow–fast family of limit periodic sets 0h with h ∈ σ¯ for X, cycles (or secondary canards), see Fig. 14.5 for details of the limit periodic sets near q. Each 0h (μ) is sent to the unique canard cycle 0 (μ) by the blow-up map . For h ∈ σ , the slow–fast cycle 0h (μ) contains the orbit γh . The two cycles 0−∞ (μ) and 0+∞ (μ) are secondary slow–fast cycles containing a part of the equator of the blow-up. To study the limit cycles bifurcating from the (secondary) canard cycles 0h , we need to compute local transitions along the flow of X¯ (that we will consider up to a multiplicative constant) near the singular points s, q1 and q2 . To this end we will consider X¯ in charts W defined by taking y¯ = −1 or x¯ = 1, and keeping, respectively, the coordinate x¯ or y¯ in some interval. We take (u, ) ¯ ∈ [0, u0 ]×[0, ¯0 ], with u0 , ¯0 > 0 and small enough.

Γ0

−1

canards without head

0

canards with head

1

v

canards with head

1

v

Γ0 −1

canards without head

0

Fig. 14.6 Family of canards, parameterized by the layer variable v, in the neighborhood of the transitory case at v = 0

314

14 Slow–fast Passage through a Jump Point

In the matching chart given by y¯ = −1 and with coordinates (x, ¯ u, ), ¯ the expression of X¯ is smoothly equivalent to ⎧ 1 ⎨ x˙¯ = −1 + x¯ 2 + 2 ¯ x¯ + O(u) 1 X¯ : u˙ = − 2 ¯ u ⎩˙ ¯ = 32 ¯ 2 .

(14.3)

In the matching chart given by x¯ = 1 and with coordinates (y, ¯ u, ), ¯ the expression of X¯ is smoothly equivalent to ⎧ ⎪ ⎨ y˙¯ = −2y¯ + ¯ X : u˙ = u ⎪ ⎩ ˙¯ = −3¯ ,

1+O(u) 1+y¯ ¯

(14.4)

if we restrict to 1 + y¯ > 0; we will in fact only use (14.4) near (0, 0, 0). (Remark that (14.4) is similar but not identical to (8.6), because in Chap. 8, a slightly different normal form for the jump point is being used.)

14.3.2 Transition at the Saddle Point s We consider a matching chart, given by {x¯ = 1}, with coordinates (y, ¯ u, ). ¯ The vector field X¯ has the expression (14.4) and the point s is located at y¯ = 0, u = 0, ¯ = 0. Looking at the expression (14.4), we see that s is a hyperbolic saddle point for X¯ with eigenvalues: −2, 1, −3. The qualitative behavior of this vector field is shown in Fig. 14.7.

ε Y D0

q2

u

s

S q1

s

Fig. 14.7 Blown-up vector field near the saddle s, and section D0

D0

14.3 Local Study Near the Jump Point q

315

Smooth normal form coordinates (Y, u, ) ¯ are given in Theorem 12.7 (by taking n = 2). We consider a compact neighborhood W˜ 0 of s = (0, 0, 0) in which X¯ can be given this normal from. Let D0 be the section, transverse to the line {Y = ¯ = 0}, contained in W˜ 0 ∩ {u = U0 } with U0 > 0. We can choose W0 = [0, u0 ] × [0, ¯0 ] × ¯ ∈ W0 \{u = [−Y0 , Y0 ] ⊂ W˜ 0 such that each trajectory of X¯ through a point (Y, u, ) 0}, reaches D0 in a positive finite time in W˜ 0 , see Fig. 14.7. This defines a transition map T0λ (Y, u, ) ¯ from W0 \{u = 0} to D0 . Later on, we will restrict T0λ to transverse sections. We can parameterize D0 by (Y˜ , ) (with Y˜ another name for Y ). The following expression of transition T0λ is given by Theorem 12.8 (by taking n = 2):

T0λ

⎧ ⎨ = u3 ¯ : ⎩Y˜ (u, , ¯ λ) =

u2 U02

   Y − ¯ (0, λ)u ln Uu0 + O(u2 ln2 u) .

(14.5)

Both the function Y˜ (u, , ¯ λ) and the O(u2 ln2 u) term in (14.5) are smooth in (u, u ln u, ) ¯ in the sense of Chap. 11. In the current chapter, it suffices to know that u2 u2 Y˜ = 2 (Y + o (1)) = 2 (y¯ + o (1)). U0 U0

14.3.3 Transition at the Semi-Hyperbolic Point q1 We look at the point q1 in a matching chart, neighborhood of q1 , given by {y¯ = −1}, with coordinates (x, ¯ u, ¯ ). The field X¯ has the expression (14.3) and the point q1 is located at x¯ = 1, u = 0, ¯ = 0. We introduce X = 1 − x¯ as local coordinate, and in the coordinates (X, u, ¯ ) the expression of the family of vector fields −2X¯ is given by ⎧ ⎨ X˙ = −4X + ¯ + 2X2 − ¯ X + O(u) ¯ − 2X : u˙ = ¯ u ⎩ ˙ ¯ = −3¯ 2.

(14.6)

Normal form coordinates (Z, , ¯ u) are given by this type of semi-hyperbolic points in Chap. 10. We will now continue working in a compact neighborhood W˜ 1 of q1 = (0, 0, 0) in the local coordinate system (Z, u, ), ¯ where the dynamics of −2X¯ are shown in Fig. 14.8. Let D1 be a section, transverse to the line {Z = ¯ = 0}, contained in W˜ 1 ∩ {u = U1 } for some U1 > 0. We can choose W1 = [−Z1 , Z1 ] × [0, U1 ] × [0, ¯1 ] ⊂ W˜ 1 such that each trajectory of −2X¯ through a point (Z, u, ) ¯ ∈ W1 , with u = 0 and ¯ = 0 (equivalently  = 0), reaches D1 in a positive finite time in W˜ 1 . This defines a transition map T1 (Z, u, , ¯ λ) from W \ { = 0} to D1 which can be continuously

316

14 Slow–fast Passage through a Jump Point

ε

Z s

S q2

D1

q1

q1

u

D1

Fig. 14.8 Dynamics (with reversed time) near the semi-hyperbolic saddle q1 , and section D1 . (The orientation of the coordinate Z follows the equator clockwise)

extended by T1 (Z, 0, ¯ , λ) = 0 and T1 (Z, u, 0, λ) = 0. Later on, we will restrict T1 to transverse sections. We can parameterize D1 by (Z˜ = Z, ), and the transition T1λ is given by the formula (12.23) of Chap. 12:

T1λ

⎧ ⎨ = u3 ¯   : 3 ⎩Z˜ = Z˜ λ (Z, u, ) ¯ = Z exp − 4A(u,¯,λ)+O(u ) ,

(14.7)

3

where A(u, ¯ , λ) = K(U1 , , λ) − K(u, , λ)

(14.8)

˜ ˜ E, λ) = (1+α(E, λ))u3 (1+O(u)) with α smooth with K = K(u,  1/3 , λ) for K(u, in (E, λ), O(u) smooth in (u, E, λ) and α(0, λ) = 0. The O(u3 ) term is smooth in (u, u ln u, , ¯ λ). A is strictly positive.

14.3.4 Transition at the Semi-Hyperbolic Point q2 We look at the point q2 in a matching chart, neighborhood of q2 , given by {y¯ = −1}, with coordinates (x, ¯ u, ). ¯ In this chart, the point q2 is located at x¯ = −1, u = 0, ¯ = 0. Then we introduce X = −x¯ − 1 as local coordinate, and in the coordinates (X, u, ¯ ) the expression of the vector field family 2X¯ is given by ⎧ ⎨ X˙ = −4X + ¯ − 2X2 + ¯ X + O(u)) 2X¯ : u˙ = −¯ u ⎩ ˙ ¯ = 3¯ 2 .

(14.9)

14.3 Local Study Near the Jump Point q

317

ε Z D2 s

S q2

D2

u q2

q1

Fig. 14.9 Dynamics near the semi-hyperbolic saddle q2 , and section D2

Again, we use the normal form coordinates (Z, ¯ , u) which are given in Chap. 10. We refer to Fig. 14.9 for the dynamic behavior. In these coordinates, we choose a compact neighborhood W˜ 2 of q2 as we did before for q1 . We consider in W˜ 2 a section C2 = {−Z2 } × [0, u2 ] × [0, ¯2 ] for some  Z2 , u2 , ¯2 > 0 and a section D2 = −Z2 , Z2 × 0, u 2 × {¯2 } for some Z2 , u 2 > 0 and ¯2 > ¯2 . These sections are chosen such that each trajectory starting at a point of C2 \ {¯ = 0} reaches D2 in a finite positive time in W˜ 2 . In this way, we define a transition map T2λ from C2 \ {¯ = 0} to D2 which is continuously extended by 0 at {¯ = 0}. We will parameterize D2 by (Z, ). The transition T2λ is given by

T2λ

⎧ ⎨ = u3 ¯   : ⎩Z˜ = −Z2 exp − A (u,¯ ,λ) .

(14.10)



¯ → (0, 0). The o(1)-term is a smooth with A (u, ¯ , λ) = ( 43 + o(1))u3 for (u, ) function in (u, u3 ln u, ¯ , λ).

 ,λ)  ,λ) Remark 14.2 We can also write A (u,¯ = 43 β(u,¯ with β(0, 0, λ) = 1. For  ¯ u = 0, this gives the following expression for the transition map near the semi¯ {u=0} : hyperbolic point q2 of X|

 4 + O(¯ ) ¯ → −Z2 exp − . 3¯ 

318

14 Slow–fast Passage through a Jump Point

14.4 Transition Maps Outside the Jump Point q We recall the choice of the transverse section D0 near the saddle point s (Sect. 14.3.2), and the transverse section D1 near the semi-hyperbolic point q1 (Sect. 14.3.3). We also recall that the slow–fast transitory cycle that we study has a canard connection p of jump type or Hopf type, and associated to it we consider a section T, just like in Sect. 13.2. In this section, we describe the two (family of) transition maps Rout η : D0 → T,

Lout η : D1 → T,

to a well-chosen section T, and for suitable small sections D0 and D1 . The section D0 is located at x = U0 > 0 in the local chart where we have a√ normal form for the jump point q in (14.1). The section D1 is located at y = − U1 < 0 in the same coordinate system, see Fig. 14.10. The transverse section D0 is parameterized by (Y˜ , ) where Y˜ = Y is a coordinate introduced for putting the saddle s on the blow-up locus in normal form coordinates, see Sect. 14.3.2. Similarly, the transverse ˜ ), where Z˜ = Z is a coordinate introduced section D1 is parameterized by (Z, for putting the semi-hyperbolic point q1 on the blow-up locus in normal form coordinates, see Sect. 14.3.3. In this section, we state some properties on Rout η and Lout ; in the next section, we compose with maps from a central section C to define η to left and right transition maps Lλ and Rλ . out The structure of both maps Rout η and Lη has been studied in full detail in Chap. 12. More precisely in Sect. 12.5, Theorem 12.10 in dealing with a jump breaking mechanism and in Sect. 12.6, Theorem 12.11 in dealing with a Hopf breaking mechanism. We will define a set of monomials in , denoted  that we will use to express smoothness of transition maps, just like in Chap. 13. In fact,  will be defined exactly as in Definition 13.1. Its definition depends on the case (Hopf or jump).

Rout η

D0

C

C D1 Lη Lout η

T

D0

D1

Rη

T

out Fig. 14.10 Sections T, transition maps Rout η and Lη . In the figure, a model scenario with a Hopf breaking mechanism is seen

14.4 Transition Maps Outside the Jump Point q

319

We proceed exactly like in Chap. 13, where we have worked toward unified expressions for the transition maps along attracting sequences, both in the jump case as in the Hopf case. In particular, the choice of the section T is precisely the out same as in that chapter (Sect. 13.2). Let us now restate the results on Rout η and Lη in a way that we will use them in the sequel of the current chapter. Similarly to Chap. 13, for the sake of unification, we work with a (, b, μ) = (, η)-family of vector fields. For the jump case, b = a (and η = λ); for the Hopf case, we write b = a¯ and η = (a, ¯ μ). We restrict this chapter to the orientable setting and will choose a parameterizaout tion of T so that the maps Rout η and Lη are orientation preserving. ˜ Proposition 14.2 In both cases I and II, the map Rout η (Y , ) has the following expression:  J˜(Y˜ , , η) + ϕ(, η), = θsf exp  

˜ Rout η (Y , )

(14.11)

where J˜ and ϕ are smooth in (, Y˜ , η), and where θsf = −1 in case I and θsf = +1 in case II. We moreover have that J˜(Y˜ , 0, η) = J (U02 Y˜ , μ) + O(b), where J (y, μ) is the slow divergence integral between D0 and p computed in Sect. 14.2. Remark 14.3 Regarding the outer map Rout η , we will only use (14.11). We remark that the transition map is orientation preserving when θsf

∂ J˜ ˜ (Y , 0) > 0. ∂ Y˜

This property can be deduced, in the limiting situation (b, ) = (0, 0), from formula (14.2). We moreover find that J˜(Y˜ , 0) = J0 (μ) + J1 (μ)U02 Y˜ + O(Y˜ 2 ) + O(b). Proposition 14.3 The map Lout η : D1 → T is defined for  > 0 small enough and ˜ ˜ Z close enough to 0. The map Lout η (Y , ) has the following expression: 

˜ Lout η (Z, )

 ˜ (U1 , , η) L ˜ + O(Z)) ˜ exp = Z(1 , 

(14.12)

where L˜ (U1 , , η) is a smooth in (, U1 , η) and the function L (U1 , μ) = L˜ (U1 , 0) is the slow divergence integral between D1 and p.

320

14 Slow–fast Passage through a Jump Point

14.5 Cyclicity of the Transitory Canard Cycle 0 In this section we want to prove Theorem 14.1, i.e. that the cyclicity of the canard cycle without head 0 is at most two in case I and at most three in case II. To this end, we will consider a displacement function η (θ, ) with some variable that we will specify in Sect. 14.5.1. The map η will describe the behavior of the dynamics near 0 , seen after performing the blow-up of the jump point q.

14.5.1 The Displacement Function η Recall the neighborhood W1 near the point q1 on the equator in the blow-up of the jump point q. We choose inside W1 a section C transverse to the trajectories ¯ This section will be transverse to the blow-up locus W1 ∩ {u = 0} and to of X. W1 ∩ {¯ = 0}. It cuts {u = 0} along a half-circle with end points (−Z1 , 0, 0) and (Z1 , 0, 0) (this half-circle is a section σ¯ transverse to the trajectories in W as   defined } × 0, u × in Sect. 14.3.1).The section C cuts {¯  = 0} along two segments {−Z 1 1  {0} and {Z1 } × 0, u 1 × {0} (see Fig. 14.11). This section is an half-open rectangle with two corners along C ∩ { = 0} at the points (−Z1 , 0, 0) and (Z1 , 0, 0). We choose coordinates (θ, ) ∈ [−θ0 , θ0 ] × [0, 0 ] on C and we will suppose that they are smooth outside the two corner points and continuous at the two corner points. It can be easily seen that such a coordinate system exists; in fact there is no need to choose an explicit coordinate θ because all the proofs will be made in different local sections on which we will use explicit coordinates. We will consider C \ { = 0} as smoothly foliated by the open intervals C = ]−θ0 , θ0 [ × {}, with  = 0.

ε

u3 ¯ = (−Z1 , 0, 0)

C

Z

u ε

(Z1 , 0, 0) q1

θ θ0

θ

θr

−θ0

Fig. 14.11 Section C near the point q1 and its intersection with a level surface of u3 ¯ , redrawn schematically in the figure to the right

14.5 Cyclicity of the Transitory Canard Cycle 0

321

The choice of  as coordinate in C implies that ]−θ0 , θ0 [ × {0 } is a side of C, which is disjoint from W1 ∩ { = 0}, contained in the level surface of 0 . Opposite to it [−θ0 , θ0 ]×{0} contains three line segments and the two corner points: the point (−Z1 , 0, 0) with coordinates (θr , 0) in C and the point (Z1 , 0, 0) with coordinates (θ , 0). We assume that the coordinate θ is chosen such that: −θ0 < θr < θ < θ0 (with this choice the maps that we will introduce later on will be orientation preserving). We call (Rη (θ, ), ) the transition map from C \ { = 0} to T along the trajectories of X¯ and (Lη (θ, ), ) the transition map from C \ { = 0} to T along ¯ We define the displacement function to be the trajectories of −X. η (θ, ) = Rη (θ, ) − Lη (θ, ). The function η (θ, ) is clearly smooth for  = 0. The second line of each of the transition maps (Rη (θ, ), ) and (Lη (θ, ), ) is given by  = uη (θ, )3 ¯η (θ, ) where (θ, ) → (Zη (θ, ), uη (θ, ), ¯η (θ, )) is the coordinate expression of the embedding of C in W1 . Clearly, for  = 0, the vector field X¯ has a periodic orbit through (θ, ) ∈ C (and hence also X,λ does) if and only if η (θ, ) = 0. Then, the cyclicity of 0 is at most 2 if, for  small enough, the map θ ∈ ]−θ0 , θ0 [ → η (θ, ) has at most 2 roots counted with their multiplicity.

14.5.2 Structure of the Transition Maps Toward T Rout η

In this section, we start the study of the compositions of Rη : C → D0 → T and Lout η

Lη : C → D1 → T, see Fig. 14.10. The function Lη is the restriction to C of the function

˜ Lη (Z, u, ) ¯ = Lout ¯ η), ) 3 η (Z(Z, u, , =u ¯

while the function Rη is the restriction of

˜ ¯ = Rout ( Y (Y, u, , ¯ η), ) Rη (Y, u, )

η

=u3 ¯

,

as long as the points of C belong to W0 . ˜ In these formulas, Y˜ (Y, u, , ¯ η) and Z(Z, u, , ¯ η) are the Y˜ -component of T0λ , ˜ defined on W0 and the Z-component of T1λ , defined on W1 , respectively.

322

14 Slow–fast Passage through a Jump Point

Remark 14.4 In the remainder of the chapter, in expressions such as the one above, we will not always repeat that  is substituted by u3 ¯ . In fact, at some parts of the proof, it is important to have expressions for transition maps that depend on (u, ¯ , ) (and other variables). The smoothness properties w.r.t. u, ¯ and  may be essentially different: to express the structure of Rη and Lη use the concept of a function ϕ(¯ , u, u ln u, s, ) which is smooth in (, ¯ , u, u ln u, s). As ¯ u3 ≡ , the expression of a function of (¯ , u, . . .) is not necessarily unique as a function that is smooth in (, ¯ , u, u ln u, , . . .). For instance, we can write u4 ¯ ln u or u3 ln u, but it is not permitted to write ¯ u3 ln u as  ln u. It is important to notice the following: if a function ϕ(¯ , u, u ln u, s, ) is smooth in (, ¯ , u, u ln u, s), then the function ϕ(¯ , u, u ln u, s, u3 ) ¯ is not in general smooth in (¯ , u, u ln u, s). For this reason we will not make in a systematic way the substitution  = u3 ¯ . ¯ ) is the transition from the This means that the map (Z, u, ) ¯ → (Lη (Z, u, ), points of W1 to the section T along the flow of the vector field and that the map (Y, u, ) ¯ → (Rη (Y, u, ), ¯ ) is the transition from the points of W0 to the section T along the flow of the vector field. It is now easy to deduce from the previous computation the structure of these two maps. Proposition 14.4 (1) For the transition map on the right, defined on W0 near s, we have that 

 Jˆ(Y, u, , ¯ , η) Rη (Y, u, ) ¯ = ϕ(, η) + θsf exp , 

(14.13)

with θsf = −1 in case I and θsf = +1 in case II, and with Jˆ(Y, u, , ¯ , η) = Jˆ0 (, η) + Jˆ1 (, η)u2 (Y + o(1)).

(14.14)

We have that Jˆ0 (0, η) = J0 (μ)+O(b) and Jˆ1 (0, η) = J1 (μ)+O(b), where the functions J0 , J1 are defined in Sect. 14.2. The functions Jˆ0 and Jˆ1 are smooth in (, η) and the function represented by o(1) tends to zero in the limit and is smooth in (, Y, u, u ln u, , ¯ η). Also the function Jˆ is smooth in (, Y, u, u ln u, , ¯ η). The function ϕ is smooth in (, η). (2) For the left transition map, defined on W1 , near q1 , we have that 

Lη (Z, u, ) ¯ = Z 1+O(e

¯  ,η)/ −A(u,¯

 ˆ L(u, , ¯ , η) , )O(Z) exp  



(14.15)

with ˆ L(u, , ¯ , η) = Lˆ 0 (, η) + ( 43 + o(1))u3.

(14.16)

14.5 Cyclicity of the Transitory Canard Cycle 0

323

The function Lˆ 0 is smooth in (, η), the o(1) remainder term is smooth in (, u, u3 ln u, ¯ , η), and Lˆ 0 (0, 0, η) = L0 (μ) + O(b), where L0 is defined in Sect. 14.2. ¯ The function A(u, ¯ , η), already defined in Sect. 14.3.3, is smooth in the ¯ ¯ , η) > 0. variables (u, u3 ln u, , ¯ η), and A(u, Proof (1) Recall the function J˜ entering in (14.11). We can write it as   J˜(Y˜ , , η) = Jˆ0 (, η) + Jˆ1 (, η)U02 Y˜ 1 + O(Y˜ ) . For this we have used the fact that Jˆ1 (0, η) = J1 (μ) + O(b) = 0 near b = 0. To obtain Rη we have to substitute Y˜ = Y˜ (Y, u, , ¯ η) in the second component of 2 the transition map T0λ . Looking at (14.5) we see that Y˜ (u, ¯ , Y, η) = u 2 (Y + U0

o(1)). The o(1) property is valid for  → 0, hence also as (u, ¯ ) → (0, 0). The substitution gives

Jˆ(Y, u, , ¯ , η) = Jˆ0 (, η) + Jˆ1 (, η)u2 (Y + o(1)) 1 + O(u2 ) = Jˆ0 (, η) + Jˆ1 (, η)u2 (Y + o(1)). It is easy to trace the required differentiability properties.  3) (see (14.7)) (2) Substituting the expression Z˜ = Z exp − 4A(u,¯,η)+o(u 3 in (14.12) we have that     ˆ L(u, ¯ , , η) ¯  ,η)/ −A(u,¯ Lη (Z, u, ¯ ) = Z 1 + O(e )O(Z) exp ,  with Lˆ = L˜ − 43 A + o(u). This gives ˆ 1 , u, , L(U ¯ , η) = L˜ (U1 , , η) − K(U1 , , η) 4 + (1 + α( 1/3 , η))u3 + o(u3) 3 where we use expression (14.8) for the function A. We write L˜ (U1 , , η) − ˆ η). Taking u = ¯ = 0 in this formula, it is easy to K(U1 , , η) = L(, ˆ verify (by taking U1 → 0) that L(0, η) = L0 (μ) + O(b). The properties of differentiability are easy to trace in this argument. 

324

14 Slow–fast Passage through a Jump Point

14.5.3 Covering of the Section C Starting at the section C, we will not work directly with the coordinate θ but we will introduce an atlas of charts on each C by choosing adapted new sections. The remark is that changing of section is equivalent to introducing a new chart. This remark is based on the following simple observation: Lemma 14.1 Let X be a vector field and C1 , C2 and T be three sections such that the following transition maps are defined: 1. A transition map P : C1 → C2 along the flow of X, 2. Transition maps L1 : C1 → T along the flow of −X, and a transition map R1 : C1 → T along the flow of X, 3. Transition maps L2 : C2 → T along the flow of −X, and a transition map R2 : C2 → T along the flow of X, Then R1 = R2 ◦ P and L1 = L2 ◦ P . We will apply this Lemma in the following way. The section C with coordinates (θ, ) will be the section C1 of Lemma 14.1. The section C2 will be one of the following three sections chosen also inside W1 : a section Cm ⊂ {¯ = ¯m }, a section Cr ⊂ {Z = −Z1 }, and a section C ⊂ {Z = Z1 } for some m > 0 and Z1 > Z1 . We choose the four sections C, Cm , Cr , C in such a way that starting at any point of C the trajectory of X¯ arrives at a point on Cm ∪ Cr ∪ C in a finite positive time. On the section Cm we choose (Z, ) as coordinates. The vector field X = ∂ −3¯ ∂∂¯ + u ∂u is tangent to the sections Cr and C and preserves the function  = ¯ u3 . We choose on Cr the time τ of the flow of −X as one coordinate. Explicitly, if ϕ X (τ, (u, )) ¯ is the flow of X on Cr , we may take (τ, ) as coordinates of the point ϕ −X (τ, ( 1/6 ,  1/2 )). The choice of the point ( 1/6 ,  1/2 ) is rather arbitrary. The sole interesting property of this choice is that the value of the function u ¯ 3 at this point is equal to . On the section C we make a similar choice, but with the vector field X. Let O m be the open set of C, domain of the transition map from C to Cm . We define in the same way the open sets O r and O , domains for the transition maps from C to Cr and C , respectively. By hypothesis, (O m , O r , O ) is an open covering of C. In C, Or covers a neighborhood of [−θ0 , θr ] × {0} and O covers a neighborhood of [θ , θ0 ] × {0}. For each  = 0 we have that Or = O r ∩ C is an open interval with end point (θ0 , ), O = O ∩ C is an open interval with end point (−θ0 , ), and the closure of Om = O m ∩ C is strictly contained in ]−θ0 , θ0 [ × {}. See Fig. 14.12.  , C  , C  be the curves representing the intersection of C , C , C with Let Cm m r r the -level. For each  = 0, the transition maps along the flow of X¯ define r (θ ), τ (θ ) from O m , O r , O diffeomorphisms preserving orientation, Z,λ (θ ), τ,λ    ,λ    m into Cm , Cr , C , respectively. The pairs (O , Zλ (θ )), (O , τλ (θ )), (Or , τλr (θ )) constitute the atlas of charts of C that we want to use. Note that the blow-up and choice of sections and diffeomorphisms as defined here is done with the blown-up vector field in original coordinates (, λ); it is only

14.5 Cyclicity of the Transitory Canard Cycle 0

325

Fig. 14.12 Section C and parameterizations in different charts Cm , Cr , C

ε

Z Cm

Cr

u

C q1

now, when considering transition maps to T that we consider parameter values (b, μ), which are equal to (a, μ) in the jump case and with (b, μ) = (a, ¯ μ) in the Hopf case; in both cases the parameters that will be kept in the notation are (b, μ). Let (Rηm , ), (Rηr , ), and (Rη , ) be the transition maps from, respectively, ¯ Let also the sections Cm , Cr , and C to the section T following the flow of X. m r (Lη , ), (Lη , ), and (Lη , ) be the transition maps from, respectively, the sections r ¯ Let m Cm , Cr , and C to the section T following the flow of −X. η (Z, ), η (τ, ), and η (τ, ) be the corresponding displacement functions defined on the sections Cm , Cr , and C , respectively.

14.5.4 Unique Maximum Properties First we state a result of which the proof is clear. Lemma 14.2 Let g(θ ) be twice differentiable function on an interval I . If each critical point θ∗ of g is such that g (θ∗ ) < 0 (local quadratic maximum), then there exists at most one critical point on that interval. The same results are true if each critical point is a local quadratic minimum. Remarks 14.5 1. The condition g (θ ) = 0 ⇒ g (θ ) < 0 is intrinsic (invariant by local change of coordinates). In this paper, we will verify the conditions of this lemma by covering the interval with open sets where we take local coordinates. 2. Under the conditions of the lemma, g has at most two roots on the interval.

326

14 Slow–fast Passage through a Jump Point

3. The easiest way to verify the conditions of the lemma is to prove that g (θ ) < 0 on the entire interval, but this is not always the case. In fact, a more general way 

to verify the lemma is to prove that α1 g (θ ) < 0, for some strictly positive function α. This allows to replace the first derivative by an equivalent function before deriving again. 

As explained above by Lemma 14.1, the function m η restricted to some  = 0 can be seen as the function η in restriction to the chart (Om , Zη (θ )) of C . The same remark can be made for the other displacement functions rη , η . Proposition 14.5 In case I , for  > 0 small enough, the functions m η (Z, ), r η (−τ, ), and η (τ, ) satisfy the conditions of Lemma 14.2. r In case I I , for  > 0 small enough, the functions − m η (Z, ), − η (−τ, ) satisfy the conditions of Lemma 14.2. r To study the maps m η , η , and η we introduce the following sections (see Fig. 14.13): 0 ⊂ {¯ 1. In W0 we introduce a section Cm  = ¯0 } for some ¯0 > 0, with coordinates 0 (Y, u) and a section Cr ⊂ {Y = −Y0 } for some Y0 > 0 (Y the normal form coordinate near s). 2. In W2 we have already introduced, for the definition of T2λ , a section C2 ⊂ {Z = −Z2 } for some Z2 > 0, and a section D2 ⊂ {¯ = ¯2 } for some ¯2 > 0 (Z the normal form coordinate near q2 ). 3. All the transition maps that we will consider are smooth diffeomorphisms. 0 , C , D has The choice of the different sections Cm , Cr , C followed by Cr0 , Cm 2 2 to be made in a coherent way:

1. The sections Cr and C must be chosen sufficiently small, let us say Cr = {−Z1 } × [0, u1 [ × [0, ¯1 [ and C = {+Z1 } × [0, u1 [ × [0, ¯1 [, with u1 > 0 and ¯1 > 0, both small enough. The permitted size will be made precise during the study of the different displacement functions. 2. The section Cm = ]−Z1 , Z1 [×[0, u1 [×{¯m } has to be chosen such that Z1 < Z1 and such that ¯m < ¯1 . These conditions imply that the three sections Cm , Cr , C ¯ We recall that define an atlas for the reference section C through the flow of X. Fig. 14.13 Sections near the jump point q

D0

0 Cm

Cr0

D2 q2

s

C2

Cm C

q1

Cr D1

14.5 Cyclicity of the Transitory Canard Cycle 0

327

C is chosen small enough, in fact inside the box made by the three other sections, inducing that the flow of X¯ goes from C toward Cm ∪ Cr ∪ C . 0 , C , D are chosen large enough such that transition maps 3. The sections Cr0 , Cm 2 2 0 , from C to C 0 , from C to C , and from D to C 0 . are defined from Cm to Cm r 2 2 r m The transition maps from Cr to Cr0 and from C to C2 preserve the planes {u = 0} and { = 0}, and are hence of the form (u, ) ¯ → (uH1 , H ¯ 2 ) for some strictly positive functions H1 , H2 . See Fig. 14.13. We can now prove Proposition 14.5. Analysis of the Difference Map m η on Cm The maps Rηm and Lm η are transitions Cm → T, decomposed in different steps: Rout η

Rηm :

0 Cm −→ Cm −→ D0 −→ T,

Lm η :

Cm

−→

Lout η

D1 −→ T.

The map Lm η is the restriction of Lη (in (14.15)) to Cm . On Cm we have that u = 1/3  1/3 ¯ ¯ 1/3 /¯m ¯ , η) = A( , ¯m , η) = A¯ 0 (U1 , η) + o(1), see 1/3 , so we can write A(u,  ¯m

Proposition 14.4. We find  Lˆ 0 (0, η) + o(1) . )O(Z)) exp  

Lm η (Z, )

= Z(1 + O(e

−(A¯ 0 +o(1))/

¯ We will take Z˜ = Z(1 + O(exp[− A0 +o(1) ])O(Z)) as a new coordinate on Cm . In  this coordinate the map Lm is linear: η

 ˆ0 (0, η) + o(1) I . = Z˜ exp  

˜ Lm η (Z, ) In the study of Rηm , u =

 1/3 1/3 ¯0

0 so that R | on Cm 0 is a map η Cm



 Jˆ0 (, η) + Jˆ1 (, η)(/¯0 )2/3 (Y + o(1)) Y → ϕ(, η) + θsf exp . 

328

14 Slow–fast Passage through a Jump Point

0 is a family of diffeomorphisms, smooth in (, Z, ˜ η): The transition from Cm to Cm

∂ ˜ ˜ ˜ (Z, ) → Hη (Z, ),  with ˜ Hη (Z, ) > 0, defined for  ≥ 0, including  = 0. ∂Z ˜ ) with Rη |C 0 is equal to Then we have that Rηm , composition of Hη (Z, m

˜ ) = ϕ(, η) Rηm (Z,

 ⎤ ˜ ) + o(1) Jˆ0 (, η) + Jˆ1 (, η)(/¯0 )2/3 Hη (Z, ⎦. +θsf exp ⎣  ⎡

˜ ): We have the following remarkable property for Rηm (Z, Lemma 14.3 For any order  of derivation k ∈ N we can take  > 0 small enough k ˜ ) > 0 for all (Z, ˜ , η) with  = 0. so that θsfk+1 ∂˜ k Rη (Z, ∂Z

Proof Writing   ˜ , η) = Jˆ0 (, η) + Jˆ1 (, η)(/¯0 )2/3 Hη (Z, ˜ ) + o(1) , K0 (Z, we have that     ˜ , η) ∂ m ˜ 1 ˆ K0 (Z, 2/3 ∂Hη ˜ . Rη (Z, ) = θsf J1 (, η)(/¯0 ) (Z, ) + o(1) exp   ∂ Z˜ ∂ Z˜ Notice that both in case I and in case II, θsf Jˆ1 = θsf (J1 (μ) + o(1) + O(b)) is strictly ˜ ) > 0, we can write positive (see (14.2)). As also ∂˜ Hη (Z, ∂Z

  ˜ , η) K1 (Z, ∂ m ˜ , Rη (Z, ) = exp  ∂ Z˜ with

 

∂

˜ ˜ ˜ ˆ Hη (Z, ) + o(1)

K1 (Z, , η) = K0 (Z, , η) −  ln  +  ln J1 (, η) ∂ Z˜ 2 +  ln(/¯0 ), 3 ˜ , η) = K0 (Z, ˜ , η) + o(1). This means that ∂ Rηm (Z, ˜ ) > 0 or simply K1 (Z, ∂ Z˜ ˜ , η) with  = 0 and also that K1 is completely similar to K0 . The only for all (Z, difference between K0 and K1 is that the value of the function represented by o(1) may differ from one expression to the other. Then we can repeat the same argument

14.5 Cyclicity of the Transitory Canard Cycle 0

by recurrence on k for any partial derivative each step.

329 ∂k Rm , ∂ Z˜ k η

with just a change of sign at 

We can hence write  ˆ 0 (0, η) + o(1) L . Z˜ exp  

˜ m η (Z, )

=

˜ ) − Rηm (Z,

(14.17)

2 ∂2 m ˜ ˜ In case I, where θsf = −1, we have that ∂˜ 2 m η (Z, ) = ∂ Z˜ 2 Rη (Z, ) < 0. The ∂Z conditions of Lemma 14.2 are trivially fulfilled and this concludes the proof in this case. 2 In case II, notice that ∂˜ 2 m η > 0. Intrinsically, this means Lemma 14.2 is ∂Z m satisfied for − η .

Analysis of the Difference Map rη on Cr The maps Rηr and Lrη are transitions Cr → T, decomposed in different steps: Rout η

Rηr :

Cr −→ C0r −→ D0 −→ T,

Lrη :

Cr

−→

Lout η

D1 −→ T.

On Cr we have that Z = −Z1 < 0. The map Lrη , restriction of Lη to Cr , is equal to   Iˆ(u, ¯ , , η) ¯ Lrη (u, ¯ ) = −Z1 1 + O(e−A(u,¯ ,η)/ ) exp .    ¯  ,η) But, as Z1 1 + O(exp[− A(u,¯ ]) > 0, we can rewrite Lrη :   ˆ L(u, , ¯ , η) +  A¯ (u, ¯ , , η) , = − exp  

Lrη (u, ) ¯

where A¯ (u, ¯ , , η) = ln Z1 + ∞ (u, , ¯ , η) with a function ∞ that is smooth and flat at  = 0. Taking into account the expression (14.16) for Lˆ and the equality  = u3 ¯ , we have that   4 ˆ ¯ ˆ + o(1) u3 + ¯ u3 A¯ (u, , L(u, , ¯ , η) +  A(u, ¯ , , η) = L0 (, η) + ¯ , η) 3   4 + o(1) u3 , = Lˆ 0 (, η) + 3

330

14 Slow–fast Passage through a Jump Point

where the o(1) → 0 as (u, ¯ ) → (0, 0) and it has the differentiability properties expressed in Proposition 14.4. Then we have ⎡ Lrη (u, ¯ ) = − exp ⎣

Lˆ 0 (, η) +



4 3



 ⎤ + o(1) u3 ⎦,

(14.18)

where the o(1) expression is a smooth function in (u, u3 ln u, , ¯ η) and Lˆ 0 (, η) is smooth in (, η). Remark 14.6 The minus sign in front of the exponential might give the wrong impression that the transition map is not orientation preserving. However, we recall the parameterization of the section Cr , i.e. the time of the flow of the vector field ∂ −X = 3¯ ∂∂¯ − u ∂u , and with respect to this parameterization, we will see that the map is indeed orientation preserving. On the other side, Rηr is the composition of the smooth regular transition Rη1 from Cr to Cr0 with the restriction of Rη to Cr0 . We have that Rη1 (u, ) ¯ = (uG (u, , ¯ η), G ¯ (u, ¯ , η)), ¯ η) and G (u, , ¯ η) both strictly positive on Cr (moreover, as Rη1 must with G (u, , 3 preserve  = u ¯ we have that G (u, , ¯ η)3 G (u, , ¯ η) ≡ 1). The restriction of Rη 0 to Cr is equal to  Jˆ0 (, η) + Jˆ1 (, η)(Y1 + o(1))u2 , (u, ) ¯ → ϕ(, η) + θsf exp  

where Jˆ0 , Jˆ1 , ϕ are smooth in (, η) and the function represented by o(1) is smooth in (, u, u ln u, ¯ , η). Composing on the right with Rη1 we obtain  ˆ  , , η) + o(1))u2 Jˆ0 (, η) + (−θsf K(¯ , = ϕ(, η)+θsf exp  

Rηr (u, ¯ )

(14.19)

ˆ  , , η) = θsf Jˆ1 (, η)Y0 (G (0, ¯ , η))2 (as Cr0 ⊂ {Y = −Y0 } where the function K(¯ for some Y0 > 0), is smooth in (, ¯ , η) and is strictly positive for any (¯ , η), in both cases I and II. The term o(1) is a new remainder which is again smooth in (, u, u ln u, ¯ , η). It follows from (14.18) and (14.19) that    D(u, , ¯ η) C(u, , ¯ η) + θsf exp , = ϕ(, η) + exp   

rη (u, ¯ )

(14.20)

14.5 Cyclicity of the Transitory Canard Cycle 0

331

with ˆ  , , η)u2 + o(u2 ) C(u, ¯ , η) = Jˆ0 (, η) − θsf K(¯ and 4 D(u, , ¯ η) = Lˆ 0 (, η) + u3 + o(u3 ). 3 These two functions C and D belong to the largest class of functions we have introduced so far: they are smooth in (, u, u ln u, ¯ , η). The o-terms appearing in C and D give asymptotic information as (u, ) ¯ → (0, 0). There is a mild abuse of notation in (14.20) where we write rη (u, ) ¯ for the displacement function. The function rη (τ, ) previously defined and used in Sect. 14.5.3 is just the composition of the map (τ, ) → (u(τ, ), (τ, ¯ )) defined by ∂ the flow of the vector field −X = 3¯ ∂∂¯ − u ∂u , with the function defined in (14.20). An important remark is that   ∂ r ¯ )), η (τ, ) = L−X rη (u(τ, ), (τ, ∂τ where L−X is the Lie derivative with respect to the vector field −X. The formula is valid for  = 0, when the functions that we consider are smooth. But we have a good control on the asymptotic properties of the Lie-derivation on functions of (u, ¯ , , η), which are smooth in (, u, u ln u, ¯ , η) in the sense previously defined. Here are three properties that we will need and that are easy to prove: 1. L−X (uk ) = −kuk , L−X (¯ k ) = 3k ¯ k , 2. L−X (o(uk )) = o(uk ). 3. L−X (O(¯ )) = O(¯ ). Using these properties, we easily obtain that ˆ  , , η)u2 + o(u2 ) L−X C = 2θsf K(¯

and L−X D = −4u3 + o(u3 ).

ˆ  , , η) > 0 for all (¯ , , η). Then we can rewrite: Let us recall that K(¯ 

ˆ  , , η)(1 + o(1))u2 , L−X C = 2θsf K(¯ L−X D = −(4 + o(1))u3.

(14.21)

We choose the Section Cr small enough to have 1 + o(1) ≥ 12 and 4 + o(1) ≥ 3 in the expressions in (14.21), in order to be able to bound these expressions away from zero uniformly. We now consider the Lie derivative of rη : L−X rη

    θsf 1 D C + L−X C exp . = L−X D exp    

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14 Slow–fast Passage through a Jump Point

Some of the factors in L−X C and L−X D can be written inside the exponentials. 2 After division by 2u , the equation gives      D + (ln 2 + o(1)) C + (ln Kˆ + o(1)) r . L−X η = −u exp + exp 2u2   Let us consider 

r1 η

We have that

C + (ln Kˆ + o(1)) = exp − 

∂2 r (τ, ) ∂τ 2 η 

same as the order of L−X



 L−X rη . 2u2

= L2−X rη (u, ¯ ) and the order of L2−X rη (u, ) ¯ is the  r1 3 η in u. Using  = u ¯ , we have

   ˆ 2 + o(u2 ) D − C + o(u2 ) Lˆ 0 − Jˆ0 + θsf Ku = −u exp + 1 = − exp − 1.   

r1 η

Observe that we have incorporated the leading factor u inside the exponential: there, in the numerator, it gives a term  ln u = ¯ u3 ln u = o(u2 ). Then 

L−X r1 η



  ˆ 2 + o(u2 ) 2θsf(Kˆ + o(1))u2 Lˆ 0 − Jˆ0 + θsf Ku = exp .  

(14.22)

If Cr is chosen small enough, the expression 2θsf (Kˆ + o(1)) in (14.22) will be < 0 on Cr in case I and > 0 in case II. This concludes the proof in this case. Analysis of the Difference Map η on C The maps Rη and L η are transitions C → T, decomposed in different steps: Rout η

Rη :

0 C −→ C2 −→ D2 −→ Cm −→ D0 −→ T,

L η :

C

−→

Lout η

D1 −→ T.

The transition map L η is completely similar to Lrη (see (14.18)). The only difference is that the section of definition is now C on which we have Z = Z1 > 0. We have   ⎤ ⎡ Lˆ 0 (, η) + 43 + o(1) u3 ⎦, L η (u, ¯ ) = exp ⎣ 

14.5 Cyclicity of the Transitory Canard Cycle 0

333

where the function represented by o(1) tends to zero as (¯ , u) → (0, 0) and is smooth in (, u, u ln u, , ¯ η) and Lˆ 0 (, η) is smooth in (, η). On the other side, the map Rη is the composition of four maps: 1. The regular transition Rη2 between C and C2 , preserving  = u3 . ¯ It is 2 given by Rη (u, ¯ ) = (uH (u, ¯ , η), ¯ H (u, , ¯ η)), with H (u, , ¯ η) > 0 and H (u, ¯ , η) > 0 on C . 2. The transition map T2λ near q2 , from C2 ⊂ {Z = −Z2 } to D2 , given by ⎡  Z˜ = T2λ (u, ) ¯ = −Z2 exp ⎣−

4 3

 ⎤ + o(1) u3 ⎦, 

where the o(1)-term is smooth in (u, u3 ln u, ¯ , η) and tends to zero as (¯ , u) → (0, 0). 0 with ˜ ) → (Y = (Hη (Z, ˜ ), ) from D 2 to Cm 3. A regular transition (Z, ∂Hη ˜ ˜ ˜ (Z, ) > 0. The function Hη is for sure smooth in (, Z, η). ∂Z

0 : {¯ 0 we have u = (/¯ 4. The restriction of Rη to Cm  = ¯0 }. On Cm 0 )1/3 , so we will simply write

 Rη (Y, ) = ϕ(, η) + θsf exp

Jˆ0 (, η) + Jˆ1 (, η)( ¯0 )2/3 (Y + o(1)) 

 .

The o(1)-term is smooth in (, Y, , ¯ η) and tends to zero as  → 0. The composition of the first two maps is equal to   ¯ , 4β(u, ¯ η)u3 ¯ = T2λ ◦ Rη2 (u, ¯ ) = −Z2 exp − Wη (u, ) , 3

(14.23)

¯ 0, η) = H (0, 0, η)3 > 0. where β¯ is smooth in (u, u3 ln u, ¯ , η) and β(0, ˜ ), ) is The composition of the last two maps, i.e. the expression Rη (Hη (Z, given by  ϕ(, η) + θsf exp

˜ ) + o(1)) Jˆ0 (, η) + Jˆ1 (, η)( ¯0 )2/3 (Hη (Z, 

 .

(14.24)

In order to use (14.24), we need to rewrite the exponent in a better form. To that end, we write ˜ ) + o(1) = Hη (Z, ˜ ) + ξ0 (, η) + ξ1 (, η)Z(1 ˜ + O(Z)), ˜ Hη (Z,

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14 Slow–fast Passage through a Jump Point

for some functions ξ0 and ξ1 that are o(1) as  → 0. We can rearrange this expression as ˜ ) + o(1) = Hη0 () + Hη1 ()Z(1 ˜ + O(Z)), ˜ Hη (Z, where we keep the property that Hη1 () > 0. The functions Hη0 and Hη1 are smooth in the monomials . Now the expression inside the square brackets in (14.24) can be rewritten as ˜ + O(Z)) ˜ J˜0 (, η) + Jˆ1 (, η)( ¯0 )2/3Hη1 ()Z(1  with J˜0 (, η) = Jˆ0 (, η) + Jˆ1 (, η)( ¯0 )2/3 Hη0 (). We still have J˜0 (0, η) = Jˆ0 (0, η) = J0 (μ) + O(b). This way, we find ˜ ), ) = ϕ(, η) Rη (Hη (Z, +θsf exp



˜ + O(Z)) ˜ J˜0 + Jˆ1 (, η)( ¯0 )2/3 Hη1 ()Z(1

 .



The transition map Rη is hence given by Rη (u, ) ¯ = ϕ(, η) + θsf exp

 ¯ C(u, , ¯ η) 

with ¯ C(u, ¯ , η) = J˜0 (, η) + Jˆ1 (, η)



 ¯0

2/3 Hη1 ()Wη (u, )(1 ¯ + O(Wη ))

and Wη given by (14.23). Using also the function D(u, , ¯ η) = Lˆ 0 (, η) +



 4 + o(1) u3 , 3

we can write the displacement function η in the following way: η

   ¯ D(u, , ¯ η) C(u, , ¯ η) − exp . = ϕ(, η) + θsf exp  

(14.25)

14.5 Cyclicity of the Transitory Canard Cycle 0

335

∂ In this case we will use the Lie derivative LX , X = u ∂u − 3¯ ∂∂¯ . First we have

¯ ) ¯ = Jˆ1 (, η) LX C(u,



 ¯0

2/3

Hˆ η1 ()LX (Wη (u, ¯ )(1 + O(Wη )),

where LX (W (1 + O(W )) has the same expression as LX (W ):   ¯ ¯ , η)u3 4 4β(u, 3 ¯ . LX Wη (u, ¯ ) = Z2 (β(0, 0, η) + o(1)) u exp − 12 3  0 3 >0

¯ It is now easy to simplify the expression of LX C(u, ), ¯ using similar ideas as in the previous case. We obtain that (remembering that sign J˜1 = θsf and Z2 > 0):   ¯ 0, η) + o(1))u3 (4β(0, ¯ ¯ ) = θsf u3  −1/3 exp − LX C(u, , 3 with functions that are smooth in (, u, u ln u, , ¯ η). Using the expression (14.25) we now have that:    ¯ ¯ 0, η) + o(1))u3 ( 43 β(0, C  −1/3 LX η =  exp − exp 3 u     D , −(4 + o(1)) exp  an expression which can be rewritten as      D˜ C˜ −1/3 − exp , LX η =  exp 3 u   with D˜ = D +  ln(4 + o(1))

and

¯ 0, η) + o(1))u3 . C˜ = C¯ − ( 43 β(0,

We introduce 

1 η

D˜ = exp − 



   C˜ − D˜ −1/3 − 1. LX η =  exp u3 

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14 Slow–fast Passage through a Jump Point

As in the previous case, it suffices to show that the Lie derivative LX 1 η is strictly negative for u, ¯ small enough and  = 0. It suffices of course to prove this property ˜ We have that for LX (C˜ − D).     ¯ 0, η) + o(1) u3 − [D +  ln(4 + o(1))] C˜ − D˜ = C¯ − 43 β(0,   ¯ 0, η) + o(1) u3 , = C¯ − D − 43 β(0, where we have used that  = ¯ u3 = o(1).u3. It follows that   ˜ = LX (C¯ − D) − 4β(0, ¯ 0, η) + o(1) u3 . LX (C˜ − D) ¯ we easily find Given that we have already computed LX D and LX C,   ˜ ¯ 0, η) + o(1) 4β(0, LX (C˜ − D) −1/3 ¯ 0, η) + o(1)). = θ  exp − − 4(1 + β(0, sf u3 3¯ When θsf = −1, this expression is clearly negative for  > 0. We conclude the proof for case I. In case II, i.e. when θsf = +1, we note that the factor  −1/3 can be rewritten as −1 u (¯ )−1/3 , and the (¯ )−1/3 can be rewritten as exp o(1) ¯ before being incorporated in the other exponential factor. The equation changes to   ˜ ¯ 0, η) + o(1) LX (C˜ − D) 4β(0, ¯ 0, η) + o(1)). − 4u(1 + β(0, = exp − 3¯ u2 (14.26) We continue the discussion of case II in the next subsection, where we use (14.26) r together with the conclusions on m η and η .

14.5.5 Proof of Theorem 14.1 In case I, there exists a θ0 > 0 such that for  > 0 small enough, the map θ → η (θ, ) is smooth and verifies the assertion of the Lemma 14.2 on I = ]−θ0 , θ0 [. As a consequence of Lemma 14.2 the proof that, in case I, the cyclicity of 0 is at most 2 (i.e. the proof of the first part of Theorem 14.1). It remains to consider case II. We can solve (14.26) using the Implicit Function Theorem: There exists a smooth curve u = (¯ , η) so that in the region u < , any critical point of η is a regular minimum; critical points in u >  are regular maxima, and a critical point located at u =  is of order 2. By deriving w.r.t. LX

14.6 Limit Cycles and their Unfoldings

337

the right hand side of (14.26) once more, it can be easily verified that such a point has a strictly negative third derivative. Written in terms of , the curve u = (¯ , η) leads to a parameterized continuous curve (u, ) ¯ = (U (, η), E(, η)) that is smooth for  > 0. Now recall that the chart C is just one of the charts parameterizing C. If we parameterize C by (θ, ), like we have done in Sect. 14.5.1 (see Fig. 14.11), we find a continuous curve ˜ θ = (, η), ˜ that is smooth for  > 0. Furthermore, (0, η) = θ , i.e. the left corner point defined in Sect. 14.5.1. To the left of this curve, all critical points of η are minima; to the right all critical points of η are maxima, and when a critical point happens to be located at the curve itself, it is a regular inflection point with negative third derivative. From this setting, it is a straightforward exercise to prove that there are at most two critical points, counting multiplicity, for  sufficiently small.

14.6 Limit Cycles and their Unfoldings In this section, we show that the bound on the number of limit cycles is actually reached, and that the vector field undergoes a generic saddle-node bifurcation in case I and a generic catastrophe of codimension 2 in case II. Essential maps like transition maps and difference maps obtained in the previous sections are smooth in (, b, μ). Given the structure of Rout η , the image under the right map of all canard trajectories is situated exponentially close to h = ϕ(, b, μ) in the section T. We recall that the coordinate h is chosen in a way that {h = 0} is the image of Z˜ = 0 under the left map Lout η , as presented in Proposition 14.3. Assumption The function ϕ(, b, μ) in Proposition 14.2 has the property ϕ(0, 0, μ0 ) = 0,

∂ϕ (0, 0, μ0 ) = 0. ∂b

(14.27)

We furthermore assume L0 (μ0 ) = J0 (μ0 ),

(14.28)

where L0 and J0 are the slow divergence integrals defined in Sect. 14.2. Based on this assumption, we will prove: Theorem 14.3 Given θ ∈ [−θ0 , θ0 ], there exists a unique value b = b∗ (θ, , μ) with b∗ (θ, 0, μ) = 0 so that the orbit through (θ, ) ∈ C of the vector field for b = b∗ and for  > 0 is a periodic orbit. As  → 0, the periodic orbit tends to a canard cycle v (μ), where v = v(θ ) is as follows: v = 0 when θ ∈ [θr , θ ], v < 0 for θ < θr and v > 0 for θ > θ . When the slow divergence integral of the canard

338

14 Slow–fast Passage through a Jump Point

cycle v (μ) is nonzero, its sign controls the hyperbolic stability of the nearby limit cycle, for  > 0 sufficiently small. In case I, a consequence of this theorem is the presence of a semi-stable cycle for some θ = θ∗ . Indeed: for θ = −θ0 , the slow divergence integral is strictly positive, and for θ = +θ0 , it is strictly negative. As a consequence, there is a change of stability somewhere in the middle. The semi-stable cycle may intersect the transverse section C at a point in any of the three charts C , Cm , Cr . When b is the only parameter involved, the location of the semi-stable cycle cannot be exactly identified. We can control the location of the semi-stable cycle with an extra parameter: Extra assumption The parameter μ is a scalar defined near μ0 ∈ R, and ∂L0 ∂J0 (μ0 ) = (μ0 ), ∂μ ∂μ

∂ϕ (0, 0, μ) = 0. ∂μ

(14.29)

∂ϕ Remark 14.7 The condition on ∂μ is used to see the parameter μ as essentially different from the breaking parameter. In the Hopf breaking mechanism, this is the case when μ does not appear in the expression of the vector field on the blow-up locus at the Hopf point. In the jump breaking mechanism, this is the case when the μ-perturbation does not break the connection between the two jump points. This condition can always be achieved by reparameterization.

As can be seen from the bifurcation diagram of the slow divergence integral in Fig. 14.4, the parameter μ can be used to shift the zero of the slow divergence integral in case I; we will prove that it can also be used to shift the location of the semi-stable cycle when it appears. Theorem 14.4 Under the assumptions and notations of Theorem 14.3, and under the extra assumption (14.29), there is a curve (b, μ) = (b2 (θ∗ , ), μ2 (θ∗ , )) so that b∗ |μ=μ2 = b2 , and so that the limit cycle through θ = θ∗ obtained in Theorem 14.3 is of order 2. The zeros of the difference map undergo a saddle-node bifurcation at θ∗ w.r.t. the bifurcation parameter b − b2 (θ∗ , ). In fact, all zeros of (θ, , b, μ) are explained through this saddle-node diagram for θ ∈ [−θ0 , θ0 ],  ∈ ]0, 0 ], μ near μ0 and b near 0. In case II, we immediately assume (14.27), (14.28), and (14.29). We will show that the two parameters (b, μ) serve as bifurcation parameters for a generic codimension-2 bifurcation of limit cycles. The location of the semi-stable cycle of order 3 (i.e. the location of the inflection point in the difference map) is fixed and cannot be shifted upon changing parameters; we again refer to Fig. 14.4. Theorem 14.5 In case II, under the assumption (14.27–14.28) and the extra assumption (14.29), there is a continuous curve (b, μ, θ ) = (b3 (), μ3 (),

3 ())

14.6 Limit Cycles and their Unfoldings

339

so that the difference map (θ, , b, μ) has, for  > 0, a zero of order 3 along that curve. Furthermore, the zeros of the difference map undergo a generic codimension 2 bifurcation at θ = 3 with respect to the bifurcation parameters (b − b3, μ − μ3 ).

14.6.1 Saddle-Node Bifurcation of Limit Cycles in Case I In this section, we prove Theorems 14.3 and 14.4. We recall that the section C is parameterized by (θ, ). In all charts Cm , C , and Cr , the difference map is given by (θ, , b, μ) = ϕ(, b, μ) + (θ, , b, μ), where  is exponentially small in . We refer to formulas (14.17), (14.20), and (14.25) for expressions in the individual charts. Using Assumption (14.27), the Implicit Function Theorem gives a manifold b = b∗ (θ∗ , , μ), so that the orbit through θ = θ∗ in the section C is a periodic one when b = b∗ , for  > 0. It is clear that when θ > θ , the periodic orbit tends to a canard with head v (μ) with v > 0, and when θ < θr , the periodic orbit tends to a round canard v (μ) with v < 0. When θ ∈ [θr , θ ], the periodic orbit tends to the shape 0 (μ). Looking at Fig. 14.4, we can restrict μ to a sufficiently small neighborhood of μ0 to find a headless canard with positive slow divergence integral and a headless canard with negative slow divergence integral. The statements concerning the stability of the periodic orbits in Theorem 14.3 follow the study of common cycles (see the results ∂ in Chap. 7), as we have that ∂θ (θ∗ , , b∗ (θ∗ , , μ), μ) > 0 for θ∗ < θr and < 0 for θ∗ > θ . (Note that is continuous at  = 0 but smooth for  > 0.) An argument based on continuity shows the presence of a zero of the derivative somewhere in between. The location corresponds to a semi-stable limit cycle. It is unique and non-degenerate, because of Theorem 14.1. Let θ = θ2 (, μ) be the location of the semi-stable cycle, and let b2 = b∗ (θ2 , , μ). We find (θ2 , b2 , , μ) = 0,

∂ ∂ 2 (θ2 , b2 , , μ) = 0, 2 (θ2 , b2 , , μ) = 0, ∂θ ∂θ

and ∂ ∂b (θ2 , b2 , , μ0 ) = 0. This describes a saddle-node bifurcation of zeros of the difference map. We observe that b∗ (θ∗ , , μ) → 0 as (, μ) → (0, μ0 ). Under the ∗ extra condition (14.29), we can use implicit differentiation to see that ∂b ∂μ = o(1) for all μ. This shows that b∗ = o(1) as  → 0, uniformly in μ.

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14 Slow–fast Passage through a Jump Point

Continuing under this extra assumption (14.29) and inspecting the derivatives of the expressions for in the various charts, we find that a zero of the derivative is given by J0 (μ) − L0 (μ) + o(1) = 0,

(u, ¯ ) → (0, 0).

(To see this we have used that b∗ = o(1) uniformly in μ.) The same form of equation is found in the three charts, so we find the same equation using (θ, ) as coordinates in the section C. Let 1 = J0 (μ) − L0 (μ) + o(1), where we have substituted b = b∗ (θ, , μ). Using (14.29) and the Implicit Function Theorem, we find the existence of a manifold b = b2 (θ∗ , ), μ = μ2 (θ∗ , ), so that along that manifold, the difference map has a zero at θ = θ∗ and furthermore this zero is of order 2. As is clear from the argumentation above, a saddle-node bifurcation of zeros of the difference map takes place at θ = θ∗ with respect to the bifurcation parameter b − b2 . This proves Theorem 14.4.

14.6.2 Proof of Theorem 14.5 In case II, under the assumptions of Theorem 14.5, we can follow the results of the previous sections in the same way, proving the existence of a manifold b = b2 (θ∗ , ), μ = μ2 (θ∗ , ), so that along that manifold, the difference map has a zero at θ = θ∗ , this time at least of order 2. Theorem 14.5 follows if we can prove the presence of a curve θ = θ3 () along which the (b2 , μ2 ) manifold describes a zero of order 3 of the difference map. The presence of a triple zero of the difference map is shown in chart C . We refer to (14.26), where we substitute b = b2 (θ∗ , ) and μ = μ2 (θ∗ , ). Along this manifold, we find that η has a zero of order at least 2, and we find that the derivative of order 2 has the same sign as LX ( 1 η ) (see the analysis of the difference map in C ). In other words, the sign of the second derivative at θ = θ∗ is given by the sign of the expression (14.26), which we repeat here for the sake of convenience:   ˜ ¯ 0, η) + o(1) LX (C˜ − D) 4β(0, ¯ 0, η) + o(1)), = exp − − 4u(1 + β(0, u2 3¯

14.6 Limit Cycles and their Unfoldings

341

where η have been substituted by (b2 , μ2 ). The variables (θ∗ , ) are replaced by (u, ¯ ). In this way, the above expression is an equation in variables (u, , ¯ ). It is clear that we can use the Implicit Function Theorem to show the existence of a curve u = U (¯ , ) that is smooth in (, ¯ ), along which the second derivative is zero. This corresponds to a continuous curve θ = θ3 () along which the triple point is found. This shows Theorem 14.5.

Chapter 15

Transitory Canard Cycles with Fast–fast Passage Through a Jump Point

15.1 Introduction In this chapter we want to study a special type of transitory canard cycle that we call fast–fast transitory canard cycle for a given slow–fast system. This is the second type of generic transitory canard cycle, mentioned in Chap. 4. We consider a slow–fast family of vector fields X,λ on an orientable surface M. We will consider a canard cycle that contains a single canard connection, called p, which can be of Hopf type or of jump type (see Chap. 6). The slow–fast vector field has at p a generic breaking mechanism w.r.t. breaking parameter a ∼ 0, with λ = (a, μ), μ in a small neighborhood of μ0 = 0 in Rm . Let us now describe the kind of transitory canard cycle that we want to study in this chapter. We will assume that, for the value (a, μ) = (0, 0), there is, at the boundary of a layer, a leaf of the fast foliation going from a repelling regular point n− to an attracting regular point n+ and that between these two regular points this leaf is tangent to a quadratic jump point j . We will assume that the canard cycle contains a single (generic) breaking mechanism at p, like in Chap. 14. The critical curve Sλ contains a regular repelling arc A− around n− , a regular attracting arc A+ around n+ , and a curve C that has a regular contact point at j . This contact point j divides C in an attracting arc C− (on the side of A− ) and a repelling arc C+ (on the side of A+ ). We can assume, using a stability argument, that the arcs A− , A+ , the point j , and the two fast trajectories between n− and j and between j and n+ are independent of the parameter λ. See Fig. 15.1. We can choose local coordinates (x, y) in the phase space around the jump point j such that j = (0, 0) for any value of the parameter λ. Using that the jump point

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_15

343

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15 Fast–fast Passage through a Jump Point

A−

C−

C+

A+

j

n−

σ

n+ fast segment that is part of the transitory cycle

local normal form coordinates

Fig. 15.1 Local situation of a fast–fast transitory canard cycle, passing through a jump point j JBM

JBM

j

j

j HBM

HBM

Fig. 15.2 Example canard cycles with a fast–fast passage through a jump point. Some of them are obtained through a jump breaking mechanism (JBM), some through a Hopf breaking mechanism (HBM)

is a quadratic contact point we can more precisely assume that in this coordinate system, the slow–fast system is written as (see (14.1))  X,λ :

x˙ = −y + x 2 y˙ = −(1 + xg(x, λ, ) + yh(x, y, λ, )),

(15.1)

where h, g are smooth. In this coordinate system the arcs C− and C+ are the sides x ≤ 0 and x ≥ 0 of the local critical curve {y = x 2 }. We see that they are located on the side y ≥ 0. The local orientation of the slow dynamics near the jump point is downward. For a = 0, and for any μ, there exists a local family of canard cycles y (μ) for y ∼ 0. Each canard cycle 0 (μ) contains the jump point j . Moreover, for μ = 0, we will assume that the slow divergence integral of 0 (0) is zero. This canard cycle is the fast–fast transitory cycle that we want to study. There is a great geometric variety for the possible fast–fast transitory canard cycles 0 (0). We have pictured some in Fig. 15.2. On the other side, the local description of the canard cycles y (μ) in a neighborhood of the fast arc between n− and n+ depends just on the orientation of the slow dynamics along the arcs A− , A+ leading to four possibilities as presented in Fig. 15.3 (In the normal form the orientation along C− , C+ is imposed and is downwards in all cases.) They are labeled: (A), (B), (C), (D). In each case the

15.1 Introduction

A− C−

345

C+ A+

(A)

(B)

(C)

(D)

Fig. 15.3 Local picture of a fast–fast passage through a jump point, 4 cases

canard cycle y (μ) contains a unique fast trajectory between points n− (y) ∈ A− and n+ (y) ∈ A+ for y < 0 and a fast trajectory between points n− (y) ∈ A− and q− (y) ∈ C− , followed by an arc on C− besides the fixed fast trajectory between j and n+ (n− = n− (0), n+ = n+ (0)). Like in previous chapters, we will introduce the parameters b and η = (b, μ); they are synonyms for a and λ = (a, μ) in case of a jump breaking mechanism, and in case of a Hopf √ breaking mechanism they are synonyms for a¯ and (a, ¯ μ) respectively, with a =  a. ¯ We will return to the parameter notations λ = (a, μ) whenever it is possible to do so. Behavior of the Slow Divergence Integrals We will compute the slow divergence integrals of y (μ) part by part, by subtracting from each other the integral along the attracting sequence and the integral along the repelling sequence. In fact, when computed separately, the slow divergence integrals of the attracting and repelling sequences are meaningful for all values of η = (b, μ) near (0, 0) and not just for b = 0. The expressions below will hence depend on η. As we assume to work with 1-layer canard cycles we just associate two slow divergence integrals on different segments. The first one J (y, η) corresponds to the sum of the slow divergence integrals along the regular arcs of the critical curve that we encounter when we follow the flow from n+ (y) until the breaking mechanism in the positive direction and the second one K(y, η) corresponds to the sum of the slow divergence integrals along the regular arcs of the critical curve that we encounter when we follow the flow from n− (y) until the breaking mechanism in the negative direction. These two integrals are strictly negative. We also have to consider the slow divergence integral along C− , that we denote by L(y, η). Based on (15.1) we can calculate, in terms of x, that  Lη (x0 ) = −

0

x0

4 4x 2dx = x03 (1 + O(x0)), 1 + O(x) 3

which is strictly negative for x0 < 0. √ If we write L in terms of y > 0, hence using x0 = − y, we get 4 √ L(y, η) = Lη (y) = − y 3/2 (1 + O( y)), 3

346

15 Fast–fast Passage through a Jump Point

and ∂Lη √ (y) = −2y 1/2(1 + O( y)). ∂y We see that Lη is C1 in y with Lη (0) =

∂Lη (0) = 0. ∂y

The (total) slow divergence integral of y (η)) is I (y, η) =

 J (y, η) − K(y, η) J (0, η) + L(y, η) − K(y, η)

y ≤ 0; y > 0,

and the assumption for the fast–fast transitory canard cycle is precisely that I (0, 0) = J (0, 0) − K(0, 0) = 0. We recall that if J (0, 0) − K(0, 0) = 0, it was proved in Chap. 7 that the cyclicity of the transitory canard cycle is one: In that case at most one limit cycle bifurcates from 0 (0) and it is hyperbolic. The four cases (A), (B), (C), (D) are distinguished by the sign of the derivatives ∂I ∂K ∂y (0, 0) and ∂y (0, 0). Looking at the orientations of the slow dynamics on A− , A+ , one easily obtains that 1. In case (A): 2. In case (B): 3. In case (C): 4. In case (D):

∂K ∂J ∂y (0, 0) > 0 and ∂y (0, 0) < 0. ∂K ∂J ∂y (0, 0) < 0 and ∂y (0, 0) > 0. ∂K ∂J ∂y (0, 0) < 0 and ∂y (0, 0) < 0. ∂I ∂J ∂y (0, 0) > 0 and ∂y (0, 0) > 0.

Exactly as in the case of a regular layer, we can introduce a notion of codimension for the slow divergence integral I at the transitory level y = 0: Definition 15.1 We say that the slow divergence integral I has a codimension equal to k ≥ 1 at the transitory level y = 0 and at the parameter value μ = 0, if and only if (J − K)(0, 0) =

∂(J − K) ∂ k−1 (J − K) (0, 0) = · · · = (0, 0) = 0 ∂y ∂y k−1

and ∂ k (J − K) (0, 0) = 0. ∂y k

15.1 Introduction

347

(Remark that L does not play a role in its definition and that I = J − K for y ≤ 0.) We can of course extend the definition for k = 0 by (J − K)(0, 0) = 0 in this case, but in this chapter we will just consider a codimension ≥ 1. Remark also that the fact that the canard cycle 0 (0) is a transitory canard could be considered as a phenomenon of codimension 1; the definition of codimension in Definition 15.1 is a notion of codimension inside the space of transitory canard cycles. The codimension of the transitory canard cycle is in that sense 1 higher than the codimension of the slow divergence integral. Let us return to the fast–fast transitory canard cycle at the value μ = 0. It is clear that the codimension is equal to one in the cases (A) and (B), which represent transitory cycles in the boundary of a dodging layer (see Sect. 4.4). Hence a codimension >1 may only occur in the cases (C) and (D), which represent transitory cycles in the boundary of a terminal layer (see Sect. 4.4). We will denote by (C)k and (D)k the corresponding cases of codimension k ≥ 1. We will have ∂J to distinguish different subcases. For k = 1, i.e. when ∂K ∂y (0, 0) = ∂y (0, 0), the different cases amount to 1. Subcase (C) 1 :

∂K ∂y (0, 0)




∂J ∂y (0, 0)

> 0 (hence

3. Subcase (C) 1 :

∂J ∂y (0, 0)




∂K ∂y (0, 0)

> 0 (hence

∂K ∂y (0,0) ∂J ∂y (0,0) ∂K ∂y (0,0) ∂J ∂y (0,0) ∂K ∂y (0,0) ∂J ∂y (0,0) ∂K ∂y (0,0) ∂J ∂y (0,0)

> 1); > 1); ∈ ]0, 1[); ∈ ]0, 1[).

In Fig. 15.4 we represent the graph of the function y → I (y, 0) in the different cases. We remark that

∂K ∂y (0,0) ∂J ∂y (0,0)

= 1 in all cases of codimension k ≥ 2.

Statement of the Result Theorem 15.1 Consider a fast–fast transitory canard cycle 0 (0) with slow divergence integral of codimension equal to k ≥ 1. Then: 1. In cases (A), (B), (C) 1 , and (D) 1 the cyclicity is at most 4. 2. In cases (C) 1 and (D) 1 the cyclicity is at most 5. 3. In all cases with k ≥ 2 the cyclicity is at most k + 4.

(C)1

(C)1

y

y

y

y

I

I

I

I

(D)1

Fig. 15.4 Slow divergence integrals in four different subcases in codimension 1

(D)1

348

15 Fast–fast Passage through a Jump Point

Remark 15.1 There are two main differences between the slow–fast transitory canard cycles treated in Chap. 14 and the fast–fast ones in the present chapter. First, the slow divergence integral of the canard cycle in this chapter may have any codimension k, while the codimension is always equal to 1 for the slow divergence integral of a slow-fast transitory canard in Chap. 14. Secondly, no precise knowledge of the regular transition map along the blow-up locus was needed in Chap. 14, because the exponential character of transition near the semi-hyperbolic points was predominant. Now, precise properties of this regular map become important. We will use that this map has a concave graph. This point is based on a particular property of the blow-up of jump points, that we have proved using Airy functions in Sect. 8.5.2.

15.2 Blow-up of the Jump Point In this section, we will work with respect to initial parameters (, λ). There will be a lot of analogies with the blow-up of the jump point in Sect. 14.3.1, but there are essential differences in terms of what part of the blow-up locus is relevant for the study: We are now interested in the passage between two hyperbolic saddles s− and s+ (the notations of the singular points on the blow-up locus are different from the notations used in Sect. 14.3.1), whereas in the previous chapter the passage between a semi-hyperbolic point and a hyperbolic saddle was the point of attention. As in Chap. 14, we will use the notations of Sect. 8.2 to blow-up the system at the jump point j . It is given by the formula: ¯  = u3 ¯ , x = ux, ¯ y = u2 y, 2 , u ∈ (R+ , 0). Here S 2 is the half-sphere {x¯ 2 + y¯ 2 + ¯ 2 = with (x, ¯ y, ¯ ¯ ) ∈ S+ + 1,  ≥ 0}, diffeomorphic to a 2-disk, and the blow-up formula defines a map  2 × [0, u ], for any small u > 0 into a neighborhood of from the blown-up space S+ 0 0 3 (j, 0) ∈ R+ = {(x, y, ) |  ≥ 0}. The vector field u1 −1 ∗ (X,λ ) defined for u > 0 extends to the blown-up space ¯ into a vector field Xλ called the blown-up vector field. It is in fact a λ-family of 2 × [0, u ]. We show in vector fields defined on the three-dimensional space S+ 0 2 × {0}, which is Fig. 15.5 the blown-up vector field in restriction to the disk S+ called the blow-up locus. The phase portrait of X¯ λ for u = 0 is independent of λ. The boundary of the blow-up locus (given by u =  = 0) is a circle  which contains four singular points of X¯ λ : two resonant hyperbolic saddles s− , s+ and two semi-hyperbolic singular points q, q . 2 × {0}, there is an open set W filled by In the interior of the blow-up locus S+ orbits of X¯ λ having s− as α-limit and s+ as ω-limit. The boundary ∂W of this open set is the union of two closed arcs. On top we have ∂Wu , closure of union of the center separatrix Sq of the semi-hyperbolic point q with α-limit q and ω-limit s+ ,

15.2 Blow-up of the Jump Point

h = +∞

349

q

q

∂W ud Sq s−

s+ σ

h = −∞

W ∂W d

Fig. 15.5 Blow-up of the jump point j

and the orbit (s− , q) ⊂ . On the bottom we have the union ∂Wud , closure of the orbit (s− , q) ⊂  and ∂Wd , closure of the orbit (s− , s+ ) (see Fig. 15.5). In W and near the saddle s− , we choose a section σ , transverse to X¯ λ , and parameterized by h ∈ R (we will identify σ with R and write h ∈ σ ). We assume that σ cuts each orbit of X¯ λ in W and we write γh for the orbit through h ∈ σ . The end points of σ¯ belong to the orbits (s− , q) and (s− , s+ ), respectively. They correspond to h = ±∞ and we choose the orientation of σ such that −∞ belongs to (s− , s+ ) and +∞ belongs to (s− , q). By denoting the orbit (s− , s+ ) as γ−∞ and by using γ∞ for the union of the orbit (s− , q), the point q and the center separatrix of q, we obtain a natural extension of the notation γh to h = ±∞. There is a full family of limit periodic sets ˜ h (μ) with h ∈ σ¯ for X¯ λ , that we call secondary slow–fast cycles. Each ˜ h (μ) is sent to the unique canard cycle 0 (μ) by the blow-up map . For h ∈ σ , the slow–fast cycle ˜ h (μ) contains the orbit γh . We call them regular secondary slow–fast canard cycles. The two cycles ˜ −∞ (μ) and ˜ +∞ (μ) are secondary slow–fast cycles containing a part of the equator of the blow-up. We call them singular secondary slow–fast canard cycles. The canard cycle ˜ ∞ (μ) is the limit of the canard cycles y (μ), for y > 0, when y → 0+ and ˜ −∞ (μ) is the limit of the canard cycles y (μ), for y < 0, when y → 0− , after lifting these canard cycles through the blowing up map. Apart from their lifted part, the canard cycles ˜ h (μ), for h ∈ [−∞, +∞], contain the set {s− } ∪ {s+ } ∪ γh . As the blow-up map  is a diffeomorphism for u = 0, the study of limit cycles bifurcating from 0 (0) reduces to the study of limit cycles of X¯ λ bifurcating from the union ∪h∈[−∞,+∞] ˜ h (0) of the secondary canard cycles. We will perform this study in three different steps: the central case when h ∈ K, an arbitrary large closed interval in σ , a (small) neighborhood of ˜ −∞ (0) and a (small) neighborhood of

350

15 Fast–fast Passage through a Jump Point

˜ ∞ (0). We will see that each of these cases needs rather different ideas. All these studies will be made in Sect. 15.5.

¯λ 15.3 Transitions Near the Singular Points of X To study the limit cycles bifurcating from the (secondary) canard cycles ˜ h (0), we need to compute local transitions along the flow of X¯ λ or −X¯ λ (that we will consider up to a multiplicative constant) near the singular points s− , s+ and q. To this end we will consider ±X¯ λ in different charts. ¯ u, ¯ ), neighborhood 1. A chart W− in the domain {x¯ = −1}, with coordinates (y, of s− = (0, 0, 0). In this chart, −X¯ λ is equivalent to the field ⎧ ⎪ ⎨ u˙ = u − X¯ λ : ˙¯ = −3¯ , ⎪ ⎩ y˙¯ = −2y¯ +

1−ug¯ 1 1−y¯ ¯ ,

(15.2)

where g¯1 (u, ¯ , λ) = g(−u, λ, u3 ¯ ) − u2 yh(−u, ¯ u2 y, ¯ λ, u3 ). ¯ W− is chosen small enough such that the function 1 − y¯ has no zero in W− . 2. A chart W+ in the domain {x¯ = 1}, with coordinates (y, ¯ u, ¯ ), neighborhood of s+ = (0, 0, 0). In this chart X¯ λ is equivalent to the field ⎧ ⎪ ⎨ u˙ = u X¯ λ : ˙¯ = −3¯ , ⎪ ⎩ y˙¯ = −2y¯ −

1+ug¯ 2 1−y¯ ¯ ,

(15.3)

where g¯2 (u, , ¯ λ) = g(u, λ, u3 ¯ ) + u2 yh(u, ¯ u2 y, ¯ λ, u3 ¯ ). W+ is chosen small enough such that the function 1 − y¯ has no zero in W+ . 3. A chart Wq in the domain {y¯ = 1}, with coordinates (x, ¯ u, ). ¯ In this domain the point q is the point with coordinates (−1, 0, 0). We choose as Wq a neighborhood of q. In this chart, as in (14.4) X¯ λ is equivalent to the field ⎧ 1 ⎨ u˙ = − 2 u¯ 3 X¯ λ : ˙¯ = 2 ¯ 2 ⎩˙ x¯ = −1 + x¯ 2 + 12 ¯ x¯ + O(u). As the equivalence is up to multiplication by the function 1 + uxg(u ¯ x, ¯ λ, u3 ) ¯ + 2 2 3 u h(ux, ¯ u , λ, u ¯ ), the neighborhood is chosen small enough such that this function has no zero in it.

15.3 Transitions Near the Singular Points of X¯ λ

351

15.3.1 Transition at the Saddle Points s± The study is completely similar at the two saddle points. So we will treat only the transition near the saddle s+ and content ourselves in giving the result for the transition near the saddle s− . In a chart W+ ⊂ {x¯ = 1}, with coordinates (y, ¯ u, ¯ ), the family of vector fields X¯ λ has the expression (15.3) and the point s+ is located at y¯ = 0, u = 0, ¯ = 0. Looking at the expression (15.3), we see that s+ is a hyperbolic saddle point for X¯ λ with eigenvalues −2, 1, −3. We observe that the linear part of X¯ λ is not diagonal. But, as the eigenvalues are 2-by-2 distinct, we can bring this linear part into a diagonal form. Explicitly this diagonalization is given by (u, ¯ , y) ¯ → (u, ¯ , y¯¯ = y¯ − ¯ ).

(15.4)

¯¯ the family of vector fields X¯ λ is given by In the coordinates (u, , ¯ y) ⎧ ⎪ ⎨ u˙ = u ¯ Xλ : ˙¯ = −3¯ , ⎪ ⎩ y˙¯¯ = −2y¯¯ −

¯¯  +y g¯ 2 (u,¯ ,y+¯ ¯¯  ) y+¯ ¯ . ¯¯  1−yy−¯

(15.5)

In Sect. 10.5 of Chap. 10, we

prove that by a local smooth coordinate change: ¯¯ → u, , ¯¯ we can reduce (15.5) to the normal form: (u, ¯ , y) ¯ Y (u, ¯ , y) ⎧ ⎪ ⎨ u˙ = u ¯ Xλ : ˙¯ = −3¯ ,  ⎪ ⎩ Y˙ = −2Y 1 + + (u2 Y, λ, ) + u¯ + (λ, ),

(15.6)

where + , + are smooth and + (u2 Y, λ, 0) ≡ 0 (because X¯ λ is linear at s+ for ¯ = 0). Let us consider the conjugacy map. We know that ¯¯ = y¯¯ + O(|(y, ¯¯ u, ¯ )|2 ). Y (u, ¯ , y)

(15.7)

But as X¯ λ is already linear for ¯ = 0, we can choose this function of the form ¯¯ = y¯¯ + ¯ O(|y| ¯¯ + |u| + |¯ |). Y (u, , ¯ y)

(15.8)

Composing (15.4) with this expression we obtain Proposition 15.1 The family of vector fields (15.3) is brought in the form (15.6) by a smooth conjugacy (u, ¯ , y) ¯ → u, , ¯ Y+ (u, , ¯ y) ¯ with

Y+ (u, ¯ , y) ¯ = y¯ − ¯ 1 + O(|y| ¯ + |u| + |¯ |) .

352

15 Fast–fast Passage through a Jump Point

We consider a compact neighborhood W˜ + of s+ = (0, 0, 0) in which the normal from (15.6) is defined. Let D+ be a section, transverse to the line {Y = ¯ = 0}, contained in W˜ + ∩ {u = U0 } with U0 > 0. We can choose W+ = [0, u0 ] × [0, ¯0 ] × ¯ ∈ W+ \ [−Y0 , Y0 ] ⊂ W˜ + such that each trajectory of X¯ λ through a point (Y, u, ) {u = 0} reaches D+ inside W˜ + in the positive finite time tu = − ln Uu0 . This defines ¯ from W+ \ {u = 0} to D+ . Later on, we will restrict a transition map Tλ+ (Y, u, ) Tλ+ to transverse sections. We can parameterize D+ by (Y, ) or by (y, ). On D+ we pass from y to y¯ by y = U02 y. ¯ Also, as  = U03 ¯ it follows that Y = y¯ +O() and then y = U02 Y +O() on D+ . In Sect. 12.3.3 of Chap. 12 (Theorem 12.8), we have obtained that the transition in Y from (Y, u, ) ¯ to D+ is given by u2 (Y, u, ) ¯ → Y˜ = 2 [Y + O(¯ u ln u)] . U0 If we now adopt the coordinate y on D+ , i.e. if we change Y˜ by y = U02 Y˜ + O(), using that  = u3 ¯ , we have as right hand side: U02

u2 [Y + O(¯ u ln u)] + O() = u2 [Y + O(¯ u ln u)] . U02

We have proved the following: Proposition 15.2 The transition Tλ+ : (Y, u, ) ¯ → (y, ) ∈ D+ is given by Tλ+ (Y, u, ) ¯ :



 = u3 ¯ = u2 [Y + O(¯ u ln u)] .

y + (u, , ¯ λ)

The function y + (u, ¯ , λ) as well as the remainder O(¯ u ln u) is smooth in the variables (u, u ln u, ¯ , λ). As said above we have for −X¯ λ in a neighborhood of s− , a similar normal form as (15.6) ⎧ ⎪ ⎨ u˙ = u − X¯ λ : ˙¯ = −3¯ ,  ⎪ ⎩ Y˙ = −2Y 1 + − (u2 Y, λ, ) + u¯ − (λ, ), where − , − are smooth and − (u2 Y, λ, 0) ≡ 0.

(15.9)

15.3 Transitions Near the Singular Points of X¯ λ

353

For the conjugacy map we have that Proposition 15.3 The family of vector fields (15.2) is brought in the form (15.9) by ¯ with a smooth conjugacy: (u, , ¯ y) ¯ → u, ¯ , Y− (u, ¯ , y)

¯ = y¯ + ¯ 1 + O(|y| ¯ + |u| + |¯ |) . Y− (u, ¯ , y) For the transition Tλ− between a point (u, ¯ , Y ) ∈ W− \ {u = 0} ( W− a neighborhood of s− in the normal coordinates (u, ¯ , u)) to a section D− ⊂ {u = U0 }, we have ¯ → (y, ) ∈ D− is given by Proposition 15.4 The transition Tλ− : (Y, u, ) Tλ− (u, ) ¯

 :

 = u3 ¯ = u2 [Y + O(¯ u ln u)] .

y − (u, ¯ , λ)

(15.10)

The function y − (u, ¯ , λ) as well as the remainder O(¯ u ln u) is smooth in the variables (u, u ln u, ¯ , λ).

15.3.2 Transition at the Semi-Hyperbolic Point q Using Theorem 10.3, we obtain normal form coordinates (Z, u, , ¯ λ), where q = (0, 0, 0) in which X¯ λ is written as ⎧ ⎨ Z˙ = −4F (u, ¯ , λ)Z X¯ λ : u˙ = −¯ u ⎩ ˙ ¯ = 3¯ 2 , with F smooth and F (0, 0, λ) = 1. Cq  = We choose Wq in this coordinate system. We consider in Wq a section      {−Zq }× 0, uq × 0, ¯q for some Zq , uq , ¯q > 0 and a section Dq = −Zq , Zq ×   0, u q × {¯q } for some Zq , u q > 0 and ¯q > ¯q . These sections are chosen such that every trajectory starting at a point of Cq \ {¯ = 0} reaches Dq in a finite positive q time in Wq . In this way, we define a transition map Tλ (u, ¯ ) from Cq \ {¯ = 0} to q Dq which continuously extends to Cq by Tλ (u, 0) = 0. We will parameterize Dq by (Z, ). Based on (12.22) in Sect. 12.2, we see that q the transition Tλ is given by  q Tλ

:

 = u3 ¯   3 Z˜ = −Zq exp − 43 β(u,¯,λ)u .

where β is a smooth function of (u, ¯ ln , ¯ ¯ 1/3 , λ) such that β(0, 0, λ) > 0.

354

15 Fast–fast Passage through a Jump Point

15.4 Regular Transitions for X¯ λ Along the Blow-up Locus The regular transition along the blow-up locus can be expressed by smooth (families of) diffeomorphisms. But, as said in the introduction, this trivial observation is no longer sufficient to prove the results stated in Theorem 15.1 about the cyclicity. It gave sufficient information in case of a slow–fast transitory canard cycles. Now we will need a property of concavity of the graph that we will present now.

15.4.1 Regular Transition Near the Interior of the Blow-up Locus We want to study the transition along the trajectories γh ⊂ W, h ∈ R, defined in Sect. 15.2. To this end we will work in three different charts: the two matching charts {x¯ = ±1}, neighborhood of s± , and the family chart {¯ = 1}. 1. The two matching charts W± ⊂ {x¯ = ±1}, neighborhood of s± . On these charts we consider the coordinates (u, , ¯ y) ¯ and alternatively the “normal” coordinates (u, , ¯ Y ) (in this case we will speak of left and right normal charts). As the domain of the two charts is supposed not to intersect, there is no problem to use the same name of coordinates on the two sides. 2. The family chart {¯ = 1}. To avoid confusion with the coordinates on the two previous charts we will use x, ˜ y, ˜ u˜ instead of x, ¯ y, ¯ u in this chart. The blowing up formulas are given by x = u˜ x, ˜ y = u˜ 2 y, ˜  = u˜ 3 . The family chart is defined for (x, ˜ y) ˜ ∈ R2 and u > 0 and small (of course we will restrict (x, ˜ y) ˜ to a compact but arbitrarily large domain in R2 ). The blown-up field X¯ λ is a u-parameter family in this chart with expression X¯ λ :



x˙˜ = −y˜ + x˜ 2 y˙˜ = −1 + O(u). ˜

(15.11)

The remainder term O(u) ˜ is a smooth function of (x, ˜ y, ˜ u, ˜ λ). We observe that the restriction X¯ of X¯ λ to {u˜ = 0} is a quadratic vector field on R2 , independent of (u, ˜ λ). The change of coordinates between the charts W+ and {¯ = 1} is given by 1

2

1

¯ u˜ = ¯ 3 u x˜ = ¯ − 3 , y˜ = ¯ − 3 y,

(15.12)

and the change of coordinates between the charts W− and {¯ = 1} is given by 1

2

1

¯ u˜ = ¯ 3 u. x˜ = −¯ − 3 , y˜ = ¯ − 3 y,

(15.13)

15.4 Regular Transitions for X¯ λ Along the Blow-up Locus

355

We now want to consider the orbits of X¯ in W. The set of orbits in W was parameterized in Sect. 15.2 by h ∈ R. Each orbit in W has s− as α-limit and s+ as ω-limit. In the two normal charts W± the vector fields ±X¯ are smoothly equivalent to X˜ :



¯˙ = −3¯ Y˙ = −2Y.

(15.14)

We can suppose that this linear field X˜ is defined in the whole plane of coordinates (¯ , Y ) and not only in the domain of the normalization. In particular we can consider for it the sections σ−1 and σ1 given by {¯ = 1} in the respective normal charts W− and W+ . We parameterize each of them by Y ∈ R and write Y ∈ σ−1 or Y ∈ σ−1 . We can consider the Y -coordinate on σ−1 as a parameterization of the space of local orbits of X¯ near s− , in W− ∩ {¯ > 0} and the same for the Y -coordinate on σ1 for the space of local orbits of X¯ near s+ in W+ ∩ {¯ > 0}. At each orbit γh in W is associated a unique Y ∈ σ−1 and a unique Y ∈ σ1 , corresponding respectively to the unique left and right local orbit contained in it. The parameterizations by Y give to these two spaces of local orbits a natural affine structure, because the normal form is unique up to a linear transformation of each coordinate ¯ , Y . For this reason, it is natural to consider the properties of the following map Definition 15.2 Given any Y ∈ R ≈ σ−1 , there is a unique orbit γh(Y ) which contains the left local orbit corresponding to this Y . Then F (Y ) is the Y -coordinate on σ1 corresponding to the right local orbit contained in γh(Y ) . It is clear that F is a smooth diffeomorphism from R to R, well defined up to affine transformations on the left and on the right. We will prove the following property for F : Proposition 15.5 The diffeomorphism F is concave, i.e.

d 2F (Y ) dY 2

< 0 for all Y ∈ R.

To prove this result we will first reinterpret the diffeomorphism F in terms of the vector field X¯ on the family chart {¯ = 1}. We consider in this chart the lines σ−X = {x˜ = −X} and σX = {x˜ = X} for any arbitrarily large X. These two curves are parameterized by y. ˜ On both the field X¯ is transverse for y˜ = X2 and points to 2 the right of y˜ < X . Moreover, each trajectory starting at a point y˜ < X2 must cut ¯ from σ1 exactly one time. This defines a transition map TX (y), ˜ along the flow of X, 2 2 (−∞, X ) ⊂ σ−1 to σ1 . Let us observe that the domain (−∞, X ) tends toward R ≈ σ−1 when X → +∞, i.e. when σ−X goes to infinity toward the left. In the chart W− , the line σ−X becomes the line {¯ = X−3 } and along this line we can now choose y¯ as parameterization with y¯ = X−2 y˜ (see (15.12)). Using (15.12) we can say the same thing for σ1 seen in the chart W+ . Then, if we choose on σ−X

356

15 Fast–fast Passage through a Jump Point

and σX the parameterization by the coordinate y¯ of the charts W± , the transition is given by a map T¯X (y) ¯ and we have the conjugacy relation: X−2 TX (y) ˜ = T¯X (X−2 y). ˜

(15.15)

This relation is only defined when the coordinate y¯ belongs to some fixed compact neighborhood of 0, which can be chosen independent of X. The domain for y˜ is then an interval with length of order X2 , which tends to cover R when X → +∞. Now, in the charts W± , we can parameterize σ±X by the normal coordinate Y . Let us call FX (Y ) the expression of the flow transition in this coordinate. As T¯X (y) ¯ this map FX (Y ) is only defined on some fixed compact neighborhood of 0. It is conjugate to the map F (Y ) along the flow of the linear vector field X˜ (15.14). As we pass from the section ¯ = 1 for F to the section ¯ = X−3 for FX , this conjugacy is explicitly given by X−2 F (Y ) = FX (X−2 Y ).

(15.16)

The domain for Y in (15.16) is an interval with length of order X2 , which tends to cover R when X → +∞. The change of the coordinates from y¯ to Y is given by (15.8) combined with (15.4). Restricted to {u = 0} these formulas give two functions Y = G± ¯ = X (y) y¯ + O(X−3 ). They permit to write the following relation between FX and T¯X : ¯ ¯ = FX ◦ G− (y). G+ X ◦ TX (y) X ¯

(15.17)

The domain for y¯ in (15.17) is some fixed compact neighborhood of 0. −2 ˜ We now introduce the two maps HX± (y) ˜ = X 2 G± X (X y). Lemma 15.1 The maps HX± (y) ˜ are defined on an interval KX in y, ˜ centered at 0, whose length is of order O(X2 ). Moreover HX± (y) ˜ = y˜ + O(X−1 ) in a C∞ uniform ± way on the compacts. This means that if K ⊂ R is any

then HX is defined

compact,

d H ± ˜

≤ M (K)X−(2 −1) on K for X large enough and, for all ≥ 2 we have

d y˜ X (y) for some M (K) > 0, depending just on K and . ¯ are defined and smooth on a fixed compact interval L in y, ¯ Proof The maps G± X (y) neighborhood of 0. Then HX± (y) ˜ is defined on the interval X2 L in y. ˜ We can write G± ¯ = y¯ + X−3 g(y, ¯ X−3 ), where g is smooth. Then X (y)   HX± (y) ˜ = X2 X−2 y˜ + X−3 g(X−2 y, ˜ X−3 ) = y˜ + X−1 g(X−2 y, ˜ X−3 ). Clearly, the function X−1 g(X−2 y, ˜ X−3 ) can be written as α.g(α 2 y, ˜ α 3 ) with α = −1 X small. This function is smooth in (α, y) ˜ and converges to 0 for α → 0 (i.e. for X → +∞). 

15.4 Regular Transitions for X¯ λ Along the Blow-up Locus

357

We can now prove Proposition 15.6 We have F ◦ HX− = HX+ ◦ TX so that TX (y) ˜ = F (y) ˜ + O(X−1 )

(15.18)

in a C∞ uniform way on the compacts. Proof The expression (15.18) is a direct consequence of the claim F ◦ HX− = HX+ ◦ TX , which is proved by a diagram chasing between the three relations (15.15), (15.16), and (15.17). Writing X2 the multiplication by X2 , (15.16) is just F = X2 ◦ FX ◦ X−2 , then, using (15.17) − −1 ¯ ◦ X−2 F = X 2 ◦ G+ X ◦ TX ◦ (GX )

and then, using (15.15) −2 −1 ◦ TX ◦ X2 ) ◦ (G− ◦ X−2 F = X 2 ◦ G+ X ◦ (X X) − −1 −2 2 = (X2 ◦ G+ ◦ X−2 ), X ◦ X ) ◦ TX ◦ (X ◦ (GX )

which is F = HX+ ◦ TX ◦ (HX− )−1 or equivalently F ◦ HX− = HX+ ◦ TX . 

It is perhaps more enlightening to look at the cubic diagram picture in Fig. 15.6 where it is for any face except possibly the upper one it is immediately clear that it is commutative. It clearly follows that then also the remaining face, i.e. the upper one, is also commutative: this is the relation F ◦ HX− = HX+ ◦ TX . Looking at the differential equation (15.11) of X¯ λ , we see that each orbit in W is a graph y( ˜ x). ˜ Moreover this function has finite limits for x˜ → ±∞, and a given orbit is characterized by the limit at −∞ for instance. Then, given y˜0 ∈ R there is a unique orbit y˜y˜0 (x) ˜ such that limx→−∞ y˜y˜0 (x) ˜ = y˜0 . It follows from ¯ Proposition 15.6 that limx→+∞ y ˜ ( x) ˜ = F ( y ˜ ). In fact, considering (15.11) we ¯ 0 y˜0 have d y˜ = −dt so that limx→−∞ y ˜ ( x) ˜ − lim y ˜ ( x) ¯ x→+∞ ¯ y˜0 y˜0 ˜ of which we already know that it is finite, is in fact proportional to the time to go from x¯ = −∞ to

15 Fast–fast Passage through a Jump Point

(y) ˜

˜ y ˜ → X −2 y

Σ−1

TX

Σ−X

(Y )

G− X

+ HX

Y → X −2 Y

− HX

Σ+1

F

(Y )

Σ+1 y ˜→  X −2 y ˜

Σ−1

(y) ˜

F¯X

(y) ¯

Σ+X

(Y )

G+ X

T¯X Σ− X

(Y )

Y → X −2 Y

358

Σ+X

(y) ¯

Fig. 15.6 Cubic commutative diagram

x¯ = +∞ along the orbit y˜y˜0 (x). ˜ This induces the following interpretation for the map F : Lemma 15.2 F (y) ˜ = y˜ − T (y) ˜ where T (y) ˜ is the finite time to travel all along the orbit in W whose y-component ˜ tends to y˜ when x˜ → −∞. Taking into account Lemma 15.2, we see that Proposition 15.5 is equivalent to the following one, which has been proved in Sect. 8.5.2 (see Proposition 8.2): 2

Proposition 15.7 The time function T (y) ˜ is convex in the sense that dd y˜T2 (y) ˜ > 0 for any y˜ ∈ R. The range of T (y) ˜ is (0, ∞), while the range of T (y) ˜ is (0, 1).

15.4.2 Regular Transition Near the Boundary of the Blow-up Locus We first consider two points e, f on the boundary ∂Wud or ∂Wd of the blowup locus, in a way that [e, f ] is a segment of trajectory of X¯ λ for u = ¯ = 0. Choose transverse sections e and f , through the points e and f , respectively. We parameterize these sections by (u, ¯ ) with u ≥ 0, ¯ ≥ 0 and u ∼ 0, ¯ ∼ 0. For u, ¯ sufficiently small there is a local transition map from e to f , defined by the flow of X¯ λ or −X¯ λ ; we denote it as H (u, ¯ ). Lemma 15.3 H (u, ¯ ) = (uU (u, , ¯ λ), E(u, ¯ , ¯ λ)) where U, E are smooth functions such that U (u, ¯ , λ) > 0, E(u, ¯ , λ) > 0 and U (u, , ¯ λ)3 E(u, ¯ , λ) ≡ 1.

15.4 Regular Transitions for X¯ λ Along the Blow-up Locus

359

Proof The form H = (uU, ¯ E) for the diffeomorphism H follows from the fact that H preserves the spaces {u = 0} and {¯ = 0}. This also implies that U > 0, E > 0. Finally, as the flow of X¯ λ preserves the function u3 ¯ = , we have u3 ¯ ≡ (uU )3 (¯ E) and hence U (u, , ¯ λ)3 E(u, ¯ , λ) ≡ 1. 

We now consider two sections in the matching chart {y¯ = −1}. This chart is parameterized by x¯ in an arbitrarily large interval, and (u, ¯ ) with u ≥ 0, ¯ ≥ 0 and u ∼ 0, ¯ ∼ 0. The part of  in this chart is the x-axis ¯ equal to {u = ¯ = 0}. There is no singular point of X¯ λ on the x-axis. ¯ Consider the sections −x¯0 ⊂ {x¯ = −x¯0 } and x¯0 ⊂ {x¯ = x¯0 }, for some arbitrarily large x¯0 > 0. In the matching chart {y¯ = −1}, the family of vector fields X¯ λ is C∞ -equivalent to ⎧ ⎪ u˙ = 12 u¯ ⎪ ⎨ X¯ λ : ˙¯ = − 32 ¯ 2 , ⎪ ⎪ ⎩ x˙¯ = − 1 ¯ x¯ + 1 + x¯ 2 − 1 ¯ x¯ + O(u), 2

2

where O(u) is a smooth function. We consider the two-dimensional family X¯ λ |u=0 given by 

¯˙ = − 32 ¯ 2 , ¯ x¯˙ = 1 + x¯ 2 − 12 ¯ x.

The time to go from (−x¯0 , 0) to (x¯0 , 0) is equal to  Tx¯0 =

x¯ 0 −x¯ 0

d x¯ = 2 arctan(x¯0 ). 1 + x¯ 2

We notice that Tx¯0 → π if x¯0 → +∞. By continuity, the time to go from (−x¯0, ) ¯ to x¯0 ∩ {u = 0} is equal to Tx¯0 = T¯ + O(¯ ). The integration of the first line of the above differential equation gives ¯ (t) =

¯ 1+ 32 t ¯

for ¯ as initial condition. Then the transition from −x¯0 ∩ {u = 0} to x¯0 ∩ {u = 0} is given by

3 ¯ → ¯ Tx¯0 (¯ ) = ¯ (1 − Tx¯0 ¯ + O(¯ 2 )). 2

(15.19)

360

15 Fast–fast Passage through a Jump Point

We now consider the transition in dimension 3. Formula (15.19) together with Lemma 15.3 implies Proposition 15.8 The transition map from − x¯0 to x¯0 is given by the family of diffeomorphisms (u, ¯ , λ) → H (u, ¯ , λ) = uU (u, , ¯ λ), E(u, ¯ , ¯ λ) , with the following properties: 

E(0, ¯ , λ) = 1 − 32 Tx¯0 ¯ + O(¯ 2 ), U (0, ¯ , λ) = 1 + 12 Tx¯0 ¯ + O(¯ 2 ).

(15.20)

15.5 Cyclicity of the Canard Cycle In this section we want to prove the first part of Theorem 15.1. To this end we will consider a displacement function λ (θ, ) which gives the behavior of the dynamics near 0 , seen after performing the blow-up of the jump point j . We recall that the transitory cycle that we study has a canard connection p of jump type or Hopf type, and associated to it we consider a section T, just like in Sect. 13.2. Finally recall that, similarly to previous chapters, we work with a (, b, μ) = (, η)-family of vector fields. For the jump case, b = a (and η = λ), for the Hopf case, we write b = a¯ and η = (a, ¯ μ). We retain the freedom to express some expression in terms of λ instead of η though, for example when we use the q expressions for Tλ± or Tλ . Expressions in terms of (, λ) can always be reinterpreted as similar expressions in terms of (, η) (albeit that smoothness is reduced to smoothness w.r.t.  1/2 ).

15.5.1 The Displacement Function η We choose in W− (neighborhood of the saddle s− with the normal coordinates (Y, u, )) ¯ a section C transverse to the trajectories of X¯ λ . This section C is similar to the section C used in Chap. 14, but now it is chosen near the saddle s− and not near the semi-hyperbolic point q1 . This section is also supposed to be transverse to the blow-up locus W− ∩ {u = 0} and to W− ∩ {¯ = 0}. It cuts {u = 0} along a half-circle with end points (−Y , 0, 0) and (Y , 0, 0) (this open half-circle is a ¯ transverse to the trajectories in W as defined in Sect. 15.2). The section section      C cuts {¯ = 0} along two segments {−Y } × 0, u × {0} and {Y } × 0, u × {0} (see Fig. 15.7). This section is a (singular) rectangle with two extra corners along C ∩ { = 0} at the points (−Y , 0, 0) and (Y , 0, 0). We choose coordinates (θ, ) ∈ [−θ0 , θ0 ] × [0, 0 ] on C and we will suppose that they are smooth outside the two extra corner points. We do not specify this; in fact there is no need to choose an explicit coordinate θ since all the proofs will be made in other local sections in which we will describe the coordinates explicitly.

15.5 Cyclicity of the Canard Cycle

361

ε

u3 ¯ =

Y (Y  , 0, 0)

C

u



ε

(−Y , 0, 0) q1

θ θ0

θ

θr

−θ0

Fig. 15.7 The section C in W−

Important to observe is that C \ { = 0} is smoothly foliated by the intervals C = [−θ0 , θ0 ] × {}, with  > 0. The choice of  as coordinate in C implies that one side of C, the one which is disjoint from W− ∩ { = 0}, is contained in the level surface of . The opposite side ]−θ0 , θ0 [ × {0} contains three other sides of C and the two extra corner points: the point (−Y , 0, 0) with coordinates (θr , 0) in C and the point (Y , 0, 0) with coordinates (θ , 0). We assume that the coordinate θ is chosen such that −θ0 < θr < θ < θ0 (with this choice the charts that we will introduce later on C will be orientation preserving). The arc ]θr , θ [ × {0} corresponds to the interior of W− ∩ {u = 0}. We call (Rη (θ, ), ) the transition map from C \ { = 0} to T along the trajectories of X¯ λ and (Lη (θ, ), ) the transition map from C \ { = 0} to T along the trajectories of −X¯ λ . The orientation of the variable h is chosen such that the maps Lη and Rη preserve the orientation. We define the displacement function as η (θ, ) = Rη (θ, ) − Lη (θ, ). The function η (θ, ) is clearly smooth for  = 0. The second line of each of the transition maps (Rη (θ, ), ) and (Lη (θ, ), ) is given by  = uη (θ, )3 ¯η (θ, ) where (θ, ) → (Yη (θ, ), uη (θ, ), ¯η (θ, )) is the embedding of C in W− . Clearly, for  = 0, the vector field X¯ λ , and hence also X,λ , has a periodic orbit through (θ, ) ∈ C if and only if η (θ, ) = 0. Then, the cyclicity of the fast–fast transitory canard cycle 0 is at most k + 1 if the map θ ∈ [−θ0 , θ0 ] → η (θ, ) has at most k + 1 roots counted with their multiplicity, for  small enough. Recall that it is indeed the cyclicity of 0 inside the initial family X,λ , even for the

362

15 Fast–fast Passage through a Jump Point

√ Hopf case because it is shown that there is no periodic orbit when |a| ≥ K , see Proposition 6.1.

15.5.2 Normal Form for Transitions Toward T Recall from Sect. 15.3.1 that we have already chosen two sections D+ , D− in respectively the neighborhoods W+ of s+ and W− of s− , contained in {u = U0 }. As said in Sect. 15.3.1 on these sections we can choose the coordinates (y, ). If the sections D− , D+ are chosen small enough, a transition Rout η (y, ), ) : D+ → T is defined, following the flow in the positive time and a transition Lout η (y, ), ) : D− → T is defined, following the flow in negative time. As proved in Chap. 14 (see Proposition 14.2) the map Rout η has the following expression:  J˜(y, η, ) , = ϕ(η, ) + θA exp  

Rout η (y, )

(15.21)

where J˜ and ϕ are smooth in the variables (, y, η). Recall that  is a set of monomials in  that we will use to express smoothness of transition maps, see Definition 13.1. Its definition depends on the case (Hopf or jump). We have J˜(y, η, ) = J (y, μ)+o (1) where J (y, μ) is the slow divergence integral of the attracting sequence leading to the breaking mechanism. The coefficient θA = ±1 is the sign of ∂J ∂y (0, μ). We have a similar result for Lout η (by also applying Proposition 14.2): 

Lout η (y, )

 ˜ K(y, η, ) = ϕ− (η, ) + θR exp , 

(15.22)

˜ η, 0) = K(y, μ), with K where K˜ and ϕ− are smooth in (, y, η) and K(y, the slow divergence integral of the repelling sequence leading to the breaking mechanism in negative time, and θR = ±1 is the sign of ∂K ∂y (0, μ). We now consider the functions + Rη (Y, u, ) ¯ = Rout ¯ ) η (yη (Y, u, ),

(15.23)

and − ¯ = Lout ¯ ), Lη (Z, u, ) η (yη (Z, u, ),

where, in the jump case, yη± (Y, u, ) ¯ := y ± (Y, u, , ¯ λ) (λ = η) are the y+ − component of respectively Tλ and Tλ as obtained in Propositions 15.2 and 15.4.

15.5 Cyclicity of the Canard Cycle

363

√ In the Hopf case, we have yη± (Y, u, ) ¯ := y ± (Y, u, , ¯ (  a, ¯ μ)). As the functions ¯ λ) are smooth in (u, u ln u, ¯ , λ), the functions yη± (Z, u, ¯ ) are smooth y ± (Y, u, , √ √ in (, u, u ln u, ¯ , η), because a =  a¯ in the Hopf case and  ∈  then. The map (Y, u, ) ¯ → (Lη (Y, u, ), ¯ ) is the transition from the points of W− to the section T along the flow of −X,λ and that the map (Y, u, ) ¯ → (Rη (Y, u, ), ¯ ) is the transition from the points of W+ to the section T along the flow of X,λ . It is now easy to deduce from the previous computation a normal form for these two maps. By substituting y + and y − in (15.21) and (15.22), we obtain the following normal forms:   J˜(u2 [Y + O(¯ u ln u)], η, ) Rη (Y, u, ) ¯ = ϕ+ (η, ) + θA exp ,  (15.24)   ˜ 2 [Y + O(¯ u ln u)], η, ) K(u ¯ = ϕ− (η, ) + θR exp Lη (Z, u, ) ,  ˜ ϕ+ , ϕ− are defined above for (15.21) and (15.22). The where the functions J˜, K, ˜ 2 [Y + O(¯ u ln u)], η, ) are smooth functions J˜(u2 [Y + O(¯ u ln u)], η, ) and K(u in (, Y, , ¯ u, u ln u, η).

15.5.3 From Global to Local Displacement Functions As already mentioned we will not work directly with the coordinate θ on the whole section C but we will introduce an atlas of charts on it. This atlas will correspond to the choice of adapted new smaller sections. We will replace the study of the global displacement function η by the study of three local ones. In the study of a displacement function, changing the section C to a new smaller one is equivalent to introducing a chart on C. We will apply Lemma 14.1 in a completely similar way as in Chap. 14. Nevertheless we repeat the construction in view of changing notations. The section C with coordinates (θ, ) will be the section C1 chosen in Lemma 14.1. The section C2 will be one of the following three m sections chosen also in W− a section C− = [−Y− , Y− ] × [0, U− ] × {¯− }, a section d u C− ⊂ {Y = −Y }, and a section C− ⊂ {Y = Y }. We choose Y < Y < Y− (Y is m, Cd , Cu associated to the section C; see above). We choose the four sections C, C− − − ¯ in such a way that starting at any point of C the trajectory of Xλ arrives at a point m ∪ C d ∪ C u in a finite positive time. On {u = 0} this means that the arc on C− − − u d m C ∩ {u = 0} is inside the rectangle bounded by the three sections C− , C− , C− . If this arc is a true circle it is sufficient to have Y < ¯− . m On the section C− we choose (Y, ) as coordinates. The vector field X = ∂ ∂ d u −3¯ ∂ ¯ + u ∂u is tangent to the sections C− and C− and preserves the function

364

15 Fast–fast Passage through a Jump Point

d  = ¯ u3 . We choose on C− the time τ of the flow of −X as one coordinate. We can d , we ¯ is the flow of −X on C− make the following explicit choice: if ϕ −X (τ, (u, ))

−X 1/6 1/2 τ, ( , ¯ ) . The choice of the may take (τ, ) as coordinates of the point ϕ point ( 1/6 , ¯ 1/2 ) is rather arbitrary. The unique interesting property of this choice u is that the value of the function u ¯ 3 at this point is equal to . On the section C− we make the same choice, but with the vector field X. m Let O m be the open set of C, domain of the transition map from C to C− . d u We define in the same way the open sets O and O , domains for the transition d u maps from C to C− and C− , respectively. By hypothesis, {O m , O u , O d } is an open d covering of C. In C, O covers a neighborhood of ] − θ0 , θd ] × {0} and O u covers a neighborhood of [θu , θ0 [ × {0}. For each  = 0 we have that Od = O d ∩ C is an open interval with end point (θ0 , ), Ou = O u ∩C is an open interval with end point (−θ0 , ) and the closure of Om = O m ∩ C is strictly contained in ]−θ0 , θ0 [ × {}. m d u , C− , C− with Let Cm , Cd , Cu be the curves representing the intersection of C− the -level. For each  = 0, the transition maps along the flow of X¯ λ define diffeod u morphisms preserving orientation: Y,λ (θ ), τ,λ (θ ), τ,λ (θ ), from Om , Od , Ou into Cm , Cd , Cu , respectively. The three pairs (Om , Yλ (θ )), (Od , τλd (θ )), (Ou , τλu (θ )) is the atlas of charts of C in which we want to work. Let (Rηm , ), (Rηd , ), and (Rηu , ) be the transition maps from respectively m d u , C− , and C− to the section T following the flow of X¯ λ . Let the sections C− m d also (Lη , ), (Lη , ), and (Luη , ) be the transition maps from respectively the m d u , C− , and C− to the section T following the flow of −X¯ λ . Let sections C− m d u η (Y, ), η (τ, ), and η (τ, ) be the corresponding displacement functions m , C d , and C u , respectively. defined on the sections C− − − From Lemma 14.1, we know that the function m η restricted to some  = 0 can be seen as the function η in restriction to the chart (Om , Yη (θ )) of C . The same remark can be made for the other displacement functions dη , uη . We cannot study directly the zeros of the displacement function η but we can ∂ study the zeros of its first derivative ∂θη . Using Rolle’s Theorem we then obtain ∂

a restriction on the number of zeros of η from the number of zeros of ∂θη . This result will be obtained by working in the three different charts O m , O u , and O d in which the zeros of ∂ dη ∂τ ,

∂ η ∂θ

are in bijective relation with the zeros of

∂ m ∂ uη η , ∂Y ∂τ ,

and

respectively. To study the maps Rηm , Rηd , and Rηu we introduce the following sections:

m ¯ we introduce a section C+ = 1. In W+ with its normal coordinates (Y, u, ), d [−Y+ , Y+ ] × [0, U+ ] × {¯+ } and a section C+ ⊂ {Y = −Y } for some ¯+ > 0 and Y+ > Y . q 2. In the neighborhood Wq of q we have already introduced for the definition of Tλ a section Cq ⊂ {Z = −Zq } for some Zq > 0, and a section Dq ⊂ {¯ = ¯q } for some ¯q > 0 (Z the normal form coordinate near q).

15.5 Cyclicity of the Canard Cycle

365

m d The choice of the different sections C+ , C+ , Cq , Dq have to be made in a way m d u coherent with the choice of C− , C− , C− already made above:

1. The section Cq must be chosen large enough such that the flow of X¯ λ goes from u to C . Next D must be chosen large enough such that the flow of X ¯ λ goes C− q q from Cq to Dq . m must be chosen large enough such that the flow of X ¯ λ goes from 2. The section C+ m m to C m . Dq to C+ and from C− + d 3. The section C+ must be chosen large enough such that the flow of X¯ λ goes from d d C− to C+ . It is easy to see that all the constraints can be fulfilled on the different sections by u d d m m , C+ ), C− , C− , C+ , the choice choosing them in the following order: Dq , Cq , (C− of sections in the parentheses being indifferent. It is important being able to choose d u d the sections C+ , C− arbitrarily small. Next we have to choose C− small enough in d function of the choice of C+ , and we have to choose ¯− small enough in function of u , C d . Finally we have to choose ¯ small enough in function of the choices of C− + − d m the choices of C+ and ¯− as well as in order to guarantee that the arc C+ ∪ {u = 0} cuts the separatrix Sq in its interior. It is also important to be able to choose ¯+ arbitrarily small such that ¯− can also be chosen arbitrarily small. We recall that we assume that the slow divergence integral of the fast–fast transitory canard cycle has a codimension equal to k (see Definition 15.1). We are now going to prove the following three results: Theorem 15.2 (The Function m η ) Choose any ¯− > 0 and consider the correm m sponding section C− with its displacement function m η . On C− , parameterized by ∂ m

Y ∈ [−Y− , Y− ] and , the derivative ∂Yη (Y, ) has at most two roots counted with ∂J their multiplicity if ∂K ∂y (0, 0)/ ∂y (0, 0) ∈ ]0, 1[ and at most one root, which must be ∂J simple, if ∂K ∂y (0, 0)/ ∂y (0, 0) ∈ ]0, 1[. This result is valid for  > 0 small enough and η ∼ 0.

Remark 15.2 When ¯− becomes smaller we cover a larger domain in W. Hence it remains to consider arbitrarily small neighborhoods of the two arcs ∂Wud and ∂Wd in the following Theorems 15.3 and 15.4. d is small enough (induced by Theorem 15.3 (The Function dη ) If the section C− ∂ d

d choosing the section C+ small enough), then the equation ∂τη (τ, ) = 0 has at most one root, which is simple, when k = 1 and at most k + 1 roots counted with d their multiplicity when k ≥ 2. This result is valid for (τ, ) ∈ C− , when  > 0 is small enough and η sufficiently near 0. u Theorem 15.4 (The Function uη ) If the section C− is small enough, then the ∂ u

equation ∂τη (τ, ) = 0 has at most one root, which is simple. This result is valid u , when  > 0 is small enough and η sufficiently near 0. for (τ, ) ∈ C−

366

15 Fast–fast Passage through a Jump Point

15.5.4 Proof of Theorem 15.2 for m η m We can now prove Theorem 15.2. The map Lm η is the restriction of Lη to C− . On m C− we have that ¯ = ¯− and u =

we have that

 1/3 1/3 . ¯−

Then substituting these relations in (15.24)



  1 ˜   2/3 1/3 Lη (Y, ) = ϕ− (η, ) + θR exp K 2/3 [Y + O( ln )], η,  ,  ¯− where Lη (Y, ) is smooth in (, Y, η). Remark 15.3 In the above equation and in the rest of this paragraph the Landau symbols “O,” “o” must be understood to be valid for a fixed strictly positive value of ¯− (and later of ¯+ also). In the normal coordinates, the transition along the trajectories of the linear vector field (15.14), from the section {¯ = E} to the section {¯ = 1} is the linear map 2 m m to C+ is equal to a smooth Y → E − 3 Y . Then, the map F¯− ¯+ (Y, ) from C− function  2  2 − 3 F¯− ¯+ (Y, ) = ¯+ F ¯− 3 Y + O(), where F is the function defined in Sect. 15.4.1. 1/3 m On C+ we have ¯ = ¯+ and u =  1/3 . Then, the restriction of the map Rη given ¯+

m is given by in (15.24) to C+

# $ 1 ˜  2/3 J ϕ− (η, ) + θA exp [Y + O( 1/3 ln )], η,  , 2/3  ¯+ 

and the map Rη (Y, ) is the composition of Y → F¯− ¯+ (Y, ) with it. This gives  ⎤  ⎡  Y J˜  2/3 [F 2/3 + O( 1/3 ln )], η,  ¯− ⎢ ⎥ Rη (Y, ) = ϕ+ (η, ) + θA exp ⎣ ⎦.  In order to simplify the expressions of Lη and Rη , we expand K˜ and J˜ at the first order in y ˜ K(y, η, ) = K˜ 0 (η, ) + K˜ 1 (η, )y + O(y 2),

15.5 Cyclicity of the Canard Cycle

367

and J˜(y, η, ) = J˜0 (η, ) + J˜1 (η, )y + O(y 2 ). We have of course K˜ 0 (η, 0) = K(0, η), J˜0 (η, 0) = J (0, η) and ∂K ∂J (0, η), J˜1 (η, 0) = (0, η). K˜ 1 (η, 0) = ∂Y ∂Y Using these expansions, we can write ⎛ Lη (Y, ) = ϕ− (η, ) + θR exp ⎝

K˜ 0 +  2/3 K˜ 1

Y 2/3 ¯−

+ O( ln )



⎞ ⎠

(15.25)

and Y

⎛ ˜ ⎞ J0 +  2/3 J˜1 F 2/3 + O( ln ) ¯− ⎠. Rη (Y, ) = ϕ+ (η, ) + θA exp ⎝ 

(15.26)

Let us now consider m m m η (Y, ) = Rη (Y, ) − Lη (Y, ). ∂ ˆ We consider the first derivative ∂Yη . Writing K(Y, η, ) and Jˆ(Y, η, ) to simplify, the numerators in the respective right hand sides of (15.25) and (15.26), we have

    ∂ η 1 ∂ Jˆ Jˆ 1 ∂ Kˆ Kˆ = θA exp − θR exp . ∂Y  ∂Y   ∂Y  ˆ Jˆ are smooth in (, Y, η), we obtain that As the functions K,  ∂ Kˆ  2/3  = 2/3 K˜ 1 1 + O( 1/3 ln ) ∂Y ¯−

and

 Y   ∂ Jˆ  2/3 = 2/3 J˜1 F 2/3 1 + O( 1/3 ln ) . ∂Y ¯− ¯−

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15 Fast–fast Passage through a Jump Point

This allows to write    Y   Jˆ 1/3 = θA J˜1 F 1 + O( ln  exp 2/3 ∂Y  ¯−     Kˆ − θR K˜ 1 1 + O( 1/3 ln ) exp . 

2/3 ∂ η  1/3 ¯−



We recall that θR K˜ 1 (0, 0) > 0 and that θA J˜1 (0, 0) > 0. We can suppose that the ˜ ˜ domains in  and η are small enough such  that the functions θR K1 and θA J1 remain Y strictly positive. We also know that F 2/3 > 0 for any Y and ¯− > 0. Then ¯−

the functions in front of the exponential in the above equation can be written as exp O (1) so that they can enter in the exponentials as O()-perturbations of Jˆ and ˆ respectively. This gives that K,     Jˆ Kˆ = exp − exp ∂Y  

2/3 ∂ η  1/3 ¯−

with new functions Jˆ and Kˆ which can now be written as Y Kˆ = K˜ 0 +  2/3 K˜ 1 2/3 + O( ln ) ¯− The equation

∂ η ∂Y

and Jˆ = J˜0 +  2/3 J˜1 F

 Y  2/3

¯−

+ O( ln ).

= 0 is equivalent to Jˆ − Kˆ = 0, which is an equation

J˜0 − K˜ 0 +  2/3 J˜1 F

 Y  2/3 ¯−

−  2/3 K˜ 1

Y 2/3

¯−

+ O( ln ) = 0.

Let us introduce the parameter function 2/3 ¯− (J˜0 − K˜ 0 ) α(η, ) = .  2/3 J˜1

The Eq. (15.27) is equivalent to 2/3

¯− F

 Y  2/3 ¯−

−

K˜ 1 Y + α(η, ) + O( 1/3 ln ) = 0. J˜1

(15.27)

15.5 Cyclicity of the Canard Cycle

369

Let us derive this expression just once. We obtain that the equation  > 0, equivalent to F

 Y  2/3 ¯−

=

∂ 2 η ∂Y 2

K˜ 1 + O( 1/3 ln ). J˜1

It follows from Proposition 15.5 that the function Y → F

= 0 is, for

(15.28) 

Y 2/3

¯−

 is strictly

decreasing. Then the Eq. (15.28) has at most a single root, which must be simple. Moreover, as the range of F is the open interval ]0, 1[, we have that (15.27) has no ∂J root if (η, ) ∼ (0, 0), when ∂K ∂y (0, 0)/ ∂y (0, 0) ∈ ]0, 1[. This concludes the proof of Theorem 15.2.

15.5.5 Proof of Theorem 15.3 for dη (a) A good representation of dη d It will be easier for the computations to define the maps Rηd , Ldη on a section Cˆ − chosen within {y¯ = −y¯0 }, for a certain y¯0 > 0, rather into {Y = −Y }. We will d make a similar choice for a section Cˆ + . This does not change the results. Also, as we have to work at the same time in the three charts {y¯ = −1}, W− ⊂ {x¯ = −1}, and W+ ⊂ {x¯ = 1}, we better pay attention to the name of coordinates in this paragraph. In the two charts W± we will write (w, y, ˜ ) ˜ instead of (u, y, ¯ ¯ ) and we will keep (u, x, ¯ ) ¯ as coordinates in the matching chart {y¯ = −1}, we will however not change the names of the three charts. The changes of charts are as follows:

1. From the matching chart {y¯ = −1} to W+ : y˜ = − x¯12 , w = xu, ¯ ˜ = x¯13 ¯ . 1 2. From the matching chart {y¯ = −1} to W− : y˜ = − x¯ 2 , w = −xu, ¯ ˜ = − x¯13 ¯ . Using y˜0 =

1 , x˜ 02

we observe that a section −x¯0 ⊂ {x¯ = −x¯0 } in the matching

chart {y¯ = −1} becomes a section {y˜ = −y˜0} in the chart W− , while a section +x¯0 ⊂ {x¯ = +x¯0 } becomes a section {y˜ = +y˜0 } in the chart W+ . From now on, d such sections will be called ±x¯0 and Cˆ − = −x¯0 , for some x¯ 0 large enough. The d d ⊂ W ∩ {y¯ = −1}. It charts are chosen such that Cˆ − ⊂ W− ∩ {y¯ = −1} and Cˆ + + follows from Proposition 15.3 that in normal coordinates we have   d ⊂ {Y = −y˜0 + ˜ 1 + O(y˜0) + O(w) + O(˜ ) }. Cˆ − It follows from Proposition 15.1 that we have    d Cˆ + ⊂ Y = −y˜0 − ˜ 1 + O(y˜0) + O(w) + O(˜ ) .

370

15 Fast–fast Passage through a Jump Point

d The y-component (15.10) of the transition Tλ− , in restriction to Cˆ − = −x¯0 , can now be written as     d (w, , ˜ λ) = w2 −y˜0 + ˜ 1 + O(y˜0 ) + O(w) + O(˜ ) + O(˜ w ln w) . y− d Using the same name y− , we can now write this function in terms of the coordinates (u, ¯ ): d y− (u, ¯ , λ)

   1 + O(u) . = −u 1 + ¯ O x¯0 2

(15.29)

We can make the same computation for the y-component of the transition Tλ+ , in d =  . If this function is written in coordinates (u, ¯ ), we get restriction to Cˆ + x¯ 0 d (u, ¯ , λ) y˜+

   1 + O(u) . = −u 1 + ¯ O x¯0 2

(15.30)

Remark 15.4 For large x¯0 , a lot of terms can be when substituting (w, ) ˜   discarded 1 to (u, ). ¯ We just keep the Landau symbols O x¯0 and O(u). d to D . This We want to consider the y-component of the transition from Cˆ − + transition is obtained by composing the regular map Hη (u, ¯ ) introduced in Proposition 15.8, with the transition Tλ+ , i.e. by substituting u = uUη , ¯ = ¯ Eη in (15.30). Taking into account (15.20), this gives a function

   1 d y+ + O(u) + O(¯ ) . (u, , ¯ λ) = −u2 1 + T (x¯0 )¯ 1 + O x¯0

(15.31)

We recall that T (x¯0 ) → π when x¯0 → +∞. d We can now consider the displacement function dη on the section Cˆ − = −x¯0 , for a x¯0 > 0 to be taken (later) large enough. Because of the assumptions on the slow divergence integral J − K, we need a Taylor expansion of the functions K˜ and J˜ at order k: ˜ K(y, , η) = K˜ 0 (, η) + K˜ 1 (, η)y + · · · + K˜ k−1 (, η)y k−1 + K˜ k (y, , η)y k , and J˜(y, , η) = J˜0 (, η) + J˜1 (, η)y + · · · + J˜k−1 (, η)y k−1 + J˜k (y, , η)y k . √ d (u, ¯ , λ). We obtain a new We substitute in the Hopf case a =  a¯ in y± d function y± (u, ¯ , η), just like we did before in (15.23). They are smooth in (, u, u ln u, ¯ , η).

15.5 Cyclicity of the Canard Cycle

371

d d ˆ ˜ − Putting K(u, ¯ , , η) = K(y (u, ¯ , η), , η) and Jˆ(u, ¯ , , η) = J˜(y+ (u, ¯ , η), , η) we see that



ˆ K(u, ¯ , , η) Ldη (u, ¯ ) = ϕ− (, η) + θR exp 



and 

Rηd (u, ) ¯

 Jˆ(u, ¯ , , η) = ϕ+ (, η) + θA exp . 

ˆ Jˆ, ϕ+ , ϕ− are all smooth in (, u, u ln u, ¯ , η). We can use (15.29) We recall that K, and (15.31) to obtain expansion of Kˆ and Jˆ. For all j ∈ N we have that   1  d j ) = (−1)j u2j 1 + ¯ O + O(u) + O(¯ ) (y− x¯0 and   1  d j ) = (−1)j u2j 1 + j T (x¯0 )¯ 1 + O + O(u) + O(¯ ) , (y+ x¯0 With this we obtain Kˆ =

k−1 ! j =0

  1  2j ˜ 1 + ¯ O (−1) Kj (, η)u + O(u) + O(¯ ) + K˜ k (u, ¯ , η)u2k x¯0 j

(15.32) and Jˆ equal to Jˆ =

k−1 ! j =0

   1 2j ˜ + O(u) + O(¯ ) (−1) Jj (, η)u 1 + j T (x¯0 )¯ 1 + O x¯0 j

(15.33) + J˜k (u, ¯ , η)u2k , where J˜k and K˜ k are new functions, smooth in (, u, u ln u, ¯ , η), that are respectively equal to (−1)k Jk (η, 0) and (−1)k Kk (η, 0) for  = 0. Finally, the displacement function dη = Rηd − Ldη is given by 

dη

Kˆ = (ϕ+ − ϕ− )(, η) + θA exp 

 − θR exp

  Iˆ 

.

(15.34)

372

15 Fast–fast Passage through a Jump Point

(b) The equation [L−X dη ](u, ¯ ) = 0. There is here a mild abuse of notation in (15.34) where we write dη (u, ) ¯ for d the displacement function. The function η (τ, ) previously defined is just the composition of the map (τ, ) → (u(τ, ), (τ, ¯ )) defined by the flow of the vector ∂ field −X = 3¯ ∂∂¯ − u ∂u , with the function defined in (15.34). An important remark is that   ∂ d η (τ, ) = L−X dη (u(τ, ), ¯ (τ, )), ∂τ

(15.35)

where L−X is the Lie derivative with respect to the vector field −X. The formula is valid for  = 0, when the functions that we consider are smooth. But we have a good control of the asymptotic properties of the Lie-derivation on functions of (u, ¯ , , η), which are smooth in (, u, u ln u, ¯ , η) in the sense previously defined. ∂ It follows from (15.35), for a given value of  > 0, that the roots of ∂τ dη (τ, ) d 3 are the roots of [L−X η ](u, ) ¯ in restriction to the level curve {u ¯ = } of the d ˆ section C− . So, we have to compute this Lie derivative and find the number of roots of the equation: [L−X dη ](u, ¯ ) = 0 d , for  > 0 considered in restriction to the level curve {u3 ¯ = } of the section Cˆ − small enough. We have     θR θA Jˆ Kˆ d ˆ ˆ [L−X η ] = L−X J exp − L−X K exp .    

As L−X is a derivation, we have L−X Jˆ =

∂ J˜ d d ∂y (y+ )L−X [y+ ](u, ¯ , η).

Remark that

d y+ ,

the two expressions√of one w.r.t. λ and one w.r.t. η that we have obtained after substituting a =  a ¯ in the Hopf case, have the same Lie derivative because √ L−X ( ) = 0. ˜ d d Now, as J˜1 = 0, we have ∂∂yJ (y+ ) = J˜1 (1 + O(y+ )) = J˜1 (1 + O(u2 )) and 

d L−X y+



  1  1 (u, ¯ , η) = 2u 1 − T (x¯0 )¯ O + O(u) . 2 x¯0 2

As θA J˜1 = |J˜1 |, if η ∼ 0,  ∼ 0, we finally have θA u2 L−X Jˆ = 2 |J˜1 |(1 + O(¯ ) + O(u2 )).  

15.5 Cyclicity of the Canard Cycle

373

In a similar way we can compute that u2 θR L−X Kˆ = 2 |K˜ 1 |(1 + O(¯ ) + O(u2 )).   And then      Jˆ Kˆ 1   d 2 2 ˜ ˜ L = | J |(1+O( )+O(u ¯ )) exp |(1+O( )+O(u ¯ )) exp −| K . −X 1 1 η 2 u2  

As |J˜1 | > 0 and |K˜ 1 | > 0, we can “enter” the functions in factor inside the exponentials in the right hand member of this equation, changing it into  ⎤ Jˆ +  ln |J˜1 |(1 + O(¯ ) + O(u2 )) ⎦ exp ⎣  ⎡

 ⎤ Kˆ +  ln |K˜ 1 |(1 + O(¯ ) + O(u2 )) ⎦. − exp ⎣  ⎡

3 2 2 2 Let us carefully regroup the terms  in this expression. As  = u ¯ = u O(u + ¯ ) 2 and J˜1 (, η) = 0, the term  ln |J˜1 |(1+O(¯ )+O(u )) in the first exponential can      be added to the second term −J˜1 (, η)u2 1+j T (x¯0 )¯ 1+O x¯10 +O(u)+O(¯ )

of the sum (15.33) for Jˆ: this regrouping amounts to an O(u) perturbation. The similar term in the second exponential can equally be reconsidered and rewritten as an O(u) perturbation. As such, there exists new functions Kˆ and Jˆ, with an expansion given by respectively (15.32) and (15.33) such that we have      1   Kˆ Jˆ d − exp . L−X η = exp 2 2u  

(15.36)

As a consequence of (15.36), the equation {[L−X dη ](u, ¯ ) = 0}, for  = 0, is equivalent to ˆ Jˆ(u, u ln u, , ¯ η) − K(u, u ln u, , ¯ η) = 0, with Kˆ and Jˆ given by respectively (15.32) and (15.33). (c) Roots of the equation {[L−X dη ](u, ¯ ) = 0} It is rather trivial to prove the case of codimension 1 when J1 (0, 0) − K1 (0, 0) = 0. We can restrict the domain of study such that J˜1 (u, , ¯ , η) − K˜ 1 (u, ¯ , , η) = 0 for

374

15 Fast–fast Passage through a Jump Point

every (u, ¯ , , η). As such Jˆ − Kˆ = (J˜0 − K˜ 0 ) + (J˜1 − K˜ 1 )u2 (1 + O(u)) has at most one root, which is simple, on any curve u3 ¯ =  for  > 0 small enough. We have proved Lemma 15.4 Assume that the codimension is equal to k = 1. Then if W is a sufficiently small neighborhood of (0, 0) in the quadrant {u ≥ 0, ¯ ≥ 0}, the equation L−X dη (u, ¯ ) = 0 has at most one root, which is simple on each curve W ∩ {u3 ¯ = } for  > 0 small enough and for η sufficiently near 0. From now on we will assume that k ≥ 2. We want to prove that the number of roots of {L−X dη (u, ) ¯ = 0} is bounded by k + 1. In the first step, we will rearrange ˆ ˆ the sum in J − K. To simplify the notations we introduce    1 + O(u) + O(¯ )) Aj = (−1)j u2j 1 + j T (x¯0 )¯ 1 + O x¯0 and    1 + O(u) + O(¯ ) , Bj = (−1)j u2j 1 + ¯ O x¯0 the functions entering in (15.33) and (15.32) for j = 1, . . . , k − 1. With these notations we have Jˆ − Kˆ = J˜0 − K˜ 0 +

k−1 ! (J˜j Aj − K˜ j Bj ) + u2k (J˜k − K˜ k ) j =1

= J˜0 − K˜ 0 +

k−1 k−1 ! ! (J˜j − K˜ j )Aj + K˜ j (Aj − Bj ) + u2k (J˜k − K˜ k ). j =1

j =1

We have  1  Aj − Bj = (−1)j (j T (x¯0 ) − 1)u2j ¯ 1 + O + O(u) + O(¯ ) . x¯0  ˜ But, as K˜ 1 =  0, we can join all the terms in the sum k−1 j =1 Kj (Aj − Bj ) to replace this sum by a unique term of the same form as the first one equal to

15.5 Cyclicity of the Canard Cycle

375

    (1 − T (x¯0 ))K˜ 1 u2 ¯ 1 + O x¯10 + O(u) + O(¯ ) that we can simply write (1 −         T (x¯0 ))K˜ 1 u2 ¯ 1 + O x¯10 + · · · or equivalently 1 − T (x¯0 ) + O x¯10 K˜ 1 u2 ¯ 1 +    · · · for now a term O x¯10 depending only on the constant x¯ 0 . Also writing simply Aj = (−1)j u2 (1 + · · · ) we obtain the following expansion: Jˆ − Kˆ = J˜0 − K˜ 0 +

k−1 !

(−1)2j (J˜j − K˜ j )u2j (1 + · · · ) + (J˜k − K˜ k )u2k

(15.37)

j =1

  1    K˜ 1 u2 ¯ 1 + · · · . − 1 − T (x¯0 ) + O x¯0 We recall that x¯0is a constant arbitrarily large. We can choose it such that the term 1 − T (x¯0 ) + O x¯10 is arbitrarily near 1 − π (of course, it suffices to have 1 −   T (x¯0 ) + O x¯10 < 0). The factors J˜j − K˜ j depend on the parameter (, η) and the assumption is thatthey  are 0 for η = 0 and  = 0. On the other hand, the coefficients 1 1 − T (x¯0 ) + O x¯0 K˜ 1 and (J˜k − K˜ k ) are nonzero for η = 0 and  = 0. To prove Theorem 15.3 for dη , we can apply Theorem 11.1 of Chap. 11 to the ∂ , the expression (15.37) and the Lie-derivation L−X . Indeed, as −X = 3¯ ∂∂¯ − u ∂u 2 4 2k 2 sequence of leading monomial: 1, u , u , . . . , u , u ¯ is non-resonant and the other assumptions in Theorem 11.1 are clearly fulfilled.

15.5.6 Proof of Theorem 15.3 for uη To obtain the transition Luη , we just have to restrict Lη given by (15.24) to {Y = Y > 0}. This gives  ˜ 2 [Y + O(¯ u ln u)], η, ) K(u . = ϕ− (, η) + θR exp  

Luη (u, ¯ )

On the other side, the map Rηu is the composition of four maps: q

u and C , preserving  = u3 ¯ . It is 1. The regular transition Gη between C− q q given by Gη (u, ¯ ) = (uG (u, ¯ , η), G ¯ (u, , ¯ η)), with G (u, , ¯ η) > 0 and u. G (u, ¯ , η) > 0 on C−

376

15 Fast–fast Passage through a Jump Point q

2. The transition map Tλ near q (see Sect. 15.3.2), from Cq ⊂ {Z = −Zq } to Dq , q q given by Z˜ = Tη (u, ¯ ), where Tη is defined by rewriting λ in terms of η in the q expression Tλ :   4β(u, ¯ , η)u3 ˜ , Z = −Zq exp − 3 where β is smooth in (, u, u3 ln u, ¯ 1/3 , ¯ ln ¯ , η) and β(0, 0, η) > 0. m ˜ ) → (Y = H,η (Z), ˜ ) from Dq to C+ 3. A smooth transition (Z, . We write ˜ = H0 (, η) + H1 (, η)Z(1 ˜ + O(Z)), ˜ H,η (Z) with H1 (, η) > 0. m , see (15.24) that we will simply write 4. The restriction of Rη to C+  Rη (Y, ) = ϕ+ (, η) + θA exp

J˜0 (, η) + J˜1 (, η)( ¯0 )2/3 (Y + o (1))

 .



The composition of the two first maps is equal to   ¯ , 4β(u, ¯ η)u3 ¯ = Tηq ◦ Gqη (u, ) ¯ = −Zq exp − Wη (u, ) , 3

(15.38)

¯ 0, η) = G (0, 0, η)3 > ¯ ¯ , η) is smooth in (, u, ¯ ln ¯ , ¯ 1/3 , η) and β(0, where β(u, 0. The composition of the two last maps is equal to ˜ ) Rη (H,η (Z),



= ϕ+ (, η) + θA exp

˜ + o (1)) J˜0 (, η) + J˜1 (, η)( ¯0 )2/3 (H,η (Z) 

 .

We can rearrange the function  = J˜0 (, η) + J˜1 (, η)



 ¯0

2/3

˜ + o (1)). (H,η (Z)

First we can write ˜ + o (1) = H0 (, η) + H1 (, η)Z(1 ˜ + O(Z)) ˜ + ξ0 (, η) + ξ1 (, η)Z(1 ˜ + O(Z)), ˜ H,η (Z)

15.5 Cyclicity of the Canard Cycle

377

where ξ0 , ξ1 = o (1), to get ˜ + o (1) = Hˆ 0 (, η) + Hˆ 1 (, η)Z(1 ˜ + O(Z)), ˜ H,η (Z) where we keep the property that Hˆ 1 (, η) > 0 (for η ∼ 0,  ∼ 0). The functions ˜ + o (1) are no longer smooth but just smooth in Hˆ 0 , Hˆ 1 , as well as H,η (Z) ˜ η). Now (, Z,   2/3 ˜ + O(Z)), ˜ (Hˆ 0 + Hˆ 1 Z(1  = J˜0 + J˜1 ¯0   2/3   2/3   ˜ . Hˆ 0 + J˜1 Hˆ 1 Z˜ 1 + O(Z) = J˜0 + J˜1 ¯0 ¯0 The parameter function Jˆ0 (, η) = J˜0 + J˜1 ( ¯0 )2/3 Hˆ 0 keeps the property that Jˆ0 (0, η) = J0 (η) and remains smooth in (, η). We can now write the composition of the two last maps as ⎡ ˜ , η) = ϕ+ (, η) + θA exp ⎢ Rη (H,η (Z), ⎣

Jˆ0 + J˜1 Hˆ 1

 2/3  ⎤  ˜ 1 + O(Z) ˜ Z ¯0 ⎥ ⎦. 

The transition map Rηu is given by Rηu (u, ) ¯ = ϕ(, η) + θA exp

¯  C(u, ¯ , η) , 

with   2/3   ¯ Wη 1 + O(Wη ) , C(u, ¯ , η) = Jˆ0 + J˜1 Hˆ 1 ¯0 where Wη = Wη (u, ¯ ) is given by (15.38). Using also the function   D(u, , ¯ η) = K˜ 0 (, η) + Y K˜ 1 (, η)u2 1 + O(¯ u ln u) , we can write the displacement function uη in the following way: uη (u, ¯ , η) = (ϕ+ − ϕ− )(, η) + θA exp

  ¯  C(u, ¯ , η) D(u, , ¯ η) − θR exp .  

We want to study LX [ uη ]. First we have   2/3   ¯ LX C(u, LX Wη (1 + O(Wη )) . ¯ , η) = J˜1 (, η)Hˆ 1 (, η) ¯0

378

15 Fast–fast Passage through a Jump Point

As     u3 ¯ , 4β(u, ¯ η)u3 1/3 ¯ exp − LX Wη = 4Zq β(η) + O(¯ ) + O(u) ,  3   ¯ ¯ 0, η), we obtain a similar expression for LX Wη (1 + O(Wη )) . with β(η) = β(0, ¯ ), ¯ using similar ideas as in the It is now easy to simplify the expression of LX C(u, previous case (in particular using that  = u3 ¯ and that sign (J˜1 ) = θA ). We obtain 

¯ ¯ , η) = θA  LX C(u,

 ˜ 3 4βu u exp − , 3

−1/3 3

(15.39)

˜ ¯ , η) = β(η) ¯ with β(u, + O(¯ ) + ou (1). We also have   LX D(u, , ¯ μ) = 2Y K˜ 1 u2 1 + O(¯ u ln u) ,

(15.40)

where β˜ and this last function are smooth in (, u, u ln u, ¯ , η). Using (15.39) and (15.40) and the same ideas as above, we get    ¯   3 ˜ C  4 βu D + O() u −1/3 LX η =  u exp − exp − exp . u2 3   We have that      ¯ 3  4βu 4β¯ 2/3 2 2/3 ˆ ¯ ˆ = J0 + u ¯ O exp − C = J0 +  O exp − 3 3¯ and we will just retain that C¯ = Jˆ0 + u2 o¯ (1),  ¯ where the term o¯ (1) = ¯ 2/3 O(exp − 43β¯ ) is smooth and flat in ¯ and for sure is smooth in (, u, u ln u, ¯ ln ¯ , ¯ 1/3 , η). Using this, we can rewrite      Jˆ0 + u2 (O(u) + o¯ (1)) D + O() u −1/3 LX η =  u exp − exp , u2    

  Jˆ0 + u2 O(u) + o¯ (1) +  ln( −1/3u) D + O() = exp − exp .  

15.6 Proof of the Main Theorem

379

We can write  ln( −1/3 u) = − 13  ln  + ¯ u3 ln u, which is a smooth function in (, u, u ln u) and of order O(u3 ln u). Finally, we obtain that the equation {LX uη = 0} is equivalent to the equation  

Jˆ0 + u2 O(u) + o¯ (1) +  ln( −1/3 u) − D + O() = 0, which can be written (again using that  = u3 ¯ ):   1 J˜0 − Kˆ 0 −  ln  − Y K˜ 1 u2 1 + O(u ln u) + o¯ (1) = 0. 3 As Y K˜ 1 = 0, this equation has at most one root, which is simple. In fact this claim is not so clear as it seems at first glance because the function  in the right hand side depends on the two variables u and ¯ and we have to consider it in restriction to the curves {u3 ¯ = }. But   LX  = 2Y K˜ 1 u2 1 + O(u ln u) + o¯ (1) , which is nonzero in a sufficiently small neighborhood of (0, 0) when we keep  > 0 and η small enough. This implies the requested property.

15.6 Proof of the Main Theorem ∂

On the intervals Om , Ou , and Od , the derivative ∂θη of the global displacement function has the same number of zeros, counted with their multiplicity, as the functions

∂ m ∂ uη η ∂Y , ∂τ ,

∂ dη m u d ∂τ , respectively. As C ⊂ O ∪ O ∪ O , we can bound ∂ η ∂θ by the sum of the bounds given in Theorems 15.2, 15.4,

and

the number of zeros of and 15.3:

∂

∂j η 1. If k = 1 and ∂K ∂y (0, 0)/ ∂y (0, 0) ∈ ]0, 1[, then ∂θ has at most 3 zeros counted with their multiplicity. ∂ η ∂J 2. If k = 1 and ∂K ∂y (0, 0)/ ∂y (0, 0) ∈ ]0, 1[, then ∂θ has at most 4 zeros counted with their multiplicity. ∂ 3. If k ≥ 2, then ∂θη has at most k + 3 zeros counted with their multiplicity.

Theorem 15.1 follows by applying Rolle’s Theorem to the displacement function η .

Chapter 16

Outlook and Open Problems

16.1 Introduction In this last chapter we want to present in a more general perspective, problems for the future, remaining open for the slow–fast systems in dimension 2. Up to now there does not exist a well-established general frame for the study of these systems. We will try to sketch a general setting, in which many points must be considered as open questions, even if a lot of particular cases have already been studied. Unless we explicitly mention the contrary, the slow–fast systems will be smooth, although some problems will be limited to analytic systems. We will just consider unfoldings and not globally defined systems. This means that we will consider slow–fast systems X,λ with (, λ) ∈ [0, 0 [×P , where 0 > 0 is arbitrarily small and P ⊂ Rp is also an arbitrarily small neighborhood of a parameter value λ0 ∈ Rp . We are mostly interested in the study of individual slow–fast cycles λ0 , belonging to the limiting equation X0,λ0 . We will then consider a slow–fast system that is restricted to a neighborhood of this cycle. A general study of common cycles was made in [DMDR11] where it was proved that a common cycle, under very general conditions, bifurcates in a single hyperbolic limit cycle (see Chap. 7). In this chapter, we will hence focus on canard cycles. First, we will explain in Sect. 16.2 how to classify these systems restricted to a neighborhood of a given canard cycle λ0 . This system represents an unfolding of λ0 . A slow–fast system may be more or less degenerate, depending on the geometric properties that we have introduced in Part I: the number of contact points, their orders, the singularities of the slow dynamics. We also have the possibility of fast connections between contact points: a fast connection between two contact points p1 , p2 (not necessarily distinct) is given by a fast orbit γ with α-limit p1 and ω-limit p2 . Also, in the case of a canard cycle, we have to take into account the slow divergence integrals associated to it. Unless we mention the contrary explicitly, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6_16

381

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16 Outlook and Open Problems

we will restrict to one-layer canard cycles. In this case, we have just to consider a single slow divergence integral unfolding I (v, λ) associated to the canard cycle. Let us notice that the singularity orders of contact points as well as the slow dynamics and its singularities does not depend just on the canard cycle but also on the 1-jet of the unfolding in the -direction. By abuse of language we will nevertheless speak of the singularities and codimensions of singularities associated to a given canard cycle. This would be natural if one considers, as in [BCDD81], that a canard cycle is associated to a singular perturbation and not to a slow–fast system. As we are considering unfoldings for λ ∼ λ0 , the above mentioned degeneracy conditions will be defined in terms of the limiting equation X0,λ0 and of the slow– fast dynamics and slow divergence integrals for the parameter value λ0 . In a precise way, each of these degeneracy conditions is expressed in terms of a finite number of equations and inequalities. In general, a local singularity of a geometrical object in dimension n is said to be of codimension c if the number of equations needed to define it is equal to c + n (even in the most simple case c = 0 corresponding to a stable singularity, we need n equations to determine its position). Here a singularity may be a contact point (n = 2), a zero of slow dynamics (n = 1), or a solution of the equation I (v, λ0 ) = 0 (n = 1). We have also the jumps between two contact points which are non-local singularities of X0,λ0 . We will define the codimension of such a connection only for the case of quadratic contact points. We will say more about these codimensions in Sect. 16.2. Contact points of X0,λ0 , zeros of the slow dynamics for λ = λ0 , jumps between contact points of X0,λ0 , or zeros of the slow divergence integral I (v, λ0 ) are called the singularities of the slow–fast unfolding of the canard cycle or of the canard cycle itself. Codimensions at different singularities may be summed up to define the codimension of the system or of the canard cycle itself. We will say more about these notions in Sect. 16.2. Knowledge of the singularities for the parameter value λ0 is the preliminary step for the study of the unfolding X,λ itself. We will consider unfoldings in Sect. 16.3. In general the given unfolding may be too degenerate to allow a direct study. Then, we will have to “replace” it by a so-called versal unfolding. Roughly speaking a versal unfolding is an unfolding which generically unfolds each singularity of X0,λ0 , i.e. in the case of slow–fast systems, singularities of contact point, singularities of slow–fast dynamics, singularities of slow divergence integrals, singularities of jumps, and which induces (i.e. factorizes) the given unfolding. Existence of such versal unfolding is in general an open question. Nevertheless, such an existence was proved by D. Panazzolo for unfoldings of two-dimensional vector fields whose most degenerate singular points are nilpotent points. Let us call N this class of unfoldings of vector fields. Slow–fast unfoldings are of course included in this class N, and we can directly apply Panazzolo’s result. Next, Panazzolo proved a desingularization result for unfoldings which are versal at each nilpotent points. For a slow–fast unfolding, the nilpotent points are just the contact points. Then, we can use the desingularization result of Panazzolo at each contact point. This desingularization has nothing to do with the slow dynamics, the jumps between contact points, or the slow divergence integrals.

16.2 Codimension

383

The idea of Panazzolo was to add to the parameter λ an extra versal parameter μ ∼ 0, whose dimension is equal to the codimension of the system. Adapted to slow–fast systems, we obtain a versal unfolding for the contact points of the limiting equation X0,λ0 . We explain below, for the case of an unfolding of a slow– fast system, how to add extra parameters in order to also unfold in a versal way any other singularity (of the slow dynamics and so on) and then to obtain a versal unfolding of the given canard cycle. This versal unfolding X˜ ,λ,μ , depends trivially on the parameter λ. We have that X,λ = X˜ ,λ,μ(,λ) where μ(, λ) is a local map with μ(0, λ0 ) = 0. We say that the initial unfolding is induced from X˜ ,λ,μ by the local map ϕ : (, λ) → (, λ, μ(, λ)). To return from X˜ ,λ,μ to X,λ we need to know the map ϕ. Fortunately, for some important questions it will be sufficient to consider the versal unfolding itself: for instance, it is clear that the cyclicity of a canard cycle λ0 in the unfolding X,λ is bounded by the cyclicity of the same cycle in the versal unfolding X˜ ,λ,μ . The principal reason to introduce versal unfoldings is that versal unfoldings are well-adapted to a desingularization by blow-up. This is in general not the case for an arbitrary unfolding. The blow-up method was recalled in Chap. 8. This method is needed to simplify the system at the contact points of the limit equation X0,λ0 . It was shown for instance that a blow-up of a generic contact point produces a new system whose singular points are irreducible in the sense that they are hyperbolic or semi-hyperbolic: the unfolding is desingularized. It has to be noticed that the blownup system is not truly of a higher dimension but is no longer a regular unfolding: it contains now singular fibers and the projection on the parameter is no longer a fibration. For more general contact points, this one-step blow-up will be insufficient, as some points of the blown-up system may not be irreducible. The idea will be to iterate the blow-up. As said above, a general theory of iterated blow-up was developed by Panazzolo in [Pan02], aiming at desingularizing the unfoldings of the class N. This theory may be directly applied to slow–fast systems. In Sect. 16.3 we will explain what we can expect from this desingularization of slow–fast systems. Let us recall that we will assume that the unfoldings are smooth, unless it is explicitly mentioned otherwise.

16.2 Codimension 16.2.1 Codimension of Contact Points In Sect. 2.2, we have defined two invariants for a contact point p of the limiting equation X0,λ0 . The contact order cλ0 (p) ≥ 2 is the contact at p between the f critical curve Sλ0 and the leaf λ0 ,p through p of the fast foliation Fλ0 (foliation which extends the fast orbits). In the simplest case cλ0 (p) = 2 we speak about a quadratic or Morse contact point. The singularity order sλ0 (p) ≥ 0 has a more

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16 Outlook and Open Problems

involved definition. We just recall that this order may be interpreted as the sum of algebraic multiplicities of the (isolated) singularities that bifurcate from contact point for  > 0. We may also roughly say that this order is the multiplicity of a singularity of the slow dynamics, “hidden” at the contact point (although, the slow dynamics is not defined at the contact point!). If sλ0 (p) = 0, the contact point is regular and the system has no singular points near p for  > 0. When sλ0 (p) = 1, we have defined in Sect. 2.2 an index equal to ±1. Recall that, for  > 0, there appears a non-degenerate singular point whose index is precisely the index at the contact point : a saddle if the index is −1, an elliptic singular point (center or focus) if the index is +1. A more elementary way to define and to work with these invariants is to use the normal form (2.1). In this normal form, we have p = (0, 0) in the local coordinates (x, y). The contact order is just the order at x = 0 of the function fλ0 (x) and the singular order is just the order at x = 0 of the function g(x, 0, λ0 ). We will now define the codimension of a contact point p. As usual, one attributes a codimension 0 at the (structurally) stable situations. In our context, a stable contact point is a regular quadratic contact point (a jump point), i.e. a contact point with contact order 2 and singularity order 0. In the normal form (2.1) such a point is characterized by the conditions: fλ0 (0) =

d 2 fλ0 dfλ0 (0) = 0, (0) = 0 dx dx 2

and

g(0, 0, λ0 ) = 0.

A regular contact point is stable as, for X0,λ with λ near λ0 , we will have a unique contact point pλ near p = pλ0 , depending smoothly on λ. Moreover, this point is still a regular quadratic contact point. In case both contact order and singularity order are finite, we define the codimension of the contact point p as cod(p) = c + s − 2, where c = cλ0 (p) and s = sλ0 (p). The equations defining a contact point of contact order c and singular order s are given in the normal form (2.1) by the c+s equations: fλ0 (0) =

d d c−1 fλ (0) = 0 fλ0 (0) = · · · = dx dx c−1 0

and g(0, 0, λ0 ) =

∂ ∂ s−1 g(0, 0, λ0 ) = · · · = s−1 g(0, 0, λ0 ) = 0, ∂x ∂x

with no equation in g when s = 0. We have also the two inequalities: dc ∂s fλ0 (0) = 0 and s g(0, 0, λ0 ) = 0. c dx ∂x

16.2 Codimension

385

16.2.2 Codimension of Jumps Between Contact Points Existence of a jump (or fast connection) between two contact points of the limiting equation X0,λ0 is a degeneracy of the unfolding, as this connection may be broken by a variation of the parameter λ. This breaking may be complicated if one of the connected contact points is non-quadratic. It would be interesting to treat this in general, but here we will restrict to the case where all the possible fast connections are between quadratic contact points. Take any such connections between quadratic contact points p1 (λ0 ) and p2 (λ0 ). Seen the precise requirements in Definition 1.1 for  = 0, these contact points persist as contact points p1 (λ) and p2 (λ), depending smoothly on λ. For λ ∼ λ0 , there also exists a smooth function j (λ) measuring the distance between the leaves of the fast foliation containing respectively p1 (λ) and p2 (λ) and having the property that there exists a fast connection between p1 (λ0 ) and p2 (λ0 ) if and only if j (λ0 ) = 0. We can define such a function by considering a section transverse to the fast connecting orbit of X0,λ0 . As two such functions are smoothly equivalent, the condition j (λ0 ) = 0 does not depend on the choice of the section and of its parameterization. More generally, let us suppose now that there exist different fast connections between two contact points and no more, with associated functions j1 (λ), . . . , j (λ). Then, these functions are such that j1 (λ0 ) = · · · = j (λ0 ) = 0. We call the number of such fast connections between contact points, the jump codimension of X0,λ0 for the value λ0 . Recall that all contact points entering in these fast connections are assumed to be quadratic.

16.2.3 Codimension of Singularities of the Slow Vector Field As explained in Chap. 3, for each λ the slow vector field is defined by the slow–fast vector field Q¯ λ on the curve Sλ \ Cλ , of regular points of the critical curve Sλ (Cλ being the set of contact points). Sλ \ Cλ is a one dimensional manifold. Then, for each p ∈ Sλ \ Cλ , we can choose a local coordinate r, on an interval neighborhood ¯ λ = qλ (r) ∂ , for a smooth family of functions of 0 ∈ R, where p = {r = 0} and Q ∂r qλ (r). We suppose that p is a singular point of the slow vector field, i.e. that qλ (0) = 0 (if not, p is a regular point of the slow dynamics). One defines the codimension codQ¯ λ (p) of the singular point p to be k − 1 where k ≥ 1 is the order of qλ at 0, k

d i.e. the minimal k ∈ N for which dr k qλ (0) = 0. For instance, p is of codimension 0 if it is a simple root. This is coherent with the fact that a simple root is stable (it persists as simple root, with a position smoothly changing in function of λ). It is clear that this notion of codimension does not depend on the choice of the local coordinate r.

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16 Outlook and Open Problems

16.2.4 Codimension of a Slow–fast Unfolding Let us consider a slow–fast unfolding X,λ , defined for λ ∼ λ0 . We assume that all possible fast connections between contact points of X0,λ0 are between quadratic contact points. The codimension cod(X,λ , λ0 ) of the unfolding is just defined as the sum of the different codimensions at all the contact points of X0,λ0 , singular points of the slow vector field Q¯ λ0 plus the number of jumps (fast connections) between contact points in X0,λ0 . In the smooth case, we are just interested in unfoldings with a finite codimension. An unfolding with finite codimension is an unfolding with a finite number of contact points for X0,λ0 and singular points for Q¯ λ0 , each of them with a finite codimension, and a finite number of jumps between contact points (assumed to be quadratic). The codimension has a property of semi-continuity in function of λ. This means the following. Let X,λ be a local family defined for λ ∼ λ0 , representing the unfolding with the same name. We can look at the different unfoldings defined by this family for λ ∼ λ0 . Let C(λ) be the codimension of X,λ at the parameter value λ. Then, if C(λ0 ) is finite it will be the same for λ ∼ λ0 . Moreover, we have that C(λ) ≤ C(λ0 ) if λ is close enough to λ0 . This follows easily from the observation of the behavior of the contact order, the singular order, the order of the singular points of the slow dynamics and the number of fast connections, in function of the parameter. If, for instance, X0,λ0 has a contact point p of finite contact order, this point “bifurcates” for λ ∼ λ0 into a finite number of contact points, and the sum of their contact orders is less than the contact order at p; a similar remark is valid for the singularity order, the order at the singular points of the slow vector field and the number of fast connections between contact points.

16.2.5 Codimension of a Canard Cycle We consider now a canard cycle  belonging to X0,λ0 . As said above, we will assume that the manifold M on which the unfolding X,λ is defined is a neighborhood of , chosen such that all contact points and singular fast connections between contact points of X0,λ0 and all singular points of the slow–fast vector field Q¯ λ0 belong to . This will always be possible if the unfolding is of finite codimension, property that we will assume here. We recall that we focus on canard cycles with one layer. For this type of canard cycle there is just one connected sequence A of slow–fast segments of which all slow segments are attracting and one connected sequence R of slow–fast segments of which all slow segments are repelling. A and R could be respectively an attracting sequence and a repelling sequence as introduced in Definition 4.8, but now we can include jumps between contact points in each sequence. We pass from A to R by a canard breaking mechanism. There are two possibilities of such mechanism. In the first one, the end point of A coincides with the entering point of R and this point

16.2 Codimension

387

is a contact point with singular order s ≥ 1 (in the simple case, one has c = 2 and s = 1, and we call it a Hopf mechanism). In the second case, the mechanism is a jump between the end point of A and entering point of R, and these two points are supposed to be quadratic contact points. The fact that each of these mechanisms is based on a phenomenon of a codimension that is at least one will imply that the existence of a canard cycle is a phenomenon that is at least of codimension 1. To close the canard cycle, we have to pass from R to A by a segment contained in a leaf of the fast foliation, with end points respectively on A and R, i.e. by an orbit in a fast layer. We have already discussed this question above in the text (Chap. 6). Let us recall that we can choose a maximal section , transverse to the fast layer. This section is diffeomorphic to a real interval which may be open, semi-closed, or closed. It is parameterized by a smooth variable v and the given canard  passes through a point of  of coordinate v0 . If v0 is in the interior of , there is a fast orbit γ (v0 ) whose limit points are regular points on A and R. We have a canard cycle (v0 ), called ordinary, containing γ (v0 ) (and passing by A and next R). If v0 is an end point of , it may be in the locus of the canard mechanism, and the canard cycle degenerates to be a single singular contact point for instance when we have a birth of a canard cycle at a singular Hopf point [DR09] or it may correspond to a transitory canard cycle (v0 ). We have studied two examples of transitory canard cycles in Chaps. 14 and 15. In the first case, the canard cycle contains a fast trajectory γ (v0 ) whose limit points are respectively a quadratic contact point and a regular point of the critical curve. In the second case the canard cycle contains a segment of leaf γ (v0 ) of the fast foliation, between two regular points of the critical curve, but this segment also contains a quadratic contact point in its interior. We have studied in Chaps. 14 and 15 what happens to the canard cycle when we “push” v0 outside : In the two cases there is a continuation of the canard, but with a change of shape. It is the reason to call it a transitory canard cycle. These two examples of transitory canard cycles are the simplest ones that we can consider. It is easy to imagine more complicated transitory canard cycles: We can mix the two cases, the contact points may be non-quadratic and so on. It is an interesting open question to try to classify in a systematic way the possibilities of transitory canard cycles. The most striking property of the 1-layer canard cycle  is that it is not isolated, since it is a member of a v-parameter family (v) of 1-layer canard cycles of X0,λ0 . We have called this family of canard cycles a layer of canard cycles and the variable v a layer variable. Moreover, the given canard  = (v0 ) may be an ordinary or a transitory canard cycle. Under the hypothesis of being of finite codimension and with the restricted choice of unfolding X,λ around , the codimension of X,λ just depends on the canard cycle. One could be tempted to define the codimension of , as being the codimension of the unfolding. But, as we have seen all along the text, a crucial role is played by the slow divergence integral I (), along  and we have to take it into account to have a good definition of the codimension for . It is possible that  contains some singular points of the slow dynamics. If the attracting sequence A contains singular points (of the slow dynamics), the slow divergence integral is equal to −∞ along A. Similarly, if R contains singular points,

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16 Outlook and Open Problems

the slow divergence integral along it is equal to +∞. As a consequence, if all the singular points on  belong to A, one has that I () = −∞. In this case, it is rather easy to show that for any λ ∼ λ0 and any positive  ∼ 0, there exists at most one limit cycle near , which is hyperbolic and attracting. A similar observation can be made if all the singular points on  belong to R. But if  contains singular points on both A and R, the value of I () itself is difficult to work with but some results have been obtained by working with the derivative of I () (see [DMD10, DMD11c, DMD08]). From now on, we will assume that  does not contain singular points of the slow dynamics. In this case, one can extend the slow divergence integral I () into a family of slow divergence integrals Iλ (v), unfolding I (), i.e. such that I ((v)) = Iλ0 (v), and in particular, I () = Iλ0 (v0 ). This function is defined for v ∈  and λ ∼ λ0 . It is smooth in v in the interior of  but in general just continuous at the end points of . In the simplest cases, as the ones we consider in Chaps. 13, 14, and 15, Iλ (v) is also smooth in (v, λ) if v is in the interior of  and λ ∼ λ0 . But in more complicated situations as in the case of contact points of contact order > 3, or existence of fast connections between contact points on the sequences A or R, the function Iλ (v) is at most piecewise smooth. Checking the precise smoothness properties of Iλ (v) is an interesting open question, probably to be studied with the help of the desingularization theory introduced below. The slow divergence integral of (v) is equal to Iλ0 (v). It is not true that Iλ (v) is always the slow divergence integral of a canard cycle for λ = λ0 , but in any case it is a sum of slow divergence integrals defined along attracting or repelling sequences of the limiting equation X0,λ . If I () = Iλ0 (v0 ) = 0, the slow divergence integral has a well-defined sign and under rather general conditions the canard cycle bifurcates into a single hyperbolic limit cycle, which is attracting if I () < 0 and repelling if I () > 0. This means that the unfolding of the canard cycle is comparable to the one of a common cycle of the same nature, respectively attracting or repelling. We do not define a notion of canard cycle codimension for this common-like case, and we will assume from now on that I () = 0. To begin, let us suppose that v0 is an interior point of , i.e. that the canard cycle is an ordinary one. The function Iλ0 (v) is smooth at v0 and we define the codimension codIλ0 (v0 ) as usual to be equal to k − 1 where k is the minimum k

d number such that du k Iλ0 (v0 ) = 0 (recall that we suppose that Iλ0 (v0 ) = 0, i.e. that k ≥ 1). We call it the codimension of the slow divergence integral of , denoted codI (). This codimension is 0 if v0 is a simple zero of Iλ0 , i.e. if Iλ0 = 0 and d du Iλ0 (v0 ) = 0. If now the canard cycle  is transitory at v0 , i.e. at an end point of , it is difficult to define a codimension codI (), in particular because the function Iλ0 (v) is in general not differentiable at v0 . It is clear that this codimension must be at least 1, because  is a particular canard cycle, the neighboring canard cycles being ordinary ones. In the case of Chap. 14, we can attribute the value 1 to codI (), since we already had to use the single condition Iλ0 (v0 ) = 0. In the case of

16.3 Desingularization of Unfoldings

389

Chap. 15, the function Iλ0 (v) is piecewise smooth at v0 but as in Definition 15.1 we can make use of a related function J − K which is smooth, permitting to define codI () = codJ −K (v0 ) + 1. It is an interesting open question to make precise a general definition for codimension of an arbitrary transitory canard cycle. Once this is done, one can define the codimension of the canard cycle  belonging to the limit equation X0,λ0 , as cod() = cod(X,λ , λ0 ) + codI (), where the unfolding X,λ is restricted along  as explained above.

16.3 Desingularization of Unfoldings 16.3.1 Generic Unfoldings We will use normal forms at contact points and at singular points of the slow vector field in order to describe generic properties of an unfolding (X,λ , λ0 ). Let us first consider a contact point p of contact order c ≥ 2 and singularity order s ≥ 0. We write the Taylor expansion at these order of the functions f , g entering in the normal form (2.1): fλ (x) = a0 (λ) + a1 (λ)x + · · · + ac (λ)x c + o(x c ) and g(x, , λ) = b0 (, λ) + b1 (, λ)x + · · · + bs (, λ)x s + o(x s ), with a0 (λ0 ) = · · · = ac−1 (λ0 ) = 0 and b0 (0, λ0 ) = · · · = bs−1 (0, λ0 ) = 0 in case s ≥ 1 (when s = 0 we do not require an extra condition). We also have ac (λ0 ) = 0 and bs (0, λ0 ) = 0. The precise value of ac (λ0 ) and bs (0, λ0 ) is not important as, for (, λ) ∼ (0, λ0 ), we can reduce ac (λ) and bs (, λ) to be ±1, by a (, λ)-linear change in (x, y). Moreover, we can eliminate the coefficient a0 (λ) by changing y in y − a0 (λ) and we can also eliminate the coefficient ac−1 (λ) by making a similar translation in x, so that we can finally assume that fλ (x) =

c−2 !

ai (λ)x i ± x c + o(x c ),

(16.1)

i=1

and g(x, , λ) =

s−1 ! i=0

bi (, λ)x i ± x s + o(x s )

(16.2)

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(with void summation if c = 2 or s = 0). We will say that the unfolding is generic at the contact point p if the map λ → (a1 (λ), . . . , ac−2 (λ), b0 (0, λ), . . . , bs−1 (0, λ)) has a maximal rank c + s − 2 at λ0 . Let us now consider a singular point p of the slow dynamics. This point belongs to the regular part of the critical curve and we can choose a local coordinate r on this regular part around p such that p = {r = 0}. Locally, the slow dynamics is written as r˙ = qλ (r) = q(r, λ). We suppose that p is a zero of codimension σ . It is equivalent to say that if we expand q at order σ at r = 0, we have q(r, λ) = q0 (λ) + · · · + qσ (λ)r σ + qσ +1 (λ)r σ +1 + o(r σ +1 ) with q0 (λ0 ) = · · · = qσ (λ0 ) = 0 and qσ +1 (λ0 ) = 0. Again, we can eliminate the coefficient qσ (λ) and reduce qσ +1 (λ) to ±1 by an affine coordinate change in r in order to get q(r, λ) =

σ! −1

qi (λ)r i ± r σ +1 + o(r σ +1 ),

i=0

where the summation is void when σ = 0. We will say that the unfolding is generic at the singular point p of the slow dynamics if the map   λ → q0 (λ), . . . , qσ −1 (λ) is of maximal rank σ at λ0 , not requiring a condition if σ = 0. Let us suppose that X0,λ0 has ≥ 1 fast connections, all between quadratic contact points. These connections may be broken when λ varies, but, since the contact points are quadratic, we can define, as explained above, for each i = 1, . . . , a smooth function ji (λ) measuring, on a transverse section, the separation of the contact points along the fast dynamics. By construction, one has ji (λ0 ) = 0, for any i. We will say that the unfolding has generic singular fast connections between contact points if the map: λ → (j1 (λ), . . . , j (λ)) has a maximal rank at λ = λ0 . Collecting the different functions associated to the contact points, the singular points of the slow dynamics and the singular fast connections, we have a map λ → A(λ) ∈ RC , where C = cod(X,λ , λ0 ). This map is smooth and verifies A(λ0 ) = 0. We will say that the unfolding (X,λ , λ0 ) is generic if this map has a maximal rank equal to C at λ = λ0 . Of course this genericity implies the required partial

16.3 Desingularization of Unfoldings

391

genericities defined above at the different contact points and so on. It moreover implies that these different genericities are independent conditions. Consider now a canard cycle  = λ0 (v0 ) with I () = Iλ0 (v0 ) = 0, such that codI () = k. Let us assume that the canard cycle is ordinary. We also have to take into account the unfolding of the smooth slow divergence integral: Iλ (v) =

k+1 !

Ii (λ)(v − v0 )i + o((v − v0 )k+1 ),

i=0

where I0 (λ0 ) = · · · = Ik (λ0 ) = 0 and Ik+1 (λ0 ) = 0, not needing a summation if k = 0. We say that the canard has a generic unfolding of the slow divergence if the map: λ → (I0 (λ), . . . , Ik−1 (λ)), has a maximal rank k at λ = λ0 , not needing a condition if k = 0. If the canard cycle is transitory, it is an open problem how to associate to it smooth defining functions, except in the two cases that we have studied in the Chaps. 14 and 15. The transitory cycle of slow–fast type treated in Chap. 14 can be considered of codimension 0 and then no generic condition has to be introduced. For the case of a transitory canard cycle of fast–fast type, as treated in Chap. 15, we refer to Definition 15.1, where associated to I also occur the smooth functions J and K. We can define the genericity of the transitory canard cycle in the same way as for an ordinary canard cycle by using J − K instead of I . Now, we can put together all the functions associated to the unfolding X,λ (supposed to be restricted as explained above) and the slow divergence integral, at least in the case of an ordinary canard cycle or fast–fast transitory case. We obtain a ¯ ¯ ¯ 0 ) = 0. smooth map A(λ) : λ → RC , where C¯ = cod(). This map verifies that A(λ ¯ We will say that  has a generic unfolding in X,λ if A has a maximal range C¯ at λ = λ0 . Practical and interesting questions are to decide whether an unfolding is generic or not and more generally how to obtain generic unfoldings. For instance, if we consider the polynomial slow–fast systems of Liénard type (classical or not), we can ask several questions, with respect to the degree of the system, for instance: What topological types of canard cycles are possible?; Is a given type of canard cycle generically unfolded if the degree is high enough?; What is the minimal degree to obtain a generic unfolding of a given type? Beside these practical questions, there is a question of a more theoretical nature: Is it possible to induce any given unfolding by a generic one. If there exists such a generic unfolding, it is called versal. We will see that existence of versal unfoldings is crucial in order to develop a desingularization theory. We devote the next section to this question.

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16.3.2 Existence of Versal Unfoldings Let p be a contact point of a slow–fast system X,λ and assume that the unfolding is generic at p. We can use the Implicit Function Theorem in order to change the parameter λ, up to a local smooth diffeomorphism at λ0 , by (a, b, ν) where a = (a1 , . . . , ac−2 ), b = (b0 , . . . , bs−1 ) and the components of ν taken among the components of λ. To simplify, we go on to call λ this new parameter. With it, the normal form is written as fλ (x) =

c−2 !

ai x i ± x c + o(x c ),

(16.3)

i=1

and g(x, , λ) =

s−1 !

bi x i ± x s + o(x s ).

(16.4)

i=0

If X,λ is any unfolding, given locally by (16.1) and (16.2), for instance a nongeneric one, we can factorize it through (16.3) and (16.4). Precisely, we introduce p the new family X˜ ,λ,μ just defined in the local normal coordinates (x, y), given by (16.3) and (16.4), where μ = (a, b) and the terms o(x c ), o(x s ) are the ones p of (16.1) and (16.2), the dependence of X˜ ,λ,μ on λ just being through these remainder terms. In restriction to the domain of normal coordinates (x, y), we p obtain an unfolding of X0,λ0 = X˜ 0,λ,0 , which is generic and which moreover locally induces X,λ , since, in the normal coordinates, we have p X,λ = X˜ ,λ,μ(,λ),

where μ(, λ) is given by the functions ai = ai (λ) and bi = bi (, λ). We will p say that X˜ ,λ,μ is a versal unfolding of X0,λ0 , in a neighborhood of p, with versal parameter μ. Strictly speaking, the property of being versal is just valid for the principal parts of (16.3) and (16.4), as the remainders depend on the given unfolding p X,λ . But these terms are not important for the properties of X˜ ,λ,μ . What we mean by the property of being versal, is the induction of the given unfolding by a “good” generic one. The same local construction is also valid at each singular point of the slow dynamics, permitting to define a local versal unfolding at each of these points. We would also want to define a versal unfolding for the singular fast connections. But, as the functions ji (λ) are not locally defined, we do not know whether a versal unfolding of such connections exists. More generally, it is an open question p to know if one can construct a global versal unfolding X˜ ,λ,A , (smooth if X,λ is smooth, analytic if X,λ is analytic), depending on a versal parameter A ∈ RC , with C = cod(X,λ0 ), whose components would be the different versal parameters defined above. An important property to be verified by this versal unfolding

16.3 Desingularization of Unfoldings

393

is that it must induce locally the given unfolding X,λ through the functions ai (λ), bi (, λ), qi (λ), ji (λ) associated to the different contact points, singularities of the slow dynamics and fast connections between contact points. We recall that a partial answer was brought by Panazzolo in [Pan02] in the analytic case, since he constructed such unfolding with A limited to the versal parameters at the contact points (in fact his construction applies to the more general analytic unfoldings that we have called the class N in the introduction). As we will see in the next section, this partial result of Panazzolo is sufficient to proceed to the desingularization of unfoldings: we do not have to blow-up the singularities of the slow dynamics neither the fast connections between contact points.

16.3.3 Blowing Up of Versal Unfoldings In Chap. 8 we have explained how to desingularize an unfolding at a quadratic contact point p of singularity order at most 1, at least when the unfolding is generic at this point. Recall that in this case, the desingularization consists of a single blowup of the unfolding at the point p. This simple desingularization is an example of what one expects to perform for general unfoldings. As an unfolding at a quadratic regular contact point p is a stable one among unfolding of slow–fast systems, it has no versal parameter and one just has to blow-up the three variables (x, y, ). In the case of a Hopf point an extra parameter had to be introduced. If one wants to generalize these simple examples, we have to limit to stable unfoldings. The reason is that blowing up a non-stable unfolding could produce too degenerate situations. A general theory of desingularization for unfoldings of class N was described by Panazzolo in [Pan02]. As recalled above, he first constructed versal unfoldings for unfoldings of class N. This construction is performed for analytic unfoldings of class N, but it works also for smooth unfoldings of finite codimension in this class. In fact the construction is even more elementary in the smooth case, as it is easier gluing smooth local vector fields than it is for analytic ones. In the same article [Pan02], Panazzolo also develops a theory of desingularization for these analytic versal unfoldings. This is based on a succession of blowing ups made at all singular points of the successive blown-up spaces that are constructed. At each step of this desingularization, including at the final one, one obtains what is called in [DR96] a foliated local vector field (FLVF). This object is no longer a parameter family of vector fields but a locally defined vector field tangent to the leaves of a singular foliation (the regular leaves are two-dimensional, but one also has singular leaves of dimension 1 and 0; see [DR96] and [Pan02] for a more details). The initial unfolding is lifted through the desingularization map into the FLVF. As in the case of the Seidenberg Theorem for a single two-dimensional vector field, it is proved that one obtains after a finite number of steps a FLVF whose singular points are all elementary in the sense that they have at least a nonzero real eigenvalue.

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16 Outlook and Open Problems

We can apply these ideas to slow–fast systems, as they belong to the class N. The initial unfolding X,λ is replaced by a versal one X˜ ,λ,μ , where the versal parameter μ has components unfolding generically each contact point. The desingularization constructs a FLVF X¯ λ (in fact, a λ-family of such objects) defined on a desingularization space E of dimension M + 3 (where M is the dimension of the versal parameter space). On this space E there is a two-dimensional singular foliation (reminiscent of the regular fibration above the parameter space, in the initial unfolding) and a FLVF X¯ λ which is tangent to this foliation. The singular points of X¯ λ are the zero-dimensional leaves and also some points on the onedimensional leaves. To fix the ideas, we can look at the simple case of a regular quadratic contact point treated in Chap. 8, in which there only occur singular points on one-dimensional leaves, and there are no zero-dimensional leaves. Precisely, in the blow-up of a quadratic regular contact point we just have one-dimensional leaf, the circle C in Fig. 16.1. The desingularized space is three-dimensional and the FLVF X¯ is not depending on a parameter as λ, as we just blow-up the variables (x, y, )). There are 4 singular points located on C. Two of these points are hyperbolic and two are semi-hyperbolic. In the general case, the situation may be much more complicated. The precise description of the singular points depends on the dimension M + 3 of E. Such a description is an open question. But, as the desingularization given by Panazzolo is rather explicit, it is perhaps not too difficult to study this question. The result could be a list of all possible types of singular points, each given with an explicit normal form. For instance, even if we have no extra parameter beyond , it is clear that by iterating the blow-up procedure we can obtain singular points at which arrive two curves of non-isolated singularities (and not just one); see [HDMD13] for instance. Normal forms of these points have already been treated in Sect. 10.3. As already said, it is an open question to generalize Panazzolo’s result, in order to find versal unfoldings including all the versal parameters (not only the ones coming from contact points, but also the other ones coming from singularities of the slow dynamics, the singular fast connections, the transitory values, and the singularities of the slow divergence integral, in the case of canard cycles). But such general versal unfoldings are not necessary to perform the desingularization theory of unfoldings. Fig. 16.1 A simple case of a blow-up: the quadratic regular contact point

16.4 Analytic Slow–fast Unfoldings of Infinite Codimension

395

For the study of canard cycles, the good strategy is perhaps to use versal unfoldings merely in order to study the return map or the difference maps along the limit periodic sets produced by blowing up the initial canard cycle. We will return to this question in Sect. 16.7 below.

16.4 Analytic Slow–fast Unfoldings of Infinite Codimension Infinite codimension may occur in a lot of different ways, as it may concern the contact points, the singular points of the slow dynamics, and the slow divergence integrals. It is clear that there is nothing reasonable to expect for smooth unfoldings of infinite codimension. But in the analytic case, we can exploit the Noetherian property of the ideals of germs of analytic functions. To take an example, let us consider the slow divergence integral. It may happen that Iλ0 (v) ≡ 0, but with Iλ (v) ≡ 0, for λ ∼ λ0 . In this case we can divide the unfolding Iλ (v) in the ideal generated by the coefficients ai (λ) of the (v − v0 )expansion at any layer value v0 . This ideal, called Bautin ideal, is an ideal in the ring of λ-analytic germs at λ0 . It is independent of the choice of the value v0 . As it is Noetherian, it is generated by a finite number of germs: ϕ1 (λ), . . . , ϕ (λ). We can write Iλ (v) = ϕ1 (λ)h1 (v, λ) + · · · + ϕ (λ)h (v, λ), where h1 , . . . , h are analytic in (v, λ). This technique has e.g. been used in [Rou98, LMR06], . . . This finite sum is very similar to the one that we can write for smooth slow divergence integral of finite codimension, where the order of the Bautin ideal replaces the codimension. The mild difference is that the above sum has a projective character in the coefficients whereas in the smooth finite codimensional case, the sum has an affine character. The infinite codimension may also occur for the unfolding itself, with possible consequences for the slow divergence integral. It is then possible to define a Bautin ideal for the coefficients of the unfolding themselves. One can find an example in [DR09]. In Chap. 13, we have seen that the way to study the cyclicity of a 1-layer canard cycle with difference map (v, , λ) (expressed through exponential-flat functions) ∂ is to use the fact that the derivative ∂v is equivalent to a function equal to δ(v, , λ) = Iλ (v) + ξ(v, , λ), where the remainder ξ is o (1) and is smooth in (, v, λ), where smoothness in  is like in Chap. 11 and  is a set of monomials in . This formulation was sufficient to treat the smooth case of finite codimension in Chap. 13. But the smoothness properties of ξ are of course insufficient to treat the analytic case. We have two levels of difficulties. The first one is just technical and can be easily overcome: In fact, in the analytic case it is possible to show that the

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16 Outlook and Open Problems

remainder is smooth in  and analytic in (v, λ). The second difficulty is that it is not clear whether δ(v, , λ) (or equivalently ξ(v, , λ)) is divisible in the Bautin ideal of Iλ (v). In the examples treated until now it was indeed the case. In fact in these examples, the condition Iλ0 (v) ≡ 0 is always equivalent to a symmetry property of the unfolding itself, property which implies the desired property of division of δ. Then, in these examples, we can divide the function δ itself in the Bautin ideal: δ(v, , λ) = ϕ1 (λ)g1 (v, λ, ) + · · · + ϕ (λ)g (v, λ, ), with gi (v, λ, ) = g¯ 1 (v, λ) + o (1) where g¯ i is analytic in (v, λ) and the remainder o (1) is smooth in (, v, λ). This finite expansion allows to obtain results on finite cyclicity. It is an open question to know whether for general analytic slow–fast systems the same relation as above always holds between the difference map and the slow divergence integral.

16.5 The Question of Finite Cyclicity for Canard Cycles A general conjecture for analytic two-dimensional vector fields is that any analytic unfolding (Xλ , ) of a limit periodic set  has a finite cyclicity. This conjecture can also be stated for smooth unfoldings of finite codimension. In the case of slow– fast systems, this conjecture reduces to ask whether any analytic (or smooth with finite codimension) unfolding (X,λ , ), where  is a canard cycle of a slow–fast family X,λ , has a finite cyclicity. We of course also have to consider limit cycles generated by a single singular contact point, as the singular Hopf points considered in [DR09]). The desingularization evoked in Sect. 16.3 is a tool to tackle this problem. As seen above, it suffices to consider stable unfoldings: following [Pan02], each analytic unfolding (or smooth of finite codimension one) is induced by a versal stable one and the cyclicity in the versal unfolding bounds the cyclicity in the given one. Again, as explained in Sect. 16.3, any versal unfolding X˜ ,λ,μ can be desingularized into a λ-family of “foliated local vector fields” (FLVF) X¯ λ , defined on a space of dimension M + 3, if M is the dimension of the versal parameter μ. If we consider an unfolding (X,λ , ), restricted to a sufficiently small neighborhood of a canard cycle , we can consider all the limit periodic sets that are obtained above , i.e. which are projected on  by the desingularization map λ : E → RM+3 , the space of variables (x, y, , μ). As in Chaps. 14 and 15, we call secondary canard cycles these limit periodic sets (any of them containing non-isolated singular points). By the desingularization result, any of these secondary canard cycles is elementary in the sense that they just contain elementary singular points (partially hyperbolic singular points).

16.6 Disorienting Canard Cycles

397

˜ roughly speaking as We can define the cyclicity of a secondary canard cycle , the upper bound of the number of limit cycles existing on each leave of the FLVF, nearby ˜ (it is easy to give a more precise definition). Using the same compactness argument as in [Rou98], one can prove that the canard cycle  has a finite cyclicity if all secondary canard cycles above it (i.e. projected on  by λ0 ) have a finite cyclicity. Then the conjecture of finite cyclicity for canard cycles is reduced to the same conjecture for elementary limit periodic sets in FLVF. Some results, recalled in this book, have already been obtained in the direction of this conjecture. In Chaps. 14 and 15 we have considered secondary canard cycles obtained from two simple type of transitory canard cycles (for the slow–fast type, the results work for smooth slow–fast unfoldings without any condition, and hence also for analytic ones; for smooth fast–fast canard cycles, the results just work for smooth unfoldings of finite codimension). The way to study a limit periodic set is to look at a local return map, or equivalently at a difference map. The reader can find some examples in Chaps. 13, 14, and 15 (see also [DR09] for instance). This map is not necessarily generic. But it is reasonable to expect that in a natural way, some versal parameters would appear in its expression. This is the case in the examples studied in the previous chapters, where we have obtained expansions on admissible monomials. One could hope that this type of versal factorization is a general property. If this were the case, we would just have to look at generic return maps, to study the question of finite cyclicity. If we want to explicitly compute the cyclicity of a canard cycle , the compactness argument is not sufficient. One has to glue up the cyclicity results obtained for the different secondary canard cycles. In Chap. 14 for the slow–fast transitory canard cycles, this leads to an optimal result. It was not yet the case for the fast–fast transitory canard cycles of Chap. 15, even if the task is perhaps not too difficult. Nevertheless this question is in general quite open: it reduces to keep the control of the limit cycles when they pass from one region to another one in the blown-up space (a similar study was made successfully by Mardesic for counting the limit cycles bifurcating in a Bogdanov–Takens unfolding [Mar98]).

16.6 Disorienting Canard Cycles On a non-orientable surface it may happen that a canard cycle is a disorienting curve in the sense defined in Sect. 13.2.2: It has arbitrarily small tubular neighborhoods homeomorphic to a Möbius band. More simply, it may happen that a common cycle is a disorienting curve. But it is easy to see that the general result recalled in Chap. 7 can be applied to it: such a disorienting common cycle bifurcates in a unique hyperbolic and disorienting limit cycle. All the questions formulated above for canard cycles (which were implicitly assumed to be orienting curves) may be as well formulated for disorienting canard cycles. A disorienting canard cycle with a nonzero slow divergence integral can be treated the same way as a

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16 Outlook and Open Problems

disorienting common cycle, with a similar result. This case may be seen as a stable one (codimension 0 case). If now the slow divergence integral is zero, we could expect bifurcations of double periodic limit cycles, corresponding to double periodic points of the Poincaré return map. As mentioned in Chap. 13, we do not know how to work with the square of the return map. In the case of an orienting canard cycle, where all bifurcating limit cycles are associated to fixed points of the return map, the method is to replace the return map by the difference function. We do not know if a similar trick exists for the square of the return map. As such, questions about bifurcations of a disorienting canard cycle, with a zero slow divergence integral, seem completely open.

16.7 Recapitulation of Open Problems and Questions We will list and comment the open questions that we have encountered in the previous sections of this chapter and also add other ones. Of course, these questions are often interlinked and the list below may be seen as rather arbitrary and surely largely incomplete.

16.7.1 Questions About Codimension (1) In order to have a synthetic view of the questions about codimension, it would be interesting to define a geometrical stratification in the space of unfoldings or in the space of canard cycles (spaces to be defined!). Each stratum would correspond to a given choice of indices for a finite set of contact points, of zeros of the slow dynamics with given codimension, fast connections between contact points, and so on. The geometric codimension of the stratum would correspond to the codimension defined in Sect. 16.3. We will have natural relations of incidence between the strata defined by the fact that some stratum belongs to the adherence of another one. The properties of this incidence relation (for instance, is the stratification verifying Whitney properties?) will be interesting to study. (2) Make a general study of fast connections between contact points, including the cases with at least one non-quadratic contact point and describe the relation with the indices of contact points. Deduce from this study a good notion of codimension of the fast connections between contact points. (3) Study general canard mechanisms, their associated breaking parameters, and the related codimensions. (4) Define a good notion of codimension for canard cycles with several layers (or breaking parameters). (5) Define a codimension for canard cycles passing through singular points of the slow dynamics.

16.7 Recapitulation of Open Problems and Questions

399

(6) Make a classification of all possible transitory canard cycles and deduce from this a good notion of codimension. (7) Determine the smoothness properties of the unfoldings of the slow divergence integral and the related codimension for canard cycles with fast connections between contact points on attracting or repelling sequences (in this case, unfoldings of the slow divergence integral are expected to be just piecewise smooth). The same question may be made for transitory canard cycles.

16.7.2 Questions About Versal Unfoldings and their Desingularization (1) Is it possible to provide versal unfoldings for any unfolding of finite codimension? (2) Obtain the complete list of all possible singular points with an explicit local normal form, occurring for FLVF (produced by desingularization of generic unfoldings). (3) We can ask more precise questions for explicit classes of unfoldings. For instance, let us consider the class of polynomial Liénard slow–fast unfoldings (classical or not). We can ask for explicit conditions for an unfolding to be stable. In terms of the degree, what type of canard cycle can be found? What is the minimal degree for the unfolding to be generic? And many more questions. . .

16.7.3 Questions About Asymptotic Properties (1) For any elementary singular point of FLVF, obtain transition results generalizing the results of Chap. 12 (with probably the use of the obtained normal forms). (2) Make a general asymptotic theory for a function δ equivalent to the derivative of the difference map associated to any secondary canard cycle in a FLVF (produced by desingularization of 1-layer canard cycle.) Perhaps, one has to generalize the notions and results of Chap. 11 and their applications made in Chaps. 13, 14, and 15. (3) A similar question may be asked for canard cycles with several layers.

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16 Outlook and Open Problems

16.7.4 Questions About Analytic Unfoldings and Canard Cycles (1) Study of analytic unfoldings with infinite codimension, of their associated Bautin ideals, of the division property in Bautin ideals. (2) A more specific question would be to prove that the function δ is divisible in the Bautin ideal associated to the slow divergence integral. (3) One can also consider analytic canard cycles with several layers.

16.7.5 Questions About the Finite Cyclicity Conjecture (1) Recall that the finite cyclicity conjecture for analytic slow–fast families (X,λ ) can be reduced to a finite cyclicity conjecture for elementary limit periodic sets in FLVF. Then the question is to prove the following conjecture: any elementary limit periodic set in an analytic FLVF has a finite cyclicity. (2) A more specific question would be to explicitly deduce the cyclicity of a canard cycle in terms of the number of types and cyclicities of the secondary canard cycles above it in the desingularization (see Sect. 16.5). (3) Study the finite cyclicity conjecture in some explicit classes of systems, for instance in the class of polynomial Liénard slow–fast systems (classical or not). We recall that the finite cyclicity property for canard cycles of classical slow– fast Liénard systems would imply a positive answer to Smale’s conjecture about classical Liénard systems (see [Sma00]).

16.7.6 Questions About Disorienting Canard Cycles (1) Define the codimension for disorienting canard cycles. (2) Study the bifurcations of a regular one-layer disorienting canard cycle with slow divergence integral zero. The question can be asked for smooth canard cycles of finite codimension, as well as for analytic ones with infinite codimension. (3) What about disorienting canard cycles with several layers or of transitory type?

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Index

Symbols -admissible, 179 2-layer ordinary canard cycle, 88

A Admissible expression, 4 Admissible monomial, 193 Analytic slow-fast unfoldings of infinite codimension, 395 ASFAM-type, 202 Asymptotically smooth, 139 Asymptotically smooth function in admissible monomials, 202 Attracting sequence, 48

B Balanced canard cycle, 76 Birth cycle, 90 Birth problem, 52 Blowing up of versal unfoldings, 393 Buffer points, 83

C Canard curve, 82 Canard explosion, 278 Canard passage, 66 Canard surface, 276 Canards without head, 60 Codimension of a canard cycle, 386 Codimension of a slow–fast unfolding, 386 Codimension of contact points, 383

Codimension of jumps between contact points, 385 Codimension of singularities of the slow vector field, 385 Complete Chebyshev system, 291 Contact order, 18 Contact point, 10 Contact point of Morse type, 25 Control curve, 82 Critical set, 3 Cyclicity, 41

D Diffeomorphisms of exponentially flat type, 204 Disorienting canard cycle, 397 Divergence Integral, 53 Dodging layer, 46

E ECT system, 291 Elementary canard connection, 47

F Fast relation function, 84 Finite cyclicity for canard cycles, 396 Finite slow divergence, 59 Flying canard, 279 FSJS cycles, 80 FSTS cycles, 79 Functions of exponentially flat type, 204

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. De Maesschalck et al., Canard Cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics 73, https://doi.org/10.1007/978-3-030-79233-6

407

408 Funnel behavior, 83

G Generalized admissible monomial, 198 Generic breaking mechanism, 68 Generic Hopf breaking mechanism, 65 Generic jump breaking mechanism, 64 Generic jump point, 22 Generic turning point, 24 Generic unfoldings, 389 Geometric invariants, 13

H Hopf canard connection, 47

I Index θ , 223 Index θA , 256 Index θC , 272 Index θfi , 255, 256 Index θJ , 254, 255 Index θ , 256

J Jump canard connection, 47 Jump point, 22

L Layer, 46 Layer variable, 46 Limit periodic set, 39 Limiting vector fields, 3

N Normally attracting, 10 Normally hyperbolic, 10 Normally repelling, 10

Index R Regular common cycle, 44 Regular contact point, 18 Regularly smooth, 197 Relaxation oscillations, 40 Repelling sequence, 48 Rescaled layer, 287 Resonant monomial, 198

S Singular contact point, 18 Singularity index, 19 Singularity order, 18, 25 Sitting canard, 280 Slow divergence integral, 55 slow-divergence integral of a slow-fast cycle, 59 Slow dynamics, 27 slow–fast cycle, 39 slow–fast family of vector fields, 3 slow–fast Hopf point, 23 slow–fast system, 3 Slow relation function, 84 Slow singularity, 30 Slow time, 5 Slow vector field, 27 Smooth function in admissible monomials (SFAM) type germ, 194, 195 Standard form, 5 Strongly common attracting cycle, 45 Strongly common repelling cycle, 45

T Tame slow–fast family of vector fields, 4 Terminal layer, 46 Topological index of canard cycle, 273 Transitory canard cycles, 50 Tunnel behavior, 83 Turning point, 24

U Unbalanced canard cycle, 76 O Ordinary canard cycle, 49 V Versal unfoldings, 392 P Position curve, 299 W Weakly C∞ -equivalent, 288