Predictability of Chaotic Dynamics : A Finite-time Lyapunov Exponents Approach [2nd ed. 2019] 978-3-030-28629-3, 978-3-030-28630-9

Table of contents :
Front Matter ....Pages i-xix
Forecasting and Chaos (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 1-31
Lyapunov Exponents (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 33-69
Dynamical Regimes and Timescales (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 71-99
Predictability (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 101-129
Chaos, Predictability and Astronomy (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 131-150
A Detailed Example: Galactic Dynamics (Juan C. Vallejo, Miguel A. F. Sanjuan)....Pages 151-188
Back Matter ....Pages 189-196

Citation preview

Springer Series in Synergetics

Juan C. Vallejo Miguel A. F. Sanjuan

Predictability of Chaotic Dynamics A Finite-time Lyapunov Exponents Approach Second Edition

Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations, and the “SpringerBriefs in Complexity” which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Editorial and Programme Advisory Board Henry D.I. Abarbanel, Institute for Nonlinear Science, University of California, San Diego, USA Dan Braha, New England Complex Systems Institute, University of Massachusetts, Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of Sciences, Budapest, Hungary Karl J Friston, Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, Systems Research, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Jürgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Ronaldo Menezes, Computer Science Department, University of Exeter, Exeter, UK Andrzej Nowak, Department of Psychology, Warsaw University, Warsaw, Poland Hassan Qudrat-Ullah, Decision Sciences, York University, Toronto, Ontario, Canada Linda Reichl, Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster, Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer, System Design, ETH Zurich, Zurich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zurich, Zurich, Switzerland Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria

Springer Series in Synergetics Founding Editor: H. Haken The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foundations of the science of complex systems. Through many enduring classic texts, such as Haken’s Synergetics and Information and Self-Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach.

More information about this series at http://www.springer.com/series/712

Juan C. Vallejo • Miguel A. F. Sanjuan

Predictability of Chaotic Dynamics A Finite-time Lyapunov Exponents Approach Second Edition

123

Juan C. Vallejo Departamento de Fisica Universidad Rey Juan Carlos Móstoles, Madrid, Spain

Miguel A. F. Sanjuan Departamento de Fisica Universidad Rey Juan Carlos Móstoles, Madrid, Spain

ISSN 0172-7389 ISSN 2198-333X (electronic) Springer Series in Synergetics ISBN 978-3-030-28629-3 ISBN 978-3-030-28630-9 (eBook) https://doi.org/10.1007/978-3-030-28630-9 1st edition: © Springer International Publishing AG 2017 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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, express 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

To my wife, Laura, and my children, Alicia and Pedro To my wife, Céline, and my daughters, Alicia and Mónica

Preface

Since the first edition of Predictability of Chaotic Dynamics in 2017, we have had numerous discussions with colleagues on the book with suggestions as to how it could be improved in a future second edition. Therefore, a substantial enlargement has taken place through the addition of new material in the form of new sections. This material expands the existing topics and provides new entry points for discussing new predictability issues on a variety of areas such as machine decision-making or partial differential equations. Furthermore, there are also additional sections that deal with the relationship of the predictability and the stiffness of the underlying equations. In addition, other new sections are devoted to present the covariant exponents in this context or to link the predictability of the dynamical systems with the analysis of their attractors and basins. Finally, the parts of the book devoted to apply these ideas to astronomy have been greatly enlarged. Consequently, these sections have been restructured in two new chapters. One of them is just a gate presenting some basic aspects of predictability in astronomy. The other one expands the model devoted to the analysis of a galactic potential, including the role of a dark matter halo. Móstoles, Madrid, Spain Móstoles, Madrid, Spain June 2019

Juan C. Vallejo Miguel A. F. Sanjuan

vii

Preface to the First Edition

In a very simplified view, the main task of a physicist is to observe nature, to build models and to derive predictions from them. But, in some fields, as for instance astronomy, one recollects the necessary information from observed objects without the possibility of having direct access to them, so one has not the possibility of altering the key parameters of the studied objects. Even worst, the timescales applicable may be out of the human timescales. That is why, a key issue in these cases is to study the subject of observation through numerical simulations. Physics has seen how the simulations have gained relevance. They usually conform to an intermediate state between theory and experimentation, confronting the theory with observations, and are also in charge of exploring the consequences of varying parameters in the physical models. With the widespread usage of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of increasing interest. We can take a model, or a set of equations describing the system, and integrate it during a certain time interval. A relevant question to answer here is: How valid is the resulting forecast? Every model has inherent inaccuracies, leading its results to deviate from the true solution. Any numerical scheme used for solving it will introduce several errors. Round-off errors are present because it is impossible to represent all real numbers exactly on a machine with finite memory. Truncation errors are present when the iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Furthermore, discretisation errors must be taken into account when the solution of the discrete problem does not coincide with the solution of the continuous problem. The reliability of the calculations is directly related to the regularity and instability properties of the modelled flow. This is an interdisciplinary scenario where the underlying physics provides the simulated models, nonlinear dynamics provides their chaoticity and instability properties and the computer sciences provide the actual numerical implementation. This book faces the problem of characterising the time, which we call predictability time, during which a numerical prediction can be considered valid by using techniques and concepts derived from nonlinear dynamics and chaos theory. ix

x

Preface to the First Edition

The resulting numerical solution of a model in principle does not match with the real one, because of the possible differences between the real problem and the model used for making predictions and because the numerical methods introduce different errors and perturbations. A system is said to be chaotic when it presents a strong sensitivity to initial conditions. It is obvious that the presence of chaos can impose certain limits to the time during which two initially close trajectories, the real one and the computed one, remain close enough. Chaos does not always imply a low predictability. An orbit can be chaotic and still be predictable in the sense that the chaotic orbit is followed, or shadowed, by a real orbit, thus making its predictions physically valid. The computed orbit may lead to right predictions despite being chaotic due to the existence of a nearby exact solution. This true orbit is called a shadow, and the existence of shadow orbits is a very strong property that allows to increase the predictability of the computed orbit. There are several books that deal with the selection of the most suitable numerical scheme for solving a given problem, others that describe different chaos indicators for characterising the presence of chaos and others that perform a thoroughly theoretical study of the underlying shadowing theories. This book aims to take a different approach, and it performs a descriptive analysis of how one can gain insight in the study of the predictability of a system. This characterisation will be done through the computation of the finite-time Lyapunov exponents and the analysis of their distributions. We will focus here on continuous dynamical systems, though obviously discrete dynamical systems are mentioned when needed as a part of the discussion. As a consequence, we present here basic concepts on continuous dynamical systems that are needed for the computation of asymptotic and finite-time Lyapunov exponents. The analysis of the finite-time Lyapunov distributions provides relevant information on different properties of a given system related to its predictability. The approach taken here is basically founded on a numerical exploration perspective. We just provide the necessary mathematical background, along with appropriate bibliographic references. We do not focus on the derivation of the algorithms, but we describe how to apply them, for the specific purpose of analysing the predictability of a given system. For this aim, we describe the required procedures for this analysis in a step-by-step way. We present in each chapter a reduced set of simple case studies, either conservative or dissipative systems, once the previous procedures have been discussed. The final chapter is devoted to apply all the techniques discussed in the book to the gravitational potential of our galaxy, as an example of a more realistic case. The contents of this book are primarily thought as a text at the postgraduate level, though it can be a useful reference for researchers working and/or interested in the field of predictability of dynamical systems. It has been designed as a self-contained book, where all needed techniques are properly described so that a reader might be able to reproduce the presented results and apply them to any problem of interest. Every chapter can be read independently of the others. There are five main chapters and one appendix. The introduction describes the fundamental ideas of

Preface to the First Edition

xi

forecasting and its relationship with the predictability of the numerical computations to provide a historical perspective of the problem. The second chapter is devoted to the numerical computation of the finite-time Lyapunov exponents and describes how their distributions provide information of the properties of a continuous dynamical system at local scales. The third chapter expands the previous ideas and presents how the distributions of finite-time Lyapunov exponents change as the different regimes of the dynamical system are reached and how these changes can be used for its characterisation. The fourth chapter is concerned with the computation of the predictability index itself, in terms of the presence of the shadowing property, as an indicator of the reliability of solving numerically a given dynamical system. The fifth chapter, as it was explained earlier, is devoted to the application of all the previous techniques to the gravitational potential of the Milky Way. Finally, there is an appendix that describes the main algorithms used in the book for computing the finite-time Lyapunov exponents, with some comments about how to implement and use them in the most efficient way. We are indebted to Ricardo L. Viana, Juergen Kurths and James A. Yorke for their fruitful comments and discussions concerning the computation of finitetime Lyapunov exponents at very small local scales. We acknowledge the support and encouragement given by our family members during the preparation of this monograph. Móstoles, Madrid, Spain Móstoles, Madrid, Spain November 2016

Juan C. Vallejo Miguel A. F. Sanjuan

Contents

1 Forecasting and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Scientific Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Forecasting and Determinism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Human Knowledge and Decision-Making . . . . . . . . . . . . . . . . . . . 1.2 Chaotic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Computer Numerical Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Solving ODEs Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Solving PDEs Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Numerical Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Reliability and Stability of Numerical Schemes . . . . . . . . . . . . . 1.3.5 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Symplectic Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Shadowing and Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 9 10 14 15 16 17 20 22 23 25 28 28

2

33 33 36 38 42 44 45 47 48 55 66 66

Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Lyapunov Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Lyapunov Exponents Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Local and Non-local Timescales: Covariant Vectors . . . . . . . . . . . . . . . . . 2.5 Finite-Time Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Distributions of Finite-Time Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The Rössler System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The Hénon-Heiles System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

xiv

Contents

3

Dynamical Regimes and Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Temporal Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Regime Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Transient Behaviours, Sticky Orbits and Transient Chaos . . . . . . . . . . . 3.4 The Hénon-Heiles System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Contopoulos System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Rössler System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Hyperbolicity Characterisation Through Finite-Time Exponents . . . 3.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 74 75 78 90 92 96 97

4

Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Numerical Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Predictability, Attractors and Basins . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Predictability and Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Predictability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Hénon-Heiles System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Contopoulos System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Rössler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 101 104 107 109 117 120 125 128

5

Chaos, Predictability and Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical Forecasting in Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Time Series Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Sitnikov Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Predictability and Stability in the Solar System . . . . . . . . . . . . . . . . . . . . . . 5.7 Stellar Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 131 135 137 138 140 144 148 148

6

A Detailed Example: Galactic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Chaos in Galactic Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Predictability in a Galactic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Role of Dark Matter Haloes in Predictability . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Galactic Orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Role of the Dark Halo Orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Role of the Dark Halo Flattening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Dark Halo Orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Flattening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 152 153 163 164 165 174 179 181 182 183 186

Contents

A Numerical Calculation of Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 The Variational Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Selection of Initial Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Practical Implementation for Building the Finite-Time Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

189 189 192 194 195 195

About the Authors

Juan C. Vallejo is an astrophysicist in the Nonlinear Dynamics, Chaos and Complex Systems Research Group at the University Rey Juan Carlos since 1999. His research has focused on analysing the impact of chaotic dynamics in computer simulations for astronomy. He worked for 20 years at the European Space Astronomy Centre in Madrid and is also working in the Joint Center for Ultraviolet Astronomy at the Universidad Complutense of Madrid. Miguel A. F. Sanjuan is full professor of Physics at the Universidad Rey Juan Carlos in Madrid, Spain, where he is the Director of the Research Group in Nonlinear Dynamics, Chaos and Complex Systems. He has been a Visiting Research Professor at the University of Tokyo, funded by the Japan Society for the Promotion of Science, a Fulbright Visiting Research Scholar at the Institute for Physical Science and Technology of the University of Maryland at College Park, Visiting Research Professor at Beijing Jiaotong University and Visiting Professor at the Kaunas Technological University. He is Honorary Professor of Sichuan University of Science and Technology (Zigong, China) and Honorary Professor of Huaqiao University (Xiamen, China). He also serves as the Editor General of the Spanish Royal Physics Society. He is a Corresponding Member of the Spanish Royal Academy of Sciences, a Foreign Member of the Lithuanian Academy of Sciences and a regular member of the Academia Europaea. He has published the monograph Nonlinear Resonances (Springer, 2015).

xvii

Acronyms

We have tried to reduce the number of abbreviations used in the text. There is a variety of notations and acronyms found in the related literature, and here, we summarise them, as well as their most frequently used symbols. AFTLE CLV DLE ELN FTLE FSLE LN LCE LCI LCN MLE ODE SDLE SVD UDV UPO Δt φ(x, t) P+ h J λ

Averaged Finite-Time Lyapunov Exponent Covariant Lyapunov Exponent Direct Lyapunov Exponent Effective Lyapunov Number Finite-Time Lyapunov Exponent Finite-Size Lyapunov Exponent Lyapunov Number Lyapunov Characteristic Exponent Lyapunov Characteristic Indicator Lyapunov Characteristic Number Maximal Lyapunov Exponent Ordinary Differential Equation Scale-Dependent Lyapunov Exponent Singular Value Decomposition Unstable Dimension Variability Unstable Periodic Orbit Finite-time Interval Length Solution of the Flow Equation Probability of Positivity Predictability Index Jacobian Matrix Asymptotic Lyapunov Exponent

xix

Chapter 1

Forecasting and Chaos

1.1 Historical Introduction 1.1.1 The Scientific Method Mankind has always been concerned with the desire of understanding the universe, knowing the ultimate reasons behind past events and having the ability of forecasting the future ones. From the earliest times, the study of natural cycles has been needed for a successful harvest. Astronomy, as one of the oldest sciences, was born with the main task of compiling the several observed phenomena in the skies. It attempted to understand the underlying mechanisms of the observations to figure out what was going to be observed in the future. In a very simplified view, the main task of a physicist today is still to observe nature, to build models from those observations and to use them for deriving predictions. Forecasting is the process of making predictions of the future based on past and present data. Thanks to the scientific models, we are able to understand nature and how it works, forecast the weather, anticipate the position of celestial bodies, send probes to them, understand the behaviour of molecules and atoms, build new materials, imagine new ways for generating energy, etc. Nevertheless, the importance of forecasting goes beyond the practical purposes and points to the essence of the scientific method itself. When analysing a problem, the chosen physical model must capture the reality as faithful as possible, because this model should lead to our ultimate understanding of the physical processes under study. Precisely, one of the key aspects of the scientific method is the possibility of making predictions using the selected model and to confront these predictions with new observations. This confrontation constitutes a test of the goodness of the model, that is, of our understanding of reality. As a consequence, a key issue is the falsifiability or refutability of the theory or inherent possibility that it can be proved to be true or false. The model is built upon on existing observations but is used for making predictions that can be confronted © Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_1

1

2

1 Forecasting and Chaos

with the reality. One must analyse whether it is possible to conceive of an experiment leading to new observations that may refute that model. And it is out of question that such a new experiment must be done. If a physical theory is not falsifiable, it cannot be considered valid, because it will not allow to prove its validity. The validity of a model, based on certain assumptions, is only of temporal nature. The fact that a theory may have successfully survived several times the refutability test does not circumvent that it may eventually fail. Past and present successes do not imply future successes. Noticeably, these failures must be seen as positive facts, because they lead to a better definition of the boundaries in the application of a physical law, and may help to refine its underlying assumptions. In some cases, the failure may lead to a complete replacement of the old model by a new set of laws, which in turn may open new ways of knowledge. A famous example of the above is that of Newton’s law of universal gravitation. When formulated in 1687, these laws allowed to successfully explain the motion of the planets already known from ancient times, from Mercury up to Saturn. Then, the planet Uranus was discovered by the British astronomer Frederick William Herschel in 1781. This was the very first planet discovered in modern times, and it was discovered by chance, because no one expected to find a new planet at that time. It had been observed several times in the past, but Herschel was the first in properly annotating and observing it, realising about the true nature of Uranus as a moving planet and not as a fixed star. Once it was realised that it was a genuine planet, not a star neither a comet, subsequent observations were performed along the following decades. They revealed substantial deviations from the tables based on predictions done by the formulation of Newton’s law of universal gravitation. So confident was the scientific community in the goodness of Newton’s laws that it was hypothesised that an unknown body was perturbing the orbit through gravitational interaction. The position of this body was predicted in 1846 by the French mathematician Urbain Jean Joseph Le Verrier.1 Some observations were done in the area predicted by Le Verrier, and the planet was finally found. The discovery of Pluto roughly followed a similar path. Careful analyses of the orbit of the recently discovered planet Neptune and, again, of Uranus, predicted the existence of another new planet. These discrepancies would ultimately lead to the discovery of Pluto by the American astronomer Clyde Tombaugh in 1930. All these successes gave strong confidence in the infallibility of Newton’s law of universal gravitation. Some years after the discovery of Neptune, Le Verrier also published in 1859 a detailed study of Mercury’s orbit and how its perihelion advanced by a small amount each orbit. But, again, there were discrepancies between the observed and the predicted data. Following the previous and successful reasoning, Le Verrier hypothesised that these discrepancies were the result of the presence of a small

1 In

1843, the British mathematician and astronomer John Couch Adams also began to work on the orbit of Uranus using the data he had, and he has been sometimes credited by the discovery of Neptune.

1.1 Historical Introduction

3

planet inside the orbit of Mercury. There was such a confidence in Newton’s classical laws, that the new planet, not yet discovered, was even given a name, Vulcan. Several claims for the intramercurial planet followed, but none of them were confirmed. Conversely, the arrival of the Einstein’s theory of relativity in 1915 predicted exactly the observed amount of advance of Mercury’s perihelion. The final confirmation for Einstein’s predictions came in 1919, with the observation of the Hyades star cluster during a total solar eclipse. The new theory indeed modified the predicted orbits of all planets when compared with the Newtonian theory. But the magnitude of these differences was only most evident for the closest planet to the Sun, Mercury. Does this mean that Newton’s laws are not valid any longer? Does this mean that Einstein’s laws are the only real truth for explaining the Universe? May it appear a new theory, not yet discovered that may replace Einstein’s theory? Both Newton’s and Einstein’s laws can be considered valid but only when one takes into account their respective applicability constraints. To take these constraints into account is one of the basis of the scientific method. But one should go further and analyse the time during the given law can be considered valid, understood as the time interval, while it provides results within the desired margin of error. The fundamental idea is that a model can return “good” results just below certain time threshold and return results very far away from the real world once we are beyond that time horizon. Therefore, one essential task is the characterisation of this time limitation. This tight link between a model and its temporal validity is what we label as predictability, and we will devote this chapter to discuss it.

1.1.2 Forecasting and Determinism Forecasting is then one of the pillars of the scientific method. Risk and uncertainty are central to forecasting and prediction. As a consequence, the estimation of the degree of uncertainty attached to a given forecast is a key issue. While qualitative forecasting techniques are somehow subjective, quantitative forecasting models may predict future data as a function of past data. Dynamical models can be seen as a way for making quantitative forecasting. A dynamical model is the generic name typically described mathematically by differential equations, which are obtained by the analysis of the physical system at a fundamental level, yet involving sufficient approximations to simplify the model [63]. They constitute a mathematical rule for the time evolution on a state space, characterised by the coordinates of the phase space that describe the state of the system at any instant. This dynamical rule specifies the immediate future of all state variables, given only the present values of the same state variables. Up to the end of the nineteenth century, science relied in the concept of determinism. Such a concept was nicely

4

1 Forecasting and Chaos

described by the French mathematician and astronomer Pierre-Simon Laplace in 1776, in his book A Philosophical Essay on Probabilities [30], We ought then to regard the present state of the universe as the effect of its anterior state and as cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it -an intelligence sufficiently vast to submit these data to analysis- it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.

And it follows, The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence. Its discoveries in mechanics and geometry, added to that of universal gravity, have enabled it to comprehend in the same analytical expressions the past and future states of the system of the world.

At the time those lines were written, it was already known that an absolute errorfree precision in the initial measurements was unreachable in practice. But it was thought that close enough initial conditions would lead to final similar solutions, close enough as well. It seems rather natural to think that with an adequate increase in numerical computational facilities, the errors could be neglected and that from a set of initial conditions known with enough precision, one could predict the future state of a dynamical system. For linear systems, such an approach can be valid. But, with the advent of the study of nonlinear systems, this previous solid scheme begun to be modified. When nonlinearity is present, deterministic dynamics can lead to very complex dynamics. And one prediction can be destroyed by an initial error, even by a very small one. Among the first works where this view was exposed, we can find the memory of the French mathematician and physicist Henri Poincaré, “On the Three-Body Problem and the Equations of Dynamics [42]”. This work was a consequence of the prize of 2500 kronas that the King Oscar II, King of Sweden and Norway, offered in 1887 to whom may provide an answer to the practical question “Is the Solar System stable?”. Poincaré’s efforts in solving this question not only led him to win the prize but led to the birth of one of the most fruitful branches of mathematics, Topology, that he called Analysis Situs. These initial ideas were later expanded in his book “Les Méthodes Nouvelles de la Mécanique Céleste”, published in three volumes between 1892 and 1899 [43]. Poincaré first established the properties of the necessary dynamical equations. Then, he applied the results to the problem of an arbitrary number of bodies under Newtonian gravitational interaction. Finally, he analysed the existence of periodic solutions, following the classic approach of developing the necessary variables as infinite series, and finding that there are series with periodic coefficients formally satisfying the equations of motion. But he did not attempt to prove the convergence of the series. Conversely, he proved the existence of periodic solutions using another

1.1 Historical Introduction

5

Fig. 1.1 The Poincaré section is built by selecting an arbitrary plane in the phase space of the problem under study. The technique relies in analysing how the trajectory intersects such a plane. Every intersection point in the plane, or surface of section, is a consequent of the trajectory. The time between consecutive crosses is variable, typically labeled as Tcross . One must take into account the sign of the velocity when crossing the plane from one subspace to the other and vice versa. One can also compute an averaged < Tcross > interval for all consequents. For a periodic orbit, this interval will be equal to the period of the motion

different approach. The existence of these solutions proved, in turn, the convergence of the series. We can find in his work the Poincaré section technique, which is nowadays commonly used in chaos theory when analysing dynamical systems. He defined for the first time what a consequent is (see Fig. 1.1) and defined an invariant curve as any curve that is its own consequent. He also introduced how the Poincaré section is associated to a map, the Poincaré map. Poincaré used these tools for presenting a detailed discussion about the possible existence of solutions of different types. When discussing the doubly asymptotic solutions in the third volume of his book [43], he constrained his analysis to the special case of zero mass of the perturbed planet, circular orbit of the perturbing planet and zero inclinations, that is, the reduced Hill’s problem. And in the section devoted to the existence of homoclinic solutions in this problem, he wrote: 397. When we try to represent the figure formed by these two curves and their intersections in a finite number, each of which corresponds to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely serrated mesh. Neither of the two curves must ever cut across itself again, but it must bend back upon itself in a very complex manner in order to cut across all the meshes in the grid an infinite number of times.

6

1 Forecasting and Chaos

And it follows, The complexity of this figure will be striking, and I shall not even try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three-body problem, and of all the problems of dynamics in general, where there is no uniform integral and where the Bohlin series are divergent.

The discovery of Poincaré means that one can find extremely complex behaviours even in systems as simplified as the reduced Hill’s problem. One trajectory starts in one point of intersection of the cross section and the following time will appear in other different point, a different consequent of the grid, and so on and so forth. But the grid is so stretched and folded that the points seem to follow a complex random pattern. The physical systems that show this behaviour of strong sensitivity to initial conditions, making them unpredictable, are nowadays called “chaotic systems”. The American meteorologist Edward Lorenz published his article “Deterministic Nonperiodic Flow” [33] in 1963, where he presented many of the ideas that today belong to the chaos theory. He started computing a simplified version of the whole set of equations that describes the Bénard convection. But even being a very simplified version, they could not be solved analytically, and a numerical approach was needed. Lorenz found that the behaviour of the trajectories was very complex, without periodic motions. He found orbits showing violent oscillations around two points, resembling a random, unpredictable motion. The object traced by these trajectories is a “strange attractor”, as it was labeled later on by the Belgian-French mathematical physicist David Ruelle and the Dutch mathematician Floris Takens in 1971 [44]. Lorenz also discovered another interesting fact. He used a computer, a Royal McBee LGP-300, for doing his analyses. Because his computer was very slow, and because he wanted to study the trajectory during a very long time interval, he started a second simulation in the middle point of a previous one. As initial point of the second simulation, he took the middle point coordinates as directly read from a printer output. He observed that the second integration behaves very similar to the first one for a while. But later on, its behaviour was completely different. He realised that the reason was that the printed numbers were a rounded version of the memoryallocated point used in the first simulation. He had done that presuming there should not be a major difference in the prediction using an initial condition or another one very close to it. He had found and described what is popularly known today as the “butterfly effect” or strong sensitivity to initial conditions that characterises any chaotic system. One year before the publication of Lorenz’s paper, 1962, the French astronomer Michel Hénon studied the motion of stars in a simplified galaxy model, with a behaviour dependent on the energy level. Up to that date, classic tools used for doing so, as the theory of perturbations, relied in the fact that all solutions were supposed to be periodic or separable in several periodic or quasi-periodic components. He

1.1 Historical Introduction

7

started to study his model with a recent graduate student, the American Carl Heiles. Again, they used a computer for doing the calculation, which was not a common tool at that time. Their results were published in the article “The Applicability of the Third Integral of Motion: Some Numerical Experiments” published in 1964 [19]. They showed how the system behaves at low-energy levels, with regular and periodic trajectories. In this case, the solutions of the original model and the solutions based on a series expansion, which is equivalent to approximate the non-integrable problem by an integrable system, were very similar. But at energy levels above certain threshold, the real solutions showed a Poincaré section based on erratic, random-like points, filling all the available phase space. We can see the clear differences between the orbits in the real system and in the approximated system, which continues providing regular solutions by comparing Figs. 1.2 and 1.3, directly taken from Ref. [12]. In 1975, the Chinese-American mathematician Tien-Yien Li and the American mathematician James A. Yorke published the classic paper “Period Three Implies Chaos” [32], where the term “chaos” was used for the first time in the dynamical systems literature. Finally, the concept of “chaos control” was born in 1990, as a branch of chaos theory pioneered by James A. Yorke, with the American physicist Edward Ott and the Brazilian physicist Celso Grebogi. Today, the chaos theory is a mature branch of physics that is successfully applied to a wide set of fields. We have constrained the discussion so far to the field of classical dynamics. And we could extend this discussion by entering in the realm of quantum mechanics. The presence of chaos may jeopardise the determinism in a practical way, but such Fig. 1.2 Poincaré section for the Hénon-Heiles system and Energy=1/6 showing solutions based on series expansion of the original model. (Taken from Ref. [12]. ©AAS. Reproduced with permission)

8

1 Forecasting and Chaos

Fig. 1.3 Poincaré section for the Hénon-Heiles system and Energy=1/6, solutions of the original model. (Taken from Ref. [12]. ©AAS. Reproduced with permission)

a presence does not forbid the existence of a deterministic universe by itself. This means that the future of the universe may be already predetermined even when we would be unable to reach any knowledge about it. The future could be already written since the Big Bang, even when we will be unable to compute it. The socalled strong determinism goes even further and claims that it is not only the history of all our universe that is written but the history of all possible universes that must be present when taken into account quantum mechanics. We have seen that classical dynamics keeps the determinism of the universe, even when the presence of chaos may also break its computability from a practical point of view. Quantum dynamics may introduce additional factors to take into account. Following the ideas of the British physicist Roger Penrose [40], quantum dynamics can be considered formed by two parts. One is purely deterministic, formed by the fully deterministic Schrödinger equation about the evolution of the wave function Ψ . But there is another process, responsible for the provision of the state vector reduction and collapse of the wave function. This later process takes the squares of the quantum amplitudes for obtaining classical probabilities, and it is this second process which is responsible for adding uncertainties and probabilities in the quantum theory. This is a probability process that is not continuous and cannot be derived from the fully deterministic process. Whether it is indeed a random process or not is still under debate. However, if that is the case, or even if there is any new unknown process linking both, it could be essentially non-algorithmic. That would mean the future would completely lose its deterministic nature and its future states will never be computed from the present ones.

1.1 Historical Introduction

9

1.1.3 Human Knowledge and Decision-Making An interesting topic for discussion related to the deterministic behaviour of computers is the transparency in decision-making. One can consider computers as entities following fixed rules, which are in principle algorithmic and predictable (if the algorithm is known). An artificial intelligence (AI) machine can take decisions following these deterministic rules in a very quick and efficient manner. AI-enabled developments have the capability to generate tremendous benefits not only for individuals but also for the society as a whole, when they address and may resolve challenges such as climate change or global health issues. However, AI comes with risks and challenges that are associated to fundamental human rights and ethics. AI decisions can allow access and deny certain services to individuals without human supervisions, and we must ensure to craft a strategy that maximises the benefits of AI while minimising its risks. We will only be able to benefit from AIs if we can trust the technology behind. Professional societies, information technologies, data analytics researchers, industry and the general public are realising the power and potential of these technologies, accompanied by a sense of unease and an awareness of their potential dangers. Computers (i.e. AI machines) can deal with very complex phenomena where there are no obvious patterns to follow. With or without a proper training, computers can extract their own rules for taking the target decisions, but the exact behaviour of those rules can be hidden to the human, and the AI machine converts itself into a black box. We can start the discussion on AI machines when applied to the field of chaos theory and how machine learning can be a powerful tool for predicting chaotic systems and can help in predicting confined systems with positive Lyapunov exponents. As example, in [38] appears a machine learning algorithm called reservoir computing to learn the dynamics of a well-known model for space-time chaos, the Kuramoto-Sivashinsky equation, which intends to model the diffusive instabilities in a laminar flame front. This equation has been shown to behave as a finite dimensional system of ordinary differential equations, thus can be considered a bridge between PDEs and dynamical systems. The solution to this equation evolves like a flame front, flickering as it advances through a combustible medium. The idea was to train the computer on data from the past evolution of the Kuramoto-Sivashinsky equation, for closely predicting how the flame-like system would continue to evolve. The proper prediction was achieved up to eight Lyapunov times into the future, the standard measurement of predictability horizon. The implemented algorithm is deterministic. The system to be analysed is also deterministic, yet a chaotic system. Remarkably, the exact rules used by the machine to predict the result are not known, and one can only observe the final prediction. Algorithms acting as black boxes make decisions that we are keen to follow as they frequently prove to be correct. Yet, there are and will be discrepancies with the results, at least from the human perspective. This is of obvious interest

10

1 Forecasting and Chaos

when computers deal with predictability in other areas apart from physics, such as economics or data mining. We are not discussing the amount of data for machine learning, neither the sophistication of the algorithms. One need to tune the algorithms with real-life data, and these data can be heavily biased. In addition, these data involve lengthy and computationally intensive adjustments of a broad set of parameters, making the training, testing and validation procedures, very complex and costly. Therefore, these machine learning rules have triggered an intense debate about how “fair” can be these algorithms. These issues are of special interest in health-care decision-making, autonomous vehicular systems and so on where discriminatory classification and even life critical decisions are taken by black box algorithms. Regarding the validation, another area for discussion is the reproducible research. Reproducible research aims at facilitating the exploitation and reuse of research results, which sometimes required complex algorithms, hardware and data. Ideally, this should lead to disclosure of used codes, in addition to the data being public. Moreover, we should deal with standardised statistical procedures and experimental testbeds. Here, we face again the black box problem. Computer programs can be deterministic systems, but the algorithms may be so complex that one cannot control their output. One of the aims of the complex system research field is to learn from the system as a whole, instead of splitting the system into its smallest components. This splitting may lead us to lose essential information, and one approach may be to try to understand the final output without such a split.

1.2 Chaotic Dynamics Dynamical systems forecast future states based on a set of initial conditions. Dynamical systems are deterministic if there is a unique well-identified consequent to every state. Conversely, they are labeled as random or stochastic if there is a probability distribution of possible consequents. Our work will focus on the study of deterministic dynamical systems with a limited number of degrees of freedom, sometimes very small. In systems with many degrees of freedom, such as those found when studying the motions of molecules in a gas, it is impossible to manage the huge amount of data required for a detailed description of a single state. One must use laws for averaged quantities and to assign different probabilities to each variable and to the state of every individual particle. Because of that, these dynamical systems with many degrees of freedom are sometimes labeled as stochastic systems, even when they are not intrinsically random processes, but subject to fully deterministic equations. They are analysed with statistical tools just because the analyses done using individual deterministic equations are too far from the practical computing capabilities. The term predictability refers to the assessment of the likely errors in a forecast, either qualitatively or quantitatively. Indeed, random processes can be considered

1.2 Chaotic Dynamics

11

predictable processes when it is possible to know the next state at the present time. One may think that deterministic systems are always predictable. But, as we have seen in the previous historical introduction, the study of nonlinear systems leads to the discovery that even in deterministic systems, the predictability can deteriorate, and very fast, with time. These systems are called chaotic systems. The chaotic systems are linked to the notion of complex systems. Conversely to common systems, where we can obtain a realistic description of the system by dissecting it into its structural component parts, complex systems require a different approach due to the nonlinear interactions. In these complex systems, at best, the result of just splitting the system in its individual components does not substantially add anything to our understanding or at worst it can even be misleading. A global view or holistic approach is then needed in these systems. A chaotic system is not a stochastic system. It is a deterministic system that shows a strong sensitivity to the initial conditions. This implies that they can be low dimensional with a very low number of degrees of freedom. Independent of that, they show a complex, random-like behaviour. So, the identification of chaotic orbits is a key element in the analysis of these systems. We should note that the presence of chaos is a concept different from the fact that an orbit is stable or not. The stability property characterises whether nearby, perturbed, orbits will remain in a neighbourhood of that orbit or will move away from it. There are two types of stability: a weaker and a stronger one. The first type is marginal stability. This is when every orbit starting in the neighbourhood of a specified orbit will remain in its neighbourhood at the same distance. The other type is the asymptotic stability. This happens when every orbit starting in the neighbourhood of a specified orbit will approach the specified orbit asymptotically. Conversely, an orbit is said to be unstable when every orbit starting in the neighbourhood of a specified orbit will leave exponentially its neighbourhood. There is a variety of criteria used for characterising the stability of an orbit, and the interested reader is pointed, for instance, to Ref. [1]. Observing a complex trajectory in the phase space of a deterministic system can be a sign of strong sensitivity to initial conditions, but it is not a synonym of chaotic motion. There are certain systems that present the multistability property. Here, the behaviour of the system cannot be predicted, but there is a final state where two or more exclusive states coexist. Alternatively, there is no final state defined, leading to functions that do not repeat values after some period, showing aperiodic or irregular motions, but without showing exponential divergence of initial nearby trajectories. Despite the above, the presence of complex motions is the obvious first step for detecting the presence of chaos. Unfortunately, plotting the orbits of a dynamical system with more than two dimensions is not a straightforward task. In dissipative systems, the trajectories tend to a certain confined region of the phase space, and this region is named “attractor”. From the Poincaré-Bendixon theorem, in continuous flows of second order, attractors can only be of four types: sources, sinks, connexion arcs and limit cycles. But in higher dimensional systems, the attractors can be more complex. They are named in these cases “strange attractors”, and they usually

12

1 Forecasting and Chaos

present a fractal geometry. The presence of strange attractors in a system is a clear sign of the intrinsic complexity of the motion. There are certain techniques such as the Poincaré section that may reduce the number of dimensions to visualise flows with a number of dimensions higher than two. The Poincaré section, when chosen carefully, can be used to accurately show the stable and unstable regions of phase space. The inspection of the areas where there are closed curves or, conversely, randomly scattered dots, or areas where the cumulants get crowded as the integration progresses, provides insight into the nature of the system. But the selection of a proper surface of section is still an issue, and the results may strongly depend on the final choice. There are other alternatives. The search for invariants is one of the most common tools for understanding the dynamics of non-integrable systems (see a classical example in [9]). Usually, the basic building blocks of the dynamics, the fixed points and associated periodic orbits, are located and studied. Later on, their surroundings are analysed, as the stable orbits are mostly surrounded by quasi-periodic orbits while unstable periodic orbits by chaotic orbits. But the complexity of the higherorder orbits makes this procedure less straightforward. Moreover, the complexity of the algorithms for searching periodic orbits increases with the number of dimensions. Independent of using symplectic schemes for the integration of a given orbit, some typical algorithms such as those based on Newton algorithms must explore a set of initial conditions in the phase space keeping the exploration within the initial energy subspace. The computation of the invariant tori and invariant manifolds also gain a high degree of complexity in realistic models. One common sign pointing to the presence of chaos is the aperiodicity of the orbit. One periodic orbit may contain one or more frequencies in resonance. That is, they are rationally related. One quasi-periodic orbit will contain at least two incommensurable frequencies, that is, the ratio of the frequencies is an irrational number. These orbits are labeled as regular orbits. A regular orbit contained in N dimensions can be decomposed into N independent periodic motions. They are subject to a separable potential, and the orbit can be described as a path on an invariant N -dimensional torus. The aperiodic orbits are orbits whose motion cannot be described as the sum of periodic motions as t → ∞. These irregular orbits cannot be decomposed into independent periodic motions, and they can move anywhere energetically permitted. Although it is generally assumed that irregular orbits and chaotic orbits are the same, in Hamiltonian systems, this has not been proven in general [6]. As a consequence of the above, one common approach for chaos detection is Fourier analysis. It starts from the fact that any periodic function can be expressed in form of a Fourier series, based on one fundamental frequency and harmonically related frequencies. Quasi-periodic functions show a power spectrum with more than one fundamental frequency and their linear combinations with integer coefficients. So, a typical signal of a regular movement is a discrete power spectrum. Conversely, aperiodic functions show a continuum spectrum. And, in lowdimensional systems, this is a good indicator of the presence of chaos.

1.2 Chaotic Dynamics

13

This approach can be extended by performing a spectral analysis of the projections of the orbit on the principal axes. For instance, we have three frequency plots for a three-dimensional system. The spectral analysis method relies on detecting resonances between frequencies on different axes to identify different families of orbits. There are different choices for the spectral analysis algorithm and the choice of which data, as positions or velocities, but the basics are the same in all cases. See [6, 37, 60] for further details. Another family of methods for detecting the presence of chaos is built upon the wavelet analysis. This technique aims at representing a signal in the time-frequency domain by performing a frequency analysis of the signal at different levels, or “scales”, of detail. Some methods focus on detecting instantaneous events or peaks, and others focus on detecting the evolution of the major frequencies of the system, the instantaneous frequencies. In Ref. [5], a method is described to detect these instantaneous frequencies called “ridges”, and in Ref. [7], this method is applied to detect chaotic behaviour for time series. Another indicator of chaos is the use of correlation functions, which indicates if a given coordinate conserves a given relationship of the previous value with the current value, once a given time τ has elapsed. This time interval is called correlation time. The correlation function, sometimes called self-correlation, will indicate how a given value in a time series is similar to its value several intervals later. A correlation function that grows faster with the number of intervals is a clear indication of chaos. Conversely, one function that decreases, or decreases after a growing transient, can indicate a regular motion. Indeed, there is a relationship between a faster-growing correlation function and a broad, almost continuum power spectrum. The correlation function can be also calculated using the concept of invariant density ρ(x), and this concept also appears frequently when searching for signs of the presence of chaos. An invariant measure is defined as a measure that is preserved by some function. The Krylov-Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration. If one initially sets in the phase space a given number of systems with a density proportional to ρ(x), that density will not vary as the flow evolves. In principle, there can be several functions holding that property. When there is only one, the system is called ergodic. This means that every trajectory goes arbitrarily close of any point of the accessible phase space. When the system is not ergodic, one can define different invariant densities, but the presence of ergodicity is a sign of the presence of chaos. There is a whole set of methods relying in the study of the deviation vectors aiming at detecting the presence of chaos in a system. Those methods rely on considering the vector between the position of the particle to be observed and the position of a slightly displaced particle. This vector is called a deviation vector. Every method aims to assign a quantitative measure for the type of orbit to these deviation vectors. The Lyapunov exponents are the best known example of this method. The Lyapunov exponents measure the averaged rate of divergence of two initially close trajectories. Actually there is not only one exponent, but there are N Lyapunov

14

1 Forecasting and Chaos

exponents in a system with a N -dimensional phase space. When “the” Lyapunov exponent is mentioned, we typically mean the largest one, pointing to the fastest growing direction. The asymptotic Lyapunov exponents provide an indication on the globally averaged chaoticity of the system during an infinite integration time. But while they measure the asymptotic divergence of infinitesimally neighbouring trajectories, it is not always possible or desirable to perform these very long integrations and the limit value. Indeed, sometimes the asymptotic limit, thus the exponents themselves, may not exist [36]. The standard definition of the Lyapunov exponent uses a very long convergence time, in principle infinite. Due to the sometimes slow convergence towards the asymptotic value, many other numerical indexes and fast-averaged indicators have been developed. We can cite, among others, the rotation index [62], the smaller alignment index [51] or its generalisation, the generalized alignment index [53], the mean exponential growth factor of nearby orbits [8], the fast Lyapunov indicator [10], the relative Lyapunov indicator [47] or the finite-time rotation number [57]. See [52] for a review. These indicators allow distinguishing among the ordered, chaotic or weak chaotic orbits and even among the resonant and nonresonant regions [58]. However, Lyapunov exponents still remain valid indicators since they are quite easy to compute numerically, and they do not depend upon the metric. More importantly, in addition of mapping the “global” degree of instability, or presence of chaos, they can also reflect the local properties of the flow. This happens when they are computed during short intervals, including the timescales of regular-like behaviour of sticky chaotic orbits near remnants of periodic orbits embedded in the chaotic sea.

1.3 Computer Numerical Explorations When discussing about the capability of calculating the future states of a system based on the past states, we have briefly mentioned the notion of computability. The exact definition of this concept deserves a long discussion, and the reader interested into the details is pointed to Ref. [41]. Here, we just define a computable operation or, equivalently, an algorithmic operation, as one operation that can be achieved by a Turing machine. Just to mention there are alternative definitions based on the ideas of the American logician Alonzo Church or those of the French mathematician Jacques Herbrand and the Austrian-American logician, mathematician and philosopher Kurt Gödel, among others. What is something remarkable is that there are some very well-defined mathematical operations that are definitively non-computable. But even constraining the discussion to the formally speaking computable problems, where there exists an algorithmic procedure for solving them, the resulting solution will not match with the real one. This is because there may be inherent differences between the actual

1.3 Computer Numerical Explorations

15

real problem and the model used for making the predictions but also because the numerical methods will introduce different errors and perturbations. We must take into account these unavoidable sources of inaccuracies and errors and face them from a practical point of view. If the system to be solved is only known approximately, it is meaningless to try to solve it with great accuracy. It can be enough to solve the approximate system and to focus on assuring that the numerical resolution will not introduce large errors, keeping them, at least, of the same order of magnitude than the error introduced by having an approximated model. But if the model does properly reflect the reality, or because of any other reason, we must take into account the different sources of errors and keep them under control. If this is not possible, due to the chaotic nature of the system, we should know at least what are the implications of their existence for the intended forecast.

1.3.1 Solving ODEs Numerically The ordinary differential equations (ODEs) were born at the same time that the infinitesimal calculus was, around the seventeenth century. Because of the central role of these equations in the application of mathematics to different sciences, Newton, Leibniz, Johann and Jacob Bernoulli made the effort in creating methods and techniques for solving them. Early attempts tried the elementary integration, based on changes of variable, algebraic manipulation and problem-specific methods, for achieving the reduction to quadratures of a given problem. It was realised soon the difficulty of this task, and more powerful schemes were introduced, such as the series methods, already used by Newton, and numerical methods [15, 16]. In general, a method is called numerical when it allows to obtain, even approximately, the solution to a mathematical problem by using a finite number of arithmetic operations: sums, subtractions, products and divisions. All numerical methods will discretise the differential system to produce a difference equation or map. Two aspects must be taken into account: how good will be the approximation to the solution and the computational cost to reach it within a desired threshold. As a very brief history on the subject, the first method to be born was raised by the Swiss mathematician Leonhard Euler in 1768 in his work “Institutionum Calculi Integralis, Volumen Primum”. Euler’s method is nowadays just used for pedagogic purposes, but it can still be applied to very simple problems. Basically, it replaces the solution by a poligonal, Taylor expansion of first order. So, this method uses the present state of the system to provide the next state. Later on, Euler also invented other methods based on Taylor expansions of higher order. The simplest Euler’s method can be enhanced in several ways. The linear multistep methods form one of these enhancements. The early linear multistep methods were developed around 1850 by the British mathematician and astronomer John Couch Adams and first appeared in the literature in 1885 [3]. These methods provide the solution in a future time using information of the recent past of the

16

1 Forecasting and Chaos

solution, instead of exclusively using the information from the present state, as Euler’s method does. Another alternative way for enhancing Euler’s method was introduced by the German mathematician Carl Runge in 1895 [45]. This was generalised later on by the also German mathematician Karl Heun [21], and, in special, by his fellow countryman Martin Wilhelm Kutta in 1901 [25]. In fact, Kutta introduced different notable methods, and one of them, the one of fourth order and weights 1/6, 2/6, 2/6 and 1/6, is so frequently used that is sometimes known as the “Runge-Kutta” method. However, all that numerical calculus was made by hand, or at most, with the aid of mechanical devices in the early days. So, the practical use of numerical approaches for solving problems was not common. Nowadays, with the widespread use of computer simulations to solve complex dynamical systems, this scenario has dramatically changed. The numerical approaches are found today everywhere, and the reliability of the numerical calculations is of increasing interest.

1.3.2 Solving PDEs Numerically Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations (PDEs) often model multidimensional systems, involving rates of change with respect to continuous variables. Conversely to the ODEs, where each configuration parameter has a single value, allowed to evolve in time, in the PDE, there is a continuous distribution of the parameters describing the system. General solutions of ODEs involve arbitrary constants, and general solutions of PDEs involve arbitrary functions. Because a PDE solution is not typically unique, additional conditions must generally be specified on the boundary of the region where the solution is defined. Moreover, one can deal with ill-posed problems, when the solution does not continuously depend on the data of the problem. The French mathematician Jacques Hadamard provided three conditions to a problem to be considered well-posed: a solution exists, it is unique, and the solution’s behaviour changes continuously with the initial conditions. The typical example for an ill-posed problem may be the inverse heat equation. Here, we aim to deduce a previous distribution of temperature from the final data. The solution is then highly sensitive to changes in the final data. Hence, we find the link with the possible chaotic nature of the system and the possible chaotic nature of the method implemented for solving it. An ill-conditioned problem is indicated by a large condition number, formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. Paired with the problem are any number of algorithms that can be used to solve the problem, and if they have backward stability, which allows it to accurately solve well-conditioned problems.

1.3 Computer Numerical Explorations

17

Not well-posed problems may be needed to be reformulated for numerical treatment. Typically, this involves including additional assumptions, such as smoothness of solution, in a process is known as regularisation, which can be seen as applying Occam’s razor on the solution where there are several solutions. The three most widely used numerical methods to solve PDEs are the finite difference methods (FDM), the finite element methods (FEM) and the finite volume methods (FVM). The FDM is the simplest method, both from a mathematical and coding perspective. This is of special interest in academia, where simple model problems are used a lot for teaching and in research as verification of advanced implementations. It approximates the solutions using finite difference equations to approximate derivatives. The FEM approach is based either on eliminating the differential equation completely (steady-state problems) or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques for ODEs. The FVM approach is similar but conserve masses by design, because the surface integrals that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume, and the flux entering a given volume is identical to that leaving the adjacent volume. Both the FEM and FVM methods have been developed to greater generality and sophistication than finite differences, can cover a large amount of problems and are considered the industry standard nowadays. All the above methods rely in the definition of a mesh, where each point has a fixed number of predefined neighbours, and this connectivity between neighbours can be used to define mathematical operators like the derivative. However, some problems may require the material to deform, move around and even be destroyed. Then, another widely used approach is to use mesh-free methods, which do not require connection between nodes of the simulation domain (mesh), but are rather based on the interaction of each node with all its neighbours. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field. There is a variety of mesh-free methods, and just for reference, one may cite one of the earliest mesh-free methods, the smoothed particle hydrodynamics (SPH). The SPH treat data points as physical particles with mass and density that can move around over time and carry the data values that typically were assigned to the nodes of the mesh.

1.3.3 Numerical Forecast We can take a model, or set of equations describing the system, and integrate it during a certain time interval. The question to answer here is: How valid is the

18

1 Forecasting and Chaos

resulting forecast? Obviously, two initial points may diverge, or not, due to the presence of strong sensitivity to initial conditions. The larger this sensitivity, the larger the likelihood that a computed orbit will diverge from the real one. Every model has inherent inaccuracies leading its results to deviate from the true solution. A given model can ignore certain variables and may rely in certain assumptions and simplifications, making their results different from the solution. Even with a perfect model, the initial conditions may be based in experimental observations, meaning unavoidable experimental errors. Another source of problems is formed by the computational issues. Computers will use float numbers, not real numbers, introducing also unavoidable errors. Finally, analytical solutions are not always possible, and numerical algorithmic computations must be done. Even the best method will diverge from the true orbit beyond certain timescales. Numerical analysis aims at designing and building practical algorithms suitable for solving problems of mathematical analysis. Direct methods compute the solution to a problem in a finite number of steps and may give the precise answer if they were performed in infinite precision arithmetic. Conversely, iterative methods start from an initial guess and converge to the exact solution by iterating successive approximations. In general, even using infinite precision arithmetic, these methods would not reach the solution within a finite number of steps. Therefore, a given tolerance residual decides when a sufficiently accurate solution has been found, and the iterations must be stopped. Whatever the method is selected, it will introduce several errors. Round-off errors are present because it is impossible to represent all real numbers exactly on a machine with finite memory. Truncation errors are committed when the iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Discretisation errors must be taken into account when the solution of the discrete problem does not coincide with the solution of the continuous problem. As a consequence, the selected algorithm should possess the property of keeping the resulting error without growing along the calculation. This is called numerical stability [35]. The stability concept is different than the accuracy and tightly related to the discussion about as to whether two initially nearby orbits will remain in a defined neighbourhood or will move away from it, due to the errors, perturbations or the noise introduced by the numerical scheme. Independent of the numerical scheme or algorithm used for solving a given set of equations modelling a system, the system itself may present or not this stability property for propagating errors. So, independent of the source of this sensitivity, the problem to solve can be labeled as an “ill-conditioned” problem, due to the strong propagation of the initial errors. Otherwise, the problem is labeled as “wellconditioned”. Obviously, one would like to deal always with well-conditioned systems and well-conditioned algorithms. This is not always possible, but it can be somehow remediated by using a high-precision software. This software uses efficient algorithms for performing, to any desired precision, the basic arithmetic operations,

1.3 Computer Numerical Explorations

19

including square and n-th roots, and even transcendental functions. This is not evident, as trascendental functions are analytic functions that do not satisfy a polynomial equation, in contrast to algebraic functions. There are several cases where the use of this software is of special interest, such as when solving ill-conditioned linear systems, large summations to execute on parallel computing, very long time simulations or even simple computations scaled up massively parallel systems. See [2] and references therein for a review on these issues. Of special importance is to use a stable method when solving a stiff equation. It seems that there is not a uniquely and accepted precise definition of stiffness, but it typically occurs when there are two or more very different scales of the independent variable on which the dependent variables are changing. When this happens, one needs to follow a variation in the solution on the shortest length scale to maintain the stability of the integration. The existence of two or more timescales in different directions of the dynamical flow, one quickly growing and one slowly growing, can lead to stiffness. In a stiff problem, the step size must be set to be extremely small even when apparently the solution is smooth. Otherwise, the problem will be numerically unstable. The stiffness phenomenon is not a property of the exact solution but is a property of the differential system itself, because the same solution can be a solution of a set of stiff equations and a set of non-stiff equations [14, 27, 28]. We can classify the methods listed in the previous paragraphs to be explicit or implicit methods. The explicit methods are those when the new state is calculated from the current state. Typical examples are the forward Euler method, the midpoint method, second-order method with two stages or the “Runge-Kutta” method of order 4. Because typically one may want an estimate of the local truncation error of a single Runge-Kutta step, there is a variety of methods called adaptive RungeKutta methods, intended to provide it. These include the Runge-Kutta-Fehlberg54 method, or the Dormand-Prince DOP853, with step size control and dense output, among many others. The implicit methods find a solution involving both the current and later state of the system. The implicit schemes usually require to solve a set of complex equations without analytical solution, so they are more complex and demand more computational resources. When the Runge-Kutta methods were performed manually, only low-order schemes were used. Nowadays, there are explicit methods of order ten and implicit of arbitrary order [13]. When we find stiffness, implicit methods must be used. Otherwise, the use of an explicit method would require impractically small time steps to keep the errors within a desired tolerance. For instance, regarding the Runge-Kutta methods, the stability function of an explicit Runge-Kutta method is a polynomial, so explicit Runge-Kutta methods can never be A-stable [22]. Because of their greater stability, implicit Runge-Kutta methods must be used when studying stiff equations. From all the above, it is obvious the need of controlling the errors. We have seen the intrinsic difficulty of this task. This is of special relevance when there exist no suitable non-trivial exact solutions against which to test the results. There are some

20

1 Forecasting and Chaos

mitigating strategies, being the simplest one to run certain checks aiming to gain knowledge about the goodness of the computations [17]. The very first selection is to check the conservation of any integral of motion, if the system conserves it. The obvious choice is the total energy. But it may be of interest to monitor any other available integral of motion as the integration progresses. Another cross-check to carry out can be to integrate time backwards the equations. This is not frequently done, and interesting conclusions can be obtained. The Burrau problem in the general three-body problem was reversed in [56], and the initial coordinates could be recovered under certain precision. And this leads to an interesting note when considering whether checking the energy is enough as a claim about the goodness of the calculations. The recovery of the initial conditions was done to merely three significant figures even when the total energy was conserved to a relative accuracy much better than 10−11 . Ideally, another check is to perform independent calculations. Numerical explorations can be considered an experimental method and, as such, should produce reproducible results. Different computations with the same initial conditions using different computers or different algorithms on the same computer should produce different results. In certain fields, this is not practical at all. We can take for as an example the weather forecast. Here, the chaotic nature of the partial differential equations that govern the atmosphere joins to the required parametrisation for certain processes such as the solar radiation, precipitation and others, and it is impossible to solve these equations exactly. For getting confidence in the numerical forecast in these cases, ensemble forecast techniques are routinely used for weather forecast. Ensemble forecasting can be seen as a form of Monte Carlo analysis, where different numerical predictions are done using slightly different initial conditions and different models or different formulations of a model. Ideally, the future dynamical system state should fall within the predicted ensemble spread, and the amount of spread should be related to the uncertainty of the forecast.

1.3.4 Reliability and Stability of Numerical Schemes We focus here on discussing the reliability of the numerical analysis of the trajectories of a given dynamical system, considered solutions of a model described by one or several equations. This reliability is related to the instability properties of local issues, such as the trajectory itself, and global issues, such as the model. But, it is also related to the properties of the numerical schemes used for solving the equations. The selection of the most adequate scheme is far from trivial. If the system to be solved is only known approximately, it is meaningless to try to solve it with great accuracy. It can be enough to solve the approximate system and to focus on assuring that the numerical resolution will not introduce large errors, keeping them, at least, of the same order of magnitude than the error introduced by having an approximated

1.3 Computer Numerical Explorations

21

model. We should know at least what are the implications of their existence for the intended forecast. There are two fundamental sources of error in any simulation carried out using numerical schemes: truncation error and discretisation. Truncation or round-off errors are tightly related to the presence of strong sensibility to initial conditions. That is, how robust is the scheme for amplifying two initially nearby conditions. This is of obvious importance when dealing with ODEs. Some numerical algorithms may damp out the initial errors in the input data; meanwhile others can increase those errors. The schemes that belong to the first case are called numerically stable, and one needs to select these robust methods that produce similar results for similar input data. Notice, as usual, that a different issue is how the modelled system is robust or not (chaotic or not) to those variations in the input data. The discretisation issue faces a different aspect of the solving process, and it is of great importance when solving PDEs. The numerical scheme solves a problem that may be is not exactly the same we aim to solve. Therefore, we can exactly solve a system, but it may happen that is different from our target system. From a different point of view, the approximate solution obtained with one numerical scheme can be viewed as the exact solution of a different problem. For PDEs, an algorithm for solving a linear evolutionary partial differential equation is stable if the total variation of the numerical solution at a fixed time remains bounded as the step size goes to zero. The Lax equivalence theorem states that for a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable. It tackles related, yet different concepts. First, consistency. The finite difference representation must converge to the PDE that we are trying to solve as the space and time steps tend to zero. Then, stability. The difference between the numerical solution and the exact solution remains bounded as we increase the number of steps. Finally, convergence. The difference between numerical solutions at a fixed point tends to zero as the discretisation tends to zero. As a rule of thumb, the space resolution (spatial steps) must be synchronisedI˙ with the time resolution (time steps) of the mesh. When dealing with ODEs, there are different definitions for considering a given scheme to be stable. A numerical solver is zero stable if small perturbations in the initial conditions do not cause the numerical approximation to diverge away from the true solution provided that the true solution of the initial value problem is bounded. A second notion of stability, or absolute stability, is concerned with the behaviour of the solution as the time goes to infinity. This definition is important when dealing with stiff ODEs. An absolutely stable numerical method is one for which the numerical solution of a stable problem behaves also in this controlled fashion. For instance, the Euler method is only conditionally stable, meaning that the step size has to be chosen sufficiently small to ensure stability. Conversely, the implicit or backward Euler method is unconditionally stable as we can choose the step size arbitrarily large, and still one gets a growth factor less than unity. Remarkably, these discussions point to the stability of the scheme. For proper accuracy, the step size must be taken reasonably small.

22

1 Forecasting and Chaos

Stability is sometimes achieved by including numerical diffusion. Numerical diffusion is a mathematical term, sometimes called numerical viscosity or numerical resistivity, when dealing with fluid equations or magnetic equations. We stress again that this means we are solving a system that is indeed different than the original one. If one discretises a problem into finite-difference equations, these discrete equations are in general more diffusive than the original differential equations. Hence, the simulated system behaves differently than the intended physical system. A physical blob of fluid moving in the discretised mesh will not move as in an ideal space, and the small variations in direction due to truncation and round-off will make the simulated motion to spread more quickly than in real systems. Sometimes the scheme aims to decrease this numerical diffusion to the minimum, to be as close to the real system as possible. However, sometimes the numerical diffusion is kept or amplified to avoid physical events that are not easily tractable, as singularities, shock waves in fluids and current sheets in plasmas.

1.3.5 Stiffness A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small, what may be not practical in a real implementation to solve the system. There is no single definition of stiffness. The basic idea is that the equation includes some terms that can lead to rapid variation in the solution, typically when there are slow variations in one spatial direction and quick variations in another direction. As typical example, the presence of multiple scales of motions in fluid dynamics often leads to stiff problems. That is, a problem is stiff if it contains widely varying timescales, and some components of the solution evolve much more rapidly than others. But there are other possibilities. A problem can be considered stiff if the step size is dictated by stability requirements rather than by accuracy requirements. Or also a problem is stiff when the explicit methods do not work or work only extremely slowly. Finally, some definitions are related to the eigenvalues of the problem. Here, a linear problem is stiff if all of its eigenvalues have a negative real part, and the stiffness ratio (the ratio of the magnitudes of the real parts of the largest and smallest eigenvalues) is large. And, more generally, a problem is stiff if the eigenvalues of the Jacobian differ greatly in magnitude. From our previous discussions, this is linked to a complex evolution of the Jacobian, thus the Lyapunov exponents. A Jacobian evolution showing some contracting directions while others are expanding or some quickly evolving directions while others are of very slow evolution may point to the link of Lyapunov exponent evolution and stiffness that we will consider later on. Interestingly, a problem can be initially not stiff, but will become stiff as the solution approaches the steady state. The A-stability property provides the condition to have an effective stiff solver. Interestingly, no explicit Runge-Kutta can be

1.3 Computer Numerical Explorations

23

A-stable. This means that an adaptive explicit method such as RK45 may initially work, but it may take longer and longer to obtain the proper solution for decreasing values of one initial parameter. However, all implicit Gauss-Legendre methods, implicit Runge-Kutta, such as the implicit midpoint rule, are A-stable. Current stiff solvers can be classified in two sets. The higher-order backward differentiation formula (BDF) methods are almost A-stable. The BDFs are a family of implicit methods for ODEs. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed times, thereby increasing the accuracy of the approximation. Gear methods are based on such BDF methods and achieve higher order along with numerical stability for stiff problems by monitoring the largest and smallest eigenvalues of the Jacobian matrix and thus assessing and dealing with the stiffness adaptively. That is, these gear methods use variable order as well as variable step size. The second class of stiff solvers are the so-called Rosenbrock methods. These are related to implicit Runge-Kutta methods and contain a wide range of characteristic timescales for dealing with the stiffness of the problem.

1.3.6 Symplectic Integrators We have seen that it can be of interest to keep constant a given quantity along the integration, such as the energy in many cases. This raises the need of integrating the equations using symplectic schemes. These are numerical methods that incorporate to the numerical schemes the geometric properties of the problem. This explains that these techniques are also found under the label of “Geometric Integration”. From a mathematical point of view, Hamiltonian systems are just a specific set of systems within the broad world of possible differential equations. But they are almost everywhere in the field of applied mathematics, being present on all those systems without dissipation or with a dissipation that can be ignored, as it happens in key systems of classical mechanics, quantum mechanics, optics, . . . One system is Hamiltonian if and only if the flow of their solutions is a symplectic transformation in the phase space. The notion of symplectic transformation is geometric. In cases with one degree of freedom, a symplectic transformation conserves the area of the flow. When there are higher number of degrees of freedom, the conserved area is multidimensional. The Poincaré’s integral invariant is the most fundamental invariant in Hamiltonian dynamics. For any phase space set, the sum of the areas of all of its orthogonal projections onto all the non-intersection canonically conjugate planes is invariant under the Hamiltonian evolution. A Hamiltonian system integrated using an ordinary numerical method will become a dissipative, non-Hamiltonian, system with a completely different longterm behaviour, since dissipative systems do have attractors. When a Hamiltonian is integrated with a conventional method, the symplectic nature of the flow is not

24

1 Forecasting and Chaos

preserved, and this is equivalent to modify, or perturb, the original system, and to solve a perturbed one, not being Hamiltonian any more. Solving the system using a symplectic scheme can be seen as solving a perturbed system but without leaving the realm of Hamiltonian problems, because we will still be solving a perturbation of the real Hamiltonian function. This is important because Hamiltonian systems are not structurally stable against non-Hamiltonian perturbations, such as those introduced by classical explicit Runge-Kutta methods, independently of using fixed or variable step sizes. Some examples on how an explicit Runge-Kutta method of order 4, with variable step size (the standard ODE45), can deviate from a real solution can be found in [39]. Historically, the first symplectic methods started from the idea that the symplectic transformations can be expressed in terms of a generation function and tried to find a generation function suitable for numerical treatment. Another approach derived from showing that the Runge-Kutta methods can contain many symplectic methods [31, 49, 55]. It can be proven that all Runge-Kutta methods which are of the GaussLegendre type satisfy the necessary and sufficient condition for a Runge-Kutta method to be symplectic. Meanwhile, there is no explicit Runge-Kutta method that satisfies this condition; the simply-implicit methods can be made to obey such a condition. See [54] and references therein for a review of all these issues. There is an alternative to symplectic integrators, which are the exact energymomentum conservation algorithms, designed to preserve these constant of motion. This scenario is not anyhow as simple as one may want. There is a theorem by Zhong and Marsden that says that there cannot exist integration schemes which are both symplectic and energy conserving for non-integrable systems: the only one to preserve both is the flow of the system itself [65]. This is impossible because the symplectic map with step h would then have to be the exact time-h map of the original Hamiltonian. Thus, a symplectic map which only approximates a Hamiltonian cannot conserve the energy. So, one can find some algorithms which conserve the energy at the expense of not being symplectic. As a rule of thumb, one may think that non-symplectic schemes behave well for small integrations; meanwhile symplectic schemes may be used for long integrations, and a given variable must be conserved. That is, symplectic algorithms have generally a better performance in qualitative long-term investigations, although the energy is only preserved on an average. In general, the lack of energy conservation is not a problem if the system is close to integrable and has less than two degrees of freedom, because there will be invariant tori in the symplectic map which the orbit cannot cross, so the energy will remain oscillating but bounded. Conversely, a non-symplectic method may allow the energy to grow without limit. But, in systems with a higher number of degrees of freedom, symplectic integrations must be handled with care. Because meanwhile the original system may still have invariant tori, the scheme may add an extra degree of freedom, and this may lead to open holes on the system through Arnold diffusion [23]. Round-off errors are also a problem for Hamiltonian systems. They introduce again a non-Hamiltonian perturbation even if one uses a symplectic integrator. Even

1.4 Shadowing and Predictability

25

when such a perturbation could be less than the one seen before when using a nonsymplectic integrator, it could still convert the system into a dissipative problem. As a consequence, chaos itself can be indeed an artefact created from the numerical integration [20, 64]. And the use of symplectic integrators does not avoid a careful analysis of the system to solve. Classical high-accuracy methods may be introduced for avoiding all the above-mentioned problems. Methods based on Taylor expansions work very fine in these cases, but the implementation needs to be carefully assessed in a problem-by-problem basis. See [11] and references therein.

1.4 Shadowing and Predictability Since the very beginning of numerical linear algebra, it was of great interest to estimate the error when solving a set of linear equations. The first approach was the progressive analysis, that treated with the difference between the real solution and the computed solution. Later on, a second approach was attempted, the so-called regressive approach. By showing that the numerical solution exactly satisfies a system very close to the one that we are solving, one can try to enclose the difference between both systems. This is not just applicable when the progressive approach is not possible. In real cases, the system to solve is indeed only known approximately. For instance, the coefficients of the system may be subject to experimental errors inherent to any measurement. As a consequence, it can be considered meaningless to try to solve these approximate systems with great accuracy. Conversely, it may be enough to solve the approximate system just assuring that the numerical resolution will not introduce but similar errors to those found in the knowledge of the coefficients of the system. The regressive approach interprets the calculated solution as a solution of a system that is a perturbation of the original problem to solve. This view gave birth to the concept of shadowing, that is, the foundation of the concept of predictability as used along this monograph. To label a system as chaotic means that due to the strong sensitivity to the initial conditions, two initial nearby trajectories will diverge with time and, at the same time, show complex behaviour. In mathematical terms, an irregular trajectory is defined as chaotic if it shows at least one positive Lyapunov exponent, the motion is bounded within certain limited region and the ω-limit set is not periodic neither composed of equilibrium points [1]. Conversely, a regular orbit will have vanishing Lyapunov exponents. These chaotic motions can be analysed from the viewpoint of the Shannon theory of information. The entropy of a system, widely used in statistical mechanics, measures the probability of finding a system in a given state. The entropy is a measure of the disorder of the system, and using the Shannon theory of information, it gives the amount of information required for guessing a given state. The computation of the Kolmogorov-Sinai entropy uses this idea for calculating the K-entropy as an average of the loss of information on the system states,

26

1 Forecasting and Chaos

calculated on all possible trajectories, per unit time. We can divide the phase space in different volume cells and calculate the corresponding state at regular time intervals. Therefore, we can compute a sequence indicating in which cell the system was at a given time. The information given by the initial conditions will be lost once a given amount of finite time has elapsed. This interval will depend on the initial precision and decreases as the K-entropy grows. The K-entropy is closely related to the Lyapunov exponent values and the presence of algorithmic complexity, found when the time sequence generated from one of its chaotic trajectories cannot be compressed by an arbitrary factor [4]. As a consequence, the loss of information in a given system is associated to the presence of positive exponents. If a system is chaotic, the reliability of the prediction is limited up to a time which is related to the largest Lyapunov exponent. This time length, inverse of the asymptotic Lyapunov exponent, will be named here as reliability time. This timescale is also named as Lyapunov time and is obviously related to the e-folding time, time interval in which an exponentially growing quantity increases by a factor of e. Finally, other authors [17] consider the inverse of the finite-time Lyapunov exponent as timescale of instability. One interesting finding is that there are real systems, such as chaotic asteroids, with Lyapunov time values that are rather small. This phenomenon was called “stable chaos” in [34]. Although those orbits have positive Lyapunov exponents, they are confined in thin chaotic layers for very long times, much longer than their respective Lyapunov times. These cases can be related to the stickiness property [59], where a chaotic orbit wanders close to the boundary of an island of stability, surrounded by a cantorus producing a barrier to the chaotic diffusion. This behaviour was already anticipated by the German physicist Werner Heisenberg, very well known by his contribution to the laws of quantum mechanics, but that in 1967 already discussed these ideas [18], So you can get this kind of surprise, a final instability after a long time of apparent stability, in nonlinear problems, and that, I think, is a very characteristic feature of nonlinear problems. Therefore one might say, to use a very simple term, that nonlinear problems have a certain kind of unpredictability. One doesn’t know how the solutions will behave after a very long time.

This is the precise definition of predictability we will be using along this book. One may model the physical problem for making predictions and solve it using a given scheme. But even being very careful with both, after a given time, one can never know where the trajectory in the phase space finally will go. The term predictability of a system is understood along this book as a measure of the goodness of the system to make predictions, or forecasting, independent of being chaotic or not. Chaos does not always imply a low predictability. An orbit can be chaotic and still be predictable, in the sense that the chaotic orbit is followed, or shadowed, by a real orbit, thus making its predictions physically valid. The computed orbit, also called pseudo-orbit, may lead to right predictions despite being chaotic because of the existence of a nearby exact solution. This true orbit is called a shadow, and the existence of shadow orbits is a very strong property (Fig. 1.4).

1.4 Shadowing and Predictability

27

Fig. 1.4 The shadowing time τ seen as the time a numerical trajectory keeps close to a true trajectory. The real orbit is called a shadow. The distance to the shadow is like an observational error, and within this error, the numerically observed dynamics can be considered reliable

The shadowing concept had a direct impact over the definition of the numerical methods [29, 50]. But the shadowing itself has a deeper impact on the dynamical systems to analyse. The shadows can exist, but it may happen that after a while, they may go far away from the true orbit. The real orbit is called a shadow, and the noisy computed solution can be considered an experimental observation of one exact trajectory. The distance to the shadow is then an observational error, and within this error, the observed dynamics can be considered reliable [48]. The shadowing times define the duration over which there exists a model trajectory consistent with the real system, and these shadowing times will be the basis to assess the predictability of our models. The shadowing can be found in hyperbolic dynamical systems, characterised by the presence of different expanding and contracting directions for the derivative. In hyperbolic systems, the angle between the stable and unstable manifolds is away from zero, and the phase space is locally spanned by a fixed number of distinct stable and unstable directions [24, 61]. The shadowing can be found even in completely chaotic systems (Anosov systems), the strongest possible version of hyperbolicity where the asymptotic contraction and expansion rates are uniform (uniform hyperbolicity). Non-hyperbolic behaviour can arise from tangencies (homoclinic tangencies) between stable and unstable manifolds, from unstable dimension variability or from both. In the case of tangencies, there is a higher but still moderate obstacle to shadowing. But in other cases, the shadowing time can sometimes be very short, as, for instance, in the so-called pseudo-deterministic systems, where the shadowing is only valid during trajectories of given lengths due to the unstable dimension variability (UDV) [26]. These issues have been recently discussed in the context of the new concept of hetero-chaos [46]. We can say that a chaotic attractor is heterogeneous when different regions of the chaotic attractor are unstable in more directions than in

28

1 Forecasting and Chaos

others. This means that when arbitrarily close to each point of the attractor there are different periodic points with different unstable dimensions. In these circumstances, we say the chaos is heterogeneous, and it is called hetero-chaos, which seems to be important for all models with high-dimensional attractors. As the authors of [46] comment, this is perhaps the unifying concept linking different phenomena observed in numerous numerical simulations of chaotic dynamical systems and physical experiments, such as unstable dimension variability (UDV), on-off intermittency, riddled basins, blowout and bubbling bifurcations. It is noteworthy to mention that it is also a major cause of shadowing to fail. They also conjectured that UDV almost always implies hetero-chaos. Indeed, there are other aspects to take into account, and even regular orbits can also have shorter than expected predictability times. In strongly stiff systems, the predictability could be lower than expected for a regular and well behaved, in appearance, orbit. In these systems, the existence of two or more timescales in different directions, one quickly growing and one slowly growing, can lead to shorten the time the computations can be physically meaningful.

1.5 Concluding Remarks The presence of chaos is a concept different from the concept of stability of an orbit of a dynamical system. The stability characterises whether a perturbed orbit will remain in a neighbourhood of the unperturbed orbit or will go away from it. The presence of chaos means the presence of strong sensitivity to initial conditions, meaning that the future evolution of two initially close trajectories may be very different. We have reviewed some sources of errors. Because we deal either with experimental data or a numerical solution, and since even the best method will diverge from the true orbit beyond certain timescales, these errors are unavoidable when solving any physical model. So, even in the context of classical mechanics, the knowledge of future states of a deterministic equation seems to be limited, and beyond certain temporal boundary, the predictability time, the results from the computations can be fully unreliable. However, the shadowing theory provides some help here. The shadowing times define the duration over which there exists a model trajectory consistent with the real system, and these shadowing times will be the basis to assess the predictability of our models.

References 1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer, New York (1996) 2. Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: mathematical physics and dynamics. Appl. Math. Comput. 218, 10106 (2012)

References

29

3. Bashford, F.: An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid with an Explanation of the Method of Integration Employed in the Tables Which Give the Theoretical Form of Such Drops, by J.C. Adams. Cambridge University Press, Cambridge (1883) 4. Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002) 5. Carmona, R., Hwang, W., Torresani, B.: Wavelet analysis and its applications. In: Practical Time-Frequency Analysis: Continuous Wavelet and Gabor Transforms, with an Implementation in S, vol. 9. Academic, San Diego (1998) 6. Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998) 7. Chandre, C., Wiggins, S., Uzer, T.: Time- frequency analysis of chaotic systems. Phys. D Nonlinear Phenom. 181, 171 (2003) 8. Cincotta, P.M., Simo, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials. Astron. Astrophys. 147, 205 (2000) 9. Flaschka, H.: The toda lattice. II. Existence of integrals. Phys. Rev. B 9, 1924 (1974) 10. Froeschlé, C., Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitivity tool. Celest. Mech. Dyn. Astron. 78, 167 (2000) 11. Gerlach, E., Skokos, C.: Comparing the efficiency of numerical techniques for the integration of variational equations. Discrete Contin. Dyn. Syst. 475 (2011) 12. Gustavson, F.G.: Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point. Astronom. J. 71, 670 (1966) 13. Hairer, E.: A Runge-Kutta methods of order 10. J. Ins. Math. Appl. 21, 47 (1978) 14. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and DifferentialAlgebraic Problems, 2nd edn. Springer, New York (1996) 15. Hairer, E., Wanner, G.: Analysis by Its History. Springer, New York (1997) 16. Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations, I, Nonstiff problems, 2nd edn. Springer, Berlin (1993) 17. Heggie, D.C.: Chaos in the N-body problem of stellar dynamics. In: Roy, A.E. (ed.) Predictability, Stability and Chaos in N-Body Dynamical Systems. Plenum Press, New York (1991) 18. Heisenberg, W.: Non linear problems in physics. Phys. Today 20, 27 (1967) 19. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964) 20. Herbst, B.M., Ablowitz, M.J.: Numerically induced chaos in the nonlinear schrodinger equation. Phys. Rev. Lett. 62, 2065 (1989) 21. Heun, K.: Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhangigen Veranderlichen. Z. Math Phys. 45, 23 (1900) 22. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996) 23. Julyan, H.E.C., Oreste, P.: The dynamics of Runge–Kutta methods. Int. J. Bifurcation Chaos 2, 427 (1992) 24. Kantz, H., Grebogi, C., Prasad, A., Lai, Y.C., Sinde, E.: Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors. Phys. Rev. E 65, 026209 (2002) 25. Kutta, W.: Beitrag zur naherungweisen Integration totaler Differenialgleichungen. Zeitschr. fur Math. und Phys. 46, 435 (1901) 26. Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Phys. D 109, 81 (1997) 27. Lambert, J.D.: The initial value problem for ordinary differential equations. In: Jacobs, D. (ed.) The State of the Art in Numerical Analysis. Academic, New York (1977) 28. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. Wiley, New York (1992) 29. Larsson, S., Sanz-Serna, J.M.: A shadowing result with applications to finite element approximation of reaction-diffusion equations. Math. Compt. 68, 55 (1999) 30. Laplace, P.S.: Marquis de, a Philosophical Essay on Probabilities. Wiley, Chapman and Hall Ltd., London (1902)

30

1 Forecasting and Chaos

31. Lasagni, F.M.: Canonical Runge-Kutta methods. ZAMP 39, 952 (1988) 32. Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985 (1975) 33. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963) 34. Milani, A., Nobili, A.M., Knezevic, Z.: Stable chaos in the asteroid belt. Icarus 125, 13 (1997) 35. Press, W.H.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007) 36. Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008) 37. Papaphilippou, Y., Laskar, J.: Global dynamics of triaxial galactic models through frequency map analysis. Astron. Astrophys. 329, 451 (1998) 38. Pathak, J., Hunt, B., Girvan, M., Lu, Z., Ott, E.: Model-free prediction of large spatiotemporally chaotic systems from data: a reservoir computing approach. Phys. Rev. Lett. 120, 024102 (2018) 39. Pavani, R.: A two degrees-of-freedom hamiltonian model: an analytical and numerical study. In: Agarwal, R.P., Perera, K. (eds.) Proceedings of the Conference on Differential and Difference Equations and Applications, vol. 905. Hindawi Publishing Corporation, New York (2006) 40. Penrose, R.: Quantum implications. Essays in Honour of David Bohm. Routledge and Keegan, London/New York (1987) 41. Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. Oxford University Press, Oxford (1989) 42. Poincaré, H.: On the three-body problem and the equations of dynamics. Acta Math. 13, 1 (1890) 43. Poincaré, H.: Les Méthodes nouvelles de la mécanique céleste. Gauthier-Villars et Fils, Paris (1892) 44. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167 (1971) 45. Runge, C.: Ueber die numerische Aflosung von Differentialgleichungen. Math. Anal. 46, 167 (1895) 46. Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018) 47. Sandor, Z., Erdi, B., Szell, A., Funk, B.: The relative Lyapunov indicator. An efficient method of chaos detection. Celest. Mech. Dyn. Astron. 90, 127 (2004) 48. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997) 49. Sanz-Serna, J.M.: Runge Kutta schemes for Hamiltonian systems. BIT, 28, 877 (1988) 50. Sanz-Serna, J.M., Larsson, S.: Shadows, chaos and saddles. Appl. Numer. Math. 13, 449 (1991) 51. Skokos, C. Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A 34, 10029 (2001) 52. Skokos, C.: The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63 (2010) 53. Skokos, C., Bountis, T.C., Antonopoulos, C.: Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method. Phys. D 231, 30 (2007) 54. Stuchi, T.J.: Symplectic integrators revisited. Braz. Jour. Phys. 32, 4 (2002) 55. Suris, Y.B.: Preservation of sympletic structure in the numerical solution of Hamiltonian systems. In: Filippov, S.S. (ed.) Numerical Solution of Differential Equations. Akademii Nauk SSSR, Ins. Prikl. Mat., Moscow (1988) 56. Szebeheley, V.G., Peters, C.F.: Complete solution of a general problem of three bodies. Astronom. J. 72, 876 (1967) 57. Szezech, Jr J.D., Schelin, A.B., Caldas, I.L., Lopes, S.R., Morrison, P.J., Viana, R.L.: Finitetime rotation number: a fast indicator for chaotic dynamical structures. Phys. Lett. A 377, 452 (2013) 58. Tailleur, J., Kurchan, J.: Probing Rare physical trajectories with Lyapunov weighted dynamics. Nature, 3, 203 (2007)

References

31

59. Tsiganis, K., Anastasiadis, A., Varvoglis, H.: Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system. Chaos Solitons Fractals, 2281 (2000) 60. Valluri, M., Merrit, D.: Regular and chaotic dynamics of triaxial stellar systems. Astrophys. J. 506, 686 (1998) 61. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000) 62. Voglis, N., Contopoulos, G., Efthymiopoulos, C.: Detection of ordered and chaotic orbits using the dynamical spectra. Celest. Mech. Dyn. Astron. 73, 211 (1999) 63. Wellstead, P.E.: Introduction to Physical System Modelling. Academic, London (1979) 64. Wisdom, J., Holman, M.: Symplectic maps for the n-body problem, a stability analysis. Astron. J. 104, 2022 (1992) 65. Zhong, G.: Marsden, Lie-Poisson Hamilton Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134 (1988)

Chapter 2

Lyapunov Exponents

2.1 Lyapunov Exponents We have seen in the previous chapter that one of the fundamental questions about the dynamics of a system is to know whether it is predictable or not. The answer to this question is tightly related to analyse if chaos is present in the dynamical flow. Two trajectories starting out very close to each other may exponentially separate with time. In a finite time, this separation may reach the scale of the accessible state space. In dissipative systems is of special interest to study these accessible spaces and the different basins of attraction of the available attracting limit sets. All points in a neighbourhood of a trajectory may converge towards the same limit set, being a fixed point, a limit cycle, a periodic or a quasi-periodic orbit or a strange attractor. In conservative systems, where there are no attractors, one typically analyses how to characterise these orbits that are constrained by the value of the conserved energy. Notably, the exponential divergence to initial conditions can be also found in these conservative systems. Lyapunov exponents are a well-known diagnostic tool for analysing the presence of chaos, or chaoticity, of a system, and their utility comes in part from the fact that their values do not depend upon the metric. The ordinary or asymptotic Lyapunov exponents describe the evolution in time of the distance between two nearly initial conditions, by averaging the exponential rate of divergence of the trajectories. They indicate the dynamical freedom of the system, because a larger exponent means a larger freedom, in the sense than small changes in the past lead to larger changes in the future. The sensitivity to the initial conditions can be quantified by computing the exponential divergence or convergence of trajectories as follows: δz(t) ≈ eλt δz(0),

© Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_2

(2.1)

33

34

2 Lyapunov Exponents

where λ is a mean rate of separation of trajectories of the system, δz(0) is the initial separation and δz(t) is the separation after time t. So, the Lyapunov exponent is defined in the following manner: λ = lim

t→∞

1 δz(t) ln . δz(0)→0 t δz(0) lim

(2.2)

One important remark to raise here is that the definition is based on the notion of distance between two phase space points. The Euclidean distance seems to be a natural choice in a continuous system, but other options may be applicable in other cases. When the phase space is multidimensional, there can be different expansion rates along the different directions of the flow. So, in a N -dimensional flow, there are N different Lyapunov exponents, each one reflecting the averaged expansion rate of phase space in one direction along a given trajectory. If we consider a small N dimensional hyper-sphere centred around the starting point or initial condition, the effect of the dynamics as the time elapses will be to distort the hyper-sphere into a hyper-ellipsoid, stretching its axes along some directions, contracting them along other directions, see Fig. 2.1.

Fig. 2.1 Evolution of an initial orthonormal basis centred around an initial point x(0) after a given interval time has elapsed. The axes can be stretched and contracted, and their orientations can change too. The finite-time Lyapunov exponent χ(Δt) quantifies these changes averaging the rate of expansion or contraction during a time interval Δt. When this time interval tends to infinity, the finite-time exponents χ(Δt) tend to the asymptotic Lyapunov exponent λ

2.1 Lyapunov Exponents

35

The ellipsoid axes will be distorted by the flow dynamics. The slopes of the flow in each direction provide means to know how the perturbation will evolve. The matrix describing these slopes is the Jacobian matrix of the flow, J t , that describes these deformations after a finite time t. By solving at the same time the flow equation and the fundamental equation of the flow (i.e. the distortion tensor evolution; see Appendix A for details), we can follow the evolution of the axes along the trajectory and, in turn, their growth rate. The eigenvectors and eigenvalues are suited to study the iterated forms of a matrix, such as the Jacobian matrix. But the stretches are not related to the Jacobian matrix eigenvalues in a simple way, and the eigenvectors of the strain tensor J T J that determine the orientation of the principal axes are distinct from the Jacobian matrix eigenvectors. This is because the strain tensor does not satisfy any multiplicative property, and the principal axes must be recomputed from scratch for each time t. When the eigenvalue of the system matrix is large, the errors in the initial condition are amplified by such a factor at local scales. At local dynamics level, knowing about the eigenvalues can help in selecting the most adequate method of integration. When the eigenvalue of the system matrix is very large, the error in the initial condition or the machine truncation can be amplified by that value. The Lyapunov exponents can be seen as a generalization of the eigenvalues at a global scale. They indicate how much a given orbit diverges at a global scale, by averaging the expansion rate of the phase space, and this average is needed to do because the expansion rate and the expansion direction change continuously. So, for a continuous dynamical flow, the asymptotic Lyapunov exponent can be defined as λ(x, v) = lim

t→∞

1 ln Dφ(x, t)v, t

(2.3)

provided this limit exists [54]. Here, φ(x, t) denotes the solution of the flow equation, such that φ(x 0 , 0) = x 0 , and D means the spatial derivative in the direction of an infinitesimal displacement v. We should note that this equation implies a dependence on the direction of the distortion that will disappear once the distortion is averaged and the dynamics tend to the fastest globally growing direction. However, since in practice the calculation is performed numerically, only a finite integration time is used instead of the infinite time defined in the equation above. So, an approximated value instead of the real one is returned. This is of course more important when working with experimental data, because of the very small number of measurements. As note aside, an interesting research topic is related to the analysis of the chaotic properties of a dynamical system through the use of Lyapunov exponents without using an absolute time, such as general relativity. In other words, we have discussed the dependence on the direction of the distortion and the dependence on the total averaging time of those distortions. When dealing with general relativity, the dynamical properties of the system may depend on the specific time parametrisation.

36

2 Lyapunov Exponents

Due to the noninvariance of Lyapunov exponents in general relativity, one may think that the presence or not of chaos may depend on the choice of the space-time coordinates. Following [49], the geodesic motion of test particles in a given gravitational field can be approached with the standard nonlinear theory methods. Conversely, the time evolution of the gravitational field itself presents a dependence of the shear between nearby trajectories on the time parametrisation. Even though, the exact values of the exponents may change, the sign of these exponents is coordinate invariant. Therefore, the presence of chaos, characterised by the existence or not of positive exponents, is still valid in general relativity.

2.2 The Lyapunov Spectrum The Lyapunov spectrum of a N -dimensional system is defined as the ordered set formed by the N possible Lyapunov exponents. For N -dimensional flows, it is possible to have N global Lyapunov exponents when a distortion tensor formed from N perturbation vectors evolves according to the flow equations. For a bounded orbit of an autonomous flow, there is always an exponent with zero value in the limiting case (otherwise the system has an equilibrium in its limit set), as is tangent to the trajectory, and there is never any divergence for a perturbed trajectory in its direction [7]. Obviously, for dissipative flows the sum of all λ exponents must be negative. Because there is always a zero value, in a two-dimensional flow, the first exponent is zero, and the second one must be negative. So, chaos is not possible in these flows, in agreement with the Poincaré-Bendixon theorem, stating that the limit sets are equilibria or periodic orbits. Conversely, in higher-dimensional systems, there are more possible combinations of λ values, and the set of all available exponent values is a convenient way for categorising asymptotic behaviours of dynamical systems. In this way, we can set a table like Table 2.1, adapted from [37, 63]. Tables like this one summarise the possible combinations of the Lyapunov spectrum in dissipative systems. Table 2.1 Characterisation of attractors of a given flow dynamics based on the values of the Lyapunov spectrum (This table is an adaptation of the one found in [37, 63])

Dimension 1 2 3 3 3 3 4 4 4

Attractor Fixed point Periodic Motion Equilibrium point Limit cyle Torus T 2 Chaotic attractor C 1 Hypertorus T 3 Chaos in T 3 Hyperchaos C 2

λ spectrum – 0− −−− 0−− 00− +0− 000− +00− ++0−

2.2 The Lyapunov Spectrum

37

The presence of chaos is given when there is at least one λ exponent larger than zero. The regime is labeled as chaotic when only one global Lyapunov exponent λ is positive. We talk about a “hyperchaotic” regime, also named sometimes as highdimensional chaos, when more than one positive Lyapunov exponent is present. This phenomenon is very important as is sometimes related to the presence of unstable dimension variability (UDV) [39], as we will discuss further. The sum of all Lyapunov exponents can be seen as the divergence of the flow, indicator of the overall expansion or contraction of the phase volume V (t). Regarding the trace of the Jacobian, one can write V˙ (t) = T r(J (t))V (t),

(2.4)

dV (t) = T r(J (t))dt, V

(2.5)

V (t) = e

t

0 (T rJ (t))dt.

(2.6)

For conservative N -dimensional systems, this sum must be zero. Because of the symplectic nature of the flow, the exponents hold the property of being in inverse additive pairs λi = −λN −i . In two-dimensional conservative systems, the two exponents are zero, and the motion is quasi-periodic and confined to a torus. But in higher dimensions, chaos is possible. Regular quasi-periodic motions imply that all Lyapunov exponents are zero. But this can also happen in dynamical systems showing irregular dynamics where the separation of nearby trajectories grows in a weaker fashion than exponential, implying zero Lyapunov exponents. These systems are sometimes referred to as weakly chaotic systems [36]. It is worthy to note that the term weak chaos can have different meanings. Some papers use this term when there is a smaller Lyapunov exponent, or two sets of trend values [50]. Others use it when the phase space dynamics is mainly regular with just a few chaotic trajectories and the dynamics is strongly dependent, in a very complex way, on the chosen initial condition [16]. In these weakly chaotic systems, there is no equivalence between time and ensemble averages. This weak ergodicity breaking means that a random sampling of the invariant distribution should not have the same contents statistically speaking as a single orbit integrated for extremely long times. We note here that analysing whether the Lyapunov exponents are zero or not can be useful for distinguishing between chaotic or nonchaotic orbits, but not for distinguishing irregular nonchaotic orbits. An irregular motion is chaotic if it is bounded, the ω-limit set does not merely consists of connecting arcs and there is at least one asymptotic positive Lyapunov exponent [1]. Conversely, a regular orbit has vanishing Lyapunov exponents. However, it is not clear whether an irregular orbit will necessarily have at least a non-zero real exponent. Although it is generally assumed that irregular orbits and chaotic orbits are the same in Hamiltonian systems, this has not been proven in general [11].

38

2 Lyapunov Exponents

Coming back to dissipative systems, the dimension of the existing attractors can be a useful indicator of its complexity. There is a variety of definitions for the term dimension. But in what concerns us, we will focus on the so-called Kaplan-Yorke dimension, DKY , or “Lyapunov Dimension”. Kaplan and Yorke defined this dimension based on the value of the Lyapunov exponents [35]. They sorted all λ values of the N -dimensional flow. Considering D as the maximum number where λ1 + λ2 + . . . λD > 0, one can define a topological dimension D, and one can consider that the dimension of the attractor lies between D and D + 1. The Lyapunov dimension, or Kaplan-Yorke dimension, is then defined as DKY = D + (λ1 + λ2 + . . . + λD )/|λD+1 |.

(2.7)

The Kaplan-Yorke conjecture is that this dimension is the same as that of the dimension of the attractor and equal to the information dimension. Any chaotic flow will have DKY > 2, and chaotic maps can have any dimension. Going further, the dimension of the Lyapunov spectrum can be used as a valid estimator of the production of entropy in a system. The term DKY can be seen then as an upper limit to the information dimension of the system, and following the theorem of Pesin, the sum of all positive exponents will return an estimation of the Kolmogorov-Sinai entropy [58]. As a final remark, when dealing with systems with a very high number of dimensions, as coupled maps, oscillator chains or disordered systems, the definition of Lyapunov spectrum can be generalised and can be defined as the ordered sequence of maximal exponents λi ’s (in descending or ascending order) given as a function of i/D, where D is the total number of exponents, equal to the dimension of the system. In these systems the existence of a limit spectrum when D → ∞ [44] is assumed. Indeed, the analysis of this limit spectrum is of interest by itself. For instance, the properties of this spectrum have been studied in [55] starting from the properties of a infinite set of matrices.

2.3 The Lyapunov Exponents Family This book analyses the Lyapunov exponents, specifically the finite-time Lyapunov exponents, and their relationship with the predictability of dynamical systems. However, it should be noted that neither the notation nor the definitions are standard in the literature when dealing with “Lyapunov” exponents. That is, the same label or similar labels refer to different concepts when reading about Lyapunov-like exponents. Since this can produce some confusion, it would be worthy to summarise some of the different notations found in the literature and to explain what is our final choice here.

2.3 The Lyapunov Exponents Family

39

As we discussed before, the natural objects in dynamics are the linearized flow described by the Jacobian J t and its eigenvalues and eigenvectors, sometimes, respectively, referred to as stability multipliers and covariant vectors or covariant Lyapunov vectors [20]. Within this viewpoint, we can visualise the Lyapunov exponents as the normal modes of the system and the solution of the flow, a linear combination of them, as in Ref. [12] and references therein. These eigenvalues of the Jacobian matrix are also generally called local Lyapunov exponents, because they are computed for a given specific point. However, the term Lyapunov exponent usually refers to an asymptotic measure that characterises the average rate of growth, or shrinking, of small perturbations to the solutions of a dynamical system. These Lyapunov exponents are the ones introduced by the Russian mathematician and physicist Aleksandr M. Lyapunov when studying the stability of nonstationary solutions of ordinary differential equations [43]. Since then, these exponents have been also named as ordinary, standard or asymptotic Lyapunov exponents. The existence of these exponents is assured when the general conditions of the Oseledec ergodic multiplicative theorem are fulfilled [52]. In this article, Oseledec refers to them as Lyapunov characteristics numbers, and Eckmann and Ruelle labeled them in [18] as characteristics exponents. Therefore, these are also typical synonyms for the ordinary Lyapunov exponents. Just to mention here that when these exponents exist, their values are independent from the initial condition. But the conditions of the Oseledec ergodic multiplicative theorem may not always be met in common systems, as indicated in [54]. These ordinary Lyapunov exponents require infinite time for proper average of the growth rates. However, we can average during non-infinite time intervals. Therefore, in this case the exponents are generically labeled as finite-time Lyapunov exponents, or FTLE, independently of the finite interval length used in their computation. This is the definition of FTLE used in this book, and it is also used in [2, 51, 61]. The notation of direct Lyapunov exponent (DLE) was used in [28], as a synonym of FTLE. A related label found in [29] is Lyapunov Characteristic Indicator, or LCI, which correlates with the previous definition of finite-time exponent. Notice that these exponents are sometimes named as effective Lyapunov exponent for large but finite intervals [27]; meanwhile the term local Lyapunov exponent is preferred when such interval is small enough. The term effective Lyapunov exponent is also used in [14], meanwhile the term local Lyapunov exponent is used in [7, 24]. In addition, we can find the term transient Lyapunov exponent, as in [19, 34], meaning intervals large enough to ensure a satisfactory reduction of the local fluctuations but still adequate to reveal slow, long timescale trends. Finally, the finite size Lyapunov exponents [9] analyse the growth of finite perturbations to a given trajectory, conversely to the analysis of the growth of infinitesimal perturbations done by the finite-time exponents. For our purposes, we will use the generic term

40

2 Lyapunov Exponents

finite-time Lyapunov exponents, independently of the length of the considered interval. We will expand this definition in Sect. 2.5. The effective exponents are named as localised in [82]. This is because these authors refine this definition and differentiate between finite-time and the finitesample exponents. When calculating the exponents from an initial set of N orthogonal vectors, corresponding to the initial deviation vectors, this set undergoes a few transient steps as their initial directions evolve under the system dynamics. After a few steps of integration and orthonormalisation, they could be considered already locally characteristic, meaning specific to a certain local flow. The finitetime refers to the case when the directions coincide with the right singular vectors of the matrix resulting from the Jacobian product and the finite-sample, to the case when they correspond to the vectors resulting from the evolution of those singular vectors some steps before starting the computations, that is, when the growing rates are calculated using the direction of the Lyapunov vectors (i.e. the globally fastestgrowing direction as provided by the successive Gram-Schmidt orthonormalisations used by the algorithm of [6]). Another related term is the averaged finite-time Lyapunov exponent (AFTLE), used in [66], referring to finite-time exponents averaged on a large set of initial conditions. This is used for estimating the duration of chaotic transients. Another widely found terminology derives from the Lyapunov Number (LN). This is defined as λ = loge (LN ), as, for instance, in [1]. In general, the symbol λ is used for the standard Lyapunov exponents, and we follow this trend. But in [76] this symbol is used for the Lyapunov Number, and σ is used instead for the standard exponent. The LN is sometimes referred to as Lyapunov characteristic number, or LCN, and the Lyapunov exponent as Lyapunov characteristic exponent, or LCE. In [4, 68], it is also used the term maximal Lyapunov exponent (MLE) as an equivalent name to the LCE. Some authors as [33] or [45] use a related name by defining short-time Lyapunov characteristic numbers as follows: χ (Δt) =

δz(Δt) 1 ln , δz(0)→0 Δt δz(0) lim

(2.8)

so the definition points to be the finite-time exponent. We note here that the LCN term is also used in [13] but pointing to χ=

1 δz(t) ln . t δz(0)

(2.9)

So, in this case, the LCN seems to be the LCE. This is also the case in [68]. This author specifies the LCN (i.e., the LCE) as λ = lim χ (t), t→∞

(2.10)

2.3 The Lyapunov Exponents Family

41

and the χ symbol is used for the so-called Effective Lyapunov Number (ELN). For making things easier, this author mentions that this ELN was named in the past Local Characteristic Number. This is the effective Lyapunov Exponent, and along this monograph, we have used the symbol χ when using finite-time calculations. The term Stretching Number is used in [22, 73] for conservative maps. This is the Lyapunov exponent calculated for infinitesimal displacements and one iteration of the map. In Ref. [14], it is generalised to flows, and the interval is considered variable and equal to the integration step, a system characteristic time or a Poincaré section time, but in any case very small. Again, this seems to be the finite-time exponent. Following this author, the mean (or first moment) of the distribution of stretching numbers is the LCN (i.e., LCE). This distribution of stretching numbers is named as generalised Lyapunov indicator in [22]. The term finite-size Lyapunov exponent, or FSLE, is used in [4, 46]. These are averages of initial perturbations computed until a given error tolerance is reached. They measure the time that the perturbation grows a given factor at mesoscales. When both the perturbation and the error are infinitesimal, we have the standard exponent. In [25], a variant of the FSLE was introduced, as the scale-dependent Lyapunov exponents, or SDLE. These are FSLE calculated using an algorithm optimised for very noisy time series. When dealing with discrete-time systems, we find also the finite-space exponents, as in [38]. They measure the averaged divergence in discrete-time systems, tending to the standard exponent when the cardinality of the discrete phase space tend to infinity. Finally, the term conditional Lyapunov exponent is used in [3, 57] in the context of chaotic systems synchronisation. They are based in the idea that the conditions given by the Oseledec theorem assuring the existence of Lyapunov exponents also assure the existence of exponents in subblocks of the tangent map matrix. These subblock exponents are, as a consequence, indicators of the degree of freedom of the different system dynamics components. In what concerns the Lyapunov spectrum, the distributions of finite-time Lyapunov exponents are sometimes referred to as Lyapunov spectra, but we will reserve such a term for the set of N asymptotic values corresponding to a N -dimensional system, keeping in mind that one can say that every distribution of a single finiteexponent shows a range or “spectrum” of values. The so-called Lyapunov vector concept is linked to the Lyapunov exponents. The Lyapunov vectors are those vectors pointing in the direction in which the infinitesimal perturbation will grow asymptotically and exponentially at an average rate given by the Lyapunov exponents. In other words, a perturbation in the direction of a Lyapunov vector implies an asymptotic growth rate not smaller than λ, and almost all perturbations will asymptotically align with the vector pointing in the fastest-growing direction. These vectors do not correspond to the eigenvectors of the Jacobian, or covariant Lyapunov vectors, which are a local entity and only require a local knowledge of the system.

42

2 Lyapunov Exponents

The concept of bred, or breeding, vectors is tightly related to this. The bred vectors are created by adding random perturbations to an unperturbed initial condition, and both trajectories, the perturbed and the unperturbed, are subtracted from time to time as the integration takes place. This difference is the bred vector, and it must be scaled to be of the same size as the initial perturbation. Afterwards, this bred vector is added again to the unperturbed trajectory to create a new perturbed initial condition. Once this “breeding” process is iterated a few times, it will lead to bred vectors dominated by the fastest-growing instabilities [31, 32].

2.4 Local and Non-local Timescales: Covariant Vectors The Lyapunov exponents are a well-known tool to compute the growth rates when one perturbation is applied to a trajectory of a dynamical system. Obviously, the resulting growth rates will depend on the direction of this perturbation, and only when we use infinite integration times we will get the desired asymptotic behaviour. From this perspective, the Lyapunov exponents quantify the growth of volumes, provide information on entropy production (Kolmogorov-Sinai entropy) and the attractor dimension (Kaplan-Yorke formula) and are norm independent. However, it might be interesting to have information not only about the growth rate values themselves but also on the corresponding directions leading to those rates. Unfortunately, when starting from a given orthonormal set of deviations, the local dynamical properties may be cleared out by the successive orthonormalisation processes, which are norm dependent. The covariant Lyapunov vectors (CLV) provide this information [26]. They are covariant with the dynamics in the sense that, once computed, they are well determined at all times just using the tangent propagator. Thus, they provide invariance at arbitrary differentiable coordinate transformations. What is more remarkable, meanwhile the dynamics of the vectors associated with the large exponents represent fast dynamics, the dynamics of the vectors associated with Lyapunov exponents near zero represent much longer-lived dynamics. Therefore, the CLV help to quantify the degree of hyperbolicity of the dynamics and to explore the temporal and spatial dynamics (see [78] and references therein). The CLV are defined through a suitable geometric construction that involves both the forward and the backward Lyapunov vectors, which in turn may be seen as a byproduct of the usual Benettin algorithm [6]. Therefore, the calculations of the CLV are somehow computationally expensive. This explains why they have just started to be widely used when efficient algorithms speeding up these complex computations have been developed [23, 26, 77]. Nowadays, they are being applied to a variety of fields, such as spatiotemporal dynamics, partial differential equations, nonautonomous differentiable dynamical systems and random dynamical systems. As with Lyapunov exponents, there is not a single name convention, and we can find different terms applied to the same concept, such as covariant Lyapunov vectors, covariant vectors or Oseledet vectors. For simplicity, we will just use

2.4 Local and Non-local Timescales: Covariant Vectors

43

here the first label. These vectors are also related to the Floquet vectors, because they coincide when the flow is stationary [77], and the adjoint covariant Lyapunov vectors [40]. The latter focus on the angles between these vectors and the original covariant vectors that are norm-independent and can be considered as characteristic numbers of the dynamics. The CLVs are not the only indicators aiming the analysis of the directions providing given growth rates. We can mention the bred vectors [67]. They are finiteamplitude perturbations in a given direction that are periodically rescaled within the original phase space. By repeating this process, we get these vectors to be dominated by the directions where one can find the fastest-growing instabilities. We also have the singular, or optimal vectors [10]. They can be seen as the finitetime normal modes, or eigenvectors, of the finite-time propagator. The rationale behind is that the most general motion of a system is a superposition of its normal modes, and the eigenvectors provide this information at very local scales. As the finite-time interval grows, the local properties of the dynamics will be cleared out, the impact of having selected the directions corresponding to the eigenvectors decreases, and one begins to analyse the dynamics at non-local scales. Finally, we can also find the forward and backward Lyapunov vectors [42]. These are vectors related to orthogonal sets of singular vectors related to the propagators operating on infinite time intervals. Analysing the orientation of these vectors, one can expect to recover the local structure of the attractor towards the dynamics is tending to. Unfortunately, they always remain orthogonal and cannot indicate the directions of the stable and unstable manifolds, or their tangencies. Moreover, these vectors are not invariant under time reversal and are not covariant with the dynamics. Another drawback of these vectors is that they are norm dependent. That is, they depend on the definition of the inner products and norms in the tangent space. Covariance means that the forward (or backward) vectors at a given point are mapped by the tangent propagators to the forward (backward) vectors at the image point. The CLV can be then considered as a generalization of the normal modes of the local dynamics. However, meanwhile the normal modes are the basis for constructing the dynamics of a system; they can move independently and are orthogonal to each other; the CLV are not orthogonal. Conversely, they are made coincident with the stable and unstable manifolds. Thus, they are invariant under time reversal, norm independent, covariant with the dynamics and do yield correct growth factors, the Lyapunov exponents. Therefore, the CLV do not coincide with the local expanding and contracting directions, given by the eigenvectors of the Jacobian. Meanwhile these eigenvectors reflect the local knowledge of the system that will be washed up as the integration time increases; the CLV are, by construction, influenced by all Jacobians along the trajectory, until certain convergence is reached. As a consequence, they identify the varying directions of specific asymptotic (or at least, global regime) growth rates and obey equivariance principles. To note that this covariance property is only true for the exact covariant vectors. Hence, the numerically computed covariant vectors that accumulate numerical errors no longer hold such a property.

44

2 Lyapunov Exponents

The CLV are not necessarily orthogonal. It can be shown that any orthonormal set of vectors will eventually converge to a well-defined basis. Indeed, the GramSchmidt vectors coincide with the eigenvectors of the backward Oseledec matrix, for time-invertible systems [40]. Therefore, these vectors may not coincide with the stable and unstable manifolds, which are not generally orthonormal. In addition, these vectors are not invariant under time reversal, neither they are covariant with the dynamics. Even when the CLV are not necessarily orthogonal, they point in the directions in which an infinitesimal perturbation will grow at an average rate given by the Lyapunov exponents. Therefore, almost all perturbations will align asymptotically with the covariant Lyapunov vector corresponding to the largest Lyapunov exponent in the system [53].

2.5 Finite-Time Exponents The standard Lyapunov exponents provide an indication on the globally averaged chaoticity of the system during an infinite integration time. But while they measure the asymptotic divergence of infinitesimally neighbouring trajectories, it is not always possible or desirable to perform these very long integrations. Indeed, sometimes the asymptotic limit, thus the exponents themselves, may not exist [54]. Moreover, in practice, all numerically computed exponents are computed over finite-time intervals that can or cannot be very long. Therefore, such values are generically named as finite-time Lyapunov exponents. Unlike the global Lyapunov exponents, which take the same values for almost every initial condition in every region if chaoticity is sufficiently strong (except for a Lebesgue measure zero set, following Oseledec theorem), the values of the exponents over finite times are generally different and may change in sign along one orbit. We have seen the variety of notations and definitions regarding the finite-time Lyapunov exponents. For our purposes, we will focus in the following definition: χ (x, v, t) =

1 ln Dφ(x, t)v, t

(2.11)

which is derived from the Eq. 2.3 for finite averaging times. Obviously, λ = χ (Δt → ∞), with the implicit dependence on the point x and the deviation vector v. The finite-time Lyapunov exponents, computed according to this Eq. 2.11, reflect the growth rate of the orthogonal semiaxes (equivalent to the initial deviation vectors) of one ellipse centred at the initial position as the system evolves. Fixing this initial point, there are several choices for the initial orientation of the ellipse axes. Due to the dependence on the finite integration time interval used in Eq. 2.11, every orientation will lead to different exponents [82]. One option is to have the axes pointing to the local expanding/contracting directions, given by the eigenvectors,

2.6 Distributions of Finite-Time Exponents

45

and at local time scales the eigenvalues will provide insight on the stability of the point. Other options are the axes pointing to the direction which may have grown the most under the linearized dynamics or pointing to the globally fastestgrowing direction. In what concerns our technique, the initial axes of the ellipse are set coincident with a randomly set of orthogonal vectors, as in Ref. [70]. See Appendix A for details. This option is selected because, as the flow evolves, the axes get orientated from the arbitrary direction as per the flow dynamics. The evolution in time of this orientation will depend on the selected finite-time interval length that in turn will reflect the dynamical flow timescales.

2.6 Distributions of Finite-Time Exponents If we make a partition of the whole integration time along one orbit into a series of time intervals of size Δt, then it is possible to compute the finite-time Lyapunov exponents χ (Δt) for each interval. This section deals with the analysis of the evolution of the shapes of the finite-time Lyapunov exponent distributions as the finite-time interval size increases. This is because by doing so we can detect when the flow leaves the local regime and reaches the global regime [70]. We can get information about the degree of chaoticity of the orbit by subtracting different spectra [74], by deriving their power spectrum via the Fourier transform [75] or by analysing their shapes and cumulants or the q-moments of the distribution. Such an approach has proved to be useful in several fields, such as galactic dynamics [56, 65], analysing chaotic fluid flows in the context of fast dynamos [21] or chaotic packet mixing and transport in wave systems [81]. The distributions of effective Lyapunov exponents can be studied from the cumulant generating function, defined as the logarithm of the moment generating function, which is itself the Fourier transform of the probability density function [27]. The first four cumulants are the mean, variance, skewness and kurtosis of the distributions. As they reflect the deviation from Gaussianity, they reflect the deviation from the fully chaotic case. The generalized exponents are associated to the order-q moments of the distributions [5, 15]. For some maps, like the Ulam map x → 4x(1−x), explicit analytical expressions can be found to such probability exponents. In such cases, the probability distributions of time-n exponents strongly deviate from the Gaussian shape, decaying with exponential tails and presenting 2n−1 spikes that narrow and accumulate close to the mean value with increasing n [2]. Such tails and spikes were described for the Hénon-Heiles system in [69]. Here, we aim to use the distributions for characterising different orbital behaviours. The shape of such distribution can serve as a valid chaoticity indicator, as it shows the range of values for χ . In principle, the shape depends on the initial condition and on the sampling interval size Δt.

46

2 Lyapunov Exponents

When considering the shapes of these distributions as a valid indicator, its evolution or stationarity is a key question. Here, we follow some of the ideas started in [33, 45], where the dependence on the sampling time and the evolution towards an invariant measure in the distributions from orbits in chaotic domains were analysed. A clear description of how these spectra characterise the dynamical state in a set of Hamiltonian prototypical cases was a motivation for our work. Many distributions belonging to typical maps have been studied, as, for instance, in [17, 60, 72], but less consideration has been given to conservative systems, where no attractors are found. Indeed, we are interested in the distributions for characterising not only the possible final invariant measure, if so, but also the orbit itself, including the unstable and the open orbits (those that will escape towards infinity). The main goal will be then to generate a set of prototypical distributions for those different orbit behaviours. The distribution of finite-time Lyapunov exponents can be normalised dividing it by the total number of intervals, thus obtaining a probability density function P (χ ) that gives the probability of getting a given value χ between [χ , χ + dχ ]. Hence, the probability of getting a positive χ (Δt) or P+ (and analogously P− ) can be defined as  ∞ P+ = P (χ )dχ . (2.12) 0

Two ways for calculating such distributions are possible. The first one is starting from a given initial condition and integrating the flow equations during an interval Δt, therefore leading to have a χ (Δt) once the integration ends. Then, the integration starts again by taking the ending point of the previous integration as starting initial point of the new cycle. The second way is taking an ensemble of initial conditions on the available phase space (or energy surface). For each initial point, the value χ (Δt) is calculated as before, but without later progression in that orbit (see, for instance, Refs. [33, 45, 64]). When the phase space is largely chaotic and the regular regions small, both distributions coincide, in agreement with the ergodic theorem. If the finite intervals are large enough, the expected shapes are a Gaussian, as the central limit theorem holds and the correlations die out. However, for small finite intervals, the shapes can be different. And when regular orbits appear, shapes can differ substantially. So, we have computed the distribution by selecting an initial point of the orbit, an arbitrary set of orthonormal vectors, integrating then the flow up to a Δt interval and resetting there the calculated effective exponent χ (Δt). This reset means that we restart the integration in this new point with the arbitrary set of orthonormal vectors. Our first approach will be to compute the finite-time exponents using the smallest possible interval lengths. Later on, the interval size will be increased in order to see how the flow modifies the distributions.

2.7 The Harmonic Oscillator

47

Several criteria for choosing a small Δt are found in the literature. The shortest interval that can be used in the case of maps is one iteration of the map. However, since this time interval is a continuous quantity in flows, several approaches are possible. It can be taken very small, although obviously it should not be smaller than the integration step. It has not been completely established yet whether these finitetime Lyapunov exponents distributions are typical or stationary when computed with short intervals Δt. See Ref. [59] for a discussion.

2.7 The Harmonic Oscillator As a very simple initial example, we will briefly analyse the behaviour of the Lyapunov exponents in the harmonic oscillator. We can take it as a representative example of the slow convergence rate of χ towards the asymptotic Lyapunov exponent λ. The equations of the harmonic oscillator are the following: 

x˙ = p p˙ = −ω2 x,

(2.13)

where ω is the frequency of the oscillations. We have selected as arbitrary initial point (−0.893978, 0.316862). We have also fixed ω2 = 0.5. Figure 2.2 plots the values of the finite-time Lyapunov exponents χ (Δt), as the finite interval Δt grows. We can see how the value slowly tends to the expected asymptotic value of λ = 0.0, but we can also see how this evolution is not monotonically decreasing even in this very simple case. Indeed, we see how the χ values are roughly constant for a while when the finite intervals are very small. Fig. 2.2 Evolution of the finite-time Lyapunov exponent χ(Δt) as the finite-time interval Δt sizes increase. For the largest lengths, the values of the finite-time exponent χ(Δt) tend to reach the (asymptotic) Lyapunov exponent λ = 0

48

2 Lyapunov Exponents

The Poincaré section diagram for the harmonic oscillator is very simple, represented by a single plotted dot, with a crossing time between consecutive consequents Tcross = T ∼ 8.8, being T the period of the oscillations. Figure 2.2 also shows how the χ values tend to the final asymptotic good value λ = 0 when the finitetime length Δt is larger than this period Tcross . So, we will analyse in the following sections the importance of this Tcross . For the computation of the finite-time exponents, we will follow the Appendix A. The flow of the harmonic oscillator system is given by  v˙ = Φ(v) =

   p f1 (v) = f2 (v) −ω2 x

(2.14)

By defining the Jacobian of the flow Φ as the matrix J = Dv Φ, containing the differential slopes in every possible direction, we have  ∂f Dv Φ =



1 ∂f1 ∂x ∂y ∂f2 ∂f2 ∂x ∂y .

,

(2.15)

In this specific simple, one-dimensional harmonic system, we have the following Jacobian:   0 1 Dv Φ = , (2.16) −ω2 0 The first issue is that the trace of the Jacobian is zero, as it should be. This reflects the conservative nature of the system. The second important point is that the Jacobian does not depend on the point of the trajectory, and, conversely, it is constant. So, regarding the distribution of values of χ (Δt), the value of χ (Δt) for a given Δt and a fixed initial condition is always constant along the orbit. As a consequence, the distribution of finite-time Lyapunov exponents for each Δt size is a single δ-Dirac, centred around the corresponding fixed finite value χ (Δ).

2.8 The Rössler System Now, we will analyse a more complex system. The selected model consists of two identical, symmetrically diffusively coupled Rössler systems. We wish to describe its behaviour with the help of its global Lyapunov exponents. This system possesses a paradigmatic behaviour in relation to the chaos-hyperchaos transition and the unstable dimension variability phenomenon, which was presented in Refs. [79, 80] in a very similar system. In addition, it is a quite meaningful physical system, as it may represent the selective diffusion of two species through a semipermeable membrane in two continuously stirred tank reactors [47].

2.8 The Rössler System

49

Our system is formed by two coupled Rössler systems. These Rössler systems, as they are named today, were introduced in the 1970s by the German biochemist Otto E. Rössler as prototype equations with the minimum ingredients for continuous-time chaos. Rössler was inspired by the geometry of flows in dimension three and the reinjection principle. Motivated by the search for chemical chaos, chaotic behaviour in chemical kinetics far from equilibrium, he invented a series of systems, the most famous of which is the following: ⎧ ⎨ x˙ = −y − z y˙ = x + ay ⎩ z˙ = b + z(x − c)

(2.17)

The phase space of this system has the minimal dimension, three, to show chaos, and it is formed by simple equations. Notably, two of them are linear, and the system just contains a single quadratic term. However, it shows a complex behaviour that strongly depends on the values of its constant parameters a, b and c. When it generates a chaotic attractor, it has a single lobe, in contrast to the Lorenz attractor, which has two lobes. When we couple two oscillators, each one formed by the above set of equations, the flow of the resulting system is as follows: ⎧ ⎪ x˙1 = −y1 − z1 ⎪ ⎪ ⎪ ⎪ y˙1 = x1 + ay1 ⎪ ⎪ ⎨ z˙1 = b + z1 (x1 − c) + d(z2 − z1 ) ⎪ x˙2 = −y2 − z2 ⎪ ⎪ ⎪ ⎪ y ˙2 = x2 + ay2 ⎪ ⎪ ⎩ z˙2 = b + z2 (x2 − c) + d(z1 − z2 ).

(2.18)

The first three coordinates (x1 , y1 , z1 ) correspond to the first Rössler oscillator. The second three coordinates (x2 , y2 , z2 ) correspond to the other one. The parameter d represents the coupling, which depends on the distance between the z-coordinates of the oscillators. The parameter a is chosen as the control parameter. We have chosen the parameters b = 2.0 and c = 4.0, in order to compare our results with those from Refs. [79, 80]. We have used a simple fourth-order Runge-Kutta method, with fixed time step 0.01 and a fourth-order/fifth-order Runge-KuttaFehlberg variable step size method as integrations schemes, both leading to the same numerical results. Figure 2.3 shows the two oscillators (x1 , y1 ) and (x2 , y2 ) for three different values of the control parameter a and the final attractor of the oscillators. Note only that the (x1 , y1 ) and (x2 , y2 ) coordinates are displayed, being the zcomponent ignored. The plots are built with a total integration time of T = 10,000. The same initial condition is fixed in all cases to be (1, 1, 0, −1, −5, 0), even when the same attractor is found when starting in the neighbourhood of this point. We see in this figure how the behaviour for both oscillators is different as the parameter a changes.

50

2 Lyapunov Exponents

4

2 0 -2 -4 6 -6-4 -2

4

2 0 -2 -4 6 -6-4 -2

4

2 0 -2 -4 -6 6 -4 -2

0

2 X1

0

2 X1

4

6

0 2 X2

4

6

0 2 X2

4

6

Y2

Y1

2 0 -2 -4 -6 -4 -2

0 2 X2

Y2

Y1

2 0 -2 -4 -6-4 -2

Y2

Y1

2 0 -2 -4 -6-4 -2

0

2 X1

Fig. 2.3 Evolution of two coupled Rössler oscillators (x1 , y1 , z1 ) and (x2 , y2 , z2 ) for three different values of the control parameter a. From top to bottom, a = 0.342, a = 0.365 and a = 0.389. These three cases are indicated in Fig. 2.7 as A, B and C. The figures show the values of (x1 , y1 ) and (x2 , y2 ). The coordinates z1 and z2 are not shown, for simplicity. Total integration time is 104 time units. A dot is plotted every 0.1 time units. The coupling strength parameter is fixed as d = 0.25. (Adapted from [71] with permission)

2.8 The Rössler System

51

Now, we will compute how the finite-time exponents χ (Δt) evolve as the intervals Δt grow. Aiming to solve the variational equation, the Jacobian of the flow J = Dv Φ, containing the differential slopes, is the following 6 × 6 matrix: ⎡ ∂f ∂f ∂f ∂f ∂f ∂f ⎤ 1

1

1

1

1

1 ⎢ ∂x ∂f2 ⎢ ∂x ⎢ 1 ⎢ ∂f3 ⎢ 1 Dv Φ = ⎢ ∂x ∂f4 ⎢ ∂x ⎢ ∂f 1 ⎢ 5 ⎣ ∂x1

∂y1 ∂f2 ∂y1 ∂f3 ∂y1 ∂f4 ∂y1 ∂f5 ∂y1 ∂f6 ∂y1

∂z1 ∂f2 ∂z1 ∂f3 ∂z1 ∂f4 ∂z1 ∂f5 ∂z1 ∂f6 ∂z1

∂x2 ∂f2 ∂x2 ∂f3 ∂x2 ∂f4 ∂x2 ∂f5 ∂x2 ∂f6 ∂x2

∂y2 ∂f2 ∂y2 ∂f3 ∂y2 ∂f4 ∂y2 ∂f5 ∂y2 ∂f6 ∂y2

∂f6 ∂x1

1

∂z2 ∂f2 ⎥ ⎥ ∂z2 ⎥ ∂f3 ⎥ ∂z2 ⎥ ∂f4 ⎥ . ⎥ ∂z2 ⎥ ∂f5 ⎥ ∂z2 ⎦ ∂f6 ∂z2

(2.19)

For this specific system, the Jacobian is ⎡ 0 ⎢1 ⎢ ⎢ ⎢z Dv Φ = ⎢ ⎢0 ⎢ ⎣0 0

−1 −1 a 0 0 (x − c − d) 0 0 0 0 0 d

⎤ 0 0 0 ⎥ 0 0 0 ⎥ ⎥ 0 0 d ⎥ ⎥. ⎥ 0 −1 −1 ⎥ ⎦ 1 a 0 r 0 (p − c − d)

(2.20)

The trace of the Jacobian matrix is less than zero, indicating the dissipative nature of the system, and the existence of the attractors is shown in Fig. 2.3. Figure 2.4 shows the evolution of χ (Δt) for the three different values of the control parameter a. Different regimes for three values of a are reflected in the different convergence curves of the global Lyapunov exponents. We can see that the time required for reaching the asymptotic Lyapunov exponent is not the same in every case and depends on the value of a. So, we have seen the time interval required for computing the asymptotic Lyapunov exponent λ, and we can consider χ (Δt = 100,000) ∼ λ. Now, we continue by calculating in detail how this asymptotic Lyapunov exponent depends on the control parameter a. We recall here that for N -dimensional flows, it is possible to have N global Lyapunov exponents when a distortion tensor formed from N perturbation vectors evolves according to the flow equations. When considering a single Rössler system, the first exponent can be just zero or positive, the second exponent is zero and the third value negative, assuring the boundness of the solution. When two oscillators are coupled, a richer set of values is present. The chaotic regime is defined when only one global Lyapunov exponent λ is positive. When there are more than one positive Lyapunov exponents, the regime is sometimes labeled as hyperchaotic regime, or, less frequently, high-dimensional chaos. We should mention here that hyperchaos should not confused with hetero-chaos [62] that was discussed in the previous chapter. As a matter of fact, a hetero-chaotic attractor can have one or more positive Lyapunov exponents, and consequently it

52

2 Lyapunov Exponents

0.05

a=0.342

χ

0 -0.05 -0.1 10

2

Δt

10

3

10

4

0.05

χ

0 -0.05 a=0.365

-0.1 10

2

Δt

10

3

10

4

0.2

χ

0.1 0 a=0.389

-0.1 -0.2 10

2

Δt

10

3

10

4

Fig. 2.4 Evolution of the system of two coupled Rössler oscillators (x1 , y1 , z1 ) and (x2 , y2 , z2 ). We see the convergence towards the global Lyapunov exponent of the four largest finite-time exponents from the total six available exponents. The remaining two exponents are always negative and do not provide additional information, so they are not displayed. The upper row corresponds to a = 0.342, Point A of Fig. 2.7. The middle row to a = 0.365, Point B of Fig. 2.7. The bottom one a = 0.389, Point C of Fig. 2.7. (Adapted from [71] with permission)

2.8 The Rössler System

53

λ ~ χ ( Δ t=100000)

(I) -Asymptotic Exponents d=0.25

0 -0.1 First Exponent Second Exponent Third Exponent Fourth Exponent

-0.2 -0.3 0.34

0.35

0.36

a

0.37

0.38

0.39

Fig. 2.5 Lyapunov bifurcation diagrams or diagram showing the variation of λ with the variation of the oscillator parameter a and fixed coupling strength d. Hyperchaos is born at around a ∼ 0.367. Only the four largest exponents from the total six are displayed. The remaining are always negative and are not shown. Asymptotic Lyapunov exponents values λ are calculated by computing χ(Δt = 100,000). (Taken from [71] with permission)

need not be hyperchaotic. Furthermore, all periodic orbits of a hyperchaotic attractor might have the same number of unstable directions, in which case it would not be hetero-chaotic. The behaviour of the asymptotic exponents and raising of hyperchaotic transition, as parameters a and d are varied, are shown in Figs. 2.5 and 2.6. In Fig. 2.5, we fix the coupling parameter d = 0.25 and vary a. Below a = 0.358, all exponents are either nearly zero or below zero. Above this number, we have the chaotic regime, where there is at least one exponent larger than zero. From a = 0.368, there are at least two exponents, and the hyperchaotic regime starts. Note also that there is a window around a = 0.381 where both exponents decrease towards zero. In Fig. 2.6, we fix a = 0.358 and vary d. For almost every coupling strength d, the system is hyperchaotic. However, there is a small interval around d ∼ 0.174, where only the first global Lyapunov exponent remains positive. This shows that the strength of the divergence is not always decreasing (or increasing) with the coupling strength. These different system regimes are displayed in Fig. 2.7, which shows the areas with no positive exponents (no chaos), just one positive exponent (chaos) and more than one positive exponent (hyperchaos). The hyperchaos arises in a complex way depending on parameters a and d. There is no general trend of the hyperchaos with the coupling parameter d, as chaos sometimes increases and sometimes decreases with this parameter.

54

2 Lyapunov Exponents

(II) - Asymptotic Exponents a=0.385

t=100000)

0.06 0.04

First Exponent Second Exponent Third Exponent Fourth Exponent

0.02

λ~

0 -0.02 -0.04 0.1

0.15

0.2

d

0.25

Fig. 2.6 Lyapunov bifurcation diagram or diagram showing the variation of λ with coupling strength d and fixed parameter a. There is a drop in the hyperchaotic regime at d ∼ 0.174. Only the four largest exponents from the total six are displayed. The remaining are always negative and are not shown. Asymptotic Lyapunov exponents values λ are calculated by computing χ(Δt = 100,000). (Taken from [71] with permission)

Fig. 2.7 Hyperchaoticity chart. The number of positive global Lyapunov exponents varies with the Rössler parameter a and the coupling strength parameter d. Dark regions (Black and dark red) mean 0 positive exponents (dark red meaning that the convergence is slower). Mid-bright regions (red and dark pink) mean only 1 positive exponent (pink meaning slower convergence). Brighter regions (clear pink and above) mean 2 positive exponents. White means 3. Slower convergence means that even with Δt = 100,000 the value has not reached the zero limiting case within machine precision, but it is already smaller than 10−4 . Points A, B, and C are the three plots of Fig. 2.4. Slicing horizontally at d = 0.25 corresponds to Fig. 2.5. Slicing vertically at a = 0.385 corresponds to Fig. 2.6. (Taken from [71] with permission)

2.9 The Hénon-Heiles System

55

2.9 The Hénon-Heiles System In the previous section, we have seen the evolution of the finite-time Lyapunov exponents towards the asymptotic value and how the spectrum of the finite-time Lyapunov exponents can trace the different regimes of the system. Now we are interested in analysing how the distributions calculated with the smallest available Δt interval characterise a given system. Even when some variability is expected when taken such small intervals, they can still serve for tracing the system. In fact, a way to determine the structure of a Lyapunov spectrum locally, that is, within some small (in principle infinitesimal) time interval, is shown in Ref. [48]. Taking the interval size as small as possible, the correlation of each value to the following one will depend only on the local orbit behaviour. We will try to find out whether the local information is enough for obtaining valid results or we should increase such interval. When the interval size is increased, it can be made equal to any time interval with physical meaning, such as the characteristic time of the system or the crossing time of the orbit with a given Poincaré section. Moreover, instead of selecting a fixed Δt, it is possible to choose a variable sampling interval, as in [68], where it is taken to be equal to the interval where the χ (Δt) reaches a temporary limit. In any case, one should keep in mind that when the size of Δt is increased, the local details are washed out. And in the limit, χ (Δt → ∞) tends to the asymptotic Lyapunov exponent, and the distribution tends to be a Dirac-δ centered at this asymptotic value. We will review the behaviour of the finite-time Lyapunov exponents in the Hénon-Heiles system. This Hamiltonian is one of the first examples used to show how very simple systems might possess highly complicated dynamics. The Hénon-Heiles system can be seen as an approximation up to cubic terms to the Toda chain, an integrable regular system typically modelling three atoms in a ring, showing how an approximation to a regular system can exhibit chaos. However, it is more frequently seen as a simple gravitational potential describing an axisymmetrical galaxy, because that is how it was first described. Since then, this model has been used as a paradigm in Hamiltonian nonlinear dynamics, as a system with two degrees-of-freedom that, in spite of its simplicity, can show a very rich fractal structure in phase space when the system is open, that is, when the orbits are unbounded for energies above a given threshold energy. Moreover, it also shows a complex behaviour in the closed case, when the energy value is below such a threshold and the orbits are bounded. That is, according to the energy of the orbit, which is related to the initial condition, different dynamical behaviours may appear and even paradigmatic examples of the pseudodeterministic models that only yield to relevant information over trajectories of reasonable length due to the unstable dimension variability [39, 41] can be found. The Hénon-Heiles system was first studied by the astronomers Michel Hénon and Carl Heiles in 1964 [30], searching if there existed two or three constants of motion in the galactic dynamics. A system with a galactic potential that is axisymmetrical and time-independent possesses a 6D phase space. As there are six variables, we can find five independent conservative integrals, some of them being isolating and

56

2 Lyapunov Exponents

other nonisolating (which are physically meaningless). The question that Hénon and Heiles tried to answer is which part of this 6D phase space is filled by the trajectories of a star after very long times. By that time, it was obvious that both the total energy ET and the z-component of the angular momentum Lz were isolating integrals, while another two were usually nonisolating. Therefore, the real target became to find a third conserved quantity. In order to solve this problem, Hénon and Heiles proposed a simple potential. Their result was that a third isolating integral may be found for only some few initial conditions. The Hénon-Heiles system is one of the so-called meridional potentials systems. Meridional plane potentials are those of the form V (x, y) = V (R, z), being R and z the cylindrical coordinates, corresponding to an axisymmetric galaxy [8]. These are relatively simple potentials that can show some of the complex behaviours which are found in more realistic galactic-type potentials. The motion in the meridional plane can be described by an effective potential: Veff (R, z) = V (R, z) +

Lz 2 , 2R 2

(2.21)

where R and z are the cylindrical coordinates. For each orbit, the energy E = E(x, y, vx , vy ) is an integral of motion. Once E is fixed, only three of the four coordinates are independent and define the initial condition for the integrator. If the energy E and the z-component of the angular momentum Lz are the only two isolating integrals, an orbit would visit all points within the zero-velocity curve, defined as E = Veff . Sometimes, there are limiting surfaces that forbid the orbit to fill this volume, implying the existence of a third integral of motion, whose form cannot be explicitly written. In this case, the particle is confined to a three-torus. Alternatively, there are some axisymmetric potentials where the orbits can indeed fill the meridional plane. These are irregular (or ergodic) orbits, which are only limited by two integrals of motion. The Hénon-Heiles Hamiltonian contains two, properly weighted, coupling terms, x 2 y and y 3 , leading to a Hamiltonian with a 2π/3 rotation symmetry and three exits in the potential well. It is written as H =

1 2 1 2 (p + py2 ) + (x 2 + y 2 + 2x 2 y − y 3 ). 2 x 2 3

(2.22)

So, as a consequence, ⎧ x˙ = p ⎪ ⎪ ⎨ y˙ = q ⎪ p˙ = −x − 2.0xy ⎪ ⎩ q˙ = −y − x 2 + y 2

(2.23)

2.9 The Hénon-Heiles System

57

This Hamiltonian has been extensively studied for the range of energy values below the escape energy, where orbits are bounded and a variety of chaotic and periodic motions exist. On the other hand, if the energy is higher than this threshold value, the escape energy Ee , the trajectories may leave from the bounded region and escape to infinity through three different exits. The exact exit will depend on the initial condition, and the sets of initial conditions that lead to every individual exit have a fairly complex structure, as we will see in another chapter; see Sect. 4.1.1. As we are dealing with a two degrees-of-freedom system, four Lyapunov exponents will exist. However, since it is a conservative Hamiltonian system, λi = −λ5−i for (i = 1, . . . , 4) and only two different values of λ are independent. One of them will be tangent to the trajectory, parallel to the velocity field, and the other one transverse to it. The tangent one is nonrelevant as it tends to zero in the limiting case. The distribution of the finite-time Lyapunov exponents can be carried out by using standard methods. The initial ellipse axes are chosen arbitrarily. A sixthorder Runge-Kutta integrator with a fixed time step equal to 10−2 provides enough accuracy for our purposes. As check on the goodness of the calculations, one can see how the property λi = −λ5−i holds along the integration. We have selected the Poincaré section with the plane x = 0 as visualisation tool. We have selected this plane as each orbit repeatedly intersects it because the symmetry of the system. The crossing time Tcross will be defined as the time between successive section crosses, independently of the sign of vx when the plane x = 0 is crossed. We have selected four initial conditions leading to four prototypical behaviours in the Hénon-Heiles system. These orbits are listed in Table 2.2. The distributions of the first finite-time exponent χ (Δt) will be calculated following the same techniques described in the previous sections. The Jacobian of the flow J = Dv Φ, containing the differential slopes, is the following 4 × 4 matrix, Table 2.2 Selected orbits for the Hénon-Heiles system Orbit H4 H1 H5

Description Regular, close to period-5 orbit Weakly chaotic, cycle orbit Weakly chaotic, ergodic orbit

Initial condition for given Energy x = 0.000000 y = −0.031900 vx = 0.307044 E = 1/8 x = 0.000000 y = −0.119400 vx = 0.388937 E = 1/12 x = 0.000000 y = −0.238800 vx = 0.426750 E = 1/8

λ 0.0

Tcross 6.2

0.015

6.8

0.044

6.8

λ is the asymptotic standard Lyapunov exponent. The notion weak or strong chaos is associated to the relatively smaller or larger value of λ. Tcross is the Poincaré section crossing time corresponding to crosses with plane x = 0, independently of the sign of vx

58

2 Lyapunov Exponents

⎡ ∂f Dv Φ =

1

∂x ⎢ ∂f ⎢ ∂x2 ⎢ ⎢ ∂f3 ⎣ ∂x ∂f4 ∂x

∂f1 ∂y ∂f2 ∂y ∂f3 ∂y ∂f4 ∂y

∂f1 ∂p ∂f2 ∂p ∂f3 ∂p ∂f4 ∂p

⎤

∂f1 ∂q ∂f2 ⎥ ⎥ ∂q ⎥ ∂f3 ⎥ . ∂q ⎦ ∂f4 ∂q

(2.24)

For this specific system, the Jacobian is then as follows: ⎡

0 0 ⎢ 0 0 Dv Φ = ⎢ ⎣(−1.0 − 2.0y) (−2.0x) (−2.0x) (−1.0 + 2.0y)

1.0 0 0 0

⎤ 0 1.0⎥ ⎥. 0⎦ 0

(2.25)

We will start our analysis with a regular, quasi-periodic orbit, found in the HénonHeiles system for the energy E = 1/8. In a periodic motion, as the harmonic oscillator, plotting the distribution of χ (Δt) values leads to a δ-Dirac peaked distribution, as the χ (Δt) values repeat periodically and are constant. In cases where the Jacobian depends on the point but the trajectory is still periodic, we will obtain Poincaré sections formed by closed curves, corresponding to the section of a torus. Regarding the shapes of the finitetime distributions, one typically obtain two-peaked distributions, corresponding to the regular motion. When the interval size increases, χ (Δt >>) → 0, and the peaks tend to merge and shift towards zero. So, we start our analysis with the regular, quasi-periodic orbit, labeled as H4 in Table 2.2. The initial conditions are x = 0.000000, y = −0.031900 and vx = 0.307044. Its Poincaré section is depicted in Fig. 2.8a, and it shows a set of ten islands, which is associated to a period-5 orbit. The five islands on the left are plotted when the x = 0 plane is crossed from the x < 0 subspace towards x > 0 and the other five on the right when returning to the x < 0 subspace. The distribution of finite-time Lyapunov exponents for an interval Δt of 0.02 and total integration time of 104 time units is the solid line in the lower panel of Fig. 2.8b. It shows ten peaks, five centred around negative values and the other five centred around positive values. In the inset panel, it has been plotted the evolution of the short-time Lyapunov exponent with time (as the integrated number of intervals Δt increases). Since it is a quasi-periodic orbit, it can be observed quasi-periodic oscillations, with five oscillations per larger period. These oscillations in χ (Δt) are shown in the smaller inset panel. Each oscillation is associated to an island in the Poincaré section, thus to a peak in the distribution. Inside each period, we can count five oscillations, so five peaks are obtained in the distribution. Between each peak, a range of values is obtained, thus leading to the spectrum of values between the main peaks. As we are dealing with an orbit near a 5-period one, only 10 peaks can be obtained. This means that there are arbitrarily finite intervals for which the orbit, on average, is repelling in one of the dimensions and other intervals for which is attracting in the same dimension. The shape of the distribution is independent of the initial condition

2.9 The Hénon-Heiles System

59

0.4

y’

0.2

0

-0.2

-0.4 -0.4

-0.2

0

y

0.2

0.4

0.8

0.6

(a)

P(x)

0.01

0.005

0 -0.1

0

0.1

0

0.01

0.02

0.03

0.04

0.4 0.2 0 0.01 -0.2 -0.4 3000

6000

P(x)

0

0.005

0

-0.4

-0.2

0 Short Time Exponent (X)

0.2

0.4

(b) Fig. 2.8 (a) Poincaré section y − y˙ with plane x = 0 of a quasi-periodic orbit of energy E = 1/8, associated to a period-5 orbit. The crossing time is Tcross ∼ 6.2 time units. Each time a point crosses the section, a different island is crossed, and the total time before repeating an island is roughly 31.5 time units. (b) The lower and larger panel shows the distribution of finite-time Lyapunov exponents, showing ten peaks both in positive and negative values, when Δt = 0.02 and the total integration time is 104 time units. The green dashed probability distribution corresponds to Δt = 10 and an integration time 106 . This case is zoomed in the upper leftmost panel. The blue dashed-dotted line represents the probability distribution when Δt = 100, and the integration time is 106 units. This case is zoomed in the upper rightmost panel. (Taken from [69] with permission)

60

2 Lyapunov Exponents

along the orbit, and longer integrations do not lead to different shapes. Here, long integration means integration times longer than 600 time units or period to plot the 5 islands. We will call to this long period “circuit time”. For times longer than this circuit time, the shapes remain similar because we are just adding more circuits to the already sampled periods. When the initial condition is moved far away from the periodic orbit, the distribution broadens but remains with a similar morphology. When the interval size increases, the range of values around which the peaks are centred is reduced, and it is shifted towards positive values, as shown in the lower panel of Fig. 2.8b as dotted lines and zoomed in the upper leftmost panel. When Δt = 10, a multipeaked distribution is still observed, since this value is larger than the crossing time but still smaller than the total circuit time, which is roughly 32 time units. This case is found as dashed-dotted in the upper rightmost panel. For larger size of time intervals, the peaks begin to merge as Δt begins to be equal to that circuit time. This behaviour is different for orbits showing some chaoticity. One example appears in Fig. 2.9a, with initial energy E = 1/12. This is the orbit labeled as H1 in Table 2.2. The initial conditions are x = 0.000000, y = −0.119400andvx = 0.388937. The solid line in Fig. 2.9b shows the corresponding distribution with an integration time of 20,000 units, and Δt = 0.02. The whole available phase space is traced, and longer integrations lead basically to the same shape. This shape does not correspond to a “typical” chaotic state, where the central limit theorem holds for a number of averaged quantities, including standard Lyapunov exponents (see [27, 53]), and the distributions can be fitted by a Gaussian, since the correlations die out. Neither does it to an intermittent system, where the shape might be a combination of a normal density and a stretched exponential tail, due to the long correlation persistence. As we are analysing the evolution or stationarity of the distributions, it is important to keep in mind the difference between stationarity, due to the dynamics at certain time, and ergodicity, time average property of the trajectories. In a nonergodic orbit, the trajectory does not cover the whole hypersurface of constant energy, so two different initial conditions cover different parts of the energy surface leading to different time averages even for times tending to infinity. In such systems there is not a unique equilibrium state but different ones depending on the starting point. Conversely, in an ergodic system, it can be reached a unique equilibrium state. And generic ensembles of initial conditions will evolve towards a given distribution, time-independent or with little variability on long time scales. One key point is the time involved in such evolution towards the final state. If the physical timescales are relevant and that time is too long for being realistic, those ensembles will not be able to be used as a valid skeleton for the observed system behaviour. So, when computing distributions from a set of initial conditions, we need to be sure they are in the same domain of the Poincaré section. If that is the case, we get again the solid histogram of Fig. 2.9b. On the other hand, the stationarity of a distribution can be defined when the statistical parameters do not change with time, and this depends on the variable dynamics along the given orbit. When the probability distribution from a single orbit is computed, the morphology may

2.9 The Hénon-Heiles System

61

0.4

y’

0.2

0

-0.2

-0.4 -0.4

-0.2

0

y

0.2

0.4

0.6

0.8

(a)

P(X)

X

0.01 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0

5e+05 N. of finite intervals

1e+06

0.005

0 -0.4

-0.3

-0.2

-0.1 0 Short Time Exponent (X)

0.1

0.2

(b) Fig. 2.9 (a) Poincaré section y − y˙ with plane x = 0 of an orbit of energy E = 1/12. The crossing time is Tcross ∼ 6.8 time units. (b) The solid line shows the probability distribution of finite-time Lyapunov exponents formed with an integration of 20,000 time units when Δt = 0.02. The dotted and dashed lines represent the probability distributions corresponding to partial 1000 time units integrations started at arbitrary points of the same orbit. These partial integrations reflects some of the different transients of Table 2.3. The inset shows the oscillating behaviour of χ(Δt) as the integration takes place. (Taken from [69] with permission)

62

2 Lyapunov Exponents

Table 2.3 Several distribution behaviours in the case E = 1/12 for the smallest interval size Δt = 0.02. The statistics are for integrations of 103 time units starting at t0 t0 0 103 2 · 103 3 · 103 12 · 103 14 · 103

Mean −0.04402 −0.01337 −0.01318 −0.01318 −0.04406 −0.016346

Std. Dev. 0.18489 0.16457 0.16437 0.16438 0.18492 0.152195

Median −0.44017 −0.01337 −0.01318 −0.01318 −0.04406 −0.016346

P+ (t0 ) 0.43455 0.67674 0.66708 0.67882 0.47806 0.700080

depend on the initial point, when the total integration time is not large enough, as several transients of different behaviour are found (see Ref. [82]). In order to catch the behaviour of the transient periods, we have computed distributions formed after integrating just 103 time units (150-times the crossing time), which are described in Table 2.3. Three of them appear in Fig. 2.9b. The characteristic time on which the orbit forgets its previous degree of instability is small (low correlation time), as they are quite different. The standard deviation of the distributions σ gives a measure of the degree in which χ deviates from the mean, being a measure of the stability or variability of the values of χ along the orbit. The probability of getting a positive value for a finite-time Lyapunov exponent is returned by Eq. 2.12, and P+ takes different values ranging from 0.4 up to 0.7 quite randomly, which indicates different behaviours, ordered at some stages, chaotic in others, as reflected in the shape of the distributions. For instance, the first transient shows two well-separated peaks, like a quasi-periodic orbit (dotted line), while the third transient shows a multipeaked distribution (dotted-dashed line). When the time evolution of the finite-time Lyapunov distributions (and the time evolution of χ (Δt) itself, as shown in the smaller panel) is compared with the way the consequents of the Poincaré section fill the available phase space, we see how each distribution corresponds to a different way of tracing the Poincaré section. If we change the interval size Δt by a small integer factor, our result is only a rescaling of the spectrum, as was shown in Ref. [65]. However, when it is increased up to, say, Δt = 1, which is still smaller than the averaged crossing time, a different multipeaked shape is obtained, as shows the solid line in the lower panel of Fig. 2.10. The local details are washed out as the interval size is larger than the crossing time, so with a Δt = 10 (dotted line), the shape is again different. This distribution is zoomed in the upper panel and two smoth peaks well fitted by Gaussians, the main one centered around positive values are observed. For even larger values of Δt = 100, a single peak Gaussian distribution is found, as plotted as solid in the upper panel of Fig. 2.10, since the central limit theorem begins to hold. Finally, for much larger values of Δt = 100, the distributions collapse to δfunctions centred around the asymptotic Lyapunov value. In addition, the chaoticity indicators vary with the interval size. The values in Table 2.4 are calculated as in Table 2.3, so here it appears the statistics for the transients with Δt = 1.

2.9 The Hénon-Heiles System

63

0.01

P(X)

0.008 0.006 0.004 0.002 0 -0.1 0.006

0

0.1

0.005

P(X)

0.004 0.003 0.002 0.001 0

-0.2

-0.1 0 0.1 0.2 Short Time Exponent (X)

0.3

0.4

Fig. 2.10 The distribution of finite-time Lyapunov exponents in the case E = 1/12 formed with an integration of 106 time units when Δt = 1 is plotted as solid trace in the lower panel. The same when Δt = 10 appears in dashed line, and is zoomed in the upper panel. In this later one, is also traced the distribution when Δt = 100. (Taken from [69] with permission)

The mean value calculated with the larger interval on each transient is different to the calculated with the smallest interval. Moreover, the values of P+ are larger, and for even larger interval sizes, the transients may vanish. However, it is remarkable that the evolution of P+ , which is an indicator of the local chaoticity, is similar in both cases. Table 2.5 shows how the total integration time for a given interval size is correlated with these indicators, showing that for the smallest interval, we obtain similar results. An explanation is that by integrating up to 2 · 104 time units, we have already passed through all possible values of the oscillating finite-time Lyapunov exponent, so even increasing the total integration time up to 2 · 105 time units, the spectrum is basically the same. For larger intervals, the statistics is poorer, as the total number of intervals taken into account is smaller, but the same reasoning can be done. When Δt = 1, we are still getting almost the same pattern in the oscillations with 2 · 104 time units or 2·105 time units, so the values are still quite similar. But with Δt = 10, the values are slightly different, as the pattern of the oscillations of the finite-time Lyapunov exponent is also slightly different.

64

2 Lyapunov Exponents

Table 2.4 Several distribution behaviours in the case E = 1/12 for interval size Δt = 1. The statistics are for integrations of 103 time units starting at t0

t0 0 103 2 · 103 3 · 103 12 · 103 14 · 103

Mean 0.07362 0.09552 0.09293 0.09117 0.06484 0.08573

Std. Dev. 0.17391 0.17119 0.17478 0.17415 0.16613 0.14284

Median 0.07362 0.09552 0.09293 0.09117 0.06484 0.08573

P+ (t0 ) 0.46000 0.74000 0.72000 0.73000 0.51000 0.75000

Table 2.5 Sensitivity of the statistics of the finite-time Lyapunov distributions in the case E = 1/12 for several integration time and interval sizes t(total time) 2 · 104 2 · 105 2 · 104 2 · 105 2 · 104 2 · 105

Δt (time) 0.02 0.02 1 1 10 10

Mean −0.04403 −0.04407 0.08553 0.08454 0.03215 0.02509

Std. Dev. 0.18490 0.18492 0.17873 0.18004 0.06258 0.06671

Median −0.04403 −0.04407 0.08553 0.08454 0.03215 0.02511

P+ (t0 ) 0.64226 0.65772 0.69000 0.71060 0.90400 0.89565

Finally, the characterisation of the distributions corresponding to chaotic orbits is discussed. We take an orbit with an initial energy E = 1/8 that almost completely fills the available phase space, as shown by the Poincaré section in Fig. 2.11a. This is the orbit labeled as H5 in Table 2.2 The initial conditions are x = 0.000000, y = −0.238800 and vx = 0.426750. The corresponding distribution is plotted as a solid line in Fig. 2.11b. The inset shows again the oscillations of χ (Δt) as the integration takes place. The same probability distribution is obtained by integrating along a single initial condition or an ensemble of initial conditions, due to the ergodicity of the system. The shape reminds the one described for attractors in [39, 60], although the tail of the peak centered around positive values extends through negative values quite smoothly, instead of showing an exponential tail. Two different transients of 103 time units are plotted as dotted and dashed lines in Fig. 2.11b. We also see that the sticky orbits, those that remain near a regular island for a long time, tend to have smaller exponents than the nonsticky orbits. During the sticky periods, when the orbit appears next to a quasi-periodic orbit torus, the distribution is clearly similar to a quasi-periodic case. However, in the chaotic regime, the peaks are broadened. With larger intervals (Δt = 10) and integration times (106 time units), an almost Gaussian-shaped distribution is obtained, centred around a positive value. This shows a morphology different from the E = 1/12 case that did not reach such Gaussian form even when Δt = 10, meaning a different dynamics, which is also manifested by the time the distribution takes to its final state. Such different morphology can be seen by comparing the solid lines of Figs. 2.9b and 2.11b. In the later case, the peak is no so clear, and the distribution is smoother, indicating that there is no larger probability of getting a value over another one. In the prior case, there is a clear peak, indicating that there is high probability of getting

2.9 The Hénon-Heiles System

65

0.4

y’

0.2

0

-0.2

-0.4

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y

(a) 0.008 0.4

X

0.2

0.006

0 -0.2 -0.4

P(X)

0

5e+05 N. of finite intervals

1e+06

0.004

0.002

0

-0.4

-0.2

0 Short Time Exponent (X)

0.2

0.4

(b) Fig. 2.11 (a) Poincaré section y − y˙ with plane x = 0 of an orbit of energy E = 1/8. The crossing time is Tcross ∼ 6.8 time units. (b) Probability distribution of finite-time Lyapunov exponents. The solid line corresponds to an integration of 20,000 time units when Δt = 0.02. The dotted and dashed ones to partial integrations of 103 time units. The double peaked one corresponds to a sticky period. The inset shows the oscillating behaviour of χ(Δt) as the integration takes place. (Taken from [69] with permission)

66

2 Lyapunov Exponents

the range of values on which the peak is constructed. So the later case indicates that there is more “chaoticity” in the sense that there are no privileged values, as in the E = 1/12 case, so there is a larger ergodicity, in the sense that the orbit is able to reach with the same probability all the available phase space. However, it should be taken into account that during certain transient periods, the behaviour is equivalent to ordered motions, as during the sticky transients (double-peaked distributions).

2.10 Concluding Remarks We have seen that, in addition to the standard calculation of the asymptotic Lyapunov exponent, we can extract more information about the dynamical system by calculating the distributions of finite-time Lyapunov exponents. We have analysed the information provided by these distributions of finite-time Lyapunov exponents. Several prototypical distribution morphologies have been plotted for different energy values. Shapes well differentiated depending on the motion type, the interval size and the integration time have been found. Our calculations have focused on the use of the smallest interval size, searching for the stationarity or evolution of the distributions. It has been observed that they characterise the motion in the different possible cases. The overall shape depends on the local orbit behaviour, as the exponents can be considered specific of a certain local flow. We have detailed the results of our analysis on a Hamiltonian system. Here, the chaotic orbits are ergodic, and the results from generating the distributions from an adequate ensemble or from a single orbit are equivalent. But we have also analysed the distributions of non-ergodic orbits. The results obtained with this approach should be valid for orbits both in conservative or non-conservative systems. The previous discussion shows some implications when the physical meaning of the system is taken into account. As the long integrations required for computing the asymptotic Lyapunov exponents may have no meaning, as, for instance, in a galactic system, since the universe evolves in a shorter time, it is reasonable to use smaller integrations. Furthermore, the smallest interval sizes can be used since they characterise the local behaviour.

References 1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer, New-York (1996) 2. Anteneodo, C.: Statistics of finite-time Lyapunov exponents in the Ulam map. Phys. Rev. E 69, 016207 (2004) 3. Araujo, T., Mendes, R.V., Seixas, J.: A dynamical characterization of the small world phase. Phys. Lett. A 319, 285 (2003)

References

67

4. Aurell, E., Boffeta, G., Crisanti, A., Paladin, G., Vulpiani, A.: Predictability in the large: an extension of the concept of Lyapunov exponent. J. Phys. A Math. Gen. 30, 1 (1997) 5. Badii, R., Heinzelmann, K., Meier, P.F., Politi, A.: Correlation functions and generalized Lyapunov exponents. Phys. Rev. A 37, 1323 (1988) 6. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Meccanica 9, 20 (1980) 7. Benzi, R., Parisi, G., Vulpiani, A.: Characterisation of intermittency in chaotic systems. J. Phys. A 18, 2157 (1985) 8. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987) 9. Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367 (2002) 10. Buizza, R., Palmer, T.: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci. 52, 1434 (1995) 11. Carpintero, D.D., Aguilar, L.A.: Orbit classification in arbitrary 2D and 3D potentials. Mon. Not. R. Astron. Soc. 298, 21 (1998) 12. Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., Whelan, N., Wirzba, A.: Chaos: Classical and Quantum. ChaosBook.org, Niels Bohr Institute, Copenhagen (2016) 13. Contopoulos, G., Voglis, N.: A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophys. 317, 317 (1997) 14. Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374 (1995) 15. Crisanti, A., Paladin, G., Vulpiani, A.: Product of Random Matrices. Springer Series in Solid State Sciences. Springer, Berlin (1993) 16. Custodio, M.S., Manchein, C., Beims, M.W.: Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems. Chaos 22, 026112 (2012) 17. Diakonos, F.K., Pingel, D., Schmelcher, P.: Analyzing Lyapunov spectra of chaotic dynamical systems. Phys. Rev. E 62, 4413 (2000) 18. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985) 19. Ershov, S.V., Potapov, A.B.: On the nature of nonchaotic turbulence. Phys. Lett. A 167, 60 (1992) 20. Ershov, S.V., Potapov, A.B.: On the concept of stationary Lyapunov basis. Phys. D 118, 167 (1998) 21. Finn, J.M., Hanson, J.D., Kan, I., Ott, E.: Steady fast dynamo flows. Phys. Fluids B 3, 1250 (1991) 22. Froeschlé, C., Lohinger, E.: Generalized Lyappunov characteristic indicators and corresponding Kolmogorov like entrophy of the standard mapping. Celest. Mech. Dyn. Astron. 56, 307 (1993) 23. Froyland, G., Huls, T., Morriss, G.P., Watson, T.M.: Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study. Phys. D 247, 18–39 (2013) 24. Fujisaka, H.: Statistical dynamics generated by fluctuations of local Lyapunov exponents. Prog. Theor. Phys. 70, 1264 (1983) 25. Gao, J.B., Hu, J., Tung, W.W., Cao, Y.H.: Distinguishing chaos from noise by scale-dependent Lyapunov exponents. Phys. Rev. E 74, 066204 (2006) 26. Ginelli, F., Poggi, P., Turchi, A., Chate, H., Livi, R., Politi, A.: Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett. 99, 130601 (2007) 27. Grassberger, P., Badii, R., Politi, A.: Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988) 28. Haller, G.: Distinguished material surfaces and coherent structures in 3D fluid flows. Phys. D 149, 248 (2001) 29. Heggie, D.C.: Chaos in the N-body problem of stellar dynamics. In: Roy, A.E. (ed.) Predictability, Stability and Chaos in N-Body Dynamical Systems. Plenum Press, New York (1991)

68

2 Lyapunov Exponents

30. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73 (1964) 31. Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2007) 32. Kalnay, E., Corazza, M., Cai, M.: Are bred vectors the same as Lyapunov vectors? EGS XXVII General Assembly, Nice, 21–26 Apr 2002, abstract 6820 33. Kandrup, H.E., Mahon, M.E.: Short times characterisations of stochasticity in nonintegrable galactic potentials. Astron. Astrophys. 290, 762 (1994) 34. Kapitakinak, T.: Generating strange nonchaotic trajectories. Phys. Rev. E 47, 1408 (1993) 35. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In: Peitgen, H.O., Walter, H.O. (eds.) Functional Differential Equations and Approximations of Fixed Points. Lecture Notes in Mathematics, vol. 730. Springer, Berlin (1979) 36. Klages, R.: Weak chaos, infinite ergodic theory, and anomalous dynamics. In: Leoncini, X., Leonetti, M. (eds.) From Hamiltonian Chaos to Complex Systems. Springer, Berlin (2013) 37. Klein, M., Baier, G.: Hierarchies of dynamical systems. In: Baier, G., Klein, M. (eds.) A Chaotic Hierarchy. World Scientific, Singapore (1991) 38. Kocarev, L., Szcepanski, J.: Finite-space Lyapunov exponents and pseudochaos. Phys. Rev. Lett. 93, 234101 (2004) 39. Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A.: Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Phys. D 109, 81 (1997) 40. Kuptsov, P.V., Parlitz, U.: Theory and computation of covariant Lyapunov vectors. J. Nonlin. Sci. 22, 727 (2012) 41. Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999) 42. Legras, B., Vautard, R.: A guide to Lyapunov vectors. In: Palmer, T. (ed.) Predictability Seminar Proceedings, ECWF Seminar, vol. 1, pp. 135–146. European Centre for MediumRange Weather Forecasts, Reading (1996) 43. Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor and Francis, London (1992) (English translation from the French 1907, in turn from the Russian 1892) 44. Lepri, S., Politi, A., Torcini, A.: Chronotropic Lyapunov analysis: (I) a comprehensive characterization of 1D systems. J. Stat. Phys. 82, 1429 (1996) 45. Mahon, M.E., Abernathy, R.A., Bradley, B.O., Kandrup, H.E.: Transient ensemble dynamics in time-independent galactic potentials. Mon. Not. R. Astron. Soc. 275, 443 (1995) 46. Mitchell, L., Gottwald, G.A.: On finite size Lyapunov exponents in multiscale systems. Chaos 22, 23115 (2012) 47. Mosekilde, E.: Topics in nonlinear dynamics: applications to physics, biology and economic. World Scientific Publishing, Singapore (1996) 48. Moser, H.R., Meier, P.F.: The structure of a Lyapunov spectrum can be determined locally. Phys. Let. A 263, 167 (1999) 49. Motter, A.E.: Relativistic chaos is coordinate invariant. Phys. Rev. Lett. 91, 23 (2003) 50. Mulansky, M., Ahnert, K., Pikovsky, A., Shepelyansky, D.L.: Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems. J. Stat. Phys. 145, 1256 (2011) 51. Okushima, T.: New method for computing finite-time Lyapuunov exponents. Phys. Rev. Lett. 91, 25 (2003) 52. Oseledec, V.I.: A multiplicative ergodic theorem. Moscow Math. Soc. 19, 197 (1968) 53. Ott, E.: Chaos in Dynamical Systems, 2nd ed. Cambridge University Press, Cambridge (2002) 54. Ott, W., Yorke, J.A.: When Lyapunov exponents fail to exist. Phys. Rev. E 78, 056203 (2008) 55. Parisi, G., Vulpiani, A.: Scaling law for the maximal Lyapunov characteristic exponent of infinite product of random matrices. J. Phys. A 19, L45 (1986) 56. Patsis, P.A., Efthymiopoulos, C., Contopoulos, G., Voglis, N.: Dynamical spectra of barred galaxies. Astron. Astrophys. 326, 493 (1997) 57. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821 (1990) 58. Pesin, Y.: Dimension Theory in Dynamical Systems, Rigourous Results and Applications. Cambridge University Press, Cambridge (1997)

References

69

59. Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999) 60. Prasad, A., Ramaswamy, R.: Finite-time Lyapunov exponents of strange nonchaotic attractors. In: Daniel, M., Tamizhmani, K., Sahadevan, R. (eds.) Nonlinear Dynamics: Integrability and Chaos. Narosa, New Delhi (2000) 61. Ramaswamy, R.: Symmetry breaking in local Lyapunov exponents. Eur. Phys. J.B. 29, 339 (2002) 62. Saiki, Y., Sanjuán, M.A.F., Yorke, J.A.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018) 63. Sandri, M.: Numerical calculation of Lyapunov exponents. Math. J. 6(3), 78–84 (1996) 64. Siopis, C., Kandrup, H.E., Contopoulos, G., Dvorak, R.: Universal properties of escape in dynamical systems. Celest. Mech. Dyn. Astron. 65, 57 (1997) 65. Smith, H., Contopoulos, G.: Spectra of stretching numbers of orbits in oscillating galaxies. Astron. Astrophys. 314, 795 (1996) 66. Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov Exponents. Chaos 20, 033117 (2010) 67. Toth, Z., Kalnay, E.: Ensemble forecasting at NCEP and the breeding method. Mon. Weather Rev. 125, 3297 (1997) 68. Tsiganis, K., Anastasiadis, A., Varvoglis, H.: Dimensionality differences between sticky and non-sticky chaotic trajectory segments in a 3D Hamiltonian system. Chaos Solitons Fractals 11, 2281 (2000) 69. Vallejo, J.C., Aguirre, J., Sanjuán, M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003) 70. Vallejo, J.C., Viana, R., Sanjuán, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008) 71. Vallejo, J.C., Sanjuán, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15, 113064 (2013) 72. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000) 73. Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A27, 4899 (1994) 74. Voglis, N., Contopoulos, G., Efthymioupoulos, C.: Method for distinguishing between ordered and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372 (1998) 75. Vozikis, C., Varvoglis, H., Tsiganis, K.: The power spectrum of geodesic divergences as an early detector of chaotic motion. Astron. Astrophys. 359, 386 (2000) 76. Weisstein, E.W.: Lyapunov characteristic exponent. From MathWorld A Wolfram Web resource. http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html 77. Wolfe, C.L., Samelson, R.M.: An efficient method for recovering Lyapunov vectors from singular vectors. Tellus A 59A, 355 (2007) 78. Xu, M., Paul, M.R.: Covariant Lyapunov vectors of chaotic Rayleigh-Benard convection. Phys. Rev. E 93, 062208 (2016) 79. Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001) 80. Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaoshyperchaos transition in coupled systems. Phys. Rev. E 64, 056235 (2001) 81. Yang, H.: Dependence of Hamiltonian chaos on perturbation structure. Int. J. Bifurcation Chaos 3, 1013 (1993) 82. Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)

Chapter 3

Dynamical Regimes and Timescales

3.1 Temporal Evolution There is a variety of relevant timescales when dealing with Lyapunov exponents. The most obvious one has been already discussed in the first chapter, Sect. 1.4. There, we discussed the inverse of the asymptotic Lyapunov exponent, also known as reliability time or Lyapunov time. This quantity is a measure of the strength of chaos, and it is related to the folding time of the system and how the information is preserved along a given trajectory. In addition to this timescale, there are also other timescales that provide useful insight in the dynamics under study. We have seen that the evolution of the finite-time exponents towards the asymptotic value present some oscillations when computed with very small time intervals. These are obvious variations at local scales because the ellipse axes are still subject to the local flow dynamics, without further evolution towards the most growing direction. Typically, these variations are washed out once the intervals are large enough to cover the dynamics of the flow as a whole. Some indicators that provide information on when the local fluctuations are abandoned and we enter in the global flow when the evolution towards the most growing direction starts are the dynamical time, Tdyn , or the Poincaré section crossing time, TCross . Certainly, this later time will depend on the selected surface of section, but it is anyways a good indicator of these timescales. Nevertheless, there are variations that break any monotonically trend in the evolution of the finite-time exponents and are a source to long-term timescales. We can cite, among others, diffusion phenomena such as Arnold diffusion, the presence of transient regular-like periods (the so-called sticky orbits) or, conversely, the presence of transient chaos. All this makes interesting to focus on the analysis of the distributions of the finite-time Lyapunov exponents that are naturally well-suited for tracing the variations of the flow as the intervals get longer. There are several studies modelling universal features of the Lyapunov spectra that are based on the properties of an infinite set of matrices [23]. This has © Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_3

71

72

3 Dynamical Regimes and Timescales

been done aiming to follow the evolution of the average as the time gets nearly infinite. The effective Lyapunov exponents as logarithms of products of n matrices behave essentially like averages of n random variables. Over short times, they are correlated, leading to a linear dependence of the cumulant generating function with n [14]. Over long times, correlations may be lost. But with intermittency or in areapreserving maps, there are still long-time correlations, different scaling properties and multifractal structures with the sampling interval Δt. We cite among others some general results in [21, 24] or [32], some numerical findings in [30] or [35] and, specifically, one Hamiltonian map scaling behaviour in [18]. Being large enough, the distribution of values will be driven uniquely by the transportation along the orbit, with no use on the linear equations of the tangent space. When dealing with effective exponents (finite but large intervals), and for hyperbolic systems, there is a simple relationship between the first and second exponents, driven by the crowding indexes. For nonhyperbolic systems, the relationship may be more complicated [14]. Usually, there is multifractality or a strong nontrivial dependence on the order q of the correlations [13]. We will focus here on the relationship between finite-time exponents when computed at “small” and “medium” intervals, where all multipliers (Lyapunov numbers) are still changing in sign and contribute to the time decay and the correlations die very slow. Instead of focusing on how the averaging evolves as the time gets longer, the emphasis here is on analysing the shapes of the distributions of the finite-time exponents and how these distributions evolve when a given orbit leaves the local regime and enters into the global one. We have described in the previous chapter of this monograph the procedure for building the distributions of the finite-time Lyapunov exponents. A key issue is the arbitrary initialisation of the axes once the integration is completed after the finite-time integration reaches the size Δt. The finite-time Lyapunov exponents reflect the growth rate of the orthogonal semiaxes (equivalent to the initial deviation vectors) of one ellipse centred at the initial position. These axes change their orientation and length as the orbit is integrated during a given finite-time Δt, following Eq. 2.11. When comparing these distributions, it is needed to analyse if the ordering of the exponents according to their magnitude is preserved. The definition given by Eq. 2.11 preserves the ordering as the axes evolve, and a Gram-Schmidt orthonormalisation takes place along Δt. But for the shortest intervals, there is not enough time for tending to the largest growth direction, and after resetting the direction of the ellipsoid axes, the locally largest exponent may or not coincide with the previously computed direction.

3.2 Regime Identification Each initial orientation will lead to different exponents [41]. One option is to have the axes pointing to the local expanding/contracting directions, given by the eigenvectors. Then, at local timescales, the eigenvalues will provide insight on the

3.2 Regime Identification

73

stability of the point. Another option is to start with the axes pointing to the direction which may have grown the most under the linearised dynamics. Yet another choice is pointing them to the globally fastest growth direction. The selected technique uses as initial axes of the ellipse a set of orthogonal vectors randomly oriented, following [36]. This initial selection will properly detect the change in the dynamics from the local to the global regime. Since there is no initial preferred orientation, the evolution of the deviation vectors will be a direct consequence of the flow timescales. The key factor to build the finite-time distributions is finding the most adequate Δt, to be large enough to ensure a satisfactory reduction of the fluctuations, but small enough to reveal slow trends. This length is different for every orbit. So, in principle, one needs to calculate the distributions for a variety of lengths of the finite intervals and observe the progressive evolution of the distribution shapes. If one uses the smallest intervals, the deviation vectors will trace the very local flow dynamics. As one selects larger intervals, the local regime of the flow is replaced by the global dynamics regime, and the vectors are oriented depending on the global properties of the flow, including any transient behaviour. Finally, with the largest interval lengths, the vectors are oriented towards the final asymptotic directions of the flow, when the dynamics reaches the final invariant state. In addition to the choice of the finite interval length and the initial directions of the axes, the total integration time used to compute the distribution is also important [35]. Because the integration time for gathering the finite-time exponents is also finite, the distributions may just reflect any transient state of the system during such integration period, instead of reflecting the global or final stationary state. For instance, a common phenomenon found in conservative systems is the existence of stickiness or trapped motions. A chaotic orbit may be confined to a torus for a while, but after a very long time, it leaves the confinement and again shows the chaotic behaviour. Depending on the total time used for gathering finite-time exponents, different dynamics will or will not be averaged. We will expand this discussion in the following Sect. 3.3, when dealing with transient behaviours. We have seen in the previous chapter that when using the very smallest interval lengths the distributions show many peaks, because the randomly oriented deviation vectors are not able to evolve during such very small intervals. When the finite-time intervals are slightly larger, the resulting finite-time exponent distributions begin to be similar to flat uniform distributions. The finite-time exponents cannot be regarded at these timescales as similar to random variables leading to Gaussian distributions. The deviation vectors have been allowed to evolve from the initially randomly selected deviation directions. However, they had not enough time to tend to the finally fastest growing directions. These distributions are then characterised by large negative kurtosis. Finally, when the finite intervals are larger, the deviation vectors are oriented to the globally fastest growth direction that may be or not be the final asymptotic behaviour. Actually, this asymptotic direction is only reached at very long intervals. The timescales, when the changes from the local to the global behaviour are detected, can be shorter than the timescales when the asymptotic behaviour is

74

3 Dynamical Regimes and Timescales

reached and the mean of the distributions tends to the asymptotic value. This implies that when the finite intervals are not large enough to reach the asymptotic regime, we may still detect changes in the distribution shapes after having entered into the global regime and consequently get insight into the predictability of the orbit. This may happen even when the mean of the distribution is not yet close to zero and the fluctuations around zero are hardly detected.

3.3 Transient Behaviours, Sticky Orbits and Transient Chaos In the previous analysis of the Hénon-Heiles system, we have seen the existence of transient periods, when the orbits behave like a regular orbit and other periods of time when the behaviour was strongly chaotic. This is a behaviour similar to the intermittency phenomenon. In intermittent systems, there is an irregular alternation of periods of time between apparently regular and chaotic dynamics or between different forms of chaotic dynamics. This produces a long correlation persistence producing exponential tails in the distributions, because the trajectory behaviour follows a given pattern and, for a while, moves away from it, before returning again. This also can produce multifractal structures as the finite-time intervals are varied [6]. Another phenomenon related to the existence of transient periods of different behaviour is the existence of sticky orbits. These were firstly reported in [9]. Some points can be initially scattered in the chaotic domain, but later they come close to a regular island and start moving in a regular way around it, as if they were following an invariant curve. Only after a while, their chaotic nature is again seen when they leave such a behaviour and enter into the chaotic domain again. During this period, the aperiodic orbits spend their time with a strictly zero Lyapunov exponent, if that would be exclusively computed during that time interval. Finally, we can cite the transient chaos phenomenon that has been widely studied by several authors (see, among others [15, 16]). The transient chaos can be found in a class of systems whose asymptotic behaviour is regular with a sensitivity to initial conditions that survives only during a finite-time interval [31]. It can also be found in Hamiltonian dynamics, when the energy is above certain threshold and the trajectories can be unbounded. This is related to the existence of chaotic saddles. The nonattracting chaotic set, also known as chaotic saddle or strange saddle, is formed by a set of orbits of Lebesgue measure zero that will never escape from a scattering region for both t → ∞ and t → −∞ [3]. Its stable manifold contains the orbits that will never escape if t → ∞, while the unstable manifold is formed by the ones that will never escape if t → −∞. The orbits that constitute the chaotic set are unstable periodic orbits of any period or even aperiodic. Furthermore, this set is formed by the intersection of their stable and unstable manifolds, each of them being a fractal set with dimension between two and three in the three-dimensional phase space. As these two manifolds are invariant sets, also their intersection is invariant, and for that reason, all orbits that start in one point belonging to the chaotic set will

3.4 The Hénon-Heiles System

75

never leave the set. In fact, the stable and unstable manifolds of the chaotic set are composed of the whole set of stable and unstable manifolds of each unstable point in the chaotic set. In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient, with the exception of a set of points of zero Lebesgue measure. Trajectories starting close to this set behave chaotically for a while, before diverging from it and settling into an attractor [22]. The existence of these transient periods is very important when building the finite-time Lyapunov exponents, and the total integration time used to compute the distribution is the third factor affecting the distributions, in addition to the choice of the finite-time interval length and the initial directions of the axes as we have seen in the previous chapter [35]. Because the integration time for gathering the finite-time exponents is also finite, the distributions may just reflect any transient state of the system during such an integration period, instead of reflecting the global or final stationary state. The characterisation of the orbit may change because of the slow convergence rate towards the asymptotic global value and the finite-time characterisation of the orbit (hyperbolic or not) can change as the integration time changes [8, 20]. This means that one should take care of potentially existing transient periods within the used integration time before the final attractor is reached (dissipative system), the particle escapes (open systems) or it experiences regular-like transient periods (sticky orbits). The distributions of finite-time Lyapunov exponents can trace these periods when selecting the proper finite-time interval lengths and a total integration time constrained to the transient. But one should always take into account that this total integration time used for building the distributions must be long enough to provide enough data points for sampling and for statistical analyses purposes.

3.4 The Hénon-Heiles System The existence of all the above-mentioned transient periods implies that any method based on averaging certain quantities must be taken with care and even fast convergence methods could not be fast enough to detect and characterise a given short-lived period. Averaging during long times may lead to wrong results or, at least, to ignore the existence of those transients. There are some techniques specifically designed to cope with these cases, as, for instance, the averaged finite-time Lyapunov exponents (AFTLE) indicator used in [31]. These are finite-time exponents averaged on a large set of initial conditions used for estimating the duration of chaotic transients. However, the ordinary finite-time Lyapunov exponents can trace easily these periods, with the only caveat of taking into account that the size of the finitetime interval and the total length of the integration time should produce a number of intervals enough for having a sufficiently good distribution shape. A reduced

76

3 Dynamical Regimes and Timescales

Table 3.1 Selected orbit with transient behaviour for the Hénon-Heiles system Orbit H0

Description Close to UPO

Initial condition for given energy x = 0.001100 y = 1.024677565117189 vy = 0.0 E = 1/4

λ –

Tcross 3.6

λ is the asymptotic standard Lyapunov exponent. Tcross is the Poincaré section crossing time corresponding to crosses with plane x = 0, independently of the sign of vx

number of intervals lead to a distribution that is useful for statistical analyses, even when sometimes the analysis of the shape is enough for identification purposes. As a representative example of the above, we can follow the same approaches used in the previous chapter with the Hénon-Heiles system and build the finite-time distributions corresponding to an orbit close to an unstable periodic orbit (UPO). This orbit will initially mimic the behaviour of the periodic orbit, but since it is not periodic itself, it will move away from the UPO and finally will escape. This can be observed as an example when the energy E = 1/4 and we choose as initial conditions x = 0.001100, y = 1.024677565117189, andvy = 0.0, which are close to the Lyapunov Orbit (Table 3.1). A UPO defines a frontier. Every orbit with an initial energy larger than the escape energy and moving outwards, if it crosses the Lyapunov Orbit, will escape from the system and will never come back (see [2]). The phase space of an example of such orbit is plotted in Fig. 3.1a. For the case of an UPO, each point must avoid all regions χ (Δt) < 0. The distribution of finite-time Lyapunov exponents is formed by two peaks, both centred around positive values. When the initial condition is slightly different from the one leading to the UPO, as the selected initial condition shown in the first row of Table 3.1, the distribution is similar to the solid line of Fig. 3.1b, where we observe two broadened peaks centred around positive values and a tail associated with the orbit once it has escaped. When the orbit is confined, the behaviour is similar to an exact UPO, and we see two peaks in figure. The value of χ (Δt) oscillates between those peaks, as shown in the solid line of the smaller panel of Fig. 3.1b, leading to the intermediate spectrum of values between the main peaks. But after having integrated 8T time units, or after roughly 1600 finite-time intervals, the particle escapes. Now the range of values of χ (Δt) no longer oscillates, but gets new values, leading to a left tail of totally different values, plotting, for instance, the smaller negative centred peak that appears at t = 35 time units. Indeed, as shown in the smaller panel of Fig. 3.1a, the motion can now follow an open track; thus the tail of the distribution extends, and several small peaks centred below −0.2 (not shown) are produced. When we consider initial conditions far away from the UPO, thus orbits with smaller escape times, the general spectrum shape is different due to the tail, as it is produced by the values once the particle has escaped. But meanwhile the orbit is confined, the shape is always quite similar. If the interval size Δt is increased, but still smaller than the escaping time, it is observed that the main peaks shift towards larger positive values and begin to merge, as shown by the dashed (Δt = 0.1) and dotted (Δt = 0.3) lines of Fig. 3.1b. As reflected in the smaller panel, the oscillation (around a larger value) of the finite-time exponents values is preserved, but it begins to disappear after a smaller number of integrated intervals.

3.4 The Hénon-Heiles System

77 (a)

0.75

0.5

x’

0.25

0 3 2.5 2 1.5 1 0.5 0 -0.5

-0.25

-0.5

-0.75 -0.75

-0.5

-0.25

0 x (b)

0

1

0.25

2

0.5

3 0.75

X

0.5 0 0

150

300

450

900 1050 600 750 Number of finite intervals

1200

1350

1500

1650

0.07 0.06

P(x)

0.05 0.04 0.03 0.02 0.01 0 -0.2

-0.1

0

0.1

0.2 0.3 0.4 Short Time Exponent (X)

0.5

0.6

Fig. 3.1 (a) Trajectory x − x˙ near an UPO, when E = 1/4. The period T is roughly 3.6 time units. (b) The solid line shows the distribution formed with an integration of 40 time units when Δt = 0.02. The rightmost two peaks are traced when the orbit is confined, before escaping after 8T time units. The dashed probability distribution is when Δt = 0.1 and the dotted one when Δt = 0.3. The smaller panel shows the oscillating behaviour of χ(Δt) as the integration takes place. (Taken from [35] with permission)

As we want to analyse further how these distributions change in shape as we vary the intervals, now we will proceed to plot additional diagrams. The diagrams seen before only display the first exponent, as returned by the algorithm described

78

3 Dynamical Regimes and Timescales

Fig. 3.2 Finite-time Lyapunov exponent distributions for the Hénon-Heiles Hamiltonian. and the unstable periodic orbit H0 listed in Table 3.1. Δt = 0.01. (Adapted from [36] with permission)

in the Appendix A. But this algorithm also returns the following exponent, and it is of interest to plot how the second exponent evolves and to check any possible dependencies between them. So, we extend the computations to the second finitetime exponent and plot in Figs. 3.2 and 3.4 the distributions and relationship (twodimensional distribution) χ1 (Δt) − χ2 (Δt) diagrams for some of the prototypical orbits of the Hénon-Heiles system previously analysed. The associated numerical indexes of these distributions are found in Table 3.2. In Fig. 3.2, we see the plots at the very local scale Δt = 0.01 for the HénonHeiles orbit H0, classified as an UPO, with no oscillation around zero. When we plot one exponent against the other one, we obtain a relationship diagram that shows a linear relationship between them, with χ1 always expanding and χ2 always contracting. We can also build the same plots for the close-to period-5 orbit of the HénonHeiles system, labeled as H4 in Table 2.2. This is shown in Fig. 3.3. Here we see a similar behaviour. Because this orbit is regular and it does not escape, we can integrate it during longer times. Therefore, we can increase the size of the finite interval having at the same time enough intervals for building a properly shaped distribution. So, the panels from top to bottom show the distributions for Δt = 0.01, Δt = 1.0, and Δt = 10.0. We previously mentioned that the Tcross was roughly 6.2 time units. We see how the distributions keep peaked below such a timescale, but begin to show a different shape above it. Obviously, for even larger intervals, they will converge towards single-peaked distributions centred around the asymptotic values. Finally, in Fig. 3.4, we see the plots corresponding to the weakly chaotic, cycle orbit, labeled as orbit H1, where we see the same behaviour. First, the distributions at very local scales correspond to chaotic movements. But once the intervals are larger than the Tcross , the shapes change, and the convergence towards the asymptotic value begins to be detected.

3.5 The Contopoulos System Aiming to generalise these χ1 –χ2 diagrams, we will plot them in an even more simpler system than the Hénon-Heiles. We will analyse another four-dimensional

3.5 The Contopoulos System

79

Table 3.2 Numerical indexes associated with the finite Lyapunov exponent distributions corresponding to the Hénon-Heiles system, for one close-to period-5 orbit and one unstable periodic orbit and several Δt sizes χ1 Δt Mean UPO 0.01 0.35 Close-to period-5 0.01 0.0014 1 0.11 10 0.077 Weakly chaotic, cycle 0.01 0.0098 1 0.074 10 0.063 χ2 Δt Mean UPO 0.01 −0.32 Close-to period-5 0.01 0.0010 1 0.0053 10 0.022 Weakly chaotic, cycle 0.01 −0.0092 1 0.00086 10 0.0042

Median

σ

k

F+

h

0.34

0.11

−1.50

1.00

56.53

0.11 0.11 0.086

0.20 0.21 0.047

−0.90 −1.22 1.52

0.51 0.62 0.92

0.0069 5.13 70.93

0.051 0.098 0.072

0.16 0.17 0.047

−0.70 −1.17 −0.31

0.62 0.64 0.87

0.78 5.05 56.37

Median

σ

k

F+

h

−0.32

0.11

−1.50

0.00

52.39

−0.0097 0.028 0.032

0.20 0.22 0.066

−0.90 −0.95 −0.81

0.48 0.53 0.61

0.0050 0.22 10.15

−0.049 0.052 0.0053

0.16 0.18 0.054

−0.70 −0.99 −0.62

0.36 0.60 0.52

0.74 0.054 2.80

σ is the standard deviation, k the kurtosis, F+ the probability of positivity, h the hyperbolicity index

phase space Hamiltonian, with two degrees of freedom, originally studied by Contopoulos in Ref. [9] and given by H =

1 2 1 (px + py2 ) + (Ax 2 + By 2 ) − xy 2 . 2 2

So, the equations of motion are the following: ⎧ x˙ = p ⎪ ⎪ ⎨ y˙ = q ⎪ p ˙ = −Ax + xy ⎪ ⎩ q˙ = −By + 2.0xy, and the corresponding Jacobian is as follows:

(3.1)

(3.2)

80

3 Dynamical Regimes and Timescales

Fig. 3.3 Finite-time Lyapunov exponent distributions for the Hénon-Heiles Hamiltonian and the close-to period-5, labeled as orbit H4 in Table 2.2. From top to bottom, Δt = 0.01, Δt = 1, and Δt = 10. (Adapted from [36] with permission)

⎡

0 0 1.0 ⎢ 0 0 0 Dv Φ = ⎢ ⎣ −A (2.0y) 0 (2.0y) −B 0

⎤ 0 1.0⎥ ⎥. 0⎦

(3.3)

0

This model represents two nonlinearly coupled oscillators. We have chosen it because in spite of its simplicity, it still provides a rich dynamical behaviour. In addition, it is a physically meaningful flow. The origins of this model are traced to the galactic dynamics field, like the Hénon-Heiles potential. It also belongs to the so-called galactic-type meridional potentials, reduced potentials on the meridian plane V (R, z) of an axisymmetric galaxy [7]. The Contopoulos system can be seen as a simpler version of the Hénon-Heiles system, as it has only one mixed higher-order term, xy 2 , which introduces the essential nonlinearity of the problem, y−axis symmetry and only two exits. The amplitude parameters are A = 1.6 and B = 0.9. Such values are chosen to be √ near the resonance A/B = 4/3 [9]. The sampled initial condition is x = 0.03744, y = 0, x˙ = 0.0480, associated with the regular motion of Ref. [10]. For this initial condition, depending on the value of the coupling parameter , different orbit types are found. We have selected three values of , namely, 4.4, 4.5 and 4.6. The energy

3.5 The Contopoulos System

81

(a)

(b)

(c)

Fig. 3.4 Finite-time Lyapunov exponent distributions for the Hénon-Heiles Hamiltonian. All panels are for the weakly chaotic, cycle orbit. (a) Δt = 0.01, (b) Δt = 1, (c) Δt = 10. (Adapted from [36] with permission)

Table 3.3 Selected orbits for the Contopoulos system Orbit C2 C1 C3

Description Weakly chaotic, close to period-2, orbit Chaotic, between two tori, orbit Chaotic, ergodic, orbit

 4.5 4.4 4.6

λ 0.0125 0.093 0.066

Tcross 7.3 7.0 7.1

 is the control parameter. λ is the asymptotic standard Lyapunov exponent. The notion weak or strong chaos is associated with the relatively smaller or larger value of λ. Tcross is the Poincaré section crossing time corresponding to crosses with the plane y = 0, independently of the sign of vy .

value is set to E = 0.00765, which in the third case is close to the escape energy, 2 given by Eescape = 18 AB . These orbits are listed in Table 3.3. 2 We will focus here on comparing, for different orbits types, the distributions generated when the axes of the ellipse centred in the initial condition are allowed or not to tend to the largest stretching direction. The first considered orbit in this system, labeled as C2 in Table 3.3, is the case with  = 4.5, a weakly chaotic, close to a period-2 orbit, with Poincaré section crossing time Tcross ∼ 7.3, which appears in Fig. 3.5a as a cross symbol. The density

82

3 Dynamical Regimes and Timescales

0.2

-0.0605

y

0.1 0

-0.001

0

0.001

-0.1

C2 -0.2 -0.04

-0.02

0

x

0.02

0.04

Fig. 3.5 Orbit C2 of the Contopoulos system, as listed in Table 3.3. This is a weakly chaotic orbit that, in appearance, is a periodic orbit. It must be zoomed in to reflect its chaotic nature. The asymptotic Lyapunov exponent λ = 0.0125. (top) Orbit in the configuration space. (bottom) Poincaré section. The cross section of another ergodic orbit (x = 0.03, y = 0, x˙ = 0.04796) has been also plotted in order to ease the visualisation of the phase portrait. (Adapted from [36] and [34] with permission)

functions for the first and the second Lyapunov exponents are plotted in Fig. 3.6, and their numerical characterisation is found in Table 3.4. The distribution computed with a very short interval Δt = 0.01 appears in Fig. 3.6a. It shows the typical shape associated with a periodic orbit. For an interval 10 times larger, we have Δt = 0.1, and the length is still below Tcross . The figures are nearly identical to the previous ones, so that they are not drawn. This similarity is not obvious, since the local ellipsoid axes have now evolved a few steps; thus they have had the possibility of relaxing in the direction that permits the largest stretching and pointing to the direction of the fastest separation. For Δt = 1, panel (b), a new peak appears in the distribution of the largest exponent χ1 , but the χ2 distribution remains the same. This means that there are different rates in the evolution towards the invariant measure. When Δt ∼ Tcross , panel (c), the χ1 distributions jump towards the positive values.

3.5 The Contopoulos System

83

(a)

(b)

(c)

(d)

(e)

Fig. 3.6 Finite-time Lyapunov exponent distributions for the Eq. 3.1 Hamiltonian and  = 4.5, weakly chaotic, close-to period-2 orbit. (a) Δt = 0.01, (b) Δt = 1, (c) Δt = 7, (d) Δt = 10, (e) Δt = 100. (Adapted from [36] with permission)

This leads to think as Tcross as a threshold separating different regimes in the distributions, tracing local and non-local behaviour. Even when the choice of the Poincaré section is somehow arbitrary, it is based in the symmetry y = 0 of the potential. Thus it makes sense that the crossing time for closing an orbit (if periodic) will lead to such a threshold. At larger intervals, Δt = 10, panel (d), the oscillations around zero begin to be lost. Finally, with Δt = 100, panel (e), we are integrating several Tcross cycles,

84

3 Dynamical Regimes and Timescales

Table 3.4 Numerical indexes associated with the finite-time Lyapunov exponent distributions corresponding to Eq. 3.1, case  = 4.5, close-to period-2 orbit, for several Δt sizes χ1 Δt 0.01 0.1 1 7 10 100 χ2 Δt 0.01 0.1 1 7 10 100

Mean 0.048 0.048 0.14 0.092 0.095 0.12

Median 0.065 0.067 0.070 0.12 0.091 0.12

σ 0.27 0.29 0.24 0.066 0.072 0.0084

k −1.45 −1.45 −1.46 0.36 0.014 11.32

F+ 0.55 0.55 0.56 0.86 0.89 1.00

1.35 1.19 4.65 42.45 37.22 3406.33

Mean −0.096 −0.076 0.082 −0.021 −0.02 −0.0041

Median −0.11 −0.095 0.063 −0.028 −0.031 −0.0038

σ 0.27 0.28 0.18 0.080 0.057 0.011

k −1.45 −1.44 −1.55 −0.74 0.38 4.77

F+ 0.39 0.42 0.56 0.38 0.22 0.19

h 2.69 1.87 4.77 6.66 12.70 68.54

h

σ is the standard deviation, k the kurtosis, F+ the probability of positivity, h the hyperbolicity index

and the distributions resemble peaks centred around the λ1 ∼ 0.0125 and λ2 ∼ 0 asymptotic Lyapunov values. The second analysed case is a chaotic orbit, between two KAM tori, given when  = 4.4. This is labeled as C1 in the Table 3.3. Quasi-periodic orbits are characterised by a linear divergence of neighbouring trajectories, all asymptotic exponents are zero, and the motion is confined within a torus. With  = 4.4, the initial condition is interesting, as it does not lead to a quasi-periodic motion but to a trajectory running on a very small chaos strip between two invariant tori. The Poincaré section of this orbit appears as an elongated lobe in Fig. 3.7. The density functions for the first and the second Lyapunov exponents are plotted in Fig. 3.8. The numerical indexes which characterise such distributions are found in Table 3.5. The main timescale to take into account seems to be again the crossing time, Tcross ∼ 7. There is another physically meaningful timescale, which is the period to roughly cover the whole Poincaré section, Tlobe ∼ 136. For the shortest interval sizes, Δt = 0.01, panel (a), and Δt = 0.1, not shown, the distributions are similar, roughly double peaked, reflecting the confined motion. When the interval is increased up to Δt = 1, panel (b), there is a change in shape for χ1 , with a morphology no longer similar to a periodic orbit. However, in the tangent direction, the χ2 distribution evolves at a different rate and is still sign flipping. Once again, as the time interval is larger than the given crossing time, for Δt = 7, panel (c), the distributions are now different. When the interval is larger than Tcross , Δt = 10, panel (d), the distributions converge to the final measure, faster

3.5 The Contopoulos System

85

0.2

y

0.1 0 -0.1

C1 -0.2 -0.04

-0.02

0

x

0.02

0.04

Fig. 3.7 Orbit C1 of the Contopoulos system, as listed in Table 3.3. This is a chaotic orbit with asymptotic Lyapunov exponent λ = 0.093. (top) Orbit in the configuration space. (bottom) Poincaré section. The cross section of another ergodic orbit (x = 0.03, y = 0, x˙ = 0.04796) has been also plotted in order to ease the visualisation of the phase portrait. (Adapted from [36] and [34] with permission)

for χ1 . With Δt = 100, panel (e), both distributions resemble peaks centred in the asymptotic Lyapunov values λ1 ∼ 0.093 and λ2 ∼ 0. The third analysed case is a chaotic, ergodic orbit, given when  = 4.6. This orbit is labeled as C3 in Table 3.3. The Poincaré section appears in Fig. 3.9. The density functions for the first and the second Lyapunov exponents are plotted in Fig. 3.10. The numerical indexes that characterise such distributions are found in Table 3.6. For the shortest intervals, Δt = 0.01, panel (a), and Δt = 0.1, not shown, both the χ1 and χ2 diagrams have widened and almost completely lost the two-peaks aspect from previous cases. For Δt = 1, panel (b), χ1 distribution changes in shape. With Δt = 7, panel (c), both distributions are almost Gaussian. This is clearly observed with Δt = 10 and Δt = 100, panels (d) (e), centering around λ1 ∼ 0.066 and λ2 values. Note, however, that even when a a Gaussian shape has been achieved quite fast at very short intervals, the peak of χ2 is not still centred in the 0 value, implying a very low convergence of the averaging process.

86

3 Dynamical Regimes and Timescales

(a)

(b)

(c)

(d)

(e)

Fig. 3.8 Finite-time Lyapunov exponent distributions for the Eq. 3.1 Hamiltonian and  = 4.4, chaotic orbit, between tori. (a) Δt = 0.01, (b) Δt = 1, (c) Δt = 7, (d) Δt = 10, (e) Δt = 100. (Adapted from [36] with permission)

This orbit is ergodic in the sense that the orbit is able to reach with the same probability all its available phase space. It is interesting to keep in mind the difference between stationarity, due to the dynamics at certain time, and ergodicity, time-averaged property of the trajectories. In a non-ergodic orbit, the trajectory does not cover the whole hypersurface of constant energy, so two different initial conditions cover different parts of the energy surface leading to different temporal averages even for times tending to infinity. In such systems, there is not a unique equilibrium state but different ones depending on the starting point. In an ergodic

3.5 The Contopoulos System

87

Table 3.5 Numerical indexes associated with the finite-time Lyapunov exponent distributions corresponding to Eq. 3.1, case  = 4.4, between tori orbit, for several Δt sizes χ1 Δt 0.01 0.1 1 7 10 100 χ2 Δt 0.01 0.1 1 7 10 100

Mean −0.034 −0.039 0.077 0.060 0.054 0.083

Median −0.033 −0.038 −0.022 0.070 0.056 0.086

σ 0.27 0.29 0.20 0.059 0.052 0.010

k −1.42 −1.42 −1.11 0.34 0.48 4.51

F+ 0.46 0.46 0.47 0.85 0.84 0.99

0.92 0.93 3.78 34.88 40.03 1472.67

Mean −0.013 0.012 0.13 0.0091 0.0053 0.0023

Median −0.014 −0.0094 0.14 0.014 0.0049 0.0010

σ 0.27 0.288 0.19 0.071 0.048 0.0010

k −1.42 −1.42 −1.53 −1.07 −0.91 4.69

F+ 0.48 0.50 0.65 0.55 0.52 0.50

h 0.37 0.29 7.58 3.66 4.54 46.74

h

σ is the standard deviation, k the kurtosis, F+ the probability of positivity, h the hyperbolicity index

Fig. 3.9 Poincaré section of Orbit C3 of the Contopoulos system, a chaotic orbit with asymptotic Lyapunov exponent λ = 0.066 that fulfils the available phase space. (Adapted from [36] with permission)

system, a unique equilibrium state may be reached. And generic ensembles of initial conditions will evolve towards a given distribution, time-independent or with little variability on long timescales. In the case of conservative systems, there are no attractors and chaotic orbits are ergodic. But note that there may be regular-like transients in the so-called sticky orbits, where the particles wander pseudo-chaotically with strictly zero Lyapunov exponent during some time around

88

3 Dynamical Regimes and Timescales

(a)

(b)

(c)

(d)

(e)

Fig. 3.10 Finite-time Lyapunov exponent distributions for the Eq. 3.1 Hamiltonian and  = 4.6, chaotic, ergodic, orbit, labeled as orbit C3. (a) Δt = 0.01, (b) Δt = 1, (c) Δt = 7, (d) Δt = 10, (e) Δt = 100. (Adapted from [36] with permission)

the KAM tori. Many authors refer then to such orbits as pseudo-ergodic ones. Such transients are the reason for the broad peaks found in the distributions. Now we will analyse again the relationship between the largest finite exponent, associated with the transversal direction (if allowed to evolve), and the second exponent, associated with the tangential one (id), when they are calculated by re-initialising arbitrarily the axes after every interval Δt. The two-dimensional distributions histograms of the second exponent against the first one conform the third box of every row in Figs. 3.6, 3.8 and 3.10.

3.5 The Contopoulos System

89

Table 3.6 Parameters associated with the finite Lyapunov exponent distributions corresponding to Eq. 3.1, case  = 4.6, chaotic orbit, for several Δt sizes χ1 Δt 0.01 0.1 1 7 10 100 χ2 Δt 0.01 0.1 1 7 10 100

Mean −0.032 −0.041 0.091 0.069 0.069 0.066

Median −0.031 −0.037 −0.00092 0.059 0.062 0.065

σ 0.29 0.31 0.22 0.082 0.068 0.030

k −1.18 −1.14 −0.39 3.87 1.27 −0.19

F+ 0.46 0.46 0.49 0.83 0.86 0.99

Mean −0.015 0.014 0.13 0.012 0.0067 0.015

Median −0.017 −0.0088 0.15 0.016 0.0057 0.013

σ 0.29 0.31 0.19 0.069 0.057 0.017

k −1.18 −1.14 −1.41 0.18 1.04 1.41

F+ 0.48 0.50 0.67 0.57 0.53 0.80

h 0.74 0.84 3.80 20.39 30.13 142.49 h 0.35 0.28 7.19 4.88 4.18 100.41

σ is the standard deviation, k the kurtosis, F+ the probability of positivity, h the hyperbolicity index

To compare these distributions, we would need the order of the exponents to be preserved. However, we need to be careful with the shortest intervals, because there is no enough time for tending to the largest growth direction. Once we have reset the direction of the ellipsoid axes, the new computed direction of fastest growth might not coincide with the previous one. In the orbit C2,  = 4.5 case, and for the smallest intervals Δt = 0.01 and Δt = 0.1, there is a linear relationship. When the local flow is expanding in one direction, it is contracting in the other one. We can observe a low probability region when both directions are contracting at the same time. For Δt = 1, the correlation is no longer linear in the χ1 contracting range. This is derived from a faster convergence rate towards the transversal direction. For the second exponent, the distribution is still like a periodic one. When Δt increases, there is a clustering of the values towards the asymptotic values. In the  = 4.4 case, the results are similar for Δt = 0.01 (panel a) and Δt = 0.1 (not shown), where the density plots are also linear and below the origin. When the interval is larger, Δt = 1, panel (b), we see a multivalued curve when there is an expansion in the tangent direction. For Δt ∼ Tcross , panel (c), the curve is now somehow more fuzzy. Now, the probability of finding both exponents expanding at the same time has increased. For Δt = 10, panel (d), the points already cluster towards the asymptotic values. In the chaotic case,  = 4.6, the relationship for the smaller intervals is also linear, and when expanding, the transversal direction contracts, and vice versa. When the finite time increases up to Δt ∼ 1, the relationship curve in the expanding

90

3 Dynamical Regimes and Timescales

tangent direction part is more complicated. For Δt = 7 and Δt = 10, there is no correlation. For Δt = 100, the curves converge to a set of points centred in the final values. So the relationship is linear, independently of the nature of the orbit (periodic, confined between tori or chaotic) at the very local timescales, where no evolution towards any direction is allowed. This may be a direct consequence of the arbitrary starting direction for one axis and the orthogonality of the second. But this is the same for Δt = 0.1, where many averaging steps have been performed and the vectors tend to seek the most rapidly growing directions. At these small intervals and after resetting the initial directions, the distributions still reflect the local nature of the flow, even when the finite values ordering could have been interleaved along the orbit. The comparison of the first and second distributions reflects that they essentially offer the same information. Finally, we can mention that the linear dependence at short intervals is related to the number of degrees of freedom of the system and the associated constraints in the Lyapunov values. Indeed, in Hamiltonian systems with more degrees of freedom, this linear relationship is no longer present even for the smallest intervals.

3.6 The Rössler System We can see a similar evolution in the shapes of the distributions of χ (Δt) as the finite-time intervals Δt grow in dissipative systems. For doing so, we will analyse again the Rössler system described in the previous chapter. We will take as example the distributions corresponding to the point B seen in Fig. 2.7. This point is located in the hyperchaotic regime, by fixing d = 0.25 and a = 0.365. The corresponding figure showing these distributions is Fig. 3.11. It shows how the distribution shapes of the first three exponents depend on the finite-time interval length Δt. As Δt increases, the distributions tend to shrink centred around the global Lyapunov exponent. The distributions show a total integration time of T = 10,000 for all Δt, with the exception of T = 100,000 when Δt = 100. The first integration time T = 10,000 is enough for a proper display of the distributions and the data analysis. Every curve contains a different but sufficient number of data points. Furthermore, the results are essentially the same as when using longer integration times. The case Δt = 100, however, requires the long integration T = 100,000, in order to have enough data points and a reliable distribution. In Table 3.7 we see this trend reflected as the evolution with Δt of the numerical indexes associated with these distributions, where the very small values of σ indicate the trend towards the asymptotic value. The necessary timescales to make the initial axes to be oriented towards the final largest growth directions can be derived from the observation of the evolution of the distributions. Actually, these timescales are different depending on the nature of the orbit. For very small timescales, we have seen that there can be a linear relationship

3.6 The Rössler System

0.08 P(X1)

0.06 0.04 0.02 -0.2

-0.1

0 X1

0.1

0.2

0.05 P(X2)

0.04 0.03 0.02 0.01 0

-0.2 -0.1 0 0.1 0.2 0.3 X2

0.05 0.04 P(X3)

Fig. 3.11 Probability density distributions P (χ) for the first three finite-time exponents corresponding to point B of Fig. 2.7, a = 0.365 and d = 0.25. These plots show how the centre and shape of the distributions depend on the finite-time interval length. As the finite-time interval Δt is increased, the distributions tend to shrink and centre around the global Lyapunov exponent. Δt = 1.0 is black dotted line. Δt = 10.0 is red dashed line. Δt = 50.0 is green dot-dashed line. Δt = 100.0 is blue continuous line. The distributions sample a total integration time of T = 10,000 for all Δt, with the exception of T = 100,000, when Δt = 100 is analysed. Table 3.7 contains the applicable parameters. (Adapted from [33] with permission)

91

0.03 0.02 0.01 0 -1 -0.8 -0.6 -0.4 -0.2 0 X3

92 Table 3.7 Numerical indexes associated with the finite-time Lyapunov exponent distributions corresponding to Fig. 3.11, for the several Δt sizes. The standard deviation is σ . The probability of positivity F+

3 Dynamical Regimes and Timescales Δt χ1 1.0 10.0 50.0 100.0 χ2 1.0 10.0 50.0 100.0 χ3 1.0 10.0 50.0 100.0

Mean

σ

F+

−0.065 0.013 0.010 0.021

0.046 0.051 0.015 0.013

0.079 0.55 0.73 0.93

0.044 0.051 0.019 0.012

0.98 0.74 0.73 0.71

0.19 0.065 0.017 0.011

0.028 0.24 0.21 0.11

0.10 0.038 0.013 0.0071 −0.44 −0.043 −0.013 −0.012

among the exponents [36]. For larger timescales, the distributions can be used to characterise the hyperbolic nature of the orbit. The tangencies among several directions can be seen as the linear dependencies between local exponents, which are not lost for increasing timescales. Indeed, we could construct a local eigenvolume that will depend on the possible dependencies among the exponents and we can follow its evolution with time. This can give us information to distinguish between chaotic and ordered orbits, as done by the GALI-k index [29], which is based on how the relationship among the different deviation vectors evolve.

3.7 Hyperbolicity Characterisation Through Finite-Time Exponents A basic requirement for shadowing is hyperbolicity. Using the Rössler flow as a working example, we will extend the discussion to the analysis of the possible nonhyperbolicity of a given flow and the relationship of nonhyperbolicity and the finite-time exponent distributions. A dynamical system is hyperbolic if the phase space can be spanned locally by a fixed number of independent stable and unstable directions, which are consistent under the operation of the dynamics [28], and the angle between the stable and unstable manifolds is away from zero [17, 37]. Hyperbolic systems are structurally stable in the sense that numerical trajectories stay close to the true ones. This phenomenon is called shadowing, as introduced earlier in Sect. 1.4. In case of nonhyperbolicity, an orbit may not be shadowed, and the computed orbit behaviour may be completely different from the true one. The nonhyperbolic

3.7 Hyperbolicity Characterisation Through Finite-Time Exponents

93

behaviour can arise from tangencies between stable and unstable manifolds, from unstable dimension variability (UDV), or from both. When the nonhyperbolicity arises only from tangencies, the trajectories may be still shadowed during long times. But in a general system, we could find unstable periodic orbits (UPO), KAM tori, KAM sticky orbits or chaotic sets. And since our system is dissipative, in addition to tangencies, an attractor may pass very close to periodic orbits with different number of unstable directions. This property of unstable periodic orbits embedded in a chaotic invariant set is called unstable dimension variability. In these pseudo-deterministic systems, where the nonhyperbolicity arises from UDV, with or without tangencies, the shadowing may be not good, meaning that the shadowing is only valid during trajectories of a given length, sometimes very short. The UDV indicates a variation with position of the dimension of the invariant set subspaces and is a major difficulty when modelling high-dimensional dynamical systems because the subspaces are not invariant along a typical chaotic trajectory. The UDV can be produced by an infinite number of UPO embedded in a chaotic invariant set, having a variation with position of the dimension of the invariant set subspaces (number of eigendirections). It was first reported in the kicked double rotor [1], where the invariant set of interest is a chaotic attractor. But UDV can also appear in nonattracting chaotic sets. Several mechanisms lead to UDV, as bubbling transition in coupled oscillators, decoherence transitions in weakly coupled or nonidentical systems, hyperchaos or extrinsic noise, with associated intermittency [5, 26, 27, 38]. Hyperchaos is a common source for UDV, but note that this cannot be the case in the two-degreesof-freedom Hamiltonians, such as the Hénon-Heiles or the Contopoulos systems. UDV seems to be common in high-dimensional dynamical systems, such as coupled maps [19] and continuous flows of coupled systems [40], but it may be also present in low dimensional systems. A sign of nonhyperbolicity and bad shadowing is then the fluctuating behaviour around zero of the finite-time exponent closest to zero [11]. This reflects in principle, the varying number of dimensions along the trajectory. So, first we need to properly identify which is the closest to zero exponent. The exponent closest to zero can be derived from the inspection of the mean m of the distributions. The identification of the exponent closest to zero among all available exponents is helpful to characterise the hyperbolicity. However, we have seen how the distribution shapes change as the Δt is varied. As a consequence, it may happen that meanwhile we are leaving the local regime, the computations will return as closest to zero a different exponent. Aiming to explore this, we can trace a diagram indicating which is the exponent closest to zero, as returned by the algorithm we have used up to now (see Appendix A). Taking again the two coupled Rössler oscillators system as an example, we can plot the index as derived from the mean of the probability density of the closest to zero exponent. We will do this for two sizes of the finite-time interval Δt. This is seen in Fig. 3.12. The index will obviously depend on the control parameters of the system, a and d in this case, and this is clearly seen in the figure.

94

3 Dynamical Regimes and Timescales

Fig. 3.12 The identification of the closest exponent to zero varies with Δt, as derived from the mean of the probability density. Top: Δt = 25 and T = 10,000. When the exponent closest to zero is the first one we compute, it appears as black. If it is the second, as red. If the third, as pink. And finally for the fourth, as white. Bottom: Δt = 100 and T = 100,000. In this case, the exponent closest to zero is only one of the three first exponents. If it is the first, it appears as black. If the second, as red. And finally, if the third, as white. (Taken from [33] with permission)

But what is more important, this index will depend on the size of Δt. This makes in turn to get a different set of structures in the diagrams. For the smaller Δt intervals, the values of the finite-time exponents have not evolved towards the final ordering. With Δt = 1.0, the directions have been already integrated 100 times, but the decorrelation has not yet taken place. For larger Δt, the distributions start to be Gaussian with a given mean centred around the global values. The top panel of Fig. 3.12 shows the index for a Δt = 25. The bottom panel corresponds to Δt = 100.0. Then, the mean of the distributions clearly tends to the global asymptotic values. As a consequence, the exponent closest to zero is the one tending to the neutral flow direction. Finally, when Δt → ∞, the distributions will tend to be a Dirac delta function centred at the global asymptotic Lyapunov exponent value. Once we have identified which is the closest to zero exponent, we need to detect its oscillations around zero. These oscillations can be detected when the positivity index P+ , or probability of getting a positive χ (Δt), as described by Eq. 2.12, is nearly 0.5.

3.7 Hyperbolicity Characterisation Through Finite-Time Exponents

95

Fig. 3.13 Probability of positivity of the closest to zero exponent, for a given oscillator parameter a and a coupling strength d. Scaled values give the distance to P+ = 0.5. Darker areas, values nearly to 0.0 are those with smaller values and P+ ∼ 0.5. This means distributions centred around zero (stretched or shrinked). Brighter areas with larger values are farther from 0.5 in positive or negative direction. From top to bottom, and from left to right, Δt = 1 and T = 10,000, Δt = 25 and T = 10,000, Δt = 50 and T = 100,000 and Δt = 100 and T = 100,000. (Adapted from [36] with permission)

How far are the positivity indexes in the parameter space a − −d from the 0.5 value? This proximity, or distance, is colour coded in Fig. 3.13. Here we can see that the darker regions are those with smaller values, meaning P+ ∼ 0.5. Conversely, the larger the values, the brighter the region and the farther from 0.5 in positive or negative directions. Areas of different behaviour of the flow, such as the upper leftmost corner, with higher coupling strengths and smaller a control values, are identified even with the shorter intervals. The finest structures however can only be resolved with the larger intervals. Note that in different regions, we have derived P+ from different closest to zero exponents, as this identification changes along the parametric phase space, as shown in Fig. 3.12. We can compare this figure with Fig. 2.7, the chart showing the hyperchaoticity of this system in the parametric space. The darker areas of Fig. 3.13, those with values of P+ ∼ 0.5, are the ones most likely linked to have UDV. Hyperchaos is a common source for the UDV. When comparing the darker areas with the highly chaotic areas, we see the darker zones roughly match with the hyperchaos areas. However, the match is not perfect, and here we may conjecture that the UDV is not

96

3 Dynamical Regimes and Timescales

fully sourced to the hyperchaos. Conversely, no area of high chaoticity matches with a high predictability area. We would like to emphasise that the exponents may fluctuate without being a clear cut of UDV [4, 39]. There are situations where the positive tails appear not due to UDV, but rather by other mechanisms such as the quasi-tangencies between the stable and unstable manifolds near a homoclinic crisis. Nevertheless, the oscillations are still a good indication of the nonhyperbolic nature of the orbit. As it was already discussed in [25], the concept of heterochaos connects many phenomena like fluctuating finite-time Lyapunov exponents and UDV. Hetero-chaotic attractors contain periodic orbits with different number of unstable directions. A typical trajectory will return near each, occasionally spending long times near them before moving on, and while near the periodic orbit of a region, it will have the same number of positive finite-time Lyapunov exponents as the periodic orbit. As it moves among the periodic orbits, its number of positive finitetime Lyapunov exponents fluctuates. Some authors have used the term UDV to mean fluctuating finite-time Lyapunov exponents. As a matter both concepts are implied by other dynamical phenomena such as riddled basins, blowout bifurcations, on-off intermittency and chaotic itinerancy. When a trajectory moves from a region where the dynamics has fewer unstable directions to a region where it has more, shadowing fails, and trajectories become unrealistic. Such a transition causes fluctuations in the number of positive finite-time Lyapunov exponents, which means that fluctuating finite-time Lyapunov exponents will be common in higher-dimensional attractors. The fluctuations of the finite-time Lyapunov exponents imply that the shadowing fails, as was established in [12]. Furthermore hetero-chaotic systems cannot have the shadowing property.

3.8 Concluding Remarks We have described here how the finite-time Lyapunov exponents and their corresponding distributions serve as valid indicators to characterise an orbit, even when the initial axes have been chosen arbitrarily. We have seen that the information provided by the first and second exponents seems to be the same when computed at very local scales. At larger intervals, but below a given threshold, when axes have been allowed to point to the largest stretching direction, both exponents still trace the flow local properties; they oscillate around zero and may trace the UDV. At larger timescales, the linear relationship between both exponents is lost, and we get globally averaged values for the whole orbit. Obviously, we reach the final asymptotic value at different timescales. In addition to the known fact that the more you integrate, the better you can estimate the asymptotic Lyapunov exponent, we have also seen that using finitetime Lyapunov exponents distributions and very short time intervals is sufficient for distinguishing the regions of different predictability behaviour.

References

97

The presence of oscillations of the closest to zero exponent is an indicator of nonhyperbolicity. This implies the necessity of the calculation of the several available exponents, as the identification of the closest one depends on the selected interval, in addition to the position in the parametric space. We have noticed for the larger intervals that the exponents tend to the global values, the closest to zero points to the neutral direction, and the oscillations may be then difficult of being clearly identified. Our methods derive from calculating distributions during certain integration times T of finite-time exponents. This method does not use global averaged quantities during long intervals, unless strictly needed. So it can be used for open systems, when the energy is high enough for allowing particles to escape. In these systems one can find unbounded orbits that present transient chaos before escaping through certain exit. Here, we need to keep in mind that the total integration time required for extracting information of the distribution should be smaller than the transient trapping time. We have also seen that by analysing how the shapes of the distributions change, we can detect the finite-time interval lengths when the change from the local to the global regime occurs. The Poincaré crossing time with the surface of section is a good estimator of these timescales, but unless the orbit is periodic, this crossing time depends on the selection of the surface of section. As a matter of fact, it is not constant in the phase space once the surface has been selected. We have qualitatively described the changes in shapes of the distributions of finite-time exponents. In the next chapter, we will use a quantitative indicator, the kurtosis, for describing these changes. The kurtosis values evolve from zero to positive values, as a consequence of the shape changes when the finite-time Lyapunov exponents leave the local flow dynamics and tend towards the global regime. The larger the positive kurtosis values, the more peaked the distributions will be. Finally, one observes the asymptotic regime of the flow at the timescales when the mean of the distributions begins to be centred around the final asymptotic value [35]. As mentioned before, the flow may experience several transient periods before reaching this final asymptotic state.

References 1. Abraham, R., Smale S.: Non-genericity of Ω-stability. Proc. Symp. Pure Math. 14, 5 (1970) 2. Aguirre, J., Vallejo, J.C., Sanjuán, M.A.F.: Wada basins and chaotic invariant sets in the HénonHeiles system. Phys. Rev. E 64, 66208 (2001) 3. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, p. 383. Springer, New York (1996) 4. Alligood, K.T., Sander, E., Yorke, J.A.: Three-dimensional crisis: crossing bifurcations and unstable dimension variability. Phys. Rev. Lett. 96, 244103 (2006) 5. Barreto, E., So, P.: Mechanisms for the development of unstable dimension variability and the breakdown of shadowing in coupled chaotic systems. Phys. Rev. Lett. 85 2490 (2000)

98

3 Dynamical Regimes and Timescales

6. Benzi, R., Parisi, G., Vulpiani, A.: Characterisation of intermittency in chaotic systems. J. Phys. A, 18, 2157 (1985) 7. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987) 8. Branicki, M., Wiggings, S.: Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time exponents. Nonlinear Process. Geophys. 17 (2010) 9. Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970) 10. Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in hamiltonian systems. Astron. Astrophys. 304, 374 (1995) 11. Davidchack, R.L., Lai, Y.C.: Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits. Phys. Lett. A 270, 308 (2000) 12. Dawson, S.P., Grebogi, C., Sauer, T., Yorke, J.A.: Obstructions to shadowing when a Lyapunov exponent fluctuates about zero. Phys. Rev. Lett. 73, 1927 (1994) 13. Grassberger, P.: Generalizations of the Hausdorff dimension of fractal measures. Phys. Lett. A 107, 101 (1985) 14. Grassberger, P., Badii, R., Politi, A.: Scaling Laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988) 15. Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Phys. D 7, 181 (1983) 16. Jacobs, J., Ott, E., Hunt, R.: Scaling of the durations of chaotic transients in windows of attracting periodicity. Phys. Rev. E 56, 6508 (1997) 17. Kantz, H., Grebogi, C., Prasad, A., Lai, Y.C., Sinde, E.: Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors. Phys. Rev. E 65, 026209 (2002) 18. Kottos, T., Politi, A., Izrailev, F.M., Ruffo, S.: Scaling properties of Lyapunov Spectra for the band random matrix model. Phys. Rev. E 53, 6 (1996) 19. Lai, Y.C., Grebogi, C., Kurths, J.: Modeling of deterministic chaotic systems. Phys. Rev. E 59, 2907 (1999) 20. Mancho, A.M., Wiggins, S., Curbelo, J., Mendoza, C.: Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems. Commun. Nonlinear Sci. 18, 3530 (2013) 21. Meiss, J.D.: Transient measures for the standard map. Phys. D 74, 254 (1994) 22. Oyarzabal, R.S., Szezech, J.D., Batista, A.M., de Souza, S.L.T., Caldas, I.L., Viana, R.L., Sanjuán, M.A.F.: Transient chaotic transport in dissipative drift motion. Phys. Lett. A 380, 1621 (2016) 23. Parisi, G., Vulpiani, A.: Scaling law for the maximal Lyapunov characteristic exponent of infinite product of random matrices. J. Phys. A 19, L45 (1986) 24. Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999) 25. Saiki, Y., Sanjuán, M.A.F.: Low-dimensional paradigms for high-dimensional hetero-chaos. Chaos 28, 103110 (2018) 26. Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E. 65, 036220 (2002) 27. Sauer, T.: Chaotic itinerancy based on attractors of one-dimensional maps. Chaos 13, 947 (2003) 28. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997) 29. Skokos, Ch., Bountis, T.C., Antonopoulos Ch.: Geometrical properties of local dynamics in Hamiltonian systems: the generalized alignment index (GALI) method. Phys. D 231, 30 (2007) 30. Smith, L.A., Spiegel, E.A.: Strange accumulators. In: Buchler, J.R., Eichhorn, H. (eds.) Chaotic Phenomena in Astrophysics. New York Academy of Sciences, New York (1987) 31. Stefanski, K., Buszko, K., Piecsyk, K.: Transient chaos measurements using finite-time Lyapunov Exponents. Chaos 20, 033117 (2010)

References

99

32. Szezech, Jr.J.D., Lopes, S.R., Viana, R.L.: Finite time Lyapunov spectrum for chaotic orbits of non integrable hamiltonian systems. Phys. Lett. A 335, 394 (2005) 33. Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15, 113064 (2013) 34. Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015) 35. Vallejo, J.C., Aguirre, J., Sanjuan, M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003) 36. Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov Exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008) 37. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000) 38. Viana, R.L., Pinto, S.E., Barbosa, J.R., Grebogi, C.: Pseudo-deterministic chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 1 (2003) 39. Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005) 40. Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaoshyperchaos transition in coupled systems. Phys. Rev. E 64 056235 (2001) 41. Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)

Chapter 4

Predictability

4.1 Numerical Predictability Dynamical systems describe magnitudes evolving in time according to deterministic rules. These magnitudes may evolve in time towards some final state, depending on the initial conditions and on the specific choice of parameters. In a broad sense, unpredictability or uncertainty refers to the difficulty in the determination of this final state, and we can see it from two different points of view, following [32]. From the first point of view, one can consider a system to be unpredictable when a set of initial conditions spreads out more than a specific diameter representing the prediction accuracy of interest. This region is usually of larger diameter than the one of the set of initial conditions. From a second point of view, uncertainty is related to probability. For practical purposes, in these unpredictable systems, the exact knowledge of the initial conditions might be considered irrelevant, because this knowledge neither heightens nor lowers the probability of reaching a given final state.

4.1.1 Predictability, Attractors and Basins We aim to discuss the difficulty of predicting the evolution on an initial condition towards a final state following certain trajectories. First, we will face unpredictability as the difficulty in forecasting such a final state, that is, to which attractor the initial conditions will tend to. Later on, we will focus on the forecast of the evolution of the trajectories towards such a final state. An attractor is a set of numerical values towards which a system tends to evolve, for a wide variety of starting conditions of the system. The system values that get close enough to the attractor values will remain close even if slightly disturbed. If a given dynamical system possesses only one attractor in a certain region of © Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_4

101

102

4 Predictability

phase space, then for any initial condition, its final destination is clearly determined. However, dynamical systems often present several attractors and, in these, elucidate which orbits tend to which attractor becomes a key issue. Indeed, the presence of strong sensitivity to initial conditions complicates any detailed prediction. But, because of the existence of robust attractors, we can tackle these complex behaviour systems. An attractor can be a point, a finite set of points, a curve, a manifold or even a complicated set with a fractal structure known as a strange attractor. The later many times imply the presence of a very complex behaviour. Fractal structures appear naturally in nonlinear dynamics, in such a way that the two concepts, fractals and chaos, are deeply related. Therefore, analysing the fractality of a system is especially useful to obtain information about the evolution of many dynamical systems modelling physical phenomena. A basin of attraction is defined as the set of initial conditions whose destination is a particular region of the phase space, that is, the basin is the set of points that, taken as initial conditions, are attracted to a specific attractor. Dissipative systems can have one or more attractors, and there are many examples of fractal behaviour in this context. Conversely, there are no attractors in conservative volume preserving systems. However, we can still define the exit areas as the final states of the system towards the trajectories will evolve. Therefore, fractal basins are equally frequent among open Hamiltonian systems. For energies below a certain threshold value, which is commonly called the escape energy, the orbits are bounded, and the test particles cannot leave the scattering region. But, if the energy is above this threshold value, several exits may appear, and it is possible to escape towards infinity through anyone of them. Hence, we can define exit basins in conservative systems in an analogous way to the basins of attraction in a dissipative system, by defining an exit basin as the set of initial conditions that lead to a specific exit. When there are two different attractors in a certain region of phase space, two basins exist, which are separated by a basin boundary. This basin boundary can be a smooth curve or a fractal curve. The plot of the fractal basins associated with a dynamical system provides a qualitative idea of the complications in the prediction of its future evolution. Indeed, the notions of chaos and fractals have played a very important role in the idea of uncertainty in physics, since the very beginning of the chaos theory, when joined to the concept of final state sensitivity [17]. The presence of fractal basin boundaries may have strong consequences for the prediction of the system. The fractal basins can be classified into three main classes [12, 15]. The first class is constituted by the locally connected fractal basins. This class includes quasicircles, as the Julia Sets and the Koch snowflake, among others, that one might consider are not very common in real physical systems. It also includes nonquasicircles that have striate structure, a continuous but nowhere differentiable boundary, and they are typical of dissipative systems. Finally, they also include sporadically fractal basins, with smooth boundaries except for a zero Lebesgue measure set, and fractal dimension larger than 1.

4.1 Numerical Predictability

103

The second class is formed by the locally disconnected fractal basins, such as intertwined basins, Wada basins and riddle basins [11]. The basins generated by the Smale horseshoe can be considered typical examples. Here we have boundaries with a Cantor-like structure. The intertwined basins have the property that every fractal region of the boundary has smooth subregions; thus fractality and smoothness are intertwined at every scale. But, complexity can arise from another different property. This is the Wada property [36]. Although it is hard to imagine, it is possible to have three or more regions sharing the same boundary. We are used to think in maps in two dimensions, where three countries may only coincide in one point. Topologically, this is not necessarily true for open sets. If we talk about basins, a basin verifies the property of Wada if any initial condition that is on the boundary of one basin is also simultaneously on the boundary of three or more basins. In other words, every open neighbourhood of a point belonging to a Wada basin boundary has a nonempty intersection with at least three different basins. A second type of basins in this class happens when a basin is riddled by another one, meaning that for every point in the first basin, it is possible to find arbitrarily close points belonging to the second [20]. The riddle basins often arise when chaotic systems are coupled. Finally, and just for completeness, intermingled basins are defined when all basins are riddled by the rest. The third class of fractal basins is formed by the so-called vanishing fractal basins [1]. Interestingly, in these basins, the boundaries are less fractal as resolution increases. They are typically found in undriven dissipative systems. The basins of attraction link a given set of initial conditions to its corresponding final states. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. Hence, a proper classification, and even numerical characterisation of this unpredictability, is clearly required. The need for quantifying this uncertainty led to the concept of basin entropy [8] that quantifies the above and allows the comparison of the uncertainty of one basin with the uncertainty of another one. For instance, the Wada basins are intuitively to be even more unpredictable than fractal basins, but we need a quantitative basis for such an statement. Because we are devoting our discussion to the difficulty in forecasting to which attractor the initial conditions will tend, we need an indicator of unpredictability, understood here as the difficulty in the determination of this final state of a system. This indicator can be the basin entropy. This concept strongly differs from other entropy definitions used in nonlinear dynamics, like the Kolmogorov-Sinai entropy, the topological entropy or the expansion entropy, which focus on the difficulty of predicting the evolution of the trajectories. The main idea for computing basin entropy is to build a grid in a given region of phase space, so that through this discretization, a partition of the phase space is obtained where each element can be considered as a random variable with the attractors as possible outcomes. Applying the Gibbs entropy definition to that set results in a quantitative measure of the unpredictability associated to the basins. In this way, we can use this basin entropy as a quantitative measure to compare different basins of attraction. A broadly use concept dealing with trajectories is the

104

4 Predictability

chaotic parameter set, such as the Lyapunov diagrams. In these plots, one visually illustrates the Lyapunov exponents for different pairs of parameters. Extending this idea, for building the basin entropy plots, we choose a given scaling box size, and then we evaluate the basin entropy associated to the corresponding basins of attraction for different parameter settings. The resulting plot of the basin entropy in a two-dimensional parameter space is called the basin entropy parameter set and quantifies the final state unpredictability of dynamical systems.

4.1.2 Predictability and Trajectories We have seen unpredictability as the difficulty in forecasting the final state of a system, understood as difficulty in guessing to which attractor the initial conditions will tend to. From now onwards, we will focus on the forecast of the evolution of the trajectories towards such a final state. Here, we will devote our discussions to the impact of the strong sensitivity to initial conditions in the notion of predictability of a system, related to how long a computed orbit is close to an actual orbit. This concept is related to, but independent of, its stability or its chaotic nature. A more detailed information about this concept can be found in the Sect. 1.4 of the first chapter. Here, we will resume the most important issues related to this concept. A system is said to be chaotic when it exhibits strong sensitivity to the initial conditions. As a consequence, if we consider two initially close trajectories, one of them the exact solution and the other one the computed solution, they may diverge exponentially from each other. The predictability aims to characterise if this numerically computed orbit may be sometimes sufficiently close to another true solution, so it may be still reflecting real properties of the model, leading to correct predictions. The real orbit is called a shadow, and the noisy solution can be considered an experimental observation of one exact trajectory. The distance to the shadow is then an observational error, and within this error, the observed dynamics can be considered reliable [23]. The shadowing property characterises the validity of long computer simulations, and how they may be globally sensitive to small errors. The shadowing distance is the local phase space distance between both of them, and the shadowing time measures how long a numerical trajectory remains valid by staying close to a true orbit. The shadows can exist, but it may happen that, after a while, they may go far away from the true orbit. Consequently, a proper estimation of the shadowing times is a key issue in any simulation and provides an indication about its predictability. The shadowing time is directly linked to the hyperbolic or nonhyperbolic nature of the orbits. Hyperbolic systems are structurally stable in the sense that the shadowing is present during long times and numerical trajectories stay close to the true ones. In the case of nonhyperbolicity, an orbit may be shadowed, though only for a very short time. After that, the computed orbit behaviour may be completely different from the true one.

4.1 Numerical Predictability

105

Nonhyperbolic behaviour can arise from tangencies (homoclinic tangencies) between stable and unstable manifolds, from unstable dimension variability or from both. In the case of tangencies, there is a higher, but still moderate obstacle to shadowing. In the so-called pseudo-deterministic systems, the shadowing is only valid during trajectories of reasonable length due to the unstable dimension variability (UDV). The UDV is reflected and quantified by the fluctuations around zero of the finitetime Lyapunov exponent closest to zero [7, 31]. Fluctuations around zero of the largest transient exponent were described for attractors of quasiperiodically forced systems in [13]. Note that there are situations where the positive tails do not appear due to UDV but rather by other mechanisms, such as the quasi-tangencies between the stable and unstable manifolds near a homoclinic crisis point. A key issue when there are strong obstacles to shadowing is the calculation of the shadowing time τ , since this constrains the predictability of the system. This is specially relevant in high-dimensional systems, where it is hard to develop a good understanding of model accuracy or error growth. When the shadowing times are very short, averaged quantities, such as the Lyapunov exponents or even faster chaoticity indicators, should be handled with care, because the computed trajectories could move away from any real trajectory before reaching the necessary averaging time that could lead to unreliable results. In addition, we have seen in Sect. 3.3 that some trajectories might possess transient phenomena. All these issues point to use finite exponents, able to capture the possible hyperbolic nature of the flow and to use them for estimating the shadowing times. Once these shadowing times have been computed, we can use them as the proper averaging times. This can be applied to any averaged indicator and is of special interest in Monte Carlo simulations, based on averaging results from many initial conditions [22]. The probability distributions for calculating the shadowing time can be justified from the statistical properties of the finite-time Lyapunov exponents [30]. The distributions with UDV present a scaling law that for small shadowing times is algebraic and exponential for large times (longer shadowing times are exponentially improbable). The shadowing time typically increases exponentially when the unstable dimension changes [9]. Then, it decreases exponentially in the hyperbolic regions, with a lower bound determined by the computer round-off. These switches occur randomly in time, so they mimic a biased random walk behaviour. The most reliable results are obtained when the transversely attracting and repelling contributions nearly counterbalance. This means to have a mean closest to zero and expansions and contractions well approximated by such an random process. This process can also be described as a diffusion process where a given trajectory has a different escape route [4]. The effective range of the interactions between escape points is associated with the largest Lyapunov exponent. The shadowing is large when the escape points are located in an unstable periodic orbit. The effects of the kicks in the pseudo-

106

4 Predictability

trajectories are included as a reflecting barrier. Such diffusion process has an equilibrium distribution leading to a shadowing time τ given by τ ∼ δ −h ,

(4.1)

where δ is the round-off precision of the computer and where the h index is h=

2m . σ2

(4.2)

The exponent h is called hyperbolicity index, and it depends on m and σ , the mean and the standard deviation of the Lyapunov exponent closest to zero. We will use this exponent as an indicator of the predictability of the orbits. The lowest predictability occurs when h is very small and there is no improvement in τ , even for large values of δ. Conversely, the larger the h index, the better the shadowing. A key issue when using the h index is the analysis of its dependence on the mean m and the standard deviation σ . This means its dependence on the shape of the finite-time Lyapunov exponent distributions, which in turn strongly depend on the selected time intervals used for building them. As we have seen, at the shortest finite intervals, the h index explores the dynamics at very local scales. Conversely, as the finite interval grows, the local details are washed out. This precise interval Δt when the distributions begin to explore the global regime will provide us with the desired predictability of the system at nonlocal scales. Hence, when computed at these intervals, the “hyperbolicity” index h can effectively be considered a “predictability” index, and this is the convention that we will follow from now onwards. This means that we will need to focus our discussions on the dependency of h with the finite interval length and on the scaling laws present that may affect to the shapes of the distributions. The scaling laws for h are derived from the first and second cumulants [10, 24]. The variance is inversely proportional to the interval in ergodic orbits [14]. We can also find algebraic powers when intermittency is present [21]. Furthermore, we can find correlations decaying more slowly than the inverse of the time interval [25]. These scaling laws are closely related to intermittency and can be considered as “intermittency in miniature”. The exponential distribution is the result of small excursions that periodically move the computed trajectory away from the true trajectory and then return towards it. The assumption is that the motion follows a biased random walk, with a drift towards a reflecting barrier. The flow sometimes goes in one direction, far away from the true solution, and sometimes moves towards it. The reflecting barrier is caused by the single-step error δ, since new errors are created at each step, so that the computed trajectory can never be expected to be closer than δ to the true trajectory.

4.2 The Predictability Index

107

4.2 The Predictability Index A sign of bad shadowing is the fluctuating behaviour around zero of the closest to zero finite-time Lyapunov exponent. Plotting the finite-time distribution and assuming both the mean m and the standard deviation σ to be very small, the shadowing time τ is given by the Eq. 4.2, where the exponent h is the predictability index. This exponent is used as an indicator of the predictability of the orbits. The lowest predictability corresponds to very small h values, while large h values are a sign of a good shadowing. One important issue is to perform the h computation using a closest to zero finite-time Lyapunov exponent, since this exponent reflects in principle the varying number of unstable dimensions along the trajectory. But, as we have seen in the previous sections, the distributions of the finite-time Lyapunov exponents, used for computing h, evolve as the Δt intervals lengths grow. We have also seen that the distributions change as the interval lengths leave the local regime and enter the global regime. Aiming to check if these variations are detected by the predictability index h, we can calculate this exponent for a variety of interval lengths in both the Hénon-Heiles system, described in Sect. 2.9, and the Contopoulos system, described in Sect. 3.5. We have computed the distributions for a variety of finite interval lengths, and the results are plotted in Fig. 4.1. In general, h grows with the interval length. For the shortest intervals, there are no Gaussian distributions, and the values cannot be regarded as random variables. The exponents oscillate and h keeps small, as the variance is small. For the nonUDV orbit of the Hénon-Heiles system, m is far from zero, and h is large. When h is computed from χ1 and is compared with that from χ2 , the results are different even when both χ1 and χ2 fluctuate and are well correlated. The biased random walk model might not be fully applicable, but as the values are accumulated along a given orbit, they provide useful information in all orbit types, when computed from the second exponent h(χ2 ). For the largest intervals, distribution shapes are Gaussian-like, the correlations die out, and the ergodic theorem might be applied. The h(χ1 ) value, conversely to the h(χ2 ) value, has a wider span of values depending on the orbit type with the larger intervals. We may conclude by observing this plot that at local timescales, there are many oscillations in h. However, as the intervals grow, it seems that is possible to differentiate between the different orbit types and the corresponding h values. For that, we need to make a quantitative analysis for detecting changes in the shapes of the distributions. We aim to detect the finite-time interval lengths when the change from the local to the global regime occurs. The Poincaré crossing time with the surface of section Tcross is a good estimator of these timescales, but unless the orbit is periodic, this crossing time depends on the selection of the surface of section. As a matter of fact, it is not constant in the phase space once the surface has been selected.

108

4 Predictability

Fig. 4.1 Hyperbolicity index, as computed from the distributions of finite-time Lyapunov exponents, corresponding to χ1 (upper diagram) and χ2 (lower diagram). Contopoulos system values are marked by diamonds. (Dotted) close to period 2, (Dashed) chaotic (between tori orbits), (Dashed dot) chaotic. Hénon-Heiles system values are marked with triangles. (isolated point) UPO, (Long Dashed) close to period 5, (Solid) weakly chaotic, cycle. (Taken from [27] with permission)

So, we will complement the Tcross with the kurtosis values of the finite-time distributions. The kurtosis is usually considered a standard indicator of the sharpness of the peak of a frequency-distribution curve. However, its definition relies on a scaled version of the fourth moment of the data, and it measures better the existence of heavy tails and not the peakedness. Because the kurtosis of the normal distribution is 3, distributions with kurtosis less than 3 are said to be platykurtic, meaning that the distribution produces fewer and less extreme outliers than it does the normal distribution. Distributions with kurtosis greater than 3 are said to be leptokurtic, as the Laplace distribution that has tails that asymptotically approach zero more slowly than a Gaussian.

4.2 The Predictability Index

109

For the shake of simplicity, we will refer as kurtosis to the actual “excess kurtosis”, defined as “kurtosis−3”. Taking into account this convention, the “kurtosis” of the normal distribution is 0. A kurtosis above zero (i.e. excess kurtosis) is typically related to the existence of higher peaks, because heavy-tailed distributions sometimes have higher peaks than light-tailed distributions, although this does not imply that the distribution is flat-topped as sometimes reported [33]. Nevertheless, we are not specifically interested in detecting the existence of peaks, but in detecting the separation from the local behaviour. As we have seen in previous sections, this is typically related to the separation of a flat-topped distribution and the creation of tails and, possibly, the convergence towards the asymptotic value. So, the kurtosis indicator seems to be a simple and useful way for detecting these changes. The kurtosis of the distributions of finite-time Lyapunov exponents will evolve from zero to positive values, as a consequence of the shape changes when the finitetime exponents leave the local flow dynamics and tend towards the global regime. The larger the positive kurtosis values, the more heavy-tailed and likely more peaked the distributions will be. Finally, one observes the asymptotic regime of the flow at the timescales when the mean of the distributions begins to be centred around the final asymptotic value [26]. As mentioned before, the flow may experience several transients before reaching this final asymptotic state.

4.2.1 The Hénon-Heiles System In this section we will apply these ideas and compute the predictability index as derived from the Eq. 4.2 in simple two degree-of-freedom Hamiltonian systems. We will check if the underlying diffusion model is valid in these conservative systems, where several asymptotic exponents are zero. The Hénon-Heiles system has been described in detail in Chaps. 2 and 3. We have selected four initial conditions leading to four prototypical behaviours of this system. These orbits can be seen in Fig. 4.2, and their corresponding initial conditions are listed in Table 4.1. The first analysed case is the orbit labeled as H1. The Poincaré section is depicted in Fig. 4.3 (top). This orbit is a weakly chaotic orbit with λ = 0.015. When considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 13.0, with a minimum value of 10.1. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 16.5, with a minimum value of 12.4. These timescales roughly indicate the change of behaviour of the finite-time Lyapunov exponent distributions as the finite-time intervals grow. In Fig. 4.3 (bottom) we have plotted the hyperbolicity index computed from the closest to zero exponent distributions and the corresponding kurtosis values, against

110

4 Predictability

0.6

H1

0.4

H2

0.4 0.2

y

y

0.2 0

0 -0.2

-0.2

-0.4 -0.4

-0.4

-0.2

0

0.2

0.4

-0.4

x

-0.2

0

0.2

0.4

0

0.2

0.4

x

0.4

H3

0.4

0.2

H4

y

0.2

y

0

0

-0.2

-0.2 -0.4 -0.4

-0.4 -0.2

0

0.2

0.4

x

-0.4

-0.2

x

Fig. 4.2 The four orbits selected for calculating their predictability in the Hénon-Heiles system The corresponding initial conditions are listed in Table 4.1. Upper left: H1, a weakly orbit with asymptotic Lyapunov exponent λ = 0.015. Upper right: H2, a sticky, chaotic asymptotically, orbit, with asymptotic Lyapunov exponent λ = 0.046. The points with a regular-like transient period t < 4000 are plotted in darker colour. Bottom left: H3, a regular orbit, linked to a period 1 orbit, with asymptotic Lyapunov exponent λ = 0.0. Bottom right: H4, a regular orbit, linked to a period 5 orbit, with asymptotic Lyapunov exponent λ = 0.0. (Taken from [29] with permission) Table 4.1 Selected orbits for the Hénon-Heiles system Orbit H1 H2

H3 H4

Description Weakly chaotic, cycle orbit Sticky, asymptotically chaotic, orbit Regular, close to period 1 orbit Regular, close to period 5 orbit

Initial condition for given Energy x = 0.000000 y = −0.119400 vx = 0.388937 E = 1/12 x = 0.000000 y = 0.095000 vx = 0.396503 E = 1/8

λ 0.015

< Tcross > 14.8

0.046

13.9

x = 0.000000 y = 0.137500 vx = 0.386627 E = 1/12 x = 0.000000 y = −0.031900 vx = 0.307044 E = 1/8

0.0

12.4

0.0

12.9

λ is the asymptotic standard Lyapunov exponent. The notion weak or strong chaos is associated to the relatively smaller or larger value of λ. < Tcross > is the averaged Poincaré section crossing time

a variety of increasing finite interval lengths Δt. The total integration time used to build the distributions is T = 105 when Δt < 50.0 and T = 106 for larger intervals sizes.

4.2 The Predictability Index

111

0.4

H1 Vy

0.2 0 -0.2 -0.4 -0.4

0

y

0.2

0.4

h

k

2 1.5 1 0.5 0 -0.5 300 250 200 150 100 50 0

-0.2

10

20

30

40

50

60

30 Δt

40

50

60

0.05 0.04 0.03 0.02 0.01 0 -0.2 -0.1 0 0.1 0.2

10

20

Fig. 4.3 Hénon-Heiles weakly chaotic orbit H1. Top: Poincaré section y−vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time Lyapunov exponent distributions as the finite-time interval length is increased. Inset: Finite-time Lyapunov exponent distribution for Δt = 25.1. The predictability index is h = 54.0. (Taken from [29] with permission)

There is a clear trend of increasing h values as the interval size is larger. The kurtosis shows a similar evolution from the most negative values towards the positive ones. The kurtosis curve crosses the zero value at Δt = 25.1. The corresponding finite-time Lyapunov exponent distribution of the closest to zero exponent for this interval size is seen in the inset of the figure. It is characterised by a mean m = 0.03 and a probability of positivity P+ = 0.8. The Δt is large enough to allow the deviation vectors to enter in the global regime of the flow, but it is not large enough to reach the asymptotic zero value. Regardless of the

112

4 Predictability

above, some oscillations around zero are already detected, and these oscillations can be considered as a good indicator of the nonhyperbolicity of the flow. The corresponding predictability index is h = 54.4. We note here that because of the small slope of the kurtosis and predictability curves, small changes of the estimation of the interval size do not lead to large variations in the predictability estimation. The second analysed case is the orbit labeled as H2 in Fig. 4.2 and Table 4.1. The corresponding Poincaré section is depicted in Fig. 4.4 (top). This is a chaotic

0.4

H2

Vy

0.2 0 -0.2 -0.4 -0.4

0

y

0.2

0.4

0.6

k

0.8 0.6 0.4 0.2 0 -0.2 -0.4 200

-0.2

150 h

100 50 0

10

20

30

40

50

60

30 Δt

40

50

60

0.03 0.025 0.02 0.015 0.01 0.005 0 -0.2-0.1 0 0.1 0.2 0.3

10

20

Fig. 4.4 Hénon-Heiles sticky, chaotic asymptotically orbit H2. Top: Poincaré section y − vy with plane x = 0 and vx > 0. A regular-like transient period t < 4000 is overlaid with darker colour. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponents distribution for Δt = 11.0. The predictability index is h = 20.9. (Taken from [29] with permission)

4.2 The Predictability Index

113

orbit with λ = 0.046. Considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 13.4, with a minimum value of 8.9. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 14.5, with a minimum value of 7.5. In Fig. 4.4 (bottom), we observe the trend of increasing kurtosis with Δt. The kurtosis zero-cross is found at Δt = 11.0. The corresponding closest to zero exponent finite-time distribution is seen in the inset of the figure. It is characterised by a mean m = 0.04 and a probability of positivity P+ = 0.7. The derived predictability index is h = 20.9. This is a worse predictability value than the previous case, yet similar in order of magnitude. We may conclude that the shadowing timescales are similar in both cases. As both orbits have positive λ values, they are chaotic, yet predictable. We have seen that the orbit H1 has a relatively small Lyapunov exponent, so a relatively long Lyapunov time. This is a prototypical behaviour for a particle being chaotic, but confined to a certain region of the available phase space. But there are chaotic orbits with positive Lyapunov exponent values that show regularlike appearance during certain transient periods. These orbits stick during these transients close to islands of stability before entering in the big chaotic sea. These periods can sometimes be very short and sometimes very long. These orbits are called sticky orbits, or confined orbits [3], because they generate confined structures in the configuration space. The sticky, chaotic asymptotically, orbit H2 presents one regular-like transient during the first 4000 time units. The Poincaré section corresponding to this period is seen in Fig. 4.5 (top). Considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 14.6, with a minimum value of 13.8. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is also Tcross = 14.6, with a minimum value of 13.2. In Fig. 4.5 (bottom), we observe the trend of increasing kurtosis with Δt. The kurtosis zero-cross is found at Δt = 19.1. The corresponding closest to zero exponent finite-time distribution is seen in the inset of the figure. It is characterised by a mean m = −0.01 and a probability of positivity P+ = 0.28. The derived predictability index is h = 31.7. This means a higher predictability during the regular-like transient when compared with the predictability value resulting from integrating beyond the transient lifetime. However, this value is lower than the value of the chaotic orbit H1. This is sourced to the selection of one of the lowest values of the available ones during the distribution shape transition, where the h values suffer several oscillations, as seen in Fig. 4.5 (bottom). But it is also sourced to the nature of the transient that, being regular in appearance, it is not a truly regular motion. The third analysed case is the orbit labeled as H3 in Fig. 4.2 and Table 4.1. We have chosen this orbit because we want to analyse the applicability of the power law to orbits with zero Lyapunov exponents in addition to the obvious two central trivially zero exponents. This is a regular orbit with λ = 0.0, where all exponents are zero because the Hénon-Heiles system is a 2 degree-of-freedom Hamiltonian system.

114

4 Predictability

-0.15

Vy

-0.2 -0.25 -0.3 -0.35

H2

0.5 0 -0.5 -1 -1.5 -2 605

h

k

-0.4 -0.2

50 40 30 20 10 0 5

-0.1

10 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.2 -0.1

0

y

0.1

0.2

15

20

25

15 Δt

20

25

0

10

Fig. 4.5 Regular-like period of the Hénon-Heiles chaotic orbit H2. The figure shows the points under the regular-like transient period t < 4000. Top: Poincaré section y −vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponents distribution for Δt = 19.1. The predictability index is h = 31.7. (Taken from [29] with permission)

The corresponding Poincaré section is depicted in Fig. 4.6 (top). Considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 12.4, with a minimum value of 11.6. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 12.4, with a minimum value of 11.7. The evolution of the predictability index h with the interval size is shown in Fig. 4.6 (bottom). The kurtosis shows the previous trend from the most negative

4.2 The Predictability Index

0.2

115

H3

Vy

0.1 0 -0.1 -0.2 -0.4

-0.35

-0.3

-0.2

-0.25

2 1.5 1 0.5 0 -0.5 300 250 200 150 100 50 0

h

k

y

10

20

30

40

50

60

30 Δt

40

50

60

0.04 0.03 0.02 0.01 0 -0.04-0.02 0 0.020.04

10

20

Fig. 4.6 Hénon-Heiles regular orbit H3. Top: Poincaré section y − vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponents distribution for Δt = 48.3. The predictability index is h = 105.5. (Taken from [29] with permission)

values towards the positive ones. The zero crossing is found at Δt = 48.3. The corresponding distribution of the closest to zero exponent is plotted in the inset of the figure. It is characterised by a mean m = 0.01 and a probability of positivity P+ = 0.8. The computed predictability index value h = 105.5 is higher than the previous cases. When one compares the predictability of this orbit with the previous cases, the obtained predictability index h is one order of magnitude larger. The biased random walk seems to be applicable to the final invariant state, even when the finite-time exponent distributions of regular orbits do not follow a normal distribution shape.

116

4 Predictability

This means that the test particle sometimes approaches the real orbit, having the machine precision as bias, in the contracting directions, and sometimes moves farther away from the real orbit in the expanding directions. The fourth analysed case in the Hénon-Heiles system is the orbit labeled as H4 in Fig. 4.2 and Table 4.1. This is a regular orbit with λ = 0.0, associated to a period 5 orbit. The corresponding Poincaré section is depicted in Fig. 4.7(top). Considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 12.9, with a minimum value of 12.3. If we consider the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 12.9, with a minimum value of 9.7.

0.2

H4 Vy

0.1 0 -0.1

h

k

-0.2 0

2 1.5 1 0.5 0 -0.5 -1 300 250 200 150 100 50 0

0.1

10 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

-0.05 0

10

20

0.2

0.3

y

0.4

0.5

30

40

50

60

30 Δt

40

50

60

0.05

20

Fig. 4.7 Hénon-Heiles regular orbit H4. Top: Poincaré section y − vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponents distribution for Δt = 32.1. The predictability index is h = 69.2. (Taken from [29] with permission)

4.2 The Predictability Index

117

The evolution of the predictability index h with the interval size is shown in Fig. 4.7 (bottom). We may also observe the previous evolution of the kurtosis from negative to positive values. Furthermore, the kurtosis zero-cross is observed at Δt = 32.1.1 The corresponding closest to zero exponent distribution is plotted in the inset of the figure. It is characterised by a mean m = −0.02 and a probability of positivity P+ = 0.2. Because of the Hamiltonian exponents pairing properties, the results are equivalent except for the reversed signs to the previously discussed cases. The computed predictability index is h = 69.2, which is in agreement with previous cases. This value indicates a better predictability for this orbit than for the previous chaotic cases, though the regular orbit labeled as M3 has still the best predictability.

4.2.2 The Contopoulos System Here, we will analyse orbits that behave in appearance like regular orbits, but they have a λ > 0; therefore they are chaotic. These orbits are found in the Contopoulos system [5] described in the Sect. 3.5 of the previous chapter. The fixed parameters of the model are the same, with amplitude parameters A = 1.6 and B = 0.9, as well as the initial condition x = 0.03744, y = 0, vx = 0.0480, and the energy value is E = 0.00765. We consider first the case when the control parameter  = 4.4. This is the orbit labeled as C1 in Table 3.1. The Poincaré section x − vx with the plane y = 0 is seen in Fig. 4.8 (top). This is a regular in appearance, very thin chaotic strip, with λ = 0.093. When we consider the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 13.8, with a minimum value of 6.0. When we consider the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 14.2, with a minimum value of 13.9. The evolution of the predictability index h with the interval size is shown in Fig. 4.8 (bottom), where we can see the evolution of the kurtosis from negative to positive values. The kurtosis zero-cross is found at Δt = 13.9. The corresponding closest to zero exponent distribution is plotted in the inset of the figure. It is characterised by a mean m = −0.01 and a probability of positivity P+ = 0.3. The computed predictability index is h = 24.2, which is in agreement with previous cases and this value is very similar to the predictability of the chaotic cases of the Hénon-Heiles system, confirming the lower predictability of the “regular” in appearance, though chaotic orbit. The second case analyses the same initial condition where we fix  = 4.5. This is the orbit labeled as C2 in Fig. 4.9 and Table 3.1. The Poincaré section x − vx with

see another zero crossing at around Δt = 9.0. This value is slightly below the Tcross range of values. But as Δt increases, the distribution returns to a flat shape again. As a consequence, the peaks appear because we are still in the local regime.

1 We

118

4 Predictability

0.04

C1

Vx

0.02 0

-0.02 -0.04 0.036 0.038 0.04 0.042 0.044 0.046

x k

0.5 0 -0.5 -1 -1.5 -2 50 5

20

15

25

0.04 0.03

h

40 30 20 10 0

10

0.02 0.01 0 -0.2 -0.1

5

10

15 Δt

20

0

0.1 0.2

25

Fig. 4.8 Contopoulos chaotic orbit C1. Top: Poincaré sections x − vx with plane y = 0 and vy > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time interval length is increased. Inset: Finite-time exponents distribution for Δt = 13.9. The predictability index is h = 24.2. (Taken from [29] with permission)

plane y = 0 is seen in Fig. 4.9 (top). This is a weakly chaotic orbit with λ = 0.0125. This orbit is very close to a periodic orbit, that means an averaged Poincaré section crossing time Tcross = 14.5, both of them for crosses of the x = 0 plane with vx > 0 and also when we consider the crosses of the y = 0 plane with vy > 0. The evolution of the predictability index h with the interval size is shown in Fig. 4.9 (bottom). The kurtosis zero-cross in the evolution of the kurtosis from negative to positive values is found at Δt = 17.15. The corresponding closest to zero exponent distribution is plotted in the inset of the figure. It is characterised by a mean m = −0.003 and a probability of positivity P+ = 0.4. The values of

4.2 The Predictability Index

119

-0.0478 -0.04785

C2

Vx

-0.0479 -0.04795 -0.048 -0.04805 0.0374

0.03742 0.03744 0.03746

0.03748

x k

8 6 4 2 0 -2 50 50.08 40 0.06 0.04 30 0.02 20 0 10 0 5

15

20

25

15 Δt

20

25

h

10

-0.05

0

0.05

10

Fig. 4.9 Contopoulos weakly chaotic orbit C2. Top: Poincaré sections x − vx with the plane y = 0 and vy > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time interval length is increased. Inset: Finite-time exponents distribution for Δt = 17.15. The predictability index is h = 11.9. (Taken from [29] with permission)

the mean and P+ reflect that the asymptotic behaviour has already been reached at these timescales with contracting and expanding oscillations around zero of equal likehood. The figure shows strong oscillations in the predictability curve h against Δt. These oscillations are related to the presence of peaks in the distributions and the non-ergodic nature of the orbit. These oscillations make the h value to have strong variations with Δt, but even with these oscillations, we can see the interval belonging to the kurtosis zero-cross. The predictability index as computed from the selected Δt is then h = 11.9. This predictability index is in agreement with previous cases, and it means a lower predictability than in previous chaotic cases.

120

4 Predictability

4.2.3 The Rössler System We have seen how to compute the predictability of an orbit by computing the predictability index h from the finite-time distributions, following Eq. 4.2, in a variety of orbits corresponding to conservative systems. Now, we will proceed to compute the predictability in a dissipative system, the Rössler system, described in the Sects. 2.8 and 3.6 of the previous chapters. The initial condition is kept the same (1, 1, 0, −1, −5, 0), as well as the selection of the control parameters, a and d. The main difference to take into account when comparing the above analysed conservative cases, and this new one is that we are dealing with orbits that will end into a chaotic attractor. In the conservative cases, every orbit may have their own timescales and evolve towards the global regime at its own pace. In the case of an attractor, all initial conditions found in its basin of attraction will end there [2]. The timescales for all those orbits will be rather similar, depending on the nature of the attractor itself. We have computed the h index for several Δt intervals, even though Eq. 4.2 is only valid just for ergodic distributions, with a Gaussian-like shape. This can be observed in Fig. 4.10. This figure plots the evolution of h(Δt) for a fixed value of d = 0.25 and several a values. When Δt is small, the short finite times prevent the convergence of the exponents towards a limiting value. For this reason, the h values do not reach a final value, and

(I) - Hyperbolicity d=0.25

500 400

h

300 200 100 0

0

20

40

Δt

60

80

100

Fig. 4.10 Hyperbolicity indexes h calculated from the distributions of the closest to zero exponent, for different Δt intervals. The control parameter d, coupling strength, is fixed to be d = 0.25. Calculations start in a = 0.34, every curve increases a in 0.05 units. Continuous curves are a < 0.365. Dashed curves are those with a > 0.365. The regimes with low and high hyperbolicity are clearly identified, but only with a large enough Δt ∼ 25 interval. At larger intervals, the hyperbolicity index is reliable enough to reflect the predictability of the system. (Adapted from [28] with permission)

4.2 The Predictability Index

121

consequently they do not allow to distinguish among different regimes. But as we can observe, for Δt values larger than 25, there are two main groups of curves h(Δt). One upper set corresponds to the values a < 0.365, which corresponds to the nonchaotic regime, and is plotted as continuous curves. The lower set, in dashed curves, corresponds to a > 0.365, containing the chaotic and hyperchaotic regimes. So, we are able to identify both regimes, and it is clear that this can be achieved starting from a given timescale. Therefore, when we compute the hyperbolicity index h at these non-local intervals, we do not observe variable results resulting from the flow local properties, and we can name h as predictability index. We can plot a similar diagram by fixing the control parameter a and keeping free the coupling d. This has been done in Fig. 4.11. This figure depicts the evolution of h with Δt for a fixed a = 0.385 and several d values. For almost every coupling strength d, the system can be considered hyperchaotic, implying low values of h. However, when d ∼ 0.175, we find only one positive exponent, implying higher values of h. This can be clearly observed for Δt values larger than 40. By observing both plots, one can conclude that there is an obvious dependence of the computed predictability not only on the combination of parameters a and d but also on the size of Δt, as expected. But what is more important, we have identified a threshold, common to all orbits (i.e. same initial condition, but different control parameters), from which the predictability is different and detects the different behaviour for every orbit. We have plotted the h index against the Δt interval lengths, for different values of the control parameters. We can reverse these plots, and we will plot now the h exponent against the control parameters, for a variety of Δt intervals.

(II) - Hyperbolicity a=0.385

80

h

60 40 20 0

0

20

40

Δt

60

80

100

Fig. 4.11 Hyperbolicity indexes h calculated from the distributions of the closest to zero exponent, for different Δt intervals. The control parameter a is fixed to be a = 0.385. Calculations start in d = 0.1; every line increases d in 0.02 units. Notice the dashed line d = 0.174 that is clearly separated from the remaining hyperchaotic cases with a large enough Δt ∼ 50 interval. At larger intervals, the hyperbolicity index is reliable enough to reflect the predictability of the system. (Adapted from [28] with permission)

122

(I) - Hyperbolicity d=0.25

800 600

h

Fig. 4.12 Hyperbolicity index h against a, calculated from the distributions of the closest to zero exponent, for different Δt intervals. The black continuous curve is Δt = 100. The red dashed is Δt = 50 and the green dotted is Δt = 25. Fixed coupling strength d = 0.25. The general trend of h decreasing with a is observed at all intervals, though for larger values of Δt, we can see the details much better. (Adapted from [28] with permission)

4 Predictability

400 200

0.33

0.34

0.35 0.36 0.37 0.38 Oscillator Parameter (a)

0.39

This is done aiming to establish the most appropriate interval length for the computation of the hyperbolicity index. Figure 4.12 plots the evolution of h(a) for different values of Δt, with a fixed value d = 0.25. It can be clearly observed that the hyperbolicity index h decreases as a increases. The black continuous curve corresponds to the larger interval size Δt = 100. The red dashed curve is Δt = 50 and the green dotted curve is Δt = 25. The larger the time interval Δt, the higher the detail in the observed structures. This is specially relevant for detecting the lowest predictability valley at around a = 0.36, which coincides with the onset of the hyperchaotic regime. We can see in the figure that the valley can be detected as from Δ = 25, even when this is more clearly seen at larger intervals. The Fig. 4.13 plots h(d) with a fixed value a = 0.385, where we can see a roughly constant low predictability h for any coupling strength d. Interestingly, even at the smallest sizes of the intervals Δt, the high predictability peak is clearly detected in this almost hyperchaotic slice. We can conclude from both figures that for Δt ∼ 50 or larger, the different regimes, and thus different predictability intervals, can be identified at sufficiently large interval lengths. Because this system is specially well suited to analyse the complex behaviour of chaotic dissipative systems, where different behaviours can be observed depending on the values of the control parameter, we extend the above results to the full parametric space a-d. We want to see if there is a general pattern with the interval Δt or, conversely, there is no general pattern and the interplay of the control parameters mix in a complex way. Aiming to do so, we can plot the h exponent computed from the closest to zero exponent in the full parametric space a − d, for different Δt values. This can be seen in Fig. 4.14. The figures Figs. 4.12 and 4.13 correspond to slices from the Fig. 4.14 generated by fixing the parameter d = 0.25 and the parameter a = 0.385, respectively.

4.2 The Predictability Index

250

(II) - Hyperbolicity a=0.385

200 150

h

Fig. 4.13 Hyperbolicity index h against d, calculated from the distributions of the closest to zero exponent, for different Δt intervals. The black continuous curve is Δt = 100. The red dashed is Δt = 50 and the green dotted is Δt = 25. We fix the parameter a = 0.385. The high predictability peak at d ∼ 0.17 is better seen with the largest Δt. (Adapted from [28] with permission)

123

100 50 0.1

0.2 0.15 0.25 Coupling strength (d)

Fig. 4.14 Predictability chart, or h exponent computed from the closest to zero exponent, for a given oscillator parameter a and coupling strength d. Darker values indicate lower values of h meaning a poor predictability. From top to bottom, and from left to right, Δt = 1 and T = 10,000, Δt = 25 and T = 10,000, Δt = 50 and T = 100,000 and Δt = 100 and T = 100,000. (Adapted from [28] with permission)

124

4 Predictability

We can observe from this figure that when we use the smaller Δt, which in principle is associated to the less reliable h predictability values, there are still regions which are identified as having different predictability behaviour. This plot allows us to identify different predictability zones even for the smaller intervals in certain areas of the parametric space. When inspecting Fig. 4.14, we can also see that two main different behaviour areas are clearly visible, since the available parametric space is divided in two behaviour regions (left and right) from Δt = 25 onwards. Indeed, some specific regions can be differentiated as having a different behaviour even at Δt = 1, although this identification is not very clear in this extreme case. This is the case of the upper-and-leftmost corner of the a-d diagram, corresponding to the higher coupling d and lower a values. This region behaves differently than the others, and this behaviour is seen even using the shortest interval. However, other regions are clearly identified only at larger Δt, when the distributions are nearly Gaussian, and both m and σ are small enough. This means that the amount of time required for reaching a Gaussian-like shape, when the h exponents are more reliable, vary with the a −d values. This amount of time is related to the correlation time. Some regions are easily identified as having a different predictability behaviour that require shorter Δt, while other regions require larger Δt. These results (predictability charts) focus on finding the interval sizes for detecting the nonhyperbolic cases of worst predictability. We can get some additional insight into the sources of the nonhyperbolicity by comparing these predictability h charts with the hyperchaoticity charts and positivity charts that were previously seen in Fig. 2.7 (Sect. 2.8) and Fig. 3.13 (Sect. 3.6), respectively. The nonhyperbolicity can arise from tangencies between stable and unstable manifolds, from UDV or from both. When UDV is present, the shadowing times can be very short. The larger or shorter shadowing times are linked to the higher or lower h index values. In turn, the presence of UDV can be reflected by the oscillations around zero of the closest to zero exponent, which can be seen in the positivity charts. Therefore, we can compare Fig. 3.13 (positivity charts) and Fig. 4.14 (predictability index charts), because the later reflect the around-zero oscillations. These comparisons provide some hints about the role of UDV in the loss of predictability. As an starting point, our system is very close to the one shown in [35] and [34], where UDV was reported to be present. So UDV is likely the source for the nonhyperbolicity, at least in the cases of worst predictability (smaller shadowing times). We see that there is a good agreement among the darkest areas of both figures, mainly the central part, where both P+ ∼ 0.5 and h are low. Now, we should be aware that at the largest intervals, the P+ is not properly detected, since the distributions are tending towards the asymptotic global value. Again, Δt ∼ 25 seems to be an adequate range for comparison. Some regions of different behaviour, as the one conforming the right part of the parametric space, are nevertheless detected with almost every Δt interval. In this region, we obtain low predictability h values. But, there are no large oscillations around zero, as reflected on how P+ deviates from 0.5.

4.3 Concluding Remarks

125

Hyperchaos is a common source for UDV. When comparing the worst predictability areas (darker areas of Fig. 4.14) with the high chaotic areas of Fig. 2.7, we see that the darker zones roughly match with the hyperchaos areas of Fig. 2.7. However, this matching is not perfect, and it indicates that hyperchaos is not always the source for UDV. Conversely, no area of high chaoticity matches with a high predictability area. When comparing the high predictability areas (brighter areas in the rightmost column of Fig. 4.14) with the less chaotic areas of Fig. 2.7 (those with none or just one single positive exponent), we note that they are similar, but not identical. This means that not all well-behaved areas have the same order of predictability. As discussed previously, these comparisons are better when using the largest Δt. But even at Δt ∼ 25 or even less, the chart can be of interest.

4.3 Concluding Remarks This chapter deals with the forecast of predictability and not with the forecast of chaoticity. Both terms are closely related, but they do not always follow the same trend. We have estimated the predictability index for a variety of prototypical orbits in several cases. Here we should clearly distinguish between reliability and predictability. The reliability time is the inverse of the asymptotic Lyapunov exponent, which means that it is connected to the possible chaotic nature of the analysed orbits. Conversely, we analyse here the predictability of the system, understood as a measure of its shadowing properties. We have seen how analysing the changes in the shapes of the distributions one can derive the predictability index. The finite-time Lyapunov exponent distributions reflect the underlying dynamics [26], and by using arbitrarily oriented deviation axes, one can detect varying the finite-time interval lengths, when there is a change from the local to the global, not yet asymptotic regime. The key issue here is that the finite-time Lyapunov exponents are computed with an initial random orientation of the ellipse axes, that is, reset once the finite interval is integrated. This allows to obtain different predictability indexes that will detect the cross from local to global regime [27]. A sign of bad shadowing is the fluctuating behaviour of the closest to zero of the available Lyapunov exponents. In a general case, there can be several exponents tending to zero. Following the methods presented in [28] for a dissipative system, one should increase the finite-time interval length and select the closest to zero for computing the predictability index. In dissipative systems, the finite interval size where there is a change from local to global is the same, because all close enough orbits end in the same attractor, evolving towards similar timescales. But in conservative systems, there are no attractors, and the finite-time interval lengths are specific to every orbit. We have calculated these lengths in conservative systems by computing the Poincaré crossing times and detecting changes in the sign of the kurtosis of the finite-time distributions.

126

4 Predictability

The presence of the oscillations of the closest to zero exponent is an indicator of nonhyperbolicity. This implies the necessity of the calculation of several available exponents, since the identification of the closest one depends on the selected interval, in addition to the position in the parametric space. We have noticed that for the larger intervals, the exponents tend to the global values, the closest to the zero points to the neutral direction and the oscillations may be then difficult to identify. The results presented here are of general interest in describing how the predictability index computed using Eq. 4.2 provides information on the system dynamics. When we calculate predictability indexes, one must take into account the timescales of the analysed system for a better interpretation of the range of values corresponding to a given model. Particularly noteworthy, the shadowing times of regular orbits are not always the same, when comparing regular orbits belonging to different models. This is because the consequences of a single-step error δt are different depending on the model. Regarding chaotic orbits, we have seen that the predictability indexes of chaotic orbits can be also different when they belong to different models. Two orbits can be chaotic, yet one may have a larger index than the other. The predictability index is related to the hyperbolic nature of the orbit and, in turn, to its energy and stiffness of the system. The existence of two or more timescales in different directions, one quickly growing, one slowly growing, can lead to stiffness, and the finite-time exponents reflect these expanding/contracting behaviours. In addition, the predictability indicators depend on the timescales when there is a change on these behaviours and the global regime is reached. Different energy values lead to different dynamical times, so to different timescales. Finite-time Lyapunov exponent techniques are indeed useful for studying those transient periods that the dynamics may suffer before ending in a final invariant state. The distributions can be built using shorter total integration times than those required for reaching the asymptotic behaviour. There is a limitation, however, when reducing the total integration time that is the number of finite-time intervals needed for having good statistics values computed from the distribution. As Δt increases, the number of intervals needed for building a well-sampled distribution and a reliable mean, deviation or kurtosis calculation, also increases. Regarding the selection of the finite-time interval sizes, we have observed that, conversely to the dissipative case, the timescales when the deviation vectors leave the local regime of the flow and begin to evolve under the global dynamics can be different and smaller than the timescales where the asymptotic regime starts. When Δt is large enough, the distributions tend to shrink and centre around the asymptotic value. In early works [6], the interval length (characteristic time) for the effective exponents was tH . We have seen that it is possible to use intervals smaller than tH for gaining insight into the properties of the flow. The method we have used indicates the most adequate interval length for estimating the predictability of a given orbit, independently of their regular or irregular nature. When there are several dynamical transients, reflected in changes in the shapes of the finite-time distributions, and, consequently, different zerocrossings in the kurtosis curves, this method returns upper limits to the orbit predictability.

4.3 Concluding Remarks

127

The dynamical times are different depending on the studied orbit and model. The predictability index values can be used for comparing the predictability of different orbits. They reflect how the shadowing time increases as the precision in the computations increases. Low predictability indexes lead to short shadowing times. Selecting an integration scheme and assuring the energy is kept constant in time (within some small error) do not imply that the computed orbit is shadowed by a real one beyond certain limits. The predictability index estimates this shadowing time duration. A given numerical scheme with certain precision can be enough when the shadowing times are large. But this may be not the case when the shadowing times are shorter. A high predictability index may indicate that high-precision time-consuming schemes are not necessary, even for chaotic orbits. Indeed, RK4 integrators provide good results even for the strongest chaotic orbits seen in the presented meridional potentials. Conversely, a low predictability index points to the use of more powerful schemes. In a general case, when these indicators are really small, large increases in precision do not mean large increases in shadowing times, and one should consider the cost of implementing more complex and time-consuming schemes. The percentages of regular and chaotic orbits in the phase space are not only a function of its spatial location but also a function of the total energy and the main parameters of the model [18], and the amount of chaotic and regular motions in a given ensemble of initial conditions is related to the forecast of its predictability. Chaos detection methods based on saturation or averaging return different values as the the saturation times vary because of the possible evolving presence of different regions of chaos, moderate or strong [16]. We have estimated the finitetime interval lengths to use in the computation of the h index, from the analysis of changes in the shapes of the distributions. When several zero-crossings are present, we have selected the zero based on the Poincaré section crossing timescales. Our method has been applied to regular and chaotic orbits, and the results point to the validity of the shadowing times returned by Eq. 4.2 even when the ergodic diffusion model may not be fully applicable in the regular cases. Our work has focused on the predictability index as an estimator of the accuracy of an orbit in some time-independent potentials. The time independence of such potentials allows the trajectories to be either periodic, regular or chaotic (strong or weak). But the only unusual transitions found are those when a chaotic trajectory behaves like a regular orbit and requires long timescales to reveal its true chaotic nature. In time-dependent potentials, one can find migrations from chaotic to regular [19]. The presented techniques are applicable both when there are changes from regular to chaotic motions or changes from chaotic to regular motions, as in the timedependent cases. This is because the predictability index presented in this chapter is computed from solving the variational equations and detecting changes in the shapes of the finite-time Lyapunov exponents as the finite-time intervals are increased.

128

4 Predictability

References 1. Aguirre, J., Viana R.L., Sanjuan M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333 (1999) 2. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An introduction to dynamical systems, p. 383. Springer, New York (1996) 3. Athanassoula, E., Romero-Gómez, M., Bosma, A., Masdemont, J.J.: Rings and spirals in barred galaxies – III. Further comparisons and links to observations. Mon. Not. R. Astron. Soc. 407, 1433 (2010) 4. Buljan, H., Paar, V.: Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. Phys. Rev. E 63, 066205 (2001) 5. Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970) 6. Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374 (1995) 7. Davidchack, R.L., Lai, Y.C.: Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbit. Phys. Lett. A 270, 308 (2000) 8. Daza, A., Wagemakers, A., Sanjuan, M.A.F., Yorke, J.A.: Testing for Basins of Wada. Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016) 9. Do, Y., Lai, Y.C.: Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability. Phys. Rev. E 69, 16213 (2004) 10. Grassberger, P., Badii, R., Politi, A.: Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988) 11. Grebogi, C., Kostelich, E., Ott, E., Yorke, J.A.: Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor. Phys. D 25, 347 (1987) 12. Hunt B.R., Ott E., Rosa, E.: Sporadically fractal basin boundaries of chaotic systems. Phys. Rev. Lett. 82, 3597 (1999) 13. Kapitaniak, T.: Distribution of transient Lyapunov exponents of quasiperiodically forced systems. Prog. Theor. Phys. 93, 831 (1995) 14. Kottos, T., Politi, A., Izrailev F.M., Ruffo S.: Scaling properties of Lyapunov spectra for the band random matrix model. Phys. Rev. E. 53, 6 (1996) 15. McDonald, S.W., Grebogi, C., Ott E., Yorke, J.A.: Fractal basin boundaries. Phys. D 17, 125 (1985) 16. Maffione, N.P., Darriba, L.A., Cincotta, P.M., Giordano, C.M.: Chaos detection tools: application to a self-consistent triaxial model. Mon. Not. R. Astron. Soc. 429, 2700 (2013) 17. Mandelbrot, B.B.: Les objets fractals: forme, hasard et dimension. Flammarion, Paris (1975) 18. Manos, T., Athanassoula, E.: Regular and chaotic orbits in barred galaxies – I. Applying the SALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc. 415, 629 (2011) 19. Manos, T., Machado, R.E.G.: Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models. Mon. Not. R. Astron. Soc. 438, 2201 (2014) 20. Ott E., Alexander, J.C., Kan, I., Sommerer, J.C., Yorke, J.A.: The transition to chaotic attractors with riddled basins. Phys. D 76, 384 (1994) 21. Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999) 22. Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E. 65, 036220 (2002) 23. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997) 24. Sepulveda, M.A., Badii, R., Pollak, E.: Spectral analysis of conservative dynamical systems. Phys. Lett. 63, 1226 (1989)

References

129

25. Tomsovic, S., Lakshminarayan A.: Fluctuations of finite-time stability exponents in the standard map and the detection of small islands. Phys. Rev. E 76, 036207 (2007) 26. Vallejo, J.C., Aguirre, J., Sanjuan, M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003) 27. Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov Exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008) 28. Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15, 113064 (2013) 29. Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015) 30. Viana, R.L., Pinto, S.E., Barbosa, J.R., Grebogi, C.: Pseudo-deterministic chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 1 (2003) 31. Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a Chaotic Process. Braz. J. Phys. 35, 1 (2005) 32. Werndl, C.: What are the new implications of Chaos for unpredictability. Br. J. Philos. Sci. 60, 195–220 (2009) 33. Westfall, P.H.: Kurtosis as Peakedness. 1905–2014, R.I.P. Am. Stat. 68, 191 (2014) 34. Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001) 35. Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaoshyperchaos transition in coupled systems. Phys. Rev. E 64, 056235 (2001) 36. Yoneyama, K.: Theory of continuous sets of points. Tohoku Math. J. 11, 43 (1917)

Chapter 5

Chaos, Predictability and Astronomy

5.1 Introduction We have seen in the previous chapters the impact of the presence of chaos in a given dynamical system. Furthermore, we have seen the importance of assessing the predictability of a given model through the use of the finite-time Lyapunov exponent distributions. The last two chapters of this book aim to apply these ideas to a particular case, a mean gravitational potential that models the Milky Way. For doing so, here we start describing how nonlinear dynamics and chaos theory tools can be applied to the field of astronomy and astrophysics. We do not aim to be exhaustive but rather show some illustrative examples to better present how these tools provide some insight in this field where numerical simulations have an essential role in providing the backbone of the theoretical models. We will present the complex motions that can be found even in a very simple model, such as the Sitnikov problem. In this way, we will better see how the predictability index may help us in the analysis of numerical models describing more realistic systems.

5.2 Numerical Forecasting in Astronomy The desire of understanding the universe is a constant in the history of humanity since the most ancient times, trying to gain knowledge on the ultimate reasons behind the observed natural cycles. The final goal was to have the ability of forecasting the future events. Astronomers played a crucial role in these tasks, in all cultures, compiling the observed phenomena in the skies and trying to figure out what was going to be observed in the future and when.

© Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_5

131

132

5 Chaos, Predictability and Astronomy

Indeed, astronomy is considered sometimes the most ancient science. From the earliest times, the study of natural cycles was needed for a successful harvest. Because those cycles were marked by the movement of the skies, the study and prediction of these cycles took place since the very beginning of the history of human mankind. Up to the end of the nineteenth century, the main task of the astronomers was patiently annotating and predicting the positions of celestial bodies, ignoring the underlying physics. The French philosopher Auguste Comte compiled in 1842 examples of knowledge which were to be uncertain to humans forever. And in this list, he cited the composition of the stars. The clear reason was that they would never be reachable by any means to be properly analysed. In his book “The Positive Philosophy” [6], he claimed: Of all objects, the planets are those which appear to us under the least varied aspect. We see how we may determine their forms, their distances, their bulk, and their motions, but we can never know anything of their chemical or mineralogical structure; and, much less, that of organised beings living on their surface . . .

Indeed, Comte referred to the planets in the quotation above. Actually, he believed that we could learn even less about the stars: On the subject of stars, all investigations which are not ultimately reducible to simple visual observations are . . . necessarily denied to us. While we can conceive of the possibility of determining their shapes, their sizes, and their motions, we shall never be able by any means to study their chemical composition or their mineralogical structure . . . Our knowledge concerning their gaseous envelopes is necessarily limited to their existence, size . . . and refractive power, we shall not at all be able to determine their chemical composition or even their density. . . I regard any notion concerning the true mean temperature of the various stars as forever denied to us.

But in 1859, the German physicist Gustav Kirchhoff discovered that the chemical composition of a gas could be deduced from its electromagnetic spectrum viewed from an arbitrary distance [17]. And in 1864, 7 years after Comte died, this method was extended to astronomical bodies by the English astronomer William Huggins, who first attached a spectrograph to a telescope. Since then, astronomy was not any longer limited to just dealing with positions and brightness compilations. Astronomy begun to deal with the study of dynamics, composition and physical conditions of any celestial body and converted onto astrophysics. Astrophysics is obviously an experimental science, subject to all the topics discussed in the first chapter of this book. However, contrary to the majority of experimental sciences, astrophysics recollects information of observed objects without the possibility of having direct access to them. It is not possible to alter the key parameters of the studied objects and to study their later evolution for proper analysis. Nevertheless, one has all available evolutionary states of the subject of study scattered along the night sky. Therefore, the task for the astrophysicist is to build a valid physical model from those scattered visions. The astrophysicist has to deal with the added difficulty that the timescales applicable to the skies are generally out

5.2 Numerical Forecasting in Astronomy

133

Fig. 5.1 Results of the pioneering simulation run by Holmberg in 1941, a collision between two galaxies, each one modelled as 37 particles. Left: the two disc galaxies approaching. Right: the two galaxies after the collision. (Image taken from Ref. [15]. ©AAS. Reproduced with permission)

of the human lives’ timescales, and the whole physical picture must be assembled from the individually observed frozen-like states. As a consequence, a key tool in astrophysics is to confront the subject of observation with the results coming from numerical simulations. Astrophysics (as many other disciplines) has seen how the simulations have gained in relevance as the numerical methods and available computing resources grow. Numerical simulations usually conform an intermediate state between theory and experimentation, confronting the theory with observations, and are also in charge of exploring the consequences of varying parameters in the physical models. The numerical simulation begun at the same time that the computers were born. In 1941 the Swedish astronomer Erik Holmberg carried out in the United States one simulation for analysing the gravitational interaction of an ensemble of 74 particles, intended to model two galaxies [15], see (Fig. 5.1). Every mass element was represented by a small light bulb, the light being proportional to the mass, and the total light along the x and y axes being measured by a combination of a photodiode and a galvanometer. The calculations and integration were done by using “hands”, in desks located in one hangar. With the advent of the programmable computer in 1946, this scenario was greatly enhanced. Detailed computations of gravitational interactions among several

134

5 Chaos, Predictability and Astronomy

Fig. 5.2 This picture plots the state of two galaxies after they have collided. It is the result of a simulation based on plain Newtonian physics and 20,000 discs and 40,000 halo particles in total. It models a collision between two galaxies. The initial setup is provided as part of the distribution of a widely used N -body engine, gadget (version 2.0.7) [48]

bodies begun to be feasible. Two decades after the pioneering work of Holmberg, the first paper on an N -body simulation was published [14]. Today, the number of simulations and particles follows the famous Moore’s law that governs the increasing available computational power [34], see (Fig. 5.2). With the widespread use of computer simulations, all the issues that impact into the reliability of the numerical calculations must be considered, including the simplifications done for getting a given model from the observations, the regularity and instability properties of studied orbits found in this model and also the instability properties resulting from the numerical scheme selected for solving the model. As discussed in the first chapter, the fundamentals of the scientific method rely on linking the observations to a given physical model and the capability of this model to explain these observations and to forecast new ones. The key issue is to determine, characterise and control (if possible) the role of the simplifications done when creating the model, with respect the real system, and the role of the process of solving the model and get numerical predictions. In this scenario, we are in an interdisciplinary field, where the astrophysics field provides the simulated models, the nonlinear dynamics field provides their chaoticity and instability properties and the computational sciences provide the actual numerical implementation. Many concepts from nonlinear dynamics and chaos theory have been successfully applied to the astronomy and astrophysics fields. See, for instance, Ref. [41] for a general review or Ref. [7] for the specific area of the dynamical astronomy. The results obtained have provided many insights. But they have also opened many other questions. These concepts have demonstrated to be of interest in many areas. We can mention, among many others, time series analyses as the study of solar activity curves [40], light curves of variable stars in the visible [4, 5, 19, 44], the X-ray domain [27, 50], pulsar emission [13] or active galactic nuclei emission [20, 25, 26]. Focusing on the celestial mechanics analyses, one can list analyses of orbits and rotations of planets, asteroids or satellites, as, for instance, [1, 23, 32, 36, 42, 51].

5.3 Time Series Analyses

135

Finally, going further to the analysis of the stellar systems, this area includes studies of the dynamics of stellar globular clusters [31] and studies of the dynamics of galaxies that we will analyse in detail in the next final chapter.

5.3 Time Series Analyses The analysis of astrophysical time series using nonlinear dynamics tools usually focuses on gaining some understanding of the hidden mechanisms generating the observed dynamics. Typically, one aims to reconstruct any hidden attractor and analyse the number of variables needed for such a reconstruction. A reduced number of variables is interpreted as a signal of a simple dynamics generating a complex behaviour. Conversely, if the attractor is embedded in a high-dimensional space, this is interpreted as a signal for a complex dynamics. A first target to apply these ideas has been the analysis of the solar cycle of the Sun by means of the studies of the periodic abundance of sunspots. The sunspots are linked to local phenomena of the magnetic field; hence they are a direct indicator of the solar activity. This is not a merely academic research. Small changes in the activity of the Sun have a large influence in the modern communication satellite networks and the surface power grid outages and blackouts and even produce changes in the weather. Aiming to forecast this activity, a traditional tool has been the comparison of past activity curves with the current ones. Nowadays, some interesting results have been obtained related to the computation of both the fractal dimension of the time series and the Lyapunov exponents. Some of these results point to monthly curves that show a low-dimensional chaotic behaviour, which is a signature of a relatively simple dynamics [37]. On the other hand, other studies have obtained very complex dynamics that must be described using up to 15 dimensions [12]. The light curves of the variable stars have been also a target of these studies. The variable stars are those that show changes, regular or not, in their apparent brightness. There is a variety of variable star types, depending on the physical source for this variation. Pulsating stars have changes in their sizes. Eclipsing binaries show these variations depending on the relative positions for the observer. And eruptive stars show abrupt changes in their luminosity, sometimes linked to flaring activity, sometimes linked to mass from another companion falling on them through a accretion disc and sometimes linked to dramatic explosions. From the point of view of nonlinear dynamics, low-dimensional chaos has been found in short-period variables. This means that a very limited set of variables is able to produce the observed complexity [18]. Similar studies have been carried out in the analyses of the generator mechanisms in other wavelengths different than visible wavelengths. Both in shorter frequencies, such as radio emission curves, or longer frequencies, such as X-ray emission curves, one can also find variability in the corresponding time series.

136

5 Chaos, Predictability and Astronomy

A frequent target for these analyses has been the X-ray emission coming from X-ray binaries. In the large mass binaries, where one of the stars is a compact object (a neutron star or a white dwarf) and the companion is a blue giant, a periodicity can be observed or quasi-periodicity in the emission curves. In the low-mass binaries, where both objects are compact bodies, this emission is typically more intermittent and chaotic, in the form of bursts. In the first case, large mass binaries, the tidal effects were analysed, and it has been observed that the emission increases with the age of the system, when the Roche lobes are filled up. Nonlinear oscillations are then present that extend the hydrogen-burning phase and increase the relaxing by tidal forces. Hence, the increase of the duration as X-ray is binary. In the second case, there is X-ray emission when matter from the white dwarf falls into the neutron star, converting into plasma that gets accelerated and heated up to millions of degrees. In which we are concerned, nonlinear oscillations have been observed, likely produced by thermal or Rayleigh-Taylor instabilities. The analysis of the stability of the equations of simple accretion disc models leads to the existence of unstable regions where stationary flows cannot exist; there are limit cycles, where the system oscillates between short-lived high-mass flows’ radiative states and low-mass convective flows. Again, the scenario is not fully settled. Some studies conclude the presence of low-dimensional chaos [49]. Other studies focus on the difficulties found in those analyses and point to the possibility of having not enough data for concluding that presence [30]. Regarding the emission coming from pulsars, small fluctuations, called timing noise or residual timing, were observed in the pulsar radio emission. But abrupt changes, or glitches, in the rotation frequencies have been also observed [45]. The associated delay plots have found low-dimensional attractors, where just two variables may explain this noise. Several mechanisms are raised for explaining these observations, as MHD moments, internal moments or interactions with accretion discs, all of them exhibiting chaotic behaviour. The time series analyses have also led to interesting results in the field of galactic astrophysics, specifically, in the series associated with the emission coming from the so-called active galactic nuclei (AGN). These are galaxies that emit a larger amount of energy than the other galaxies. Therefore, in addition to normal thermal mechanisms, the presence of supermassive compact objects (black holes) is needed to explain the large amount of released energy. A phenomenon commonly found in many AGNs is a strong variability in Xray [26]. Indeed, this variability can show the red-noise property, which is itself a generalisation of the 1/f noise. This emission can be produced by white noise, but also by deterministic chaos. Some proposed models are exponential decay of shots in an accretion disc, produced by random collisions or time-dependent Compton diffusion, sourced to inhomogeneities in the accretion flow and flares in ultraviolet range. Nonlinear techniques do not allow to discriminate which model is better, but help in constraining the parameters used by each one.

5.4 Celestial Mechanics

137

5.4 Celestial Mechanics One of the most remarkable applications of nonlinear dynamics and chaos theory is the study of the motions of celestial objects, and their stability, as already discussed in the first chapter. As starting point, one can take the study of the stability of the N -body problem subject to gravitational interaction. As simple as it seems, the task of predicting the individual motions of a group of celestial objects interacting with each other gravitationally is not an easy one. We can follow here the words from the Canadian astronomer Scott Tremaine: The gravitational N-body problem is a simple problem that remains fascinating and incompletely understood after three centuries of intense study by generations of illustrious physicists and mathematicians including Laplace, Lagrange, Gauss and Poincaré. It inspired the modern subjects of nonlinear dynamics and chaos theory, and remains one of the oldest unsolved problems in physics.

Indeed, even in the simplest possible case, the three-body problem is not integrable. It has 9 degrees of freedom, and, as consequence, 18 constants of motion are needed (or, following the integrability theorem of the French mathematician Joseph Liouville, 9 conserved quantities in involution). The German mathematician and astronomer Heinrich Bruns showed in 1887 that there is no independent algebraic constant but those already discovered by the Italo-French mathematician and astronomer Joseph-Louis Lagrange, who in 1772 reduced the problem to a 6 degrees of freedom problem. Therefore, it is not possible to find analytically a general solution, and consequently, it was proposed to find approximated solutions based on series. The Canadian-American astronomer Simon Newcomb in 1876 and the German mathematician Karl Weierstrass in 1878 found a series with proper coefficients coming from planet motions. But it remained to show the convergence of these series. In the case that the convergence of these series may be found, the planetary motions would be periodic or quasi-periodic. That is, there would be just only a finite number of frequencies and their corresponding linear combinations. Conversely, if those series were divergent, nobody could forecast the long-term behaviour of planetary orbits [43]. A critical issue with these series is that there is an infinite number of coefficients with very small divisors. To assure the convergence in this case is far from trivial.1 The problem was so difficult to face that, when King Oscar II, King of Sweden and Norway, was seeking how to promote a prize for solving a difficult mathematical problem, Weierstrass proposed the challenge of assuring the convergence of the solutions of the N -body problem. Hence, the prize of 2500 kronas to whom may provide an answer to the practical question “Is the Solar System stable?” was born.

1 Indeed,

only in 1942 the German mathematician Carl Siegel could show the convergence for one series of these characteristics.

138

5 Chaos, Predictability and Astronomy

We have seen in Chap. 1 that Henri Poincaré won this prize with his work written in the memory “On the Three-Body Problem and the Equations of Dynamics” [39]. Poincaré first established the properties of the necessary dynamical equations. Then, he applied the results to the problem of an arbitrary number of bodies under Newtonian gravitational interaction. Finally, he analysed the existence of periodic solutions, following the classic approach of developing the necessary variables as infinite series and finding that there are series with periodic coefficients formally satisfying the equations of motion. But he did not attempt to prove the convergence of the series. Conversely, he proved the existence of periodic solutions using another different approach. The existence of these solutions proved, in turn, the convergence of the series. He approached the reduced, or restricted, three-body problem. Here, two massive main bodies drive the system, and a third body, called planetoid, has a much smaller mass and does not affect the other companions. Indeed, Poincaré worked mostly on the planar, circular, restricted three-body problem or PCR3BP, where the primary bodies move on a circular orbit of the two-body problem. A third body of negligible mass moves in the plane under the influence of the gravitational forces of the primaries. He found that, among the solutions to this problem, there are special points that nowadays are named as homoclinic points. The existence of homoclinic points leads to the existence of chaos, as proven by the American mathematician Stephen Smale in 1967. The Hill problem is another particular case of the restricted three-body problem, where the negligible body is a satellite of the body of lesser mass. This problem is named after the the American mathematician G.W. Hill that in the nineteenth century studied a simple approximation for the motion of the Moon around the Earth, with perturbations by the Sun. In general, the Hill problem means a model for motions in which there are two nearby bodies moving in nearly circular orbits about another much larger body at a great distance.

5.5 Sitnikov Problem The motions in these simple problems can be indeed very complex. There is an interesting example in the restricted case of the Hill problem when the trajectory of the planetoid crosses perpendicularly the orbital plane defined by the primary bodies. The Sitnikov problem [46] is one of the simplest dynamical systems in celestial mechanics that shows strong chaotic behaviour. The configuration of this system is formed by two pointlike bodies of equal masses moving on symmetrical elliptical orbits around their common centre of mass, with the third body of negligible mass moving along a line, perpendicular to the orbital plane of the primaries, going through their barycentre. Because the planetoid has no effect on the motion of the primaries, the solution of the two-body problem is needed for the determination of the motion of the test particle.

5.5 Sitnikov Problem

139

In the specific case, when the orbits of the primaries are circles, the problem is an integrable conservative Hamiltonian. An analytical solution can be found using elliptic integrals [28], and one can get a solution that restricts to harmonic oscillations. When the orbits of the primaries are ellipses, we have the Sitnikov problem. In its original classical formulation in 1961 [47], as generalisation of the MacMillan problem, the equation of motion for the massless body turns out to depend only on the time t and the eccentricity of the primaries e. The motion of the planetoid is described by the following equation: z¨ = −

z (z2

+ d 2)

3/2

,

(5.1)

where z gives the position of the planetoid along the axis of symmetry and d is the distance from one of the binary particles to the centre of mass given by d=

1 (1 − e cos(t))2 . 2

(5.2)

These equations are written in units chosen such as the total mass of the binary system is unity, the period of the binary is 2π and the gravitational constant G = 1. When the eccentricity is not zero, we have a nonautonomous, non-integrable and chaotic system. The relativistic effects have been also added to the problem, showing that the presence of chaos holds in this case [22]. But, even when we restrict ourselves to the classical formulation given by Eq. 5.1, and despite the apparent simplicity of this mathematical description, the complexity of its possible motions makes the Sitnikov problem a unique dynamical problem. There exist bounded and unbounded motions in the Sitnikov problem. In the open, unbounded case, this problem can be considered an example of a scattering process when a particle approaches the dynamical system from infinity, interacts with the system and, later on, leaves it. The existence of oscillatory motions in the restricted three-body problem that is unbounded was already showed in [47]. We can have small oscillations around the barycentre. But, interestingly, we can select an arbitrary series of numbers, and it is always possible (following the techniques seen in [35]) to find an initial condition that makes the planetoid cross the plane defined by the primaries at intervals coincident with that arbitrary series [10]. As scattering problem, the Sitnikov problem was studied in [21]. We can find here the transient chaos phenomenon. Once the test particle escapes to infinity, it can escape using different exits. Therefore, one can determine the basins of escape that show a complex topology, in an analogous way, have been described in the previous chapters for the Hénon-Heiles system. In the case when the planetoid is confined, this problem also offers a wide set of possible solutions. A representative set of these initial conditions is listed in Table 5.1. The corresponding phase space portraits are seen in Figs. 5.3 and 5.4.

140

5 Chaos, Predictability and Astronomy

Table 5.1 Some representative initial conditions for the Sitnikov problem and the values for the corresponding Lyapunov exponents and predictability indexes. The C initial conditions lead confined orbits. The E initial conditions lead the test particle to escape. See details in the text Orbit C1 C2 C3 E1 E2

Ecc. 0.01 0.61 0.61 0.50 0.50

z˙ 0 1.00 1.00 1.95 2.05 2.10

λ 0.00 0.00 0.10 – –

h(Δ = 0.1) 1.3 0.28 0.07 – –

h(Δ = 1.0) 0.17 1.04 1.23 – –

h(Δ = 5.0) 10.45 5.04 28.675 – –

h(Δ = 10.0) 33.64 17.55 49.94 – –

5.6 Predictability and Stability in the Solar System Another interesting question to answer is why we observe the “apparent” stability in our Solar System. The Russian mathematician Andrey Kolmogorov proposed in 1954 that, even when there are no conservation laws, it is possible to find stable orbits. Moreover, the major part of orbits is stable. This result was generalised by the Russian mathematician Vladimir Arnold and the German-American mathematician Jürgen Moser, and it is known now as KAM theorem. In general, the existence of perturbations destroys the conservation laws that make a system to be integrable. But the KAM theorem states that for small enough perturbations, the major part of the orbits is stable, quasi-periodic and not far away from the unperturbed system. Obviously, there are other chaotic orbits, and the regular stable orbits are surrounded by a sea of chaotic orbits. When the system is weakly perturbed, the regularity is just partially destroyed. That is, because the small masses of the planets are compared with the mass of the Sun, the major part of the orbits is stable, and the Solar System will remain as observed today. However, the KAM theorem does not prove if one specific orbit is or not stable. Indeed, simulations run in the past decades show that the chaos is in fact present in the Solar System. In these long-term simulations, the key issue is to analyse the characteristic time between dramatic changes in eccentricity of a given body. The fact that we do not know any general solution to the N -body problem does not imply that there are no particular solutions. Lagrange found already in 1772 two solutions of interest to the three-body problem. One consists on the three bodies moving in the vertexes of an equilateral triangle rotating around the centre of mass of the system. Another solution relies on the three bodies aligned on a rotating imaginary line. Regarding the restricted three-body problem, these positions correspond to five fixed positions. Three positions are aligned with the primaries (L1, L2, L3) and are unstable. Two other points are stable and form equilateral triangles in the orbit plane (L4, L5). These orbits are not mere theoretical positions. One can find real examples as the zodiacal light or the “gegenschein”. These phenomena are sourced to the sunlight dispersed by dust in the Solar System accumulated around these Lagrange points. The zodiacal light is a weak light pyramid reflecting off dust in the Solar System

5.6 Predictability and Stability in the Solar System Fig. 5.3 Phase space portraits corresponding to the initial conditions C1, C2 and C3 of the Sitnikov problem, leading to the confined orbits listed in Table 5.1

141

142 Fig. 5.4 (Upper and middle panels) Phase space portraits corresponding to the initial conditions E1 and E2 of the Sitnikov’s problem, leading to the unbounded orbits listed in Table 5.1. (Bottom panel). Time series formed by the z(t) evolution in time. The crossing times when z(t) = 0.0 do not follow any regular pattern before the particle escapes

5 Chaos, Predictability and Astronomy

5.6 Predictability and Stability in the Solar System

143

that can be observed after the evening twilight, and before the dawn, following the ecliptic line or apparent trajectory of the Sun in the sky. So, it has been discovered that there are dust clouds at L4 and L5. The “gegenschein” is a name coming from the German meaning “countershine”, and it is a phenomenon more difficult to observe. It is a small diffuse and weak light sourced to dust in L2, hence, moving in opposite direction to the Sun at the same velocity. Another case related to the Lagrange points L4 (preceding the primary) and L5 (after the primary) in the Solar System is the Trojan asteroids bound to Jupiter or the Trojan asteroids bound to the Earth. However, only one is confirmed to have this nature. Going further, there are some current efforts in detecting trojans in extrasolar systems [24]. There are also some practical applications in the field of flight dynamics of spacecrafts. A widely used location for a space telescope is the L2 point. Here, a space observatory can target the major part of the sky without eclipses by the Earth, with a minimum spent fuel. Some examples of missions located (or to be located) in L2 are the infrared telescope Herschel, the astrometric surveyor GAIA or the visible and near-infrared James Webb Space Telescope. Of course, contrary to these examples, solar observatories will be more suitably located in L1, where an uninterrupted view of the Sun can be achieved. This is the case for SOHO, for instance. Finally, there are no planned missions to locate a spacecraft in L3. Notably, L4 and L5 correspond to hilltops of the gravitational potential surface; meanwhile L1, L2 and L3 correspond to saddles. Therefore, a satellite located right in the theoretical point is not stable and will wander off. In fact, satellites located in L2 follow the so-called halo orbits around L2. These halo orbits (or their generalisation the Lissajous orbits that are not periodic) around L2 are selected to be large enough so that the spacecraft can always see the Sun around the Earth without any blockage. Another case of great interest regarding the nonlinear dynamics being applied to the Solar System is the analysis of the asteroids. The orbits of these bodies, located in the asteroid belt between the orbits of Mars and Jupiter, are not uniformly distributed. Notably, some major semiaxes tend to group around certain values, as the Hilda group, close to the 3 : 2 resonance, or the group of Trojan asteroids, in 1 : 1 resonance with Jupiter, already mentioned before. Conversely, other groups of asteroids tend to go farther away from certain values, leading to the so-called Kirkwood gaps. Many of those gaps are related again to Jupiter resonances. That is, a body at that distance will periodically encounter Jupiter, and their mutual interaction would increase as the time goes. However, other observed gaps were not properly explained until tools from nonlinear dynamics and chaos theory were applied. This is the case for the 3 : 1 resonance. At this distance, an asteroid is under chaotic regime, and can suffer dramatic changes in eccentricity, crossing the orbit of another planet, and can even be expelled from the Solar System, leading to the gap generation. These tools have also been applied to the analysis of the stability of the rotation axes of the planets. The ancient Greek astronomer Hipparchus already discovered

144

5 Chaos, Predictability and Astronomy

that the Earth’s rotation axis was not always pointing to the same position of the sky. Conversely, this axis describes a precession path in the sky with a period of roughly 26,000 years. This motion is generated by the combined forces exerted by the Sun and the Moon over the non-completely spherical Earth. The rotation axis of the Earth currently points to a certain star, the so-called polar star. But this is a casual pointing that will change with time. These precession motions have direct consequences on the climate of the Earth. In 1941, the Serbian astronomer M. Milankovitch formulated, based on previous works from Urbain Le Verrier, the hypothesis that the Quaternary glaciation was due to changes in the amount and location of solar radiation reaching the Earth, which in turn were sourced to secular variations of the orbit of the Earth and its rotation axis tilt [33]. This hypothesis does not fully agree with current observations and detailed climatic records. Therefore, some additional mechanisms for explaining these discrepancies are needed. However, this hypothesis triggered many efforts for understanding the stability of the Earth’s orbital parameters. As a consequence, it was discovered that the tilt of the Earth’s rotation axis can be chaotic, even when only for a small range of angles. Noteworthy, this small range of values can be amplified by other factors, such as the variable albedo from the polar caps. Luckily, the presence of the Moon is a major factor for increasing the stability of the tilt. In other words, in planets without the presence of a major satellite, these oscillations could be very large. For instance, the rotation orientation of Venus is retrograde, which means a tilt of almost 180 degrees. This is an orientation reversed with respect to the remaining planets of the Solar System. A possible move into chaotic regime in the past may explain its current value [9]. Regarding Mars, the angular rotation speed of the precession and the orbital motion have close values, making resonances and large variations in the obliquity possible. However, this scenario is not settled at all [11]. Finally, a classic study of the role of chaos in celestial mechanics is the study of the attitude of Hyperion, one of the satellites of Saturn. This body has an orbit with a high eccentricity value but is still regular and predictable. What makes this body quite remarkable is that the orientation of its axes along its orbit is fully chaotic and changing, as clearly seen when the corresponding Poincaré cross sections are plotted [1].

5.7 Stellar Systems The presence of chaos and the efforts in characterising it in the field of the stellar systems modelling (clusters and galaxies) is the area where we have decided to focus on our main example. We have selected this area because it directly connects the efforts done in solving the N -body problem that we have seen can be traced to the original work done by Poincaré and the birth of the chaos theory, with the studies

5.7 Stellar Systems

145

done by Hénon and Heiles in 1964, and the discovery of how simple dynamics can lead to very complex behaviour. Regarding small stellar systems, such as globular clusters, the studies of the presence of chaos in the dynamics of these stellar systems led to find irregularities in oscillations of these systems [2]. The star clusters are groups of hundreds or thousands of old stars orbiting around galaxies, at roughly 50,000 light years. Numerical models show chaotic oscillations that appear after the initial collapse. Independently of the formation mechanism, there are bifurcations once a given number of stars is used in the models. When the phase space is reconstructed, an attractor is usually found, and as the number of stars increases, typical perioddoubling bifurcations occur, which can show the appearance of deterministic chaos. However, this chapter will focus on the study of galactic systems. Since the early work of Hénon and Heiles, there is huge evidence of the existence of non-integrable potentials, with both regular and chaotic orbits. The latter are interesting orbits, because they serve to populate regions of the phase space that would be empty otherwise. Their presence may also trigger starbursts, where the star formation rate is dramatically amplified. Other consequences are mass inflows towards the central areas or bulge formation because of vertical heating processes. There is no strict agreed definition on what a galaxy is, even when everybody agrees in which you know is a galaxy when you see it. One may think that a galaxy is simply a large group of stars. A few stars may be a group and a hundred or more, a cluster. The point where a cluster becomes a galaxy is the point that is still under debate, specially after the discovery of the ultracompact dwarfs or ultra-faint dwarf spheroidal galaxies that are similar to galaxies in mass but similar to clusters in appearance. A valid definition would be a self-gravitating system composed of three components: stars (in broad sense, including compact objects), dark matter and interstellar medium (ISM). Regarding the stars, it is expected that a galaxy will host a variety of different types of stars. Regarding the ISM, it includes a variety of material such as hot ionised gas, warm gas, cold gas and dust. And finally, it seems it should be held together by the dark matter. We will not discuss here on the composition of the dark matter, which is currently unknown and under debate. The presence of the dark matter is proven because of its gravitational influence. But it does not emit any type of known radiation, making it undetectable to current devices. Dark matter could be composed of stellar remnants, black holes, brown dwarfs or even more exotic elementary particles. Classically, astronomers classified galaxies according to their morphology. Following the scheme initially conceived by the American astronomer Edwin Hubble [16], they divided the galaxies in three classes. Spirals, which consist of a flattened disc, with stars forming a spiral structure, and a central concentration of stars known as the bulge. Sometimes they present a bar-like in the disc, and they are known as barred spiral galaxies. Elliptical, which presents smooth and featureless light distributions and appears as ellipses in images. These are mainly triaxial systems. And finally, irregulars, which embrace all those that do not fit in the above.

146

5 Chaos, Predictability and Astronomy

This morphological classification is somehow subjective, because there is no clear border among the classes, and it may be a continuous evolution from one type to the other. This is the case of lenticular galaxies, for instance, consisting of a bright bulge with extended disc but no spiral structure. Moreover, galaxies are not isolated entities, and the collisions between them are very frequent. Many merger or galaxy pairs are also found, with complex shapes and interesting phenomena (cannibalism) leading to double-core galaxies, for instance. As a consequence, nowadays astronomers classify galaxies according to measured physical parameters, as kinematics information (velocity curves), photometric information (bright curves or surfaces), mass flows, star formation rate and energy production. The main goal of the galactic dynamics is modelling the dynamics and morphology of the observed galaxies. The most important aspect to keep in mind is that galaxies are mutable entities which evolve. In addition to the frequent collisions between galaxies, the components of a typical galaxy are not fixed along their lifetime. The stars within a galaxy can be in any of their evolutionary states, which includes main sequence stars, white dwarfs, neutron stars and compact massive objects as black holes. This means that the stars are born, live and die during the galaxy lifetime, and this implies in turn an interchange of mass between the three components (stellar material, dark matter and ISM). Evolution means that gas can convert into stars. And when stars die, they are converted into gas and may become collapsed objects. The collapsed objects can accrete the gas, or modify mass flows, which lead to gas condensations, which will create, in turn, new stars. Depending on their composition, we find Population I stars, the current population, and Population II stars, the previous generation, much poor in metal composition. The presence of changes with time leads to the problem of relaxation or evolution towards a stationary state. This is a key issue when considering the temporal timescales applicable to a dynamical model of a galaxy. Indeed, the time spent for relaxation could be as long as the timescales of the evolution of the universe. As a consequence, the galaxies can be considered complex systems. Indeed, following a complex system approach as the one applied to environmental problems, one would be tempted to find similar parameters to those of living beings, because as these ones, galaxies are born, live, are interdependent of other individuals during their lives and, finally, die. In galactic astronomy, as any other science, one makes observations, establishes a model and tries to make predictions based on it. The important point here is that one cannot modify the observed parameters when studying galaxies, but conversely, one is constrained to observe a variety of galaxies apparently frozen in a different stage of their evolution. As a consequence, the resolution of realistic enough models is a fundamental aspect in galactic astronomy, and quite often, scientists working in this field are seen as model constructors. One approximation to the problem is to solve the gravitational N -body problem. The galaxy is modelled as a self-gravitating model containing many pointlike masses associated with the star component. Additional masses model the gas and

5.7 Stellar Systems

147

the dark matter. The problem will consist in solving a 6N dimensional system. One obstacle that any N -body simulation has to face is that even the most modest galaxy contains around 1011 stars, not counting the gas and dark matter. This number of stars is well above the capabilities of current N -body computational facilities that are capable of dealing with roughly 108 particles at most. This means that the gravitational potential must be artificially modified and smoothed by adding some scaling parameters to the model. One alternative is to describe the galaxy by a mass distribution function, also called fine distribution function. This distribution function provides the star masses within a phase space volume at a given time. The self-consistent approach derives the potential that a given particle sees from this function by solving the noncollisional Boltzmann equation. This equation provides the evolution in time of the distribution function as the sum of the external forces and the diffusion of particles. When modelling galaxies one must take into account that the stars within galaxies are extremely far away one from another, so they can be considered noncollisional systems. The dynamics can be considered as formed by independent trajectories within a global potential, where the motion of each star is just driven by a continuous smooth potential. A dynamical model usually describes a given system by a mathematical expression, with a potential like a function of the distance from the centre of the galaxy. The system to solve is then a 6-dimensional system. The Jeans theorem specifies that the density of the distribution can be calculated using only integrals of motion, based on the ergodic property of the orbits. The first studies on galactic dynamics focused on integrable or near-integrable systems, searching for invariants in equilibrium and isolated systems. It was observed that those systems supported both regular and chaotic orbits. Some of these studies analysed the standard map [38], and others searched for generic properties using simple and numerically reasonable potentials, imposing certain axisymmetric properties [29]. Later on, more sophisticated self-consistent models were built, with timedependent and time-independent solutions to the non-collisional Boltzmann equation. Analytic, non-trivial solutions such as Stäckel integrable models are of interest [3, 8], but in the general case, it is mandatory to use the numerical calculus approach. For our purposes, triaxial gravitational potentials are of special interest. Initial studies of galactic dynamics used standard time-independent Jeans modelling, derived from the Boltzmann equation. Meanwhile Jeans solutions are simple; they just support regular orbits and did not reproduce the observed mass distributions and light profiles. Therefore, more general models were introduced later on. Among them, triaxial models allow a better modelling of the observed time-evolving features and allow the presence of chaotic orbits. We will detail these aspects in the following chapter.

148

5 Chaos, Predictability and Astronomy

5.8 Concluding Remarks In this chapter we have briefly listed how nonlinear dynamics and chaos theory tools can be applied to the field of astronomy and astrophysics. We did not aim to be exhaustive, but rather provide some illustrative examples to better present how these tools provide some insight in the field. Astrophysics is an experimental science but, in general, recollects information of observed objects without the possibility of having direct access to them. As a consequence, the confidence in the results coming from numerical simulations is of the outmost interest, as these simulations become more widely used as the available computing resources grow. In this scenario, we have an interdisciplinary field, where the astrophysics field provides the simulated models, the nonlinear dynamics field provides their chaoticity and instability properties and the computational sciences provide the actual numerical implementation. We have reviewed just a few of the most common applications in this field. First, we have seen the analysis of time series using nonlinear dynamics tools. These analyses usually aim to gain some understanding of the hidden mechanisms generating the observed dynamics, trying to provide some thresholds to the dimensions of the analysed model. However, we will focus on the applications of nonlinear dynamics and chaos theory to the celestial mechanics field. We already saw in the first chapter how these analyses gave born to the chaos theory, when trying to study the stability of the Solar System. Indeed, the motions in even a very simple problem like the Sitnikov problem, where a planetoid crosses perpendicularly the orbital plane defined by two primary bodies, can lead to arbitrarily complex motions. If this simple example can lead to this complexity, it may be thought that the analyses of more realistic models such as galaxies and N-body systems will be useless from a practical point of view. However, the predictability index presented in the previous chapter can help a lot in these analyses as we will see in the following chapter.

References 1. Boyd, P.T., Mindlin, G.B., Gilmore, R., Solari, H.G.: Topological analysis of chaotic orbits: revisiting Hyperion. Astrophys. J. 431, 425 (1994) 2. Breeden, J.L.: Chaos in core oscillations of globular clusters. Astrophys. J. 448, 672 (1995) 3. De Bruyne, V., Leewin, F., Dejonghe, H.: Approximate third integrals for axisymmetric potentials using local stackel fits. Mon. Not. R. Astron. Soc. 311, 297 (2000) 4. Buchler, J.T., Goupil, M.J.: A mechanism for the irregular variability of supergigant stars. Astron. Astrophys. 190, 137 (1988) 5. Cannizo, J.K., Goodings, D.A.: Chaos in SS cygni? Astrophys. J. 334, L31 (1988) 6. Compte, A.: The Positive Philosophy, Book II, Chapter 1. George Bell and Sons, London (1842)

References

149

7. Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin (2002) 8. Contopoulos, G., Vandervoort, P.O.: A rotating Staeckel potential. Astrophys. J. 389, 118 (1992) 9. Correia, A.C.M., Laskar, J.: Lon-term evolution of the spin of Venus: II numerical simulations. Icarus 163, 24 (2003) 10. Dvorak, R.: Numerical results to the Sitnikov-problem. Celest. Mech. Dyn. Astron. 56, 7180 (1993) 11. Edvardsson, S., Karlsson, K.G.: Spin axis variations of Mars: numerical limitations and model dependencies. Astron. J. 135, 1151 (2008) 12. Hanslmeier, A., Brajsa, R.: The chaotic solar cycle: I analysis of cosmogenic 14C-data. Astron. Astrophys. 509, A5 (2010) 13. Harding, A.K., Shinbrot, T.: A chaotic attractor in timing noise from the Vela pulsar. Astrophys. J. 353, 588 (1990) 14. von Hoerner, S.: Die numerische integration des n-Körper-Problemes für Sternhaufen. I. Z. Astroph. 50, 184 (1960) 15. Holmberg, E.: On the clustering tendencies among the nebulae. II. A study of encounters between laboratory models of stellar systems by a new integration procedure. Astrophys. J. 94, 385 (1941) 16. Hubble, E.: Extragalactic nebulae. Astrophys. J. 64, 321 (1926) 17. Kirchhoff, G.: Ueber die Fraunhofer’schen Linien (On Fraunhofer’s lines), Monatsbericht der Kniglichen Preussische Akademie der Wissenschaften zu Berlin (Monthly report of the Royal Prussian Academy of Sciences in Berlin), 662–665 (1859) 18. Kiss, L.L., Szatmary, K.: Period-doubling events in the light curve of R Cygni: evidence for chaotic behaviour. Astron. Astrophys. 390, 585 (2002) 19. Kollath, Z., Buchler, J.R., Serre, T., Mattei, J.: Analysis of the irregular pulsations of AC Herculis. Astron. Astrophys. 329, 147 (1998) 20. Konig, M., Timmer, J.: Analysing X-ray variability by linear state space models. Astron. Astrophys. Suppl. Ser. 124, 589 (1997) 21. Kovacs, T., Erdi, B.: Transient chaos in the Sitnikov problem. Celest. Mech. Dyn. Astron. 105, 289 (2009) 22. Kovacs, T., Bene, G.Y., Tel, T.: Relativistic effects in the chaotic Sitnikov problem. Mon. Not. R. Astron. Soc. 414, 2275 (2011) 23. Laskar, J., Joutel, F., Robutel, P.: Stabilization of the Earth’s obliquity by the Moon. Nature 361, 615 (1993) 24. Laughlin, G., Chambers, J.E.: Extrasolar Trojans: the viability and detectability of planets in the 1:1 resonance. Astron. J. 600, 124 (2002) 25. Lawrence, A., Papadakis, I.: X-ray variability of active galactic nuclei: a universal power spectrum with luminosity dependent amplitude. Astrophys. J. 414, L85 (1993) 26. Lehto, H.J., Mc Hardy, I.M.: AGN X-ray light curves -shot noise or low dimensional attractor? Mon. Not. R. Astron. Soc. 261, 125 (1993) 27. Lochner, J.C., Swank, J.H., Szymkowiak, A.E.: A search for a dynamical attractor in Cygnus X-1. Astrophys. J. 337, 823 (1989) 28. MacMillan, W.D.: An integrable case in the restricted problem of three bodies. Astrophys. J. 27, 11 (1911) 29. Magnenat, P.: Asymptotic orbits and instability zones in dynamical systems. Astron. Astrophys. 77, 332 (1979) 30. Mannatil, M., Gupta, H., Chakraborty, S.: Revisiting evidence of chaos in X-Ray light curves: the case of GRS1915+105. Astrophys. J. 833, 208 (2016) 31. Merrit, D., Fridman, T.: Triaxial galaxies with cusps. Astrophys. J. 460, 136 (1996) 32. Mikkola, S., Innanen, K.: Solar system chaos and the distribution of asteroid orbits. Mon. Not. R. Astron. Soc. 277, 497 (1995) 33. Milankovitch, M.: Canon of insolation and the ice-age problem (Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem). Academie royale serbe, Editions speciales, 132 Belgrad (1941)

150

5 Chaos, Predictability and Astronomy

34. Moore, G.E.: Cramming more components onto integrated circuits. Electronics 38, 114–117 (1965) 35. Moser, J.: Stable and random motion in dynamical systems. Ann. Math. Stud. 77, 199 (1973) 36. Muller, P., Dvorak, R.: A survey of the dynamics of main belt asteroids. Astron. Astrophys. 300, 289 (1995) 37. Orzaru, C.M.: On the dimension of the solar activity attractor. IAUS 157, 910 (1993) 38. Pfenniger, D.: Numerical study of complex stability. Astron. Astrophys. 150, 97 (1985) 39. Poincaré, H.: On the three-body problem and the equations of dynamics. Acta Math. 13, 1 (1890) 40. Qin, Z.: A nonlinear prediction of the smoothed monthly sunspot numbers. Astron. Astrophys. 310, 646 (1996) 41. Regev, O.: Chaos and Complexity in Astrophysics. Cambridge University Press, New York (2006) 42. Ritcher, P.H.: Harmony and complexity: order and chaos in mechanical systems. Lecture on “The emergence of complexity in Mathematics, Physics, Chemistry and Biology” Plenary session of the Pontifical Academy of Sciences, Rome (October 1992) 43. Roy, A.E.: Orbital Motion. Ed. Adam Hilger, Bristol (1982) 44. Serre, T., Kollath, Z., Buchler, J.R.: Search for low dimensional nonlinear behavior in irregular variable stars. Astron. Astrophys. 311, 833 (1996) 45. Seymour, A.D., Lorimer, D.R.: Evidence for chaotic behaviour in pulsar spin-down rates. Mon. Not. R. Astron. Soc. 428, 983 (2013) 46. Sitnikov, K.: The existence of oscillatory motions in the three-body problems. Dokl. Akad. Nauk. USSR 133, 303 (1960) 47. Sitnikov, K.: The existence of oscillatory motions in the three-body problem. Sov. Phys. Dokl. 5, 647 (1961) 48. Springel, V.: The cosmological simulation code GADGET-2. Mon. Not. R. Astron. Soc. 364, 1105 (2005) 49. Sukova, P., Grzedzielski, M., Janiuk, J.: Chaotic and stochastic processes in the accretion flows of the black hole X-ray binaries revealed by recurrence analysis. Astron. Astrophys. 586, A143 (2016) 50. Timmer, J., Schwarz, U., Voss, H.U., Kurths, J.: Linear and nonlinear time series analysis of the black hole candidate Cygnus X-1. Phys. Rev. E 61, 1342 (2000) 51. Varvoglis, H., Voyatzis, G., Scholl, H.: Spectral analysis of asteroidal trajectories in the 2:1 resonance. Astron. Astrophys. 300, 591 (1995)

Chapter 6

A Detailed Example: Galactic Dynamics

6.1 Introduction We have seen the impact of the presence of chaos in a given dynamical system. Furthermore, the analysis of the finite-time Lyapunov exponent distributions allows the computation of the predictability indicator, which characterises the presence of shadowing in a given orbit. Our purpose now is to apply these ideas to a particular case in the field of galactic astronomy, modelling the Milky Way. As a matter of fact, the predictability of this model depends on several physically meaningful parameters associated with the shape of the potential. We will focus on exploring the effect of the dark halo shapes on the predictability of computed orbits in a Milky Way mean field model. We will also present the sources for the low predictability found in the less predictable orbits from the analysis of the distributions of the finite-time Lyapunov exponents. This analysis shows that the lowest predictability may be related to the presence of a strong unstable dimension variability. These analyses also reveal that not all chaotic orbits have the same predictability and that the predictability of some orbits is more sensitive than others to the orientation and shape of the dark halo. Furthermore, we will see the importance of using the proper interval lengths. This corresponds to the timescales when the flow dynamics leaves the local regime and enters the global regime. This can be done by computing those distributions and analysing the evolution of their shapes as we vary the finite-time interval.

© Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9_6

151

152

6 A Detailed Example: Galactic Dynamics

6.2 Chaos in Galactic Astronomy Galactic models are usually near-integrable. In general for non-integrable systems, there is a lack of a general proof of ergodicity for irregular orbits, and the Jeans theorem is not applicable any longer. When the number of integrals changes from one point to another, the phase space population is not uniform, and the orbits can be semi-ergodic, as it happens when the stickiness phenomenon is present. The Jeans theorem must be reformulated to take into account time dependencies and turbulent flows. Indeed, the non-collisional hypothesis is not valid when we take into account the masses interchange with the ISM. Nowadays galactic models are modelled following a full Boltzmann equation, adding discreteness effects such as dynamical friction and noise. This noise can appear in the form of random kicks or in the form of multiplicative noise, when it is dependent both on velocity and position. The gas clouds can be modelled in a continuous approximation by a NavierStokes equation, but in a discrete approximation, they can be seen as points around the galaxy which suffer dissipation and drag. For modelling this dissipation, one can use the Chandrasekhar dynamical friction or dissipation laws proportional to v n [10, 23]. Rotating triaxial models, barred spiral galaxies, axisymmetric galaxies with bulges with strong density peaks or inter-actuating galaxies, are examples where the presence of chaos has been confirmed. As a general result, the regular orbits build the skeleton of the mass distribution function generating the potential. And the chaotic orbits populate regions of phase space which are not accessible to the regular orbits. These chaotic orbits can support some observed structures such as the bars and explain the creation of bulges and thick discs. They can also explain some phenomena as the disc heating and the mass flow through the bars, accelerating the relaxation process towards a stationary state [8, 19, 23, 24]. Semi-ergodic orbits can lead to turbulences in the stellar flow (velocities field). These chaotic orbits seem to be more dependent on the perturbations. So, the differences between the smooth potential and the real model can grow in times shorter than the relaxation time [22]. An unsolved problem is the fact that chaotic confined orbits cannot be confined because of the Arnold diffusion. This diffusion can be accelerated because of triaxiality or noise. So, the number of available orbits could be reduced. The variety of shapes of dark matter haloes indicates that their structure plays an important role in the dynamics of the galaxies. In general, as the galaxy disc masses are larger, the amount of chaos increases. Spherically symmetric haloes reduce this chaoticity. But nearly all computed perturbations, such as density concentrations, flattening and departures from ellipticity, increase the degree of stochasticity in these potentials [27, 37]. Triaxial haloes introduce nonlinear coupling that affects the chaoticity. The studies of the role of triaxial dark haloes are of special interest. One of the predictions of the cold dark matter (CDM) models is that galaxy-scale dark matter haloes are described by triaxial density ellipsoids. The characteristic axial ratios of

6.3 Predictability in a Galactic System

153

these haloes are typically predicted to be far from spherical [15]. Depending on the degree of triaxiality, the phase space of a logarithmic potential can be occupied to a large extent by chaotic orbits [3, 4, 20, 36, 38, 39]. Observations of individual galaxies have not yet fully confirmed this point, as gravitational lensing and X-ray observations are sensitive just to the integral of the density profile along the line of sight, without providing full three-dimensional information. So simulations are a key element in these studies, and, as a consequence, the analysis of the predictability of these models is of interest.

6.3 Predictability in a Galactic System In the previous chapters, we have seen how one can derive the predictability index from the distributions of the finite-time Lyapunov exponents and by analysing how their shapes evolve with the interval length Δt. This has been done in simple conservative systems, such as the Hénon-Heiles and the Contopoulos system and also in dissipative systems, such as the one formed by two nonlinearly coupled Rössler oscillators. We will show now how the computations of the predictability index behave when they are applied to a more realistic, three degree-of-freedom, 6-dimensional system. When dealing with high-dimensional systems, some issues must be taken into account. In systems with 2 or less degrees-of-freedom, regular and non-regular orbits are separated by impenetrable barriers, the KAM tori, leading to islands of regularity embedded into a surrounding chaotic sea. According to the KAM theorem, these tori will survive under small perturbations if their frequencies are sufficiently incommensurable [16]. Resonant tori may be strongly deformed even under small perturbations, however, leading to a complicated phase space structure of interleaved regular and chaotic regions. Where tori persist, the motion can still be characterised in terms of N local integrals. Where tori are destroyed, the motion is chaotic, and the orbits move in a space of higher dimensionality than N . In systems with more than 2 degrees-of-freedom, like the selected potential, the chaotic sea contains a hidden non-uniformity because the motion can diffuse through invariant tori, reaching arbitrarily far regions. Within the chaotic sea, there are cantori, leaking or fractured KAM tori, associated with the breakdown of integrability. These cantori are just partial barriers that, over short times, divide the chaotic orbits into two types: confined and unconfined. The confined ones are chaotic orbits which are trapped near the regular islands and, for a while, exhibit regular-like behaviour. Conversely, the unconfined orbits travel unmixed through the whole allowed sea. Furthermore, the cantori are partial barriers, allowing one orbit to change from one class to the other, via the intrinsic diffusion or Arnold diffusion. This is a very slow phenomenon, with typical timescales longer than the age of the universe. In 6-dimensional phase space systems, the sticky transients are not present, cantori appears, and the Arnold diffusion produces the ultimate merging of all

154

6 A Detailed Example: Galactic Dynamics

orbits. But this diffusion seems to be very small. Strong local instability does not mean diffusion in phase space. And some chaotic trajectories may require very long timescales to reveal its asymptotic nature. They can have very short Lyapunov times, but they cannot show the expected significant orbital changes at long times [2, 7, 26]. In these cases, the dynamics can be considered as regular motion from an astronomical point of view during the applicable timescales. As a consequence, the fact that two regions in phase space are connected does not mean that all the areas in that volume will be accessed on comparable timescales. This long lifetime transient, unconfined orbit, is sometimes called near-invariant distribution, as it uniformly populates the filling region. It is remarkable that even when the true equilibrium corresponds to a uniform distribution through both cavities, at physically meaningful timescales, the quasi-equilibrium may have one cavity uniformly populated, while the other one is, essentially, empty. The selected model is a model of our galaxy (i.e. the Milky Way), based on a mean field potential. This selection could be seen as too simplistic. The gravitational N -body simulation is a common tool to study the evolution of the galaxies and the formation of their features. The galaxy is modelled as a self-gravitating system containing stars, gas particles and dark matter, all of them modelled as pointlike masses. The self-consistency of these models captures very well the necessary details of the galactic dynamics; however, the available computational resources impose a limit to the number of particles to be taken into account. This usually implies an artificial smoothing of the potential and a proper handling of the required scaling parameters. As an alternative, another approach that can be taken is the use of simulations based on a single mean field potential. As there are no collisions among particles, the dynamics of a galaxy can be considered to be formed by independent trajectories within the global potential where the motion of each star is just driven by a continuous smooth potential. A dynamical model describes the potential as a function of the distance from the centre of the galaxy. Some potentials are derived at specific snapshots of the N -body simulations, and some others are selected to physically represent desired characteristics of the galaxies. There is a considerable number of realistic galactic models in the literature that capture and describe several observed features in galaxies such as bars, spirals or rings. See, among others, [8, 21, 25, 35]. In this chapter, we have selected the potential described in [15] and references therein. This is a smooth fixed gravitational time-independent potential that models the Milky Way but focuses on the parameters controlling the shape and orientation of a triaxial dark halo. It consists of a Miyamoto-Nagai disc [18], a Hernquist spheroid and a logarithmic halo. This potential provides enough information to be considered to be a realistic one. It does not take the gravitational influence of a rotating galactic bar into account, but it is considered sufficient because it reproduces the flat rotation curve for a Milky Way-type galaxy and can be easily shaped to the axial ratios of the ellipsoidal isopotential surfaces. By selecting different values of the model parameters, it will

6.3 Predictability in a Galactic System

155

allow focusing on their effect on the predictability of the model. These control parameters are the orientation of the major axis of the triaxial halo and the flattening. The dynamical system to solve is a particle (star) subject to a potential built upon three components: V = Φdisc + Φsphere + Φhalo .

(6.1)

The respective contribution of every component to the gravitational potential is given by: Φdisc = −α 

GMdisc , √ R 2 + (a + z2 + b2 )2

(6.2)

GMsphere , r +c

(6.3)

Φsphere = −α

2 2 Φhalo = vhalo ln (C1 x 2 + C2 y 2 + C3 xy + (z/qz )2 + rhalo ),

(6.4)

where the various constants C1 , C2 and C3 are given by:  C1 =  C2 =

cos2 φ sin2 φ + q12 q22 cos2 φ sin2 φ + 2 q2 q12 

C3 = 2 sin φ cos φ

 ,

(6.5)

,

(6.6)



1 1 − 2 2 q1 q2

 .

(6.7)

It should be noted that there is no symmetry in the potential and V (φ) = V (−φ) because of the sign dependence in the xy coupling factor C3. When φ = 0, q1 is aligned with the galactic X-axis and Eq. 6.4 reduces to 2 2 ln ((x/q1 )2 + (y/q2 )2 + (z/qz )2 + rhalo ). Φhalo = vhalo

(6.8)

The results with φ = 0 are then comparable with non triaxial, purely logarithmic potentials. When φ = 90, q1 is aligned with the galactic Y -axis, and it takes the role of q2. The parameter α could range from 0.25 up to 1.0 and following [13, 15] is fixed to 1.0. We also adopt Mdisc = 1.0 · 1011 Msun , Msphere = 3.4 · 1010 Msun , α = 1.0, a = 6.5 kpc, b = 0.26 kpc, c = 0.7 kpc, rhalo = 12 kpc. We have also fixed vhalo = 128 km/s (leading to a local standard of rest (LSR) of 220 km/s). The time units are in Gyr with these parameters values.

156

6 A Detailed Example: Galactic Dynamics

Table 6.1 Selected orbits for analysing the predictability index in a 3 d.o.f. Milky Way system Orbit Orbit type M1 Regular

Control parameter λ φhalo = 0.0 0.0

M2

φhalo = 0.0

0.14

φhalo = 90.0

0.099

φhalo = 0.0

5.86

M3 M4

Initial condition x = 10.0 y = 0.0 z = 0.0 vx = 0.0 vy = 200.0 vz = 0.0 Chaotic x = 10.0 y = 0.0 z = 10.0 vx = 0.0 vy = 45.0 vz = 0.0 Chaotic x = 10.0 y = 0.0 z = 10.0 vx = 0.0 vy = 200.0 vz = 0.0 Strongly chaotic x = 5.0 y = 0.0 z = 0.5 vx = 0.0 vy = 100.0 vz = 0.0

λ is the asymptotic standard Lyapunov exponent. The notion weak or strong is associated with the relatively smaller or larger value of λ

The control parameters of this model are the orientation of the major axis of the triaxial halo φ and its flattening. This flattening is introduced along the three axes by the parameters q1 , q2 and qz . The qz represents the flattening perpendicular to the galactic plane, while q1 and q2 are free to rotate in the galactic plane at an angle φ to a right-handed galactocentric X, Y coordinate system. We follow the parameter settings of [15] and, without loss of generality, q2 = 1.0, q1 = 1.4 and qz = 1.25. Regarding the particle initial conditions, we use stars with velocities within the halo kinematics range [5, 6]. These initial conditions, and the values of the control parameter φ, corresponding to the four analysed orbits, are listed in Table 6.1. The initial velocity vector in all cases is contained into the z = 0 plane, meaning vz = 0.0, and is normal to the x axis, meaning vx = 0.0. We just select for every initial condition the velocity modulus, |v| = vy . Massless particles subject to the selected gravitational potentials are integrated using the standard variational method described in the Appendix A to compute the finite-time Lyapunov exponents. We solved at the same time the flow equations and the fundamental equations or evolution of the distortion tensor, associated with the initial set of deviation vectors used for the exponent computation. The detailed equations are not listed here for the sake of conciseness, but are easily derived following the same methods described for the remaining simpler systems. We use as integrator the well-known and reliable Dop853 algorithm described in [11]. The Lyapunov exponents follow the pairing property and that the energy value is constant throughout the computation, typically having a percentual error of 10−8 for the potential. We will analyse the four orbits shown in Fig. 6.1, labeled as M1, M2, M3 and M4. They are listed in Table 6.1. The first orbit, M1, is a regular orbit, selected for comparing the timescales of this model with the previously analysed meridional potentials. The following orbits, M2, M3 and M4, are chaotic orbits. These are confined within some phase space domain for a while but, afterwards, can escape from those domains. As a consequence of these transients, the distribution shapes vary depending on the selection of the finite-time lengths. The first analysed case is a regular orbit, characterised by λ = 0.0 and confined into the disc plane z = 0 for the whole integration. This is the orbit labeled as M1 in

6.3 Predictability in a Galactic System

157

Fig. 6.1 Four orbits selected for calculating the predictability in a Milky Way-type potential. The initial conditions and halo orientation values are listed in Table 6.1. Upper left: M1, a regular orbit confined to the disc, with asymptotic Lyapunov exponent λ = 0.0. Upper right: M2, a chaotic orbit out of the disc plane, with asymptotic Lyapunov exponent λ = 0.14. Bottom left: M3, a chaotic orbit out of the disc plane, with asymptotic Lyapunov exponent λ = 0.099. Bottom right: M4, a strongly chaotic orbit, inner and close to the disc plane, with asymptotic Lyapunov exponent λ = 5.86. (Taken from [29] with permission)

Fig. 6.1 and in Table 6.1. We have selected it in order to compare the predictability timescales in this model with respect to the meridional potentials seen before. This is of interest because a single-step error δt may have different consequences in every model, and the shadowing times for regular orbits in different models are not necessarily similar. The corresponding Poincaré section y − vy with plane x = 0 is seen in Fig. 6.2 (top). When considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 0.50, with a minimum value of 0.44. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 0.50, with a minimum value of 0.46. The evolution of the kurtosis and predictability index with the interval size are shown in Fig. 6.2 (bottom). Contrary to the regular orbits seen in the meriodional potential cases, the kurtosis does not show a simple trend as the interval length grows, and there is a set of different zero crossings starting around Δt = 0.06. These oscillations at small interval lengths below the Tcross range of values are sourced to the fluctuations of the shapes of the distributions when the intervals are very small [32]. The Tcross indicates when the global regime is reached, and the kurtosis zero-

158

6 A Detailed Example: Galactic Dynamics

60

M1

40

Vy

20 0 -20 -40 -60

6 5 4 3 2 1 0 0.01

9

8.5

9.5

y

0.1

10

0.1

11

10 0.03 0.025 0.02 0.015 0.01 0.005 0

10 1 0.01

10.5

1

h

k

8

Δt

1

0 2 4 6 8

10

Fig. 6.2 Milky Way regular orbit M1, confined into the disc plane, with asymptotic Lyapunov exponent λ = 0.0. Top: Poincaré sections y − vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finitetime length is increased. Inset: Finite-time exponents distribution for Δt = 0.6. The predictability index is h = 5.03. (Taken from [29] with permission)

cross corresponding to these scales is seen at Δt = 0.6. The corresponding closest to zero exponent distribution is plotted in the inset of Fig. 6.2. It is characterised by a mean m = 6.53 and a probability of positivity P+ = 0.99. The mean and the probability of positivity indicate that we have detected the global regime, but we are still far away from the asymptotic regime. The predictability index is h = 5.03. Note that this is a very low predictability value when one compares it with the values seen in the meridional potentials, both for regular and chaotic orbits. This indicates that one must handle with care long

6.3 Predictability in a Galactic System

600

159

M2

400

Vy

200 0 -200 -400 -600 -20

-10

0

10

20

y k

5 4 3 2 1 0 0.01 10

0.1

1

10

0.04

1 h

0.03

0.1

0.02 0.01

0.01 0.001 0.01

0 -20-15-10 -5 0 5

0.1

Δt

1

10

Fig. 6.3 Milky Way chaotic orbit M2, out of the disc plane, with asymptotic Lyapunov exponent λ = 0.14. Left: Poincaré sections y − vy with plane x = 0 and vx > 0. Right: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponent distribution for Δt = 0.6. The predictability index is h = 1.31. (Taken from [29] with permission)

integrations in this potential. The shadowing time of a regular orbit can be large or small depending on the analysed potential, because of the different dynamical times. And when the shadowing times are very low, one should use higher precision schemes, even when the gain in shadowing time may be small in the extreme cases. The next analysed initial condition corresponds to the star labeled as M2 in Fig. 6.1 and Table 6.1. The Poincaré section y − vy with plane x = 0 corresponding to this orbit is seen in in Fig. 6.3 (top). This is a chaotic orbit characterised by λ = 0.14. We have selected this orbit because it is a chaotic orbit that initially

160

6 A Detailed Example: Galactic Dynamics

remains in a limited domain of the phase space but then fills up a larger domain of the available phase space, as seen in Fig. 6.3 (top). When considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 0.61, with a minimum value of 0.32. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 0.53, with a minimum value of 0.34. When considering the crosses of the z = 0 plane with vz > 0, the averaged Poincaré section crossing time is Tcross = 0.41, with a minimum value of 0.04. The evolution of the kurtosis and predictability index h with the interval size are shown in Fig. 6.3 (bottom). Contrary to previous models, there is not a simple increasing trend of kurtosis with Δt. Instead, there is a set of different zero crossings. We observe a zero crossing in the kurtosis curve at around Δt = 0.035, but this value is well below the Tcross range of values. There is also a zero crossing at a very large interval size (not shown in the figure), when the asymptotic regime is reached. The kurtosis zero-cross corresponding to the timescales, when the global regime of the flow is reached, is seen at Δt = 0.6. The corresponding closest to zero exponent distribution is plotted in the inset of the figure. It is characterised by a mean m = −2.8 and a probability of positivity P+ = 0.08. The mean and probability of positivity indicate that we have detected the global regime, but we are still very far away from the asymptotic regime. The predictability index is h = 1.31. Note that this is a very low predictability value when compared with previous cases, indicating that some care must be taken when performing long integrations using this potential. Indeed, taking into account the kurtosis oscillations, we may consider that we have taken an upper limit for the value of the predictability, and within certain transients, the predictability of the orbit may be even worse. The following initial condition is the orbit labeled as M3 in Fig. 6.1 and Table 6.1. The Poincaré section of this orbit is seen in Fig. 6.4 (top). This orbit is characterised by λ = 0.099. The motion is then chaotic, with some transient periods spent in the external lobes of the section. When considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 0.87, with a minimum value of 0.71. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 0.90, with a minimum value of 0.76. When considering the crosses of the z = 0 plane with vz > 0, the averaged Poincaré section crossing time is Tcross = 0.86, with a minimum value of 0.62. The evolution of the predictability index h with the interval size is shown in Fig. 6.4 (bottom). The zero crossing of the kurtosis within the range of values indicated by the Poincaré crossing time Tcross is found at Δt = 1.01. The corresponding finite-time distribution is plotted in the inset of the figure. It is characterised by a mean m = 0.83 and a probability of positivity P+ = 0.8. The derived predictability index is h = 2.06. Similar to the previous case, we can consider this value as an upper limit to the predictability of the orbit, since the orbit may suffer transient periods with an even worse predictability.

6.3 Predictability in a Galactic System

161

300 200

M3

Vy

100 0 -100 -200 -300 -20

-10

0

10

y

20

k

3 2 1 0 0.1

1

10

10 h

0.04

1

0.03 0.02 0.01

0.1

0

0.1

Δt

1

-2 0 2 4

10

Fig. 6.4 Milky Way chaotic orbit out of the disc plane, M3, with asymptotic Lyapunov exponent λ = 0.099. Top: Poincaré sections y − vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finite-time length is increased. Inset: Finite-time exponent distribution for Δt = 1.01. The predictability index is h = 2.06. (Taken from [29] with permission)

The fourth analysed condition is the orbit labeled as M4 in Fig. 6.1 and Table 6.1. This is a star close to the disc plane; therefore, it is located in an inner region than the previous orbits. The Poincaré section of this orbit is seen in Fig. 6.5 (top). This orbit is characterised by λ = 5.86. The motion is then strongly chaotic. When considering the crosses of the x = 0 plane with vx > 0, the averaged Poincaré section crossing time is Tcross = 0.18, with a minimum value of 0.12. When considering the crosses of the y = 0 plane with vy > 0, the averaged Poincaré section crossing time is Tcross = 0.19, with a minimum value of 0.16. When considering the crosses of the

162

6 A Detailed Example: Galactic Dynamics

300

M4

200

Vy

100 0 -100 -200

k

-300

4 3 2 1 0 0.01

0

1

2

3

y

0.1

4

1

h

10

6

10 0.04 0.03

1

0.02

0.1 0.01 0.01

5

0.01 0 -200 -100 0

0.1

Δt

1

10

Fig. 6.5 Milky Way strongly chaotic orbit out of the disc plane, M4, with asymptotic Lyapunov exponent λ = 5.86. Top: Poincaré sections y −vy with plane x = 0 and vx > 0. Bottom: Evolution of the kurtosis k and predictability index h of the finite-time exponent distributions as the finitetime length is increased. Inset: Finite-time exponent distribution for Δt = 0.07. The predictability index is h = 0.18. (Taken from [29] with permission)

z = 0 plane with vz > 0, the averaged Poincaré section crossing time is Tcross = 0.095, with a minimum value of 0.06. The evolution of the predictability index h with the interval size is shown in Fig. 6.5 (bottom). The zero crossing of the kurtosis within the range of values indicated by the Poincaré crossing time Tcross is found at Δt = 0.07. The corresponding finite-time distribution is plotted in the inset of the figure. It is characterised by a mean m = −33.55 and a probability of positivity P+ = 0.043. We are again far away from the timescales when the asymptotic dynamics is

6.4 Role of Dark Matter Haloes in Predictability

163

reached. The derived predictability index is h = 0.18. This is a very low value when compared with the previous cases, in agreement with the relatively high Lyapunov asymptotic exponent.

6.4 Role of Dark Matter Haloes in Predictability We have seen in the previous section that one prediction of the cold dark matter (CDM) models is that galaxy-scale dark matter haloes are described by a triaxial density ellipsoid and that triaxial models must be handled with particular care. The nonlinear coupling introduced by dark triaxial haloes increases the degree of chaoticity and may affect the goodness of the computed orbits. We are going to focus on the characterisation of the predictability of the computed orbits in the field of simulations of self-consistent models based on a single mean potential. A gravitational N -body simulation is a common tool to study the evolution of galaxies and formation of their features. However, the available computational resources impose a limit on the number of particles that can be taken into account. This usually implies an artificial smoothing of the potential and a proper handling of the required scaling parameters. As an alternative, another approach that might be taken is the use of simulations based on a single mean field potential. As there are no collisions among particles, the dynamics of a galaxy can be considered to be formed by independent trajectories within the global potential where the motion of each star is only driven by a continuous smooth potential. A dynamical model typically describes in a mathematical way the potential as a function of the distance from the centre of the galaxy. Some potentials are derived at specific snapshots of the N -body simulations, and others are selected to physically represent desired characteristics of the galaxies. Predictability times have been calculated and applied to the field of N -body simulations in [12], where an iterative refinement method was applied to simulate noisy trajectories and to estimate the shadowing times. Our work focuses on analysing the predictability times in galactic systems using mean field potentials. These times can be estimated from the analysis of the distributions of finite-time Lyapunov exponents. By analysing the evolution of the shapes of the finite-time Lyapunov exponent distributions as the finite-time interval size increases, we can detect when the flow leaves the local regime and reaches the global regime [32]. At this point, we can estimate the predictability index using the obtained distribution and identify the orbits with the shortest predictability times [29]. The main goal of the sections to follow is to analyse the role of the shapes and orientations of triaxial dark matter haloes on the predictability of the computed orbits. We aim to analyse here the correlation between chaos and low predictability values and how these low values depend on the dark halo parameter values. We also aim to detect the parameters leading to the lowest predictability values and to analyse the possible sources for these lowest values.

164

6 A Detailed Example: Galactic Dynamics

6.4.1 Galactic Orbits We applied the finite-time Lyapunov exponent technique to a set of representative orbits and identified those with the lowest predictability. We computed their respective predictability indexes as one control parameter of the model is varied. We checked the role that the orientation and flattening of the dark halo have on the predictability and timescales of every orbit. Triaxial galaxies have four main orbit families: box orbits and three tube orbits (short-axis tubes, inner long-axis tubes and outer long-axis tubes). The orbit structure and number of orbits belonging to each family are different in cusp, core, main body and outer part (the halo), reaching from box orbits in the central core to tube orbits outside the core region and boxlets and stochastic orbits (but generally a small fraction) at the largest radii or halo region. It is noteworthy that different combinations of orbits with distinct shapes can produce the same triaxial density distributions, so that there is a high degree of non-uniqueness in the distribution functions consistent with a given mass model. We start by selecting a representative set of initial conditions and analysed how their chaotic nature and predictability changed with the dark halo parameters. We selected the initial conditions listed in Table 6.2. We used stars with velocities within the halo kinematics range [5, 6]. The initial velocity vector in all cases is contained in the z = 0 plane, meaning V z = 0.0, and is normal to the x-axis, meaning V x = 0.0. We selected in every initial condition the velocity modulus, |v| = V y. For labeling purposes, the set of initial conditions was divided into orbits with high initial z position, or out of disc H-orbits, and those with a lower initial z position, or close to disc L-orbits.

Table 6.2 Selected representative orbits for analysing the role of the dark halo in a 3 d.o.f. Milky Way system given by the potential Eq. 6.1. The dominant behaviour label refers to the intensity of the chaos for most halo parameters Orbit HF1 (M2) HF2 (M3) HC1 HC2 LF1 LF2 LC1 LC2

Initial location High z, far from centre, low velocity High z, far from centre, high velocity High z, close to centre, low velocity High z, close to centre, high velocity Low z, far from centre, low velocity Low z, far from centre, high velocity Low z, close to centre, low velocity Low z, close to centre, high velocity

(x0 , y0 , z0 ), (10.0, 0.0, 10.0)

(V x0 , V y0 , V z0 ) (0.0, 50.0, 0.0)

Dominant behaviour Chaotic

(10.0, 0.0, 10.0)

(0.0, 200.0, 0.0)

Mainly regular

(5.0, 0.0, 10.0),

(0.0, 50.0, 0.0)

Chaotic

(5.0, 0.0, 10.0)

(0.0, 200.0, 0.0)

Mainly regular

(10.0, 0.0, 0.5)

(0.0, 50.0, 0.0)

Chaotic

(10.0, 0.0, 0.5)

(0.0, 200.0, 0.0)

Regular

(5.0, 0.0, 0.5)

(0.0, 50.0, 0.0)

Highly chaotic

(5.0, 0.0, 0.5)

(0.0, 200.0, 0.0)

Regular

6.4 Role of Dark Matter Haloes in Predictability

165

In addition, we have selected three initial conditions that have been already described in our previous section. Specifically, orbit HF1 corresponds to orbit M2, and orbit HF2 corresponds to orbit M3 (orbit LC1 was labeled M4, but referred to a different velocity).

6.4.2 Role of the Dark Halo Orientation This section presents the variation in chaos, predictability and flow timescales with dark halo orientation φ. The triaxiality is fixed in this section to be qz = 1.25, following [15, 29]. 6.4.2.1

Chaos

The possible chaotic nature of an orbit is reflected by a positive asymptotic Lyapunov exponent λ. An irregular motion is chaotic when it is bounded, the ω-limit set does not merely consist of connecting arcs, and there is at least one asymptotic positive λ [1]. Conversely, a regular orbit has vanishing Lyapunov exponents. The notion of a given orbit being weakly or strongly chaotic is here associated with the respective lower or higher value of λ. Figure 6.6 shows the variation in highest asymptotic Lyapunov exponent, or maximal Lyapunov exponent, with dark halo orientation φ for the selected orbits. The total integration time was T = 106 Gyr, long enough to reach convergence towards the final asymptotic state.

7 6 5

HF1 HF2 HC1 HC2 LF1 LF2 LC1 LC2

λ

4 3 2 1 0

0

45

90 φ

135

180

Fig. 6.6 Variation in the highest asymptotic Lyapunov exponent λ, or maximal Lyapunov exponent, with dark halo orientation φ. The flattening is fixed to be qz = 1.25. The red curves, lowest velocities, show higher values of λ, meaning stronger chaos. The blue curves, highest velocities, are mainly regular. The L-orbits, closest to the disc (dashed blue), remain regular at any orientation; the H-orbits (continuous blue) convert into chaotic at orientations around 90 degrees. (Taken from [30] with permission)

166

6 A Detailed Example: Galactic Dynamics

The orbits with lower velocities, light red lines (LC1, LF1, HC1 and HF1), seem to be the most strongly chaotic orbits (LC1 the most). All these low-velocity orbits have high values of the highest asymptotic Lyapunov exponents without any apparent dependence on the dark halo orientation. The orbits with higher velocities, dark blue lines (LC2, LF2, HC2 and HF2), are mainly regular. Of these, the H-orbits, out of the disc, blue continuous lines (HC2 and HF2), seem to be more affected by the halo orientation, while the regular nature of the orbits close to the disc, blue dashed lines (LC2 and LF2), is not affected by the halo orientation. The following figures are intended to illustrate these changes from a descriptive point of view. Figure 6.7 shows the trajectories in the physical configuration space (x, y, z) integrated during T = 50 Gyr for the orbits of Table 6.2, with a dark halo orientation of φ = 0 and a flattening parameter qz = 1.25. This flattening was chosen for comparison with previous works. The figure also shows the corresponding Poincaré sections y − vy with plane x = 0 and vx > 0, integrated during T = 5000 Gyr. Spherical-like and short-axis tube-like orbits in the physical space are visible, with regular behaviour at the highest velocities, LC2, LF2, HC2 and HF2. Figure 6.8 shows the same plots when the dark halo orientation changes to φ = 90. This figure illustrates the evolution of the shapes of the different types of orbits when the dark halo is varied. The orbits with lower velocities (LC1, LF1, HC1 and HF1) show stronger chaos. The H-orbits that previously were regular (HF2 and HC2) are now chaotic. Conversely, the L-orbits remain regular.

6.4.2.2

Predictability

We aim to show the variation in predictability with orientation of the dark halo. The predictability of an orbit is related to, but independent of, its stability or its chaotic nature. We are interested here in analysing whether the predictability evolves in the same way the chaoticity does when the dark halo orientation φ is varied. Figure 6.9 shows this evolution. The values in the predictability index h were derived from finite-time Lyapunov exponent distributions generated by gathering the finite-time intervals Δt during a total time of T = 105 Gyr. Such a long integration leads to very good statistics, even when shorter integrations are enough because of the typically much shorter Poincaré crossing times, hence the shorter applicable timescales. The top panel shows the predictability index for the L-orbits. As before, orbits with higher velocities (LC2 and LF2) have higher values of the h index, meaning better predictabilities. This agrees with Fig. 6.6, which shows that they have the lowest asymptotic Lyapunov exponents. But this top panel also shows that the good predictability of these orbits depends more strongly on the orientation of the halo than the chaoticity dependence seen in Fig. 6.6. Moreover, orbits with the lowest h predictabilities, those with lower velocities (LC1 and LF1), depend less strongly for h on φ. The predictability indexes h corresponding to the H-orbits are shown in the bottom panel of Fig. 6.9. Here, orbits with higher velocities (HC2 and HF2) show a

6.4 Role of Dark Matter Haloes in Predictability

LF1

167

800 10

600 400

5

-5

-15

-10 -5

0

X

5

-5 -10 10 15 -15

0

200

Z Vy

0

-10 15 10

5

0 -200 -400 -600

Y

-800

-15

0

-5

-10

l = 1.55 h = 0.17 LF2

10

15

80

-5 0

X

5

-5 10 15 -15-10

0

5

60 40 20

Vy

0.6 0.4 0.2 0.0 Z -0.2 -0.4

-15 -10

5

Y

0 -20

-0.6 15 10

-40 -60

Y

-80

7.5

8.0

8.5

9.0

9.5

10.0 10.5 11.0

Y

l = 0.00 h = 0.21 LC1

800

-6 -4 -2

X

0

2

4

6 -6

-4

-2

0

4

2

600 400 200

Vy

4 3 2 1 0 Z -1 -2 -3 -4 6

0 -200 -400 -600

Y

-800

-6

-4

-2

0

2

4

6

Y

l = 6.03 h = 0.12 LC2

30 0.6

-6 -4 -2

X

0

2

4

6 -6

-4

-2

0

l = 0.00 h = 0.75

2

4

Y

20 10

Vy

0.4 0.2 0.0 Z -0.2 -0.4 -0.6 6

0 -10 -20 -30

4.4

4.6

4.8

5.0

5.2

Y

Fig. 6.7 Physical trajectories and the corresponding Poincaré sections y − vy with plane x = 0 and vx > 0 for the orbits listed in Table 6.2. The orientation is φ = 0.0; the flattening is qz = 1.25. The asymptotic Lyapunov exponent λ characterises the chaos intensity. The predictability index h

168

6 A Detailed Example: Galactic Dynamics

HF1

800 15

-20-15 -10 -5

600 400 200

Vy

10 5 0 Z -5 -10 -15 20 15 10 5

0 -200 -400 -600

0 -5 Y 0 5 -10 -15 10 15 -20 20 X

-800 -20 -15 -10

-5

l = 1.38 h = 0.23 HF2

-15 -10

5

10

15

20

5

10

15

20

80

-5 0

X

5

60 40 20

Vy

20 15 10 5 0 Z -5 -10 -15 -20 20 15 10 05 -5 Y -10 -15 10 15 -20

0 -20 -40 -60 -80

-20

-15

-10

-5

HC1

0

Y

l = 0.00 h = 0.46

-15 -10

0

Y

800 15 10

-5

X

0

-5 -10 5 10 15 -15

0

5

600 400 200

Vy

5 0 Z -5 -10 -15 15 10

0 -200 -400 -600

Y

-800

-15

-10

-5

0

5

10

15

5

10

15

Y

l = 1.66 h = 0.21 HC2

30 15

5 -8 -6 0 -4 -2 -5 Y 0 2 -10 4 6 X 8 -15

l = 0.00 h = 0.41

20 10

Vy

10 5 0 Z -5 -10 -15 15 10

0 -10 -20 -30

-15

-10

-5

0

Y

Fig. 6.7 (continued) characterises the predictability. There are spherical-like and tube-like orbits in the physical space, with regular behaviours at the highest velocities (LF2, HF2, LC2 and HC2). (Taken from [30] with permission)

6.4 Role of Dark Matter Haloes in Predictability

LF1

169

800

-15 -10 -5

0

X

5

-5 -10 10 15 -15

0

5

600 400 200

Vy

0.8 0.6 0.4 0.2 0.0 Z -0.2 -0.4 -0.6 -0.8 15 10

0 -200 -400 -600

Y

-800

-15

-10

-5

l = 2.26 h = 0.18 LF2

-15 -10

0

5

10

15

Y

4

-5

X

0

5

10 15 -10

-5

0

5

3 2 1

Vy

0.6 0.4 0.2 0.0 Z -0.2 -0.4 -0.6 10

0 -1 -2 -3

Y

-4

8.75 8.80 8.85 8.90 8.95 9.00 9.05 9.10

Y

l = 0.00 h = 0.53 LC1

800

-6 -4 -2

X

0

2

4

6 -6

-4

-2

0

2

4

600 400 200

Vy

4 3 2 1 0 Z -1 -2 -3 -4 6

0 -200 -400 -600

Y

-800

-6

-4

-2

0

2

4

6

Y

l = 5.98 h = 0.17 LC2

2.0 0.6 0.4

-6 -4 -2

X

0

2

4

6 -6

-4

-2

0

l = 0.00 h = 0.48

2

Y

4

1.0 0.5

Vy

0.2 0.0 Z -0.2 -0.4 -0.6 6

1.5

0.0 -0.5 -1.0 -1.5 -2.0

4.73

4.74

4.75

4.76

Y

Fig. 6.8 Effect of the halo orientation. Physical trajectories and the corresponding Poincaré sections y − vy with plane x = 0 and vx > 0 for the orbits listed in Table 6.2. The orientation is φ = 90.0; the flattening is qz = 1.25. The orbits with lower velocities, LC1, LF1, HC1 and HF1, previously chaotic, remain chaotic. Regarding the higher velocities, H-orbits HF2 and HC2, which

170

6 A Detailed Example: Galactic Dynamics

HF1

800

-20-15 -10 -5

600 400 200

Vy

20 15 10 5 0 Z -5 -10 -15 -20 20 15 10 5

0 -200 -400 -600

0 -5 Y -10 0 5 -15 10 15 -20 20 X

-800 -20 -15 -10

-5

l = 1.29 h = 0.22 HF2

5

10

15

20

5

10

15

20

5

10

15

20

300 15 10

0 5 10 15 X

200 100

Vy

5 0 Z -5 -10 -15 20 15 510

-20-15 -10 -5

0 -100 -200

0 -5 Y -10 -15 20 -20

-300

-20

-15

-10

-5

HC1

0

Y

l = 0.39 h = 0.25

-15 -10

0

Y

800 15 10

-5

X

0

-5 -10 5 10 15 -15

0

5

600 400 200

Vy

5 0 Z -5 -10 -15 15 10

0 -200 -400 -600

Y

-800

-20

-15

-10

-5

0

Y

l = 1.67 h = 0.21 HC2

300

-15 -10 -5 0

X

5

-5 -10 10 15 -15

0

l = 0.43 h = 0.03

5

200

5

100

0 Z -5 -10 -15 15 10

Y

Vy

15 10

0 -100 -200 -300

-15

-10

-5

0

5

10

15

Y

Fig. 6.8 (continued) were previously regular, are now chaotic. L-orbits LC2 and LF2, previously regular, remain regular. The predictabilities remain the same for low velocities, but they change towards lower values for the highest velocities (except for LF2). (Taken from [30] with permission)

6.4 Role of Dark Matter Haloes in Predictability

2 LF1 LF2 LC1 LC2

1.5 h

1 0.5 0

0

90 φ

45

180

135

2 0.1

HF1 HF2 HC1 HC2

1.5 0.05

h

Fig. 6.9 Dependence of the predictability index on the dark halo orientation φ. The flattening is fixed to qz = 1.25. A higher value means a better shadowing and thus a better predictability. (top panel) L-orbits, close to the disc orbits. (bottom panel) H-orbits, initial conditions with higher z. The inset zooms on the orientation angles around 90 degrees. In this interval we see that HC2 and HF2, the blue orbits with higher initial velocities, have very low predictabilities. Specifically, HC2 has a close to zero predictability at φ = 115. This orbit is plotted in Fig. 6.10. (Taken from [30] with permission)

171

1 0 60

75

90

105 120 135 150

0.5 0 0

45

90 φ

135

180

strong dependence on φ. They are again the orbits with better (higher) h values, and for certain orientations, they also show very low values of h for other orientations. These are shown in the inset of Fig. 6.9 (bottom). This inset shows that the values of h can sometimes be extremely low. For instance, the orbit HC2 at around φ ∼ 115. This case of very low h is shown in Fig. 6.10. Here we show the trajectories in the physical configuration space (x, y, z) for the HC2 orbit and the corresponding Poincaré section y − vy with plane x = 0 and vx > 0. Standard double-precision computations may set a value of δ = 10−16 that with an h value as low as 10−4 may lead to predictability times as short as τ ∼ 1 Gyr. Some refined integration schemes may be implemented in cases like this. Interestingly, the h index follows a very similar pattern for HC2 and HF2. This means that the dependence with φ does not depend on the initial distance to the centre. As for L-orbits, the H-orbits with the lowest velocities (HC1 and HF1) show on average the poorest predictabilities, without any strong dependence on φ.

172

6 A Detailed Example: Galactic Dynamics

Fig. 6.10 One of the orbits with lowest predictabilities is HC2 when the halo has an orientation of φ = 115.0 and a flattening of qz = 1.25. This trajectory in the physical configuration space (x, y, z) is shown in the upper panel, and the corresponding Poincaré section y − vy with plane x = 0 and vx > 0 is shown in the bottom panel. This orbit has λ = 0.0 and h = 10−4 , corresponding to a case of a regular orbit with a very low predictability, linked to a very stiff problem. (Taken from [30] with permission)

Figure 6.6 shows that only HF2 and LC2 have a chaotic or regular nature depending on the halo orientation, which changes from regular to chaotic and back at around φ ∼ 90. Figure 6.9 shows that both orbits have low predictabilities for the same range of φ ∼ 90 values. However, Fig. 6.9 also shows a dependence of the predictability h on φ out of the φ ∼ 90 interval. The predictability h of these orbits also varies for a broader range of orientations, when h has higher values, even for the orientations leading to regular-like behaviours, with highest zero Lyapunov exponents.

6.4.2.3

Timescales

The distribution shapes of the finite-time Lyapunov exponents change as the finitetime interval sizes increase. The critical size of the interval Δt used to calculate h corresponds to the size when the distribution changes from a negative kurtosis to a

6.4 Role of Dark Matter Haloes in Predictability

0.3 LF1 LF2 LC1 LC2

0.25 0.2 Δt

Fig. 6.11 Variation in φ of the critical Δt interval length when the behaviour of the distributions change. The flattening is fixed to qz = 1.25. This critical interval length reflects the timescales when the dynamical flow reaches the global regime. (top) L-orbits, close to the disc orbits. (bottom) H-orbits, initial conditions with higher z. (Taken from [30] with permission)

173

0.15 0.1 0.05 0

45

90 φ

135

180

0.3 0.25

Δt

0.2 0.15 HF1 HF2 HC1 HC2

0.1 0.05

0

45

90 φ

135

180

positive kurtosis. This coincides with the timescale when the flow leaves the local dynamics and enters the global dynamics, causing the variational ellipse axes to orient themselves towards the most growing direction. The kurtosis can have many other zero crossings at very short intervals, when the distribution shapes vary strongly. Among them, we can choose the very first zero crossing in the kurtosis at an interval beyond the Poincaré section crossing time as an indicator of the proper timescale. The variation in this critical Δt with the orientation of the dark halo is presented in Fig. 6.11, the top panel corresponding to the L-orbits and the bottom panel to the H-orbits. We show that the L-orbits with different velocities can have similar timescales. The L-orbits close to the axis (LC1 and LC2) seem to have the smallest Δt, with no apparent dependence on the orientation of the halo. Conversely, the higher values of Δt correspond to the L-orbits far from the axis (LF1 and LF2), showing a stronger dependence on φ. The bottom panel corresponds to the H-orbits. As before, the H-orbits with different velocities have similar same timescales. The longest Δt intervals correspond to HF2 and HC2 (higher velocities), but these timescales are not very different from

174

6 A Detailed Example: Galactic Dynamics

the low-velocity stars. Regarding the initial distance to the centre, all timescales are similar as well. There is no dependence on the orientation φ in any case. This is interesting because such a dependence was observed for HF2 and HC2 when we plotted the predictability index. When we compare the two panels, the L-orbits have shorter interval lengths than the H-orbits. These results can be compared with the h dependence seen in Fig. 6.9. The timescales of the H-orbits do not depend strongly on the halo orientation, but they showed stronger dependencies in the plots of their chaotic nature and their predictability. The variability in h values can be directly correlated to the variability in Δt values, as is evident when we compare Fig. 6.9. The same variation in Δt may lead to different values of h in different orbits, however, as a consequence of Eq. 4.2 and the dependence on the specific characteristic of the orbit. We can therefore identify the orbits with the lowest predictabilities and the sources of these low values (see Sect. 6.5).

6.4.3 Role of the Dark Halo Flattening We analysed the dependence of the maximal Lyapunov exponent (as a chaos indicator) and the predictability indexes on the dark halo orientation. Since we have found that the strongest influence of the orientation appears at around φ = 90, now we fixed the halo orientation to that value, and this section presents the variation in chaoticity and predictability when we varied the flattening, with qz ranging from 1 to 1.8, to the potential with a rotated halo (φ = 90).

6.4.3.1

Chaos

Figure 6.12 plots the dependence of the maximal Lyapunov exponent as the qz parameter evolves. The most strongly chaotic orbits are those with the lower velocities, as in Fig. 6.6. A clear change from a regular to chaotic nature of the orbits for the orbits HC2 and HF2 (blue dark continuous line) is found, which are the two orbits with initial conditions out of the disc and with the highest velocity. From Figs. 6.6 and 6.12, we can conclude that these orbits are the most dependent on the halo parameters, both φ and qz . Following Figs. 6.7, 6.8 and 6.13 illustrates the above by showing the trajectories in the physical configuration space (x, y, z) integrated during T = 50 Gyrs, for the orbits of Table 6.2, with a dark halo orientation of φ = 90 and a flattening parameter qz = 1.4, increased with respect to previous cases. This figure also shows the corresponding Poincaré sections y − vy with plane x = 0 and vx > 0, integrated during T = 5000 Gyr.

6.4 Role of Dark Matter Haloes in Predictability

175

7 6 5

0.2

4 λ

0.1

3

0

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

HF1 HF2 HC1 HC2 LF1 LF2 LC1 LC2

2 1 0

1

1.1 1.2

1.3 1.4 1.5 1.6 1.7 1.8 qz

Fig. 6.12 Variation in the maximal Lyapunov exponent with the dark halo flattening qz . The halo orientation is fixed to φ = 90. The higher exponent values, corresponding to the most strongly chaotic orbits, are found with the lowest initial velocities (red curves). The inset shows the range of qz values where the HC2 and HF2 orbits are chaotic; otherwise, they are regular. (Taken from [30] with permission)

6.4.3.2

Predictability

The variation in predictability index h of the orbits with the flattening qz is shown in Fig. 6.14. The top panel corresponds to the L-orbits. It shows no apparent dependence of the predictability h on qz . The predictability is higher; thus, the shadowing is better, for the orbits with higher velocities (LC2 and LF2, in dark colour), in agreement with the previous section. The bottom panel shows the H-orbits, and we see a non-uniform curve, without a clear trend (increasing or decreasing) of h with qz . The two panels show a similar average range of predictability indexes of the L-orbits and the H-orbits. The inset in Fig. 6.14 zooms in the range of qz values where h is close to zero. This figure also shows for the orbit HC2 a high peak of the h value around qz = 1.45. A large h means very good predictability for this orbit. However, this orbit also shows very low predictability values when the flattening qz is below 1.3, as shown by the inset. This behaviour agrees with the dependence of the dark halo orientation φ seen in previous section. The orbit with the lowest predictability is found at qz = 1.32. Figure 6.15 presents this case, showing in the top panel the trajectories in the physical configuration space (x, y, z), and the corresponding Poincaré section y−vy with plane x = 0 and vx > 0 in the bottom panel. We conclude that HC2 is very sensitive to the dark halo shape and orientation. Nevertheless, the HF2 orbit, which also showed a dependence of h on the halo orientation, does not seem to have such a dependence on qz .

176

6 A Detailed Example: Galactic Dynamics

LF1

600

-15 -10

-5

0

X

-5 -10 5 10 15 -15

0

5

400 200

Z Vy

5 4 3 2 1 0 -1 -2 -3 15 10

0 -200 -400

Y

-600 -15

-10

-5

5

10

15

Y

l = 2.39 h = 0.13 LF2

-15

0

4

-10 -5

0

X

5 10

15 -10

-5

0

5

3 2 1

Vy

0.6 0.4 0.2 0.0 Z -0.2 -0.4 -0.6 10

0 -1 -2 -3 -4

Y

8.75 8.80 8.85 8.90 8.95 9.00 9.05 9.10

Y

l = 0.00 h = 0.53 LC1

800

-6 -4 -2

0

X

2

4

6 -6

-4

-2

0

2

4

600 400 200

Z Vy

4 3 2 1 0 -1 -2 -3 -4 6

0 -200 -400 -600 -800

Y

-6

-4

-2

0

2

4

6

Y

l = 5.97 h = 0.12 LC2

2.0 0.6 0.4

-6 -4 -2 0

X

2

4

6 -6

-4

-2

0

2

4

Y

1.5 1.0 0.5

Vy

0.2 0.0 Z -0.2 -0.4 -0.6 6

0 -0.5 -1.0 -1.5 -2.0

4.73

4.74

4.75

4.76

Y

l = 0.00 h = 0.48 Fig. 6.13 Effect of the halo flattening. Physical trajectories and the corresponding Poincaré sections y − vy with plane x = 0 and vx > 0 for the orbits listed in Table 6.2. The orientation is φ = 90.0; the flattening has been increased with respect to Fig. 6.8 to be qz = 1.4. The L-orbits,

6.4 Role of Dark Matter Haloes in Predictability

HF1

177

800

15 510

-20-15 -10 -5

600 400 200

Vy

20 15 10 5 0 Z -5 -10 -15 -20 20

0 -200 -400 -600

0 -5 Y -10 0 5 -15 10 15 -20 20 X

-800 -20 -15 -10

-5

l = 1.28 h = 0.25 HF2

-15 -10

0

5

10

15

20

5

10

15

20

Y

300

-5

X

0

200 100

Vy

20 15 10 5 0 Z -5 -10 -15 -20 20 15 10 5

0 -100 -200

0 -5 -10 Y 5 -15 10 15 -20

-300

-20 -15

-10

-5

l = 0.00 h = 0.29 HC1

0

Y

800

-15 -10 -5

0

X

5

-5 -10 10 15 -15

0

5

15

600

10 5 0 Z -5 -10 -15 15 10

400

Vy

200 0 -200 -400 -600 -800

Y

-15

-10

-5

5

10

15

5

10

15

Y

l = 1.70 h = 0.20 HC2

0

300

5 -8 -6 0 -4 -2 -5 0 2 -10 Y 4 6 8 -15 X

200 100

Z Vy

20 15 10 5 0 -5 -10 -15 -20 15 10

0 -100 -200 -300

-15

-10

-5

0

Y

l = 0.00 h = 0.91 Fig. 6.13 (continued) close to disc, with high velocities remain similar. The predictability of orbits with low velocities slightly decreases. The H-orbits with low velocities remain similar. The predictability of orbits with high velocities increases, and the chaos decreases. (Taken from [30] with permission)

178

6 A Detailed Example: Galactic Dynamics

2 LF1 LF2 LC1 LC2

h

1.5 1 0.5 0

1

1.2

1.4 qz

1.6

1.8

2 HF1 HF2 HC1 HC2

1.5 0.2

h

Fig. 6.14 Variation in predictability index with the flattening of the halo, qz , with a dark halo orientation fixed to φ = 90. A higher value means a better shadowing and thus a better predictability. (top) L-orbits, close to the disc orbits. (bottom) H-orbits, initial conditions with higher z. The inset zooms in some flattening values leading to very low predictabilities for the blue orbits with high initial velocities, coincident with the chaotic nature of the orbits with the flattening parameter values seen in Fig. 6.12. The orbit with the lowest predictability is HC2 when qz = 1.32. (Taken from [30] with permission)

0.1

1

0

1

1.1

1.2

1.3

1.4

0.5 0

6.4.3.3

1

1.2

1.4 qz

1.6

1.8

Timescales

Figure 6.16 shows the variation in critical Δt, when the distributions change, with the flattening parameter qz . The top panel corresponds to the L-orbits, and it shows that the intervals are longer for orbits far from the axis (LF1 and LF2) than for orbits close to the axis (LC1 and LC2). This agrees with the variation in Δt with the halo orientation shown in Fig. 6.11. The intervals Δt are roughly constant with the qz parameter, in contrast with the dependence of Δt on φ shown in Fig. 6.11. This uniformity correlates with the absence of a dependence of h on qz shown in Fig. 6.14. The bottom panel of Fig. 6.16 corresponds to the H-orbits. The critical timescales of these orbits are longer than for the previous L-orbits. We see a stronger fluctuation of Δt with qz , but without any clear trend.

6.5 Analysis of the Results

179

Fig. 6.15 Another case of low predictability is shown in the orbit HC2 with a dark halo orientation of φ = 90.0 and a flattening of qz = 1.32. This orbit corresponds to the case with the lowest predictability seen in the inset of Fig. 6.14. This trajectory in the physical configuration space (x, y, z) is shown in the top panel, and the corresponding Poincaré section y − vy with plane x = 0 and vx > 0 is shown in the bottom panel. This orbit has λ = 0.39 and h = 0.02. The orbit has moderate chaos, and the predictability grows, though it remains very low. (Taken from [30] with permission)

6.5 Analysis of the Results In the previous sections, we have shown the dependence of the predictability index h on the dark halo orientation φ and the flattening qz . When the h values are very low, following Eq. 4.2, the predictability times τ can be very short. The lowest values of h are found in the bottom panels of Figs. 6.9 and 6.14 and orbit out of the disc with the highest velocities, HF2 and HC2. This section analyses the possible sources for the low predictability values found on these orbits. The lowest predictability indexes h are associated with the nonhyperbolic nature of the dynamical flow. A dynamical system is hyperbolic when the phase space can be spanned locally by a fixed number of independent stable and unstable directions, which are consistent under the operation of the dynamics, and when the angle between the stable and unstable manifolds is different from zero [14, 33]. In hyperbolic regions, the shadowing theory guarantees the existence of a nearby true

180

0.25

LF1 LF2 LC1 LC2

0.2 Δt

Fig. 6.16 Variation depending on qz of the critical Δt interval length when the behaviour of the distributions changes. The orientation is fixed to φ = 90. This critical interval length reflects the timescales when the dynamical flow reaches the global regime. (top) L-orbits, close to the disc orbits. (bottom) H-orbits, initial conditions with higher z. (Taken from [30] with permission)

6 A Detailed Example: Galactic Dynamics

0.15 0.1 0.05 1

1.2

1.4 qz

1.6

1.8

0.25

Δt

0.2 0.15 0.1

HF1 HF2 HC1 HC2

0.05 1

1.2

1.4 qz

1.6

1.8

trajectory. The finite-time Lyapunov exponents oscillate around zero because the shadowing distance changes from exponential increases to exponential decreases, mimicking a random walk. In a nonhyperbolic region, a normally expanding direction converts itself into a contracting direction, causing an excursion away from the reflecting barrier. The finite-time exponent values strongly deviate from zero, and a breakdown in the shadowing occurs. Nonhyperbolic behaviour can arise from tangencies, homoclinic tangencies, between stable and unstable manifolds, from unstable dimension variability or from both. For tangencies, there is a higher but still moderate obstacle to shadowing. But in the so-called pseudo-deterministic systems, the shadowing is only valid during trajectories of short lengths because of the unstable dimension variability (UDV). We have already discussed that the UDV is reflected and quantified by the fluctuations around zero of the finite-time Lyapunov exponent closest to zero [9, 34] and that these fluctuations are then a good indicator of the nonhyperbolic nature of the orbit. A typical source of UDV is the hyperchaos phenomenon, defined as the presence of more than one positive asymptotic Lyapunov exponent. There is hyperchaos in

6.5 Analysis of the Results

181

this system because every time there is one positive Lyapunov exponent, the second one is also positive (and the third one is zero, as expected). We must emphasise that there are situations where the distributions of finite-time Lyapunov exponents show positive tails that are not due to the presence of UDV. Other different mechanisms, such as the quasi-tangencies between the stable and unstable manifolds near a homoclinic crisis point, can also produce those tails. Regardless of the above, oscillations still reflect both expanding and contracting directions, and it remains a useful index to compute. The strength of the fluctuations can be derived by computing the probability of the positivity P+ of the distributions. This index can be calculated as follows. The distribution of finite-time Lyapunov exponents can be normalised by dividing it by the total number of intervals, thus obtaining a probability density function P (χ ) that gives the probability of obtaining a given value χ between [χ , χ + dχ ]. Hence, the probability P+ for obtaining a positive χ (Δt) can be defined as  P+ =

∞

P (χ )dχ .

(6.9)

0

This value provides the probability of obtaining an expanding value of the finitetime Lyapunov exponent once the distribution is fixed. There is an equivalent definition of P− for the contracting exponent values, and the sum of both is unity. As we are interested in the analysis of how strong the oscillations are around zero, we define here the parameter d0.5 . This indicates how far from the 0.5 value is P+ . The closer d0.5 is to 0, the stronger UDV may be present [28].

6.5.1 Dark Halo Orientation Here we analyse the variation in distance d0.5 with the dark halo orientation φ. Figure 6.17 (top panel) plots this variation for the L-orbits. There is no strong dependence of this distance on the dark halo orientation φ, but for LF2, which was also the orbit with the strongest variation in predictability h on φ (seen in Fig. 6.9). The orbit LF2 also shows the strongest oscillations around zero, reflected in the shortest distance d0.5 ≈ 0.10 with φ ≈ 40. At this orientation, the predictability of LF2 was low, but not much lower than the h values of the remaining L-orbits. We conclude that a weak UDV does not modify the predictability of the orbit very much. We show in Fig. 6.17 that the distances are sometimes far from 0.5, meaning a P+ ∼ 1. This can occur when the changes from the local to global behaviour occur when the critical finite-intervals are not long enough to reach the asymptotic regime and the mean of the closest to zero exponent may not be so close to zero. Regarding the H-orbits, the strongest dependence of the distance is found in HF2 and HC2 orbits, as expected from Fig. 6.9 (bottom). We observe in these cases that

182

0.5

d0.5

0.4 0.3 0.2 LF1 LF2 LC1 LC2

0.1 0 0

45

90 φ

180

135

0.5 HF1 HF2 HC1 HC2

0.4 d0.5

Fig. 6.17 One of the sources of low predictability are oscillations around zero of the closest to zero finite-time Lyapunov exponent. The existence of both expanding and contracting directions is stronger when d0.5 ≈ 0.0. (top) Variation in d0.5 for the L-orbits, close to disc orbits. (bottom) Variation in d0.5 for the H-orbits, orbits with higher z. The flattening is fixed to be qz = 1.25 in both panels. The H-orbits show the lowest values, with the orbits of higher initial velocities having the closest to zero values. These strong oscillations around zero are the origin of the very low predictabilities shown in Fig. 6.9. (Taken from [30] with permission)

6 A Detailed Example: Galactic Dynamics

0.3 0.2 0.1 0 0

45

90

135

180

φ the distance can reach the zero value around φ ≈ 90. This means a very strong UDV, a good source for the very low predictability of these orbits. Some points for HF2 and HC2 are also around φ ≈ 10 with short d0.5 distances, that is, with strong UDV oscillations. The associated h values are not so low in these cases, however. As for the L-orbits, we may conclude that only when the distance d0.5 is really close to zero, the UDV is strong enough to lower the orbit predictability. This agrees with the comparison of Figs. 6.9 and 6.17, and there can be orbits with high h predictabilities even for relatively short distances d0.5 .

6.5.2 Flattening We now analyse the variation in distance d0.5 with the dark halo flattening qz , with the dark halo orientation fixed to φ = 90. The top panel of Fig. 6.18 plots this variation for the L-orbits. There is no strong variation of this distance with qz . The

6.6 Concluding Remarks

0.5 0.4 d0.5

Fig. 6.18 Variation in distance d0.5 with the dark halo flattening, qz . The orientation is fixed to φ = 90. (top) Variation in the L-orbits, close to disc orbits. (bottom) Variation in the H-orbits, orbits with higher z. The L-orbits show the higher values. The H-orbits show the lowest values, but they are never low enough to produce low predictabilities. (Taken from [30] with permission)

183

0.3 0.2 LF1 LF2 LC1 LC2

0.1 0

1

1.2

1.4 qz

1.6

1.8

0.5 HF1 HF2 HC1 HC2

d0.5

0.4 0.3 0.2 0.1 0

1

1.2

1.4 qz

1.6

1.8

orbits LC1 and LC2, those close to the centre, show some fluctuating behaviour, but the values of d0.5 are never close enough to zero to have very low predictabilities. Regarding the H-orbits, the bottom panel of Fig. 6.18 shows the variation of the distance d0.5 with qz for these orbits. There is a wider spread of values of d0.5 as qz varies. Following previous discussions, only the closest to zero values will lead to very low predictabilities. The lowest values of d0.5 are seen for HC2 around qz ≈ 1.32. This seems to agree with the low h values seen in Fig. 6.14.

6.6 Concluding Remarks As in the previous chapter, we deal here with the forecast of predictability, and not with the forecast of chaoticity. The predictability of a system is understood here as a measure of its shadowing properties, reflecting the time during the computed orbit that is followed by a real, physically meaningful orbit. We note that this time

184

6 A Detailed Example: Galactic Dynamics

length is different from another widely used index, the reliability time, which is the inverse of the asymptotic Lyapunov exponent, which considers the possible chaotic or regular nature of one orbit. The finite-time Lyapunov exponent distributions reflect the underlying dynamics [31], and by using arbitrarily oriented deviation axes, one can detect varying finitetime interval lengths, when there is a change from the local to the global, not yet asymptotic regime. This allows to obtain different predictability indexes that will detect the cross from a local to a global regime [32]. The fluctuating behaviour of the closest to zero of the available Lyapunov exponents is a sign of bad shadowing and indicator of nonhyperbolicity. We have applied the techniques described in the previous chapter, and we have examined the effect of the dark halo shapes and orientation on the predictability of the computed orbits in a mean field modelling the Milky Way. However, it must be noted that our technique can be also used in any other potential modelling of other types of galaxies. Therefore, the discussions will be very similar. Simple galactic models are integrable systems, but when perturbations are added, they loose integrability. Irregular orbits are characterised by exponential sensitivity to perturbations, and the predictions in non-integrable potentials can be rapidly inaccurate, as perturbations such as granularity or neighbour galaxies are present and are amplified because of the strong dependence on initial conditions. This can occur with rotation, bars, asymmetries or bulges with density peaks, for instance. In our case, the triggering factor was the triaxiality in the dark halo, but the results can be considered of general interest for other models. We have computed both the asymptotic Lyapunov exponents λ and the predictability index h for a representative sample of orbits. We used a technique that is based on computing the distribution of finite-time Lyapunov exponents to identify the orbits with the lowest predictabilities. We can conclude that both terms are closely related, but they do not always follow the same trend. We have shown that not all chaotic orbits have the same values of the predictability index and that some chaotic orbits show good predictability behaviour, while others show a very poor behaviour. We have also shown that not all regular orbits have the same predictabilities. Certain orbits at given values of the control parameters can lead to all Lyapunov exponents to be zero. But, despite of this, low predictability values can be present at the same time. Conversely, we have shown that there are orbits that are more strongly chaotic (with no dependence on the halo orientation) but can be predictable and show a good shadowing behaviour. When the dark halo orientation φ is varied and the flattening remains fixed, the variation in λ with φ from zero to positive is only seen in the orbits with highest velocities and initial z out of the disc. These are orbits that are regular but with φ around 90◦ . Finding regular orbits out of the disc that depend the most on φ may be considered obvious. Nevertheless, the evolution of the changes in their predictabilities is less straightforward. We therefore have also analysed the variation in predictability index h with the dark halo orientation. The predictability of the most chaotic orbits, usually those with slower velocities, does not show a strong dependence on the dark halo

6.6 Concluding Remarks

185

orientation. However, the variation of the predictability with φ for all orbits is wider than the variation in chaoticity, confirming that chaoticity is different from predictability. We have also shown that the orbits with higher predictability values, usually those with higher velocities, depend most strongly on φ. For these high-velocity orbits, those out of the disc have the lowest values of h, for instance, HC2 and HF2. A similar analysis was carried out by adding a variation in flattening of the halo qz to the fixed rotation φ = 90◦ . The orbits with initial z out of the disc and highest velocities are the orbits whose chaos depends most upon the halo flattening. However, in contrast to the previous case, the flattening seems to produce a weaker effect on the predictability than the orientation, except for the HC2 orbit, which again shows the lowest value. The effect of the dark halo orientation φ on the predictability is stronger than the effect of the flattening qz . Interestingly, orbits HC2 and HF2 have the shortest predictability times for certain halo parameters. Even when they have high values of the maximal Lyapunov exponent, however, they are not the orbits with the highest values. Because predictability is different from chaoticity, some strongly chaotic orbits can be more predictable than others with weaker chaos. We have also shown the evolution of the timescales where the flow leaves the local dynamics with the dark halo parameters. The longest timescales can be seen with the orbits with initial z out of the disc. Finally, we analysed the presence of UDV as source for low predictability. We have shown that the lowest predictability indexes are linked to the strongest oscillations around zero of the finite-time exponents. A low predictability does not mean that a model is incorrect, but it means that for a given range of parameters, the model may lead to very short predictability times. As a general conclusion, we have shown that certain areas in the parametric space must be taken with care. When fitting observed quantities with certain model parameters, we may be fitting them in areas where the validity of the model predictions may be very short and refined numerical schemes may be needed. The predictability index is linked to the hyperbolic or nonhyperbolic nature of the orbit. This is related, in turn, to its energy and stiffness. Different energy values lead to different dynamical times and consequently to different timescales. The existence of two or more timescales in different directions, one quickly growing, one slowly growing, can lead to stiffness. The finite-time Lyapunov exponents reflect these expanding or contracting behaviours, and the predictability indexes depend on the timescales when these behaviours change. We have shown very low predictabilities, sometimes as low as h ∼ 10−4 . Following Eq. 4.2, we can take a typical value for the round-off error δ ∼ 10−16 , and these low values of h lead to predictability times as short as 1 Gyr. A given numerical scheme with certain precision can be enough when the shadowing times are long and high-precision time-consuming schemes are not necessary. However, for low predictability orbits, more powerful schemes may be necessary, but we note that in the extreme cases with very short predictability times, strong increases in precision do not mean strong increases in shadowing times. Therefore, in those

186

6 A Detailed Example: Galactic Dynamics

cases where the shadowing times are really short, one must balance the cost of implementing more complex and time-consuming schemes and the relatively small gains in shadowing time. Our technique uses finite-time interval lengths shorter than the Hubble time tH in the computations. This allowed us to apply it to transient behaviours and unbounded orbits, or potential scattering problems, because it does not use long integration times in principle. One limitation is found when using very short integration times. In this case, the number of finite-time Lyapunov intervals needed to obtain good statistics may be not large enough. For the analysed cases, we used total integration times of up T ≈ 105 , but shorter integration times can still be used for the typical Δt interval lengths that we have obtained. The physical meaning of using total integration times of some orders of magnitude larger than tH may be debated. Simple simulations that consider static potentials should be constrained to times of about 5 Gyr [17], because longer integrations lasting several times the age of the universe should take into consideration that the galaxies may have evolved and disappeared. The rationale is that our long integrations can be read as the sum of individual integrations, each one sized shorter than tH . Considering that the initial conditions are reset after every finite interval, the galaxy model can be considered valid. We have worked on a reduced set of representative initial conditions and checked how their chaotic nature and predictability can change with the dark halo parameters. An interesting work to extend these results is to calculate the predictability indexes using a complete exploration of different initial conditions on a given potential, including a detailed model of a galactic bar, and the analysis of the variation in percentages of high- and low predictability orbits.

References 1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems, p. 383. Springer, New York (1996) 2. Cachucho, F., Cincotta, P.M., Ferraz-Mello, S.: Chirikov diffusion in the asteroidal three-body resonance (5, -2, -2). Celest. Mech. Dyn. Astron. 108, 35 (2010) 3. Caranicolas, N.D., Zotos, E.E.: The evolution of chaos in active galaxy models with an oblate or a prolate dark halo component. Astron. Nachr. 331, 330 (2010) 4. Carpintero, D., Muzzio, J.C., Navone, H.D.: Models of cupsy triaxial stellar systems – III. The effect of velocity anisotropy on chaoticity. Mon. Not. R. Astron. Soc. 438, 287 (2014) 5. Casertano, S., Ratnatunga, K.U., Bahcalli, J.N.: Kinematic modeling of the galaxy. II – two samples of high proper motion stars. Astrophys. J. 357, 435 (1990) 6. Chiba, M., Beers, T.C.: Structure of the galactic stellar halo prior to disk formation. Astrophys. J. 549, 325 (2001) 7. Cincotta, P.M., Giordano, C.M.: Topics on diffusion in phase space of multidimensional Hamiltonian systems. In: New Nonlinear Phenomena Research, p. 319. Nova Science Publishers, Inc., New York (2008) 8. Contopoulos, G., Harsoula, M.: 3D chaotic diffusion in barred spiral galaxies. Mon. Not. R. Astron. Soc. 436, 1201 (2013)

References

187

9. Davidchack, R.L., Lai, Y.C.: Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits. Phys. Lett. A 270, 308 (2000) 10. Habib, T., Kandrup, H.E., Mahon, M.E.: Chaos and noise in Galactic potentials. Astrophys. J. 480, 155 (1997) 11. Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems, 2nd ed. Springer, Berlin (1993) 12. Hayes, W.B.: Shadowing-based reliability decay in softened n-body simulations. Astrophys. J. 587, L59–L62 (2003) 13. Johnston, K.V., Spergel, D.N., Hernquist, L.: The disruption of the sagittarius dwarf galaxy. Astrophys. J. 451, 598 (1995) 14. Kantz, H., Grebogi, C., Prasad, A., Lai, Y.C., Sinde, E.: Unexpected robustness-against-noise of a class of nonhyperbolic chaotic attractors. Phys. Rev. E, 65, 026209 (2002) 15. Law, D.R., Majewski, S.R., Johnston, K.V.: Evidence for a triaxial Milky Way dark matter halo from the sagittarius stellar tidal stream. Astrophys. J. Lett. 703, L67 (2009) 16. Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics, 2nd ed. Applied Mathematical Sciences, vol. 38. Springer, New York (1992) 17. Martinez-Valpuesta, I., Shlosman, I.: Why buckling stellar bars weaken in disk galaxies. Astrophys. J. 613, 613L (2004) 18. Miyamoto, M., Nagai, R.: Three dimensional models for the distribution of mass in galaxies. Publ. Astron. Soc. Jpn. 27, 533 (1975) 19. Olle, M., Pfenigger, D.: Vertical orbital structure around the Lagrangian points in barred galaxies. Link with the secular evolution of galaxies. Astron. Astrophys. 334, 829 (1998) 20. Papaphilippou, Y., Laskar, J.: Global dynamics of triaxial galactic models through frequency map analysis. Astron. Astrophys. 329, 451 (1998) 21. Pfenniger, D.: The 3D dynamics of barred galaxies. Astron. Astrophys. 134, 373 (1984) 22. Pfenniger, D.: Relaxation and dynamical friction in non-integrable stellar systems. Astron. Astrophys. 165, 74 (1986) 23. Pfenniger, D.: Dissipation in barred galaxies: the growth of bulges and central mass concentrations. Astrophys. J. 363, 391 (1990) 24. Pfenniger, D., Firedli, D.: Structure and dynamics of 3D N-body barred galaxies. Astron. Astrophys. 252, 75 (1991) 25. Skokos, C., Patsis, P.A., Athanassoula, E.: Orbital dynamics of three-dimensional bars – I. The backbone of three-dimensional bars. A fiducial case. Mon. Not. R. Astron. Soc. 333, 847 (2002) 26. Tsiganis, K., Varvoglis, H., Hadjidemetriou, J.D.: Stable chaos in high-order Jovian resonances. Icarus 155, 454 (2002) 27. Udry, S., Pfenniger, D.: Stochasticity in elliptical galaxies. Astron. Astrophys. 198, 135 (1988) 28. Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15 113064 (2013) 29. Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015) 30. Vallejo, J.C., Sanjuan, M.A.F.: Role of dark matter haloes on the predictability of computed orbits. Astron. Astrophys. 595, A68 (2016) 31. Vallejo, J.C., Aguirre, J., Sanjuan M.A.F.: Characterization of the local instability in the HenonHeiles Hamiltonian. Phys. Lett. A 311, 26 (2003) 32. Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008) 33. Viana, R.L., Grebogi, C.: Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462 (2000) 34. Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a chaotic process. Braz. J. Phys. 35, 1 (2005) 35. Wang, Y., Zhao, H., Mao, S., Rich, R.M.: A new model for the Milky Way bar. Mon. Not. R. Astron. Soc. 427, 1429 (2012)

188

6 A Detailed Example: Galactic Dynamics

36. El-Zant, A.A., Hassler, B.: Dynamics of galaxies with triaxial haloes. New Astron. 3, 493 (1998) 37. El-Zant, A.A., Shlosman, I.: Dark Halo shapes and the fate of stellar bars. Astrophys. J. 577, 626 (2002) 38. Zotos, E.E.: Classifying orbits in galaxy models with a prolate or an oblate dark matter halo. Astron. Astrophys. 563, 19 (2014) 39. Zotos, E.E., Caranicolas, N.D.: Revealing the influence of dark matter on the nature of motion and the families of orbits in axisymmetric galaxy models. Astron. Astrophys. 560, 110 (2013)

Appendix A

Numerical Calculation of Lyapunov Exponents

A.1 The Variational Equation There is a large variety of numerical schemes for calculating the Lyapunov exponents. Each method possesses certain advantages and disadvantages, and a detailed comparison can be found in Ref. [12]. The most common methods can be divided in two main categories: those based on the direct calculation of the distance between trajectories and those based on solving the variational equation. The first family of methods, named as differences or direct, derives from the definition of exponents as indicators of the divergence between points initially close enough. These schemes start by selecting two very close points, separated by a distance d0 . The system is iterated and the new separation is computed. The logarithm of the new and old separation is calculated, and the scheme is repeated, averaging the results. For flows, the distance between trajectories can be easily computed as the phase space distance, but it would be worth a discussion on which norm could be used for the computation of the distance. For maps, the distance can be also calculated after a given number of iterations. These methods imply a renormalisation process. That is, the distance should be set again to be the initial one after every certain number of steps. The reason is that when dealing with orbits within an attractor, the orbits do not diverge at a certain timescale and may even begin to converge. For flows, the renormalisation is done every t time units. This means the existence of a scale factor that when multiplied by the distance d(t) will return the original distance d0 . The asymptotic Lyapunov exponent can be obtained from averaging the logarithm of the new and old separation after applying the scale factor. For the calculation of all exponents and not only the largest one, the normalisation process is somehow more complex [9]. Basically, the computation of the second exponent can be done by considering the evolution of a two-dimensional surface

© Springer Nature Switzerland AG 2019 J. C. Vallejo, M. A. F. Sanjuan, Predictability of Chaotic Dynamics, Springer Series in Synergetics, https://doi.org/10.1007/978-3-030-28630-9

189

190

A

Numerical Calculation of Lyapunov Exponents

evolving with e(λ1 +λ2 )t . So, λ2 can be computed once we have λ1 . The remaining exponents can be derived following the same idea. One major disadvantage of the differences method is the selection of a practical initial distance d0 and also a proper selection of the renormalisation period t. Since the strict definition requires infinitesimal deviations, to take finite initial deviations may lead to wrong results. It may happen that starting close to a limit cycle of exponent zero, the solution could be deviated towards initial conditions where the flow may converge. Conversely, we could be under saturation effects, when the points are as farther from the attractor as possible, and they cannot move away any farther, so the distance may keep roughly constant. As a consequence of the above, nowadays it is more common to use the variational methods. The finite-time Lyapunov exponents are computed by solving the variational equation that reflects the growth rate of the orthogonal semiaxes (equivalent to the initial deviation vectors) of one ellipse centred at the initial position as the system evolves [1]. The variational equation is essential when analysing the stability of orbits, evolution of phase space volumes under the dynamics, stable and unstable manifolds and, obviously, Lyapunov exponents. We define here Φ(x, t) as the solution of the flow equation. The time evolution of a phase-space point subject to a given flow dynamics is given by flow equations x˙ = Φ(x, t). Without loss of generality, we can give an example based on the equations for a three-dimensional continuous flow: ⎧ ⎨ x˙ = f1 (x, y, z) (A.1) y˙ = f2 (x, y, z) ⎩ z˙ = f3 (x, y, z) with initial condition x0 = (x0 , y0 , z0 ). Once the initial condition x0 is fixed, we can integrate the flow during a given time t, and the initial point will follow certain trajectory in phase space, ending in a final point x. Imagine that we add a small perturbation to x0 in say the x-direction. Evidently, the resulting initial perturbed condition vector will evolve towards a different point x  . The same can be said if we perturbed the initial condition in other directions (y and z, respectively). The slopes of the flow in each direction provide a means to know how the perturbation will evolve. It may be kept constant, enlarged, shrank or even both, as it happens when the perturbation points out diagonally from a saddle point. The matrix describing these slopes is the Jacobian matrix of the flow Φ, J , that describes the evolution of deformations after a finite-time t. So, J = Dv Φ contains the differential slopes in every possible direction:

A

Numerical Calculation of Lyapunov Exponents

⎡ ∂f1 J = Dv Φ =

∂x ⎢ ∂f2 ⎣ ∂x ∂f3 ∂x

191 ∂f1 ∂y ∂f2 ∂y ∂f3 ∂y

∂f1 ⎤ ∂z ∂f2 ⎥ ∂z ⎦ . ∂f3 ∂z

(A.2)

We can use the Jacobian J for analysing how the perturbations, or variations, evolve under the flow dynamics. The variations in each direction [δx ], [δy ] and [δz ] are defined as vectors that will track the perturbation along each direction, as follows: ⎡ ⎤ δxx [δx ] = ⎣δxy ⎦ , δxz

(A.3)

⎡ ⎤ δyx [δy ] = ⎣δyy ⎦ , δyz

(A.4)

⎡ ⎤ δzx [δz ] = ⎣δzy ⎦ . δzz

(A.5)

The variation [δ] is then defined as the 9-component tensor: ⎤ ⎡ δxx δyx δzx [δ] = ⎣δxy δyy δzy ⎦ , δxz δyz δzz

(A.6)

and the variational equation is then ˙ = J [δ] = Dv Φ[δ]. [δ]

(A.7)

Equivalently ⎡

˙ δxx ˙ = ⎣δxy ˙ [δ] ˙ δxz

˙ δyx ˙ δyy ˙ δyz

∂fx ∂fx ⎤ ⎡ δxx ∂y ∂z ∂fy ∂fy ⎥ ⎣ δxy ⎦ ∂y ∂z ∂fz ∂fz δxz ∂x ∂y ∂z

⎤ ⎡ ∂fx ˙ δzx ∂x ∂fy ˙ ⎦=⎢ δzy ⎣ ∂x ∂fz δ˙zz

⎤ δyx δzx δyy δzy ⎦ . δyz δzz

(A.8)

So, in practical terms, the variational equation is encoded by writing ∂fx ∂fx ⎤ ⎡ ⎤ δxx ∂y ∂z ∂fy ∂fy ⎥ ⎣ ⎦ ∂y ∂z ⎦ δxy , ∂fz ∂fz δxz ∂x ∂y ∂z

⎤ ⎡ ∂fx ˙ δxx ∂x ∂fy ˙ ⎦=⎢ [δ˙x ] = ⎣δxy ⎣ ∂x ∂fz ˙ δxz ⎡

(A.9)

192

A

Numerical Calculation of Lyapunov Exponents

for solving the evolution of the three-dimensional variation δx . Following the same approach we have ∂fx ∂fx ⎤ ⎡ ⎤ δyx ∂y ∂z ∂fy ∂fy ⎥ ⎣ ⎦ ∂y ∂z ⎦ δyy , ∂fz ∂fz δyz ∂x ∂y ∂z

⎤ ⎡ ∂fx ˙ δyx ∂x ∂fy ˙ ⎦=⎢ [δ˙y ] = ⎣δyy ⎣ ∂x ∂fz ˙ δyz ⎡

(A.10)

for solving the evolution of the 3-dimensional variation δy , and similarly, ∂fx ∂fx ⎤ ⎡ ⎤ δzx ∂y ∂z ∂fy ∂fy ⎥ ⎣ ⎦ ∂y ∂z ⎦ δzy , ∂fz ∂fz δzz ∂x ∂y ∂z

⎡ ⎤ ⎡ ∂fx ˙ δzx ∂x ∂fy ˙ ⎦=⎢ [δ˙z ] = ⎣δzy ⎣ ∂x ∂fz δ˙zz

(A.11)

for solving the evolution of the three-dimensional variation δz . Therefore, we need to solve the variational equation and the system equation at the same time, working with the so-called augmented vector. This means that for an n-dimensional system, we have (n + n2 ) variables. The first n variable corresponds to n components of the “physical” n-dimensional vector, x, and the following n2 variables are required for solving the evolution of the n-variations, each one being a n-dimensional vector.

A.2 Selection of Initial Perturbations By solving at the same time the flow equation and the fundamental equation of the flow, that is, the distortion tensor evolution, we can follow the evolution of the vectors, or axes, along the trajectory and, in turn, their growth rate. This method is described in [1, 3]. The key point is that solving the variational equation implies solving the flow equations. So, for the augmented vector, it must be selected a suitable initial condition, visualised as the initial axes lengths and directions of an initial deviation vector. This is a key issue when dealing with finite integrations, because depending on this selection, the evolution of the distortion vectors will be different. Obviously, there are several choices for the initial orientation of the ellipse axes. Due to the dependence on the finite integration time interval used in Eq. 2.11, every orientation will lead to different exponents [17]. A first simple choice may be the identity matrix, but this option does not seem to reflect any property of the flow. To allow the flow to point this initial selection to the most growing directions, and to obtain proper averaged indexes, we should integrate during long times.

A

Numerical Calculation of Lyapunov Exponents

193

Another suitable option is to have the axes pointing to the local expanding/contracting directions, given by the eigenvectors of the Jacobian matrix. At local timescales the eigenvalues will provide insight on the stability of the point. Furthermore, these finite-time exponents can trace the stable and unstable manifolds, the later with a time backwards integration [5, 7]. Note that in turn, the angle of both manifolds provides the nonhyperbolic nature of the system. Another way of doing it could be to point the axes towards the direction which may have grown the most under the linearized dynamics or to point them to the globally fastest-growing direction. Another possibility is to select the initial deviation vectors, by using the singular vectors linked to the Singular Value Decomposition (SVD) of the Jacobian matrix. The SVD takes into account that every matrix can be written in form of the product of three matrices. The first one is formed by the left singular vectors (gene coefficient vectors, visualised as a “hanger” matrix). The second one is a diagonal matrix containing the so-called singular-values (mode amplitudes, visualised as a “stretcher” matrix). The third one is formed by the right singular vectors (expression level vectors, visualised as an “aligner” matrix). The options listed above provide interesting insights on the behaviour of the dynamics of the flow at local scales when integrated during small finite-time intervals. They are selected for pointing to directions that are already known “a priori” to express these local properties. As the integrations are larger, the local properties are washed out, and the deviation vectors will end following the averaged global properties of the flow. The axes will tend towards the fastest-growing direction, maybe at exponential rate, making their computation difficult to tackle. Because of that, the most commonly used methods use a Gram-Schmidt orthonormalisation process [2, 15]. By annotating the vector magnitudes before the normalisation, we can calculate all Lyapunov exponents as defined in Sect. 2.1. However, sometimes we are not merely interested in the final asymptotic values but, conversely, in looking for properties of the flow before the asymptotic values are returned. This is achieved by using the finite-time Lyapunov exponents, as described in Sects. 2.3 and 2.5. It is worth to note that by selecting “a priori” directions, we may be already favouring the evolution towards the most growing directions. Therefore, some information about the timescales taken by the system for such an evolution may be lost. As a consequence, there is also another interesting choice for the initial axes of the ellipse, that is, to arbitrarily set them coincident with a random set of orthogonal vectors. This is the option selected along this book, used in, for instance, Refs. [13, 14]. With this selection, as the flow evolves, the axes get orientated from the arbitrary chosen direction as per the flow dynamics. Because this initial orientation will not favour any initial privileged direction, the growth rates of the axes of the ellipse will depend naturally on the flow timescales and will begin to point to globally growing directions once a necessary finite-time interval elapses.

194

A

Numerical Calculation of Lyapunov Exponents

Obviously, as the finite-time interval grows above theses timescales, the asymptotic regime will begin to appear. This algorithm returns the nearly asymptotic Lyapunov values ordered from the largest to the smallest when large enough time intervals are used. Conversely, there will be just a simple relationship among the exponents when using very local timescales [13]. But when using intermediate interval sizes, the returned values will characterise a given orbit.

A.3 Other Methods The two methods described earlier are the most commonly used. But it is worthy to mention the existence of additional algorithms and methods. We refer the interested reader to follow, for instance, Refs. [8, 12]. There are the so-called QR methods, based on the factorisation of the matrix resulting from the QR decomposition of the Jacobian, writing it as the product of an orthogonal matrix Q and an upper triangular matrix R. These methods seem to be more adequate for the computation of LCEs, but not for the computation of FTLEs, because they introduce certain errors that are only cleared as the integration time increases. There are also methods based on the SVD decomposition mentioned in the previous section. In general, both methods could not be so convenient in nonhyperbolic systems. In these systems, the FTLEs can accumulate around zero and the decomposition to have an almost degenerate spectrum. Certainly, there are corrections to the QR method that can be applied to degenerated cases. And, indeed, there are modifications for proper handling Hamiltonian systems, incorporating their symplectic nature [10]. Nevertheless, some of these modifications are only effective in systems with one or two degrees of freedom, and they are not very suitable for systems with a higher number of degrees of freedom. In any case, every method has certain advantages and disadvantages, and some additional description can be found in [11, 12]. A final remark could be given regarding the computation of Lyapunov exponents in time series. In these cases some specific methods are required [6, 16]. If we do not have an appropriate knowledge of the fundamental equations of the system, having only experimental data available, these methods can be classified in two families: direct methods and tangent space methods. The direct methods are based on searching for time series in the neighbourhood of the initial point and rely on computing the necessary comparisons. The tangent space methods perform the computation by predicting the Jacobian using the available time series. See [4] and references therein.

A

Numerical Calculation of Lyapunov Exponents

195

A.4 Practical Implementation for Building the Finite-Time Distributions If we make a partition of the whole integration time along one orbit into a series of finite-time intervals of size Δt, then it is possible to compute the finite-time Lyapunov exponents χ (Δt) for each interval. The distribution is built by integrating the augmented vector under the flow dynamics up to a selected Δt interval. We fix the initial point of the orbit, as desired, and as initial perturbation an arbitrary set of orthonormal vectors, as described in the previous Sect. A.2. We will keep it for later use. A small note could be raised here related to this orthonormalisation process. A widely used algorithm is the Gram-Schmidt process. This has been considered as inherently numerically unstable, that is, very sensitive to round-off errors. We can alleviate this by avoiding divisions by small numbers and column pivoting appropriately. We should aim to more numerically stable processes by using some of the available modified versions of the Gram-Schmidt algorithms or by using Householder transformations. When integrating, at each integration step, we propagate the variations and calculate how the logarithm of their norms evolve following Eq. 2.11 and Sect. 2.5. When the selected finite-time interval size value is reached, we save the value of the calculated finite-time exponent χ (Δt) and start the cycle again, resetting the augmented vector. The new initial condition is the current point of the trajectory, when we have stopped. The new initial perturbation will be the one we selected previously. This is done to assure that we are comparing how the same perturbation evolves along the trajectory points. We repeat the above process until the total integration time is reached. This integration time could be as long as needed, embracing any transient period, and going farther, or stopping the integration once the transient has ended. This distribution of finite-time Lyapunov exponents can be normalized dividing it by the total number of intervals, thus obtaining a probability density function P (χ ), that gives the probability of getting a given value χ between [χ , χ + dχ ]. There is a huge amount of available programming languages and mathematical packages for solving the dynamical flow equation and the variational equation and computing the finite-time or asymptotic infinite Lyapunov exponents. Some of them have already implemented many of the required algorithms, and the final choice for using a given language or package, commercial or free is up to the user.

References 1. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. p 383. Springer, New-York (1996) 2. Bay, J.S.: Fundamentals of Linear State Space Systems. McGraw-Hill, Boston (1999)

196

A

Numerical Calculation of Lyapunov Exponents

3. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Meccanica 9, 20 (1980) 4. McCue, L.S., Troesch, A.W.: Use of Lyapunov exponents to predict chaotic vessel motions fluid mechanics and its applications. Contemp. Ideas Ship Stab. Capsizing Waves Part 5 97, 415 (2005) 5. Doerner, R., Hubinger, B., Martiensen, W., Grossmann, S., Thomae, S.: Stable manifolds and predictability of dynamical systems. Chaos Solitons Fractals 10, 1759 (1999) 6. Eckmann, J.P., Oliffson, S., Ruelle, D., Ciliberto, S.: Lyapunov exponents from time series. Phys. Rev. 34, 4971 (1986) 7. Joseph, B., Legras, B.: On the relation between kinematic boundaries, stirring and barriers. J. Atmos. Sci. 59, 1198 (2002) 8. Okushima, T.: New method for computing finite-time Lyapunov exponents. Phys. Rev. Lett. 91, 25 (2003) 9. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989) 10. Partovi, H.: Reduced tangent dynamics and Lyapunov spectrum for Hamiltonian system. Phys. Rev. Lett. 82, 3424 (1999) 11. Ramasubramanian, K., Sriram, M.S.: Lyapunov spectra of Hamiltonian systems using reduced tangent dynamics. Phys. Rev. E 62, 4850 (2000) 12. Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of Lyapunov spectra with different algorithms. Phys. D 139, 72 (2000) 13. Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008) 14. Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015) 15. Wolf, A.: Chapter 13, Chaos. In: Quantifying Chaos with Lyapunov Exponents. Princeton University Press, Princeton (1986) 16. Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining Lyapunov exponents from a time series. Phys. D 16, 285 (1985) 17. Ziehmann, C., Smith, L.A., Kurths, J.: Localized Lyapunov exponents and the prediction of predictability. Phys. Lett. A 271, 237 (2000)