Bifurcation Dynamics in Polynomial Discrete Systems [1st ed.] 9789811552076, 9789811552083

Table of contents :
Front Matter ....Pages i-xi
Quadratic Nonlinear Discrete Systems (Albert C. J. Luo)....Pages 1-92
Cubic Nonlinear Discrete Systems (Albert C. J. Luo)....Pages 93-166
Quartic Nonlinear Discrete Systems (Albert C. J. Luo)....Pages 167-256
(2m)th-Degree Polynomial Discrete Systems (Albert C. J. Luo)....Pages 257-334
(2m + 1)th-Degree Polynomial Discrete Systems (Albert C. J. Luo)....Pages 335-428
Back Matter ....Pages 429-430

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Nonlinear Physical Science

Albert C. J. Luo

Bifurcation Dynamics in Polynomial Discrete Systems

Nonlinear Physical Science Series Editors Albert C. J. Luo , Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA Dimitri Volchenkov , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Advisory Editors Eugenio Aulisa , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Jan Awrejcewicz , Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Lodz, Poland Eugene Benilov , Department of Mathematics, University of Limerick, Limerick, Limerick, Ireland Maurice Courbage, CNRS UMR 7057, Universite Paris Diderot, Paris 7, Paris, France Dmitry V. Kovalevsky , Climate Service Center Germany (GERICS), Helmholtz-Zentrum Geesthacht, Hamburg, Germany Nikolay V. Kuznetsov , Faculty of Mathematics and Mechanics, Saint Petersburg State University, Saint Petersburg, Russia Stefano Lenci , Department of Civil and Building Engineering and Architecture (DICEA), Polytechnic University of Marche, Ancona, Italy Xavier Leoncini, Case 321, Centre de Physique Théorique, MARSEILLE CEDEX 09, France Edson Denis Leonel , Departmamento de Física, Sao Paulo State University, Rio Claro, São Paulo, Brazil Marc Leonetti, Laboratoire Rhéologie et Procédés, Grenoble Cedex 9, Isère, France Shijun Liao, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Josep J. Masdemont , Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain Dmitry E. Pelinovsky , Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada Sergey V. Prants , Pacific Oceanological Inst. of the RAS, Laboratory of Nonlinear Dynamical System, Vladivostok, Russia

Laurent Raymond Marseille, France

, Centre de Physique Théorique, Aix-Marseille University,

Victor I. Shrira, School of Computing and Maths, Keele University, Keele, Staffordshire, UK C. Steve Suh , Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA Jian-Qiao Sun, School of Engineering, University of California, Merced, Merced, CA, USA J. A. Tenreiro Machado Porto, Portugal

, ISEP-Institute of Engineering, Polytechnic of Porto,

Simon Villain-Guillot , Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, Talence, France Michael Zaks Germany

, Institute of Physics, Humboldt University of Berlin, Berlin,

Nonlinear Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications. Topics of interest in Nonlinear Physical Science include but are not limited to: • • • • • • • •

New findings and discoveries in nonlinear physics and mathematics: Nonlinearity, complexity and mathematical structures in nonlinear physics: Nonlinear phenomena and observations in nature and engineering: Computational methods and theories in complex systems: Lie group analysis, new theories and principles in mathematical modeling: Stability, bifurcation, chaos and fractals in physical science and engineering: Discontinuity, synchronization and natural complexity in physical sciences: Nonlinear chemical and biological physics

More information about this series at http://www.springer.com/series/8389

Albert C. J. Luo

Bifurcation Dynamics in Polynomial Discrete Systems

With 61 figures

123

Albert C. J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL, USA

ISSN 1867-8440 ISSN 1867-8459 (electronic) Nonlinear Physical Science ISBN 978-981-15-5207-6 ISBN 978-981-15-5208-3 (eBook) https://doi.org/10.1007/978-981-15-5208-3 Jointly published with Higher Education Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from Higher Education Press © Higher Education Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is the second part for bifurcation and stability of nonlinear discrete systems. The first part mainly presents a local theory for monotonic and oscillatory stability and bifurcations of nonlinear discrete systems, and such monotonic and oscillatory stability and bifurcations on specific eigenvectors of the corresponding linearized discrete systems are discussed. In this book, the bifurcation dynamics of one-dimensional polynomial nonlinear discrete systems is presented and bifurcation trees caused by period-doubling and monotonic saddle-node bifurcations are discussed for forward and backward polynomial discrete systems. The mechanism of bifurcation trees caused by monotonic saddle-node bifurcations is determined. The appearing and switching bifurcations of simple and higher-order period-1 fixed-points are discussed in this book. From this book, one will find more interesting research results in nonlinear discrete systems. This book consists of five chapters. In Chap. 1, a global bifurcation theory for quartic polynomial discrete systems is presented, and the bifurcation trees through period-doubling and monotonic saddle-node bifurcations are discussed for forward and backward quadratic discrete systems. In Chap. 2, a global bifurcation theory for cubic polynomial discrete systems is discussed, and the bifurcation trees through the period-doubling and monotonic saddle-node bifurcation are also presented, which is different from the quadratic discrete system. In Chap. 3, a global bifurcation theory for quartic polynomial discrete systems is presented for extension to the (2m)th degree polynomial discrete systems. The bifurcation and stability of the (2m)th and (2m+1)th degree polynomial discrete systems are presented in Chaps. 4 and 5 as a general theory of stability and bifurcations for polynomial nonlinear discrete systems. Finally, the author hopes the materials presented herein can last long for science and engineering. Some typos and errors may exist in the book, which can be corrected by readers during reading. Herein, the author would like to thank all supporting people during the difficult time period. Edwardsville, IL, USA

Albert C. J. Luo

vii

Contents

1 Quadratic Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . . . 1.1 Linear Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Forward Quadratic Discrete Systems . . . . . . . . . . . . . . . . . . . . 1.2.1 Period-1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . 1.2.2 Period-1 Switching Bifurcations . . . . . . . . . . . . . . . . . . 1.3 Backward Quadratic Discrete Systems . . . . . . . . . . . . . . . . . . . 1.3.1 Backward Period-1 Appearing Bifurcations . . . . . . . . . . 1.3.2 Backward Period-1 Switching Bifurcations . . . . . . . . . . 1.4 Forward Bifurcation Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Period-2 Appearing Bifurcations . . . . . . . . . . . . . . . . . . 1.4.2 Period-Doubling Renormalization . . . . . . . . . . . . . . . . . 1.4.3 Period-n Appearing and Period-Doublization . . . . . . . . . 1.4.4 Period-n Bifurcation Trees . . . . . . . . . . . . . . . . . . . . . . 1.5 Backward Bifurcation Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Backward Period-2 Quadratic Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Backward Period-Doubling Renormalization . . . . . . . . . 1.5.3 Backward Period-n Appearing and Period-Doublization . 1.5.4 Backward Period-n Bifurcation Trees . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 7 7 21 28 28 39 44 44 53 62 71 75

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75 79 82 92 92

2 Cubic Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . 2.1 Period-1 Cubic Discrete Systems . . . . . . . . . . . . . . . 2.2 Period-1 to Period-2 Bifurcation Trees . . . . . . . . . . . 2.3 Higher-Order Period-1 Switching Bifurcations . . . . . 2.4 Direct Cubic Polynomial Discrete Systems . . . . . . . . 2.5 Forward Cubic Discrete Systems . . . . . . . . . . . . . . . 2.5.1 Period-Doubled Cubic Discrete Systems . . . . 2.5.2 Period-Doubling Renormalization . . . . . . . . . 2.5.3 Period-n Appearing and Period-Doublization .

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93 93 104 114 117 121 121 128 138

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ix

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Contents

2.5.4 Sampled Period-n Appearing Bifurcations . . . . . . . . . . . 2.6 Backward Cubic Nonlinear Discrete Systems . . . . . . . . . . . . . . 2.6.1 Backward Period-2 Cubic Discrete Systems . . . . . . . . . 2.6.2 Backward Period-Doubling Renormalization . . . . . . . . . 2.6.3 Backward Period-n Appearing and Period-Doublization . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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147 150 150 153 157 166

3 Quartic Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . 3.1 Period-1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 3.2 Period-1 to Period-2 Bifurcation Trees . . . . . . . . . . . . . . . . . . . 3.3 Higher-Order Period-1 Quartic Discrete Systems . . . . . . . . . . . 3.4 Period-1 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Simple Period-1 Switching Bifurcations . . . . . . . . . . . . 3.4.2 Higher-Order Period-1 Switching Bifurcations . . . . . . . . 3.5 Forward Quartic Discrete Systems . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Period-2 Quartic Discrete Systems . . . . . . . . . . . . . . . . 3.5.2 Period-Doubling Renormalization . . . . . . . . . . . . . . . . . 3.5.3 Period-n Appearing and Period-Doublization . . . . . . . . . 3.6 Backward Quartic Discrete Systems . . . . . . . . . . . . . . . . . . . . . 3.6.1 Backward Period-2 Quartic Discrete Systems . . . . . . . . 3.6.2 Backward Period-Doubling Renormalization . . . . . . . . . 3.6.3 Backward Period-n Appearing and Period-Doublization . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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167 167 179 188 200 201 215 223 224 227 231 239 239 243 247 256

4 (2m)th-Degree Polynomial Discrete Systems . . . . . . . . . . . . . 4.1 Global Stability and Bifurcations . . . . . . . . . . . . . . . . . . . 4.2 Simple Fixed-Point Bifurcations . . . . . . . . . . . . . . . . . . . 4.2.1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . . . 4.2.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . 4.2.3 Switching-Appearing Bifurcations . . . . . . . . . . . . . 4.3 Higher-Order Fixed-Points Bifurcations . . . . . . . . . . . . . . 4.3.1 Appearing Bifurcations . . . . . . . . . . . . . . . . . . . . 4.3.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . . . . 4.3.3 Appearing-Switching Bifurcations . . . . . . . . . . . . . 4.4 Forward Bifurcation Trees . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Period-Doubled (2m)th-Degree Polynomial Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Renormalization and Period-Doubling . . . . . . . . . . 4.4.3 Period-n Appearing and Period-Doublization . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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257 257 276 276 282 289 294 294 305 311 318

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318 321 326 334

5 (2m + 1)th-Degree Polynomial Discrete Systems 5.1 Global Stability and Bifurcations . . . . . . . . . 5.2 Simple Fixed-Point Bifurcations . . . . . . . . . 5.2.1 Appearing Bifurcations . . . . . . . . . .

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335 335 354 354

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Contents

5.2.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . 5.2.3 Switching-Appearing Bifurcations . . . . . . . . . . 5.3 Higher-Order Fixed-Point Bifurcations . . . . . . . . . . . . 5.3.1 Higher-Order Fixed-Point Bifurcations . . . . . . 5.3.2 Switching Bifurcations . . . . . . . . . . . . . . . . . . 5.3.3 Switching-Appearing Bifurcations . . . . . . . . . . 5.4 Forward Bifurcation Trees . . . . . . . . . . . . . . . . . . . . . 5.4.1 Period-Doubled ð2m þ 1Þth -Degree Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Renormalization and Period-Doubling . . . . . . . 5.4.3 Period-n Appearing and Period-Doublization . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

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369 374 379 379 398 407 410

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410 415 419 428

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Chapter 1

Quadratic Nonlinear Discrete Systems

In this Chapter, the global stability and bifurcation of quadratic nonlinear discrete systems are discussed. Appearing and switching bifurcations of simple period-1 fixed-points are discussed. Period-1 and period-2 bifurcation trees with global stability are presented for forward and backward quadratic discrete systems. The period-2 appearing and period-doubling renormalizations of quadratic discrete systems are discussed, and the period-n appearing and period-doublization in quadratic discrete systems are presented. Similarly, backward period-2 appearing, backward period-doubling renormalization, and backward period-n appearing and period-doublization are also discussed. The forward and backward period-n bifurcations are presented, and the corresponding bifurcation dynamics can be determined as well.

1.1

Linear Discrete Systems

In this section, the stability and stability switching of fixed-points in linear discrete systems are discussed. The monotonic and oscillatory sink and source fixed-points are discussed. Definition 1.1

Consider a 1-dimensional linear discrete system xk þ 1 ¼ xk þ AðpÞxk þ BðpÞ

ð1:1Þ

where two scalar constants AðpÞ and BðpÞ are determined by a vector parameter p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:2Þ

(i) If AðpÞ 6¼ 0, there is a fixed-point of

© Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3_1

1

2

1 Quadratic Nonlinear Discrete Systems

xk ¼ a1 ðpÞ ¼ 

BðpÞ ; with a0 ðpÞ ¼ AðpÞ; AðpÞ

ð1:3Þ

and the corresponding discrete system becomes xk þ 1 ¼ xk þ a0 ðxk  a1 Þ:

ð1:4Þ

(ii) If AðpÞ ¼ 0, Eq. (1.1) becomes xk þ 1 ¼ xk þ BðpÞ:

ð1:5Þ

(ii1) For BðpÞ 6¼ 0, the 1-dimensional linear discrete system is called a constant adding discrete system. (ii2) For BðpÞ ¼ 0, the 1-dimensional linear discrete system is called a permanent invariant discrete system. For kpk ! kp0 k ¼ b, if the following relations hold AðpÞ ¼ a0 ¼ e ! 0; BðpÞ ¼ ea1 ðpÞ ! 0;

ð1:6Þ

then there is an instant fixed-point to the vector parameter p xk ¼ a1 ðpÞ:

ð1:7Þ

Theorem 1.1 Under assumption in Eq. (1.6), a standard form of the 1-dimensional discrete system in Eq. (1.1) is xk þ 1 ¼ xk þ f ðxk Þ ¼ xk þ a0 ðxk  a1 Þ

ð1:8Þ

(i) If j1 þ a0 ðpÞj\1 (or j1 þ df =dxk jxk ¼a1 j\1), then the fixed-point of xk ¼ a1 ðpÞ is stable. Such a stable fixed-point is called a sink or a stable node. (i1) If a0 ðpÞ 2 ð1; 0Þ (or df =dxk jxk ¼a1 2 ð1; 0Þ), then the fixed-point of xk ¼ a1 ðpÞ is monotonically stable. Such a stable fixed-point is called a monotonic sink or a monotonically stable node. (i2) If a0 ðpÞ 2 ð2; 1Þ (or df =dxk jxk ¼a1 2 ð2; 1Þ), then the fixed-point xk ¼ a1 ðpÞ is oscillatorilly stable. Such a stable fixed-point is called an oscillatory sink or an oscillatorilly stable node. (i3) If a0 ðpÞ ¼ 1 (or df =dxk jxk ¼a1 ¼ 1), then the fixed-point of xk ¼ a1 ðpÞ is invariantly stable. Such a stable fixed-point is called an invariant sink. (ii) If j1 þ a0 ðpÞj [ 1 (or j1 þ df =dxk jxk ¼a1 j [ 1), then the fixed-point of xk ¼ a1 ðpÞ is unstable. Such an unstable fixed-point is called a source or an unstable node.

1.1 Linear Discrete Systems

3

(ii1) If a0 ðpÞ 2 ð0; 1Þ (or 1 þ df =dxk jxk ¼a1 2 ð1; 1Þ), then the fixed-point of xk ¼ a1 ðpÞ is monotonically unstable. Such a stable fixed-point is called a monotonic source or a monotonically unstable node. (ii2) If a0 ðpÞ 2 ð1; 2Þ (or 1 þ df =dxk jxk ¼a1 2 ð1; 1Þ), then fixed-point of xk ¼ a1 ðpÞ is oscillatorilly unstable. Such a stable fixed-point is called an oscillatory source or an oscillatorilly unstable node. (iii) If a0 ðpÞ ¼ 0 (or 1 þ df =dxk jxk ¼a1 ¼ 1), then the flow in the neighborhood of fixed-point xk ¼ a1 ðpÞ is invariant. Such an invariant point is called a monotonic saddle switching. (iv) If a0 ðpÞ ¼ 2 (or 1 þ df =dxk jxk ¼a1 ¼ 1), then the flow in the neighborhood of fixed-point xk ¼ a1 ðpÞ is flipped. Such an invariant point is called an oscillatory saddle switching. Proof

Let yk ¼ xk  a1 and yk þ 1 ¼ xk þ 1  xk . Thus, Eq. (1.8) becomes yk þ 1 ¼ ð1 þ a0 Þyk :

The corresponding solution is yk ¼ ð1 þ a0 Þk y0 where y0 ¼ x0  a is an initial condition. (i) If ja0 ðpÞ þ 1j\1, we have lim ðxk  a1 Þ ¼ lim yk ¼ lim ð1 þ a0 Þk y0 :

k!1

(i1)

k!1

k!1

If a0 ðpÞ 2 ð1; 0Þ, we have 0\1 þ a0 \1 and lim ðxk  a1 Þ ¼ lim yk ¼ lim ð1 þ a0 Þk y0 ¼ 0

k!1

k!1

k!1

) lim xk ¼ a1 : k!1

(i2)

Thus, the fixed-point of xk ¼ a1 ðpÞ is monotonically stable. Such a fixed-point is also called a monotonic sink. If a0 ðpÞ 2 ð2; 1Þ, we have 1\1 þ a0 \0 and lim ðxk  a1 Þ ¼ lim yk ¼ lim j1 þ a0 jk ð1Þk y0 ¼ 0

k!1

k!1

k!1

) lim xk ¼ a1 : k!1

(i3)

Thus, the fixed-point of xk ¼ a1 ðpÞ is oscillatorilly stable. Such a fixed-point is also called an oscillatory sink. If a0 ðpÞ ¼ 1, we have 1 þ a0 ¼ 0 and

4

1 Quadratic Nonlinear Discrete Systems

ðxk  a1 Þ ¼ yk ¼ ð1 þ a0 Þk y0 ¼ 0 ) xk ¼ a1 ðk ¼ 1; 2; . . .Þ: Thus, the fixed-point of xk ¼ a1 ðpÞ is invariant. Such a fixed-point is also called an invariant sink. So the fixed-point of xk ¼ a1 ðpÞ is stable. The fixed-point is called a sink or stable node. (ii) If ja0 ðpÞ þ 1j\1, we have (ii1)

If a0 ðpÞ 2 ð0; 1Þ, we have 1 þ a0 [ 1 and lim ðxk  a1 Þ ¼ lim yk ¼ lim ð1 þ a0 Þk y0 ¼ 1

k!1

k!1

k!1

) lim xk ¼ 1: k!1

(ii2)

Thus, the fixed-point of xk ¼ a1 ðpÞ is monotonically unstable. Such a fixed-point is also called a monotonic source. If a0 ðpÞ 2 ð1; 2Þ, we have 1 þ a0 \  1 and lim ðxk  a1 Þ ¼ lim yk ¼ lim j1 þ a0 jk ð1Þk y0 k!1 k!1
1; k ¼ 2l ! 1; ¼ 1; k ¼ 2l þ 1 ! 1;
1; k ¼ 2l ! 1; ) lim xk ¼ k!1 1; k ¼ 2l þ 1 ! 1: k!1

Thus, the fixed-point of xk ¼ a1 ðpÞ is oscillatorilly unstable. Such a fixed-point is also called an oscillatory source. So the fixed-point of xk ¼ a1 ðpÞ is unstable. The fixed-point is called an oscillatory source or an oscillatorilly unstable node. (iii) If a0 ðpÞ ¼ 0, we have 1 þ a0 ¼ 1 and lim ðxk  a1 Þ ¼ lim yk ¼ lim ð1 þ a0 Þk y0 ¼ y0 :

k!1

k!1

k!1

So the fixed-point of xk ¼ a1 ðpÞ is invariant. The fixed-point is called a monotonic saddle switching. (iv) If a0 ðpÞ ¼ 2, we have 1 þ a0 ¼ 1 and lim ðxk  a1 Þ ¼ lim yk ¼ lim ð1Þk y0 ¼

k!1

k!1

k!1




y0 ; k ¼ 2l; y0 ; k ¼ 2l þ 1:

1.1 Linear Discrete Systems

5

So the fixed-point of xk ¼ a1 ðpÞ is flipped. The fixed-point is called an oscillatory saddle switching. ■

The theorem is proved.

As in Luo (2010, 2012), the theory for the positive and negative mappings in discrete systems are used herein. If the discrete system in Eq. (1.1) is a positive mapping for xk ðk ¼ 1; 2; . . .Þ via x0 , then the corresponding negative mapping is from xk þ 1 ¼ xk þ AðpÞxk þ BðpÞ

ð1:9Þ

for xk ðk ¼ 1; 2; 3; . . .Þ via x0 with the corresponding stability determined by dxk =dxk þ 1 jxk þ 1 : Such a negative mapping is equivalent to the following mapping xk ¼ xk þ 1 þ AðpÞxk þ 1 þ BðpÞ

ð1:10Þ

for xk ðk ¼ 1; 2; 3; . . .Þ via x0 with the corresponding stability determined by dxk þ 1 =dxk jxk : Such a linear discrete system with the negative mapping is called a linear backward discrete system. The linear discrete system with a positive mapping is called a linear forward discrete system. Definition 1.2

Consider a 1-dimensional, linear, backward discrete system xk ¼ xk þ 1 þ AðpÞxk þ 1 þ BðpÞ

ð1:11Þ

where two scalar constants AðpÞ and BðpÞ are determined by a vector parameter p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:12Þ

(i) If AðpÞ 6¼ 0, there is a fixed-point of xk ¼ a1 ðpÞ ¼ 

BðpÞ ; with a0 ðpÞ ¼ AðpÞ AðpÞ

ð1:13Þ

and the corresponding backward discrete system becomes xk ¼ xk þ 1 þ a0 ðxk þ 1  a1 Þ:

ð1:14Þ

(ii) If AðpÞ ¼ 0, Eq. (1.1) becomes xk ¼ xk þ 1 þ BðpÞ:

ð1:15Þ

For BðpÞ 6¼ 0, the 1-dimensional backward discrete system is called a constant adding discrete system. For BðpÞ ¼ 0, the 1-dimensional backward discrete system is called a permanent invariant discrete system.

6

1 Quadratic Nonlinear Discrete Systems

(iii) For kpk ! kp0 k ¼ b, if the following relations hold AðpÞ ¼ a0 ¼ e ! 0; BðpÞ ¼ ea1 ðpÞ ! 0;

ð1:16Þ

then there is an instant fixed-point to the vector parameter p xk ¼ a1 ðpÞ:

ð1:17Þ

Theorem 1.2 Under assumption in Eq. (1.16), a standard form of the 1-dimensional, linear, backward discrete system in Eq. (1.11) is xk ¼ xk þ 1 þ f ðxk þ 1 Þ ¼ xk þ 1 þ a0 ðxk þ 1  a1 Þ

ð1:18Þ

(i) If jð1 þ a0 ðpÞÞ1 j\1 (or jð1 þ df =dxk þ 1 jxk þ 1 ¼a1 Þ1 j\1), then the fixed-point of xk ¼ a1 ðpÞ is stable. Such a stable fixed-point is called a sink or a stable node. (i1) If a0 ðpÞ 2 ð1; 0Þ (or ð1 þ df =dxk þ 1 jx ¼a1 Þ1 2 ð1; 0Þ), then the kþ1 fixed-point of xk ¼ a1 ðpÞ is oscillatorilly stable. Such a stable fixed-point is called an oscillatory sink or an oscillatorilly stable node. (i2) If a0 ðpÞ 2 ð0; 1Þ (or ð1 þ df =dxk þ 1 jx ¼a1 Þ1 2 ð0; 1Þ), then fixedkþ1 point xk ¼ a1 ðpÞ is monotonically stable. Such a stable fixed-point is called a monotonic sink or a monotonically stable node. (ii) If jð1 þ a0 ðpÞÞ1 j [ 1 (or jð1 þ df =dxk þ 1 jxk þ 1 ¼a1 Þ1 j [ 1), then the fixedpoint of xk ¼ a1 ðpÞ is unstable. Such an unstable fixed-point is called a source or an unstable node. (ii1) If a0 ðpÞ 2 ð0; 1Þ (or ð1 þ df =dxk þ 1 jx ¼a1 Þ1 2 ð1; 1Þ), then the kþ1 fixed-point of xk ¼ a1 ðpÞ is monotonically unstable. Such a stable fixedpoint is called a monotonic source or a monotonically unstable node. (ii2) If a0 ðpÞ 2 ð1; 2Þ (or ð1 þ df =dxk þ 1 jx ¼a1 Þ1 2 ð1; 1Þ), then kþ1 fixed-point xk ¼ a1 ðpÞ is oscillatory unstable. Such a stable fixed-point is called an oscillatory source or an oscillatorilly unstable node. (iii) If a0 ðpÞ ¼ 0 (or ð1 þ df =dxk þ 1 jxk þ 1 ¼a1 Þ1 ¼ 1), then the discrete flow in the neighborhood of fixed-point xk ¼ a1 ðpÞ is invariant. Such an invariant point is called a monotonic saddle switching. (iv) If a0 ðpÞ ¼ 2 (or ð1 þ df =dxk þ 1 jxk þ 1 ¼a1 Þ1 ¼ 1), then the discrete flow in the neighborhood of fixed-point xk ¼ a1 ðpÞ is flipped. Such an invariant point is called an oscillatory saddle switching. Proof The proof is similar to the proof of Theorem 1.1. This theorem is proved.

1.1 Linear Discrete Systems

Invariant

7

Switch

Invariant

Flip

x = a1

xk∗

xk∗ = a1

xk∗ oSO

|| p ||

Switch

Flip

∗ k

a0 < −2

oSI

a0 = −2

mSI

a0 = −1

mSO

a0 = 0

a0 > 0

oSI

|| p ||

a0 < −2

(i)

oSO

a0 = −2

mSO

a0 = −1

a0 = 0

mSI

a0 > 0

(ii)

Fig. 1.1 Stability of single fixed-point in the 1-dimensional linear discrete system. (i) Positive mapping (forward), (ii) Negative mapping (backward). Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The stability switches are labelled by solid circular symbols. (Flip: flipped switch for oscillatorilly stable to unstable fixed-point; Switch: for monotonically stable and unstable fixed-points; Invariant: invariant sink; mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink.)

To illustrate the stability of fixed-points, one fixed-point of xk ¼ a1 ðpÞ changes with a vector parameter p. The stability of such a fixed-point is determined by the constant a0 ðpÞ: The stability switching is at the boundary p0 2 @X12 with a0 ¼ 0: The stability of the fixed-points for the positive (forward) and negative (backward) maps in the 1-dimensional discrete system is presented in Fig. 1.1i and ii, respectively. The stable and unstable portions of the fixed-point are presented by the solid and dash curves, respectively.

1.2

Forward Quadratic Discrete Systems

In this section, the stability of fixed-points in 1-dimensional quadratic nonlinear discrete systems are discussed. The upper-saddle-node and lower-saddle-node appearing and switching bifurcations are presented. The nonlinear discrete systems with positive and negative mappings will be discussed. As in Luo (2010, 2012), the discrete system with a map with a positive (forward) iteration is called a positive (forward) discrete system, and the discrete system with a map with a negative (backward) iteration is called a negative (backward) discrete system.

1.2.1

Period-1 Appearing Bifurcations

For one of the simplest nonlinear discrete systems, consider a positive (forward) quadratic nonlinear discrete system first.

8

1 Quadratic Nonlinear Discrete Systems

Definition 1.3 Consider a 1-dimensional quadratic nonlinear discrete system as xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ

ð1:19Þ

where three scalar constants AðpÞ 6¼ 0; BðpÞ and CðpÞ are determined by a vector parameter p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:20Þ

(i) If D ¼ B2  4AC\0 for

p 2 X1  Rm ;

ð1:21Þ

then the quadratic discrete system does not have any fixed-points. The discrete flow without fixed-points is called a non-fixed-point discrete flow. (i1) If a0 ðpÞ ¼ AðpÞ [ 0, the non-fixed-point discrete flow is called a positive discrete flow. (i2) If a0 ðpÞ ¼ AðpÞ\0, the non-fixed-point discrete flow is called a negative discrete flow. (ii) If D ¼ B2  4AC [ 0

for

p 2 X2  Rm ;

ð1:22Þ

then the quadratic discrete system has two different simple fixed-points as xk ¼ a1 and xk ¼ a2 ;

ð1:23Þ

and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ; where a0 ¼ AðpÞ; a1;2

ð1:24Þ

pffiffiffiffi BðpÞ  D with a1 \a2 : ¼ 2AðpÞ

ð1:25Þ

p ¼ p0 2 @X12  Rm1 ;

ð1:26Þ

(iii) If D ¼ B2  4AC ¼ 0

for

then the quadratic nonlinear discrete system has a double repeated fixed-point, i.e., xk ¼ a1 and xk ¼ a1 ;

ð1:27Þ

1.2 Forward Quadratic Discrete Systems

9

with the corresponding standard form of xk þ 1 ¼ xk þ a0 ðxk  a1 Þ2 ; where a0 ¼ Aðp0 Þ;

and a1 ¼ a2 ¼ 

ð1:28Þ Bðp0 Þ : 2Aðp0 Þ

ð1:29Þ

Such a discrete flow with the fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is called a monotonic saddle discrete flow of the second order. (iii1) If a0 ðpÞ [ 0, then the fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is a monotonic upper-saddle of the second-order. (iii2) If a0 ðpÞ\0; then the fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is a monotonic lower-saddle of the second-order. (iv) The fixed-point of xk ¼ a1 for two fixed-points vanishing or appearance is called a monotonic saddle-node appearing bifurcation of the second-order at a point p ¼ p0 2 @X12 , and the bifurcation condition is D ¼ B2  4AC ¼ 0:

ð1:30Þ

(iv1) If a0 ðpÞ [ 0, the bifurcation at xk ¼ xk þ 1 ¼ a1 ðpÞ for two fixed-points appearance or vanishing is called a monotonic upper-saddle-node appearing bifurcation of the second-order. (iv2) If a0 ðpÞ\0, the bifurcation at xk ¼ xk þ 1 ¼ a1 ðpÞ for two fixed-points appearance or vanishing is called a monotonic lower-saddle-node appearing bifurcation of the second-order.

Theorem 1.3 (i) Under a condition of D ¼ B2  4AC\0;

ð1:31Þ

a standard form of the quadratic nonlinear discrete system in Eq. (1.19) is xk þ 1 ¼ xk þ a0 ½ðxk 

1B 2 1 Þ þ 2 ðDÞ 2A 4A

ð1:32Þ

with a0 ¼ AðpÞ, which has a non-fixed-point flow. (i1) If a0 ðpÞ [ 0, the non-fixed-point discrete flow is called a positive discrete flow. (i2) If a0 ðpÞ [ 0, the non-fixed-point discrete flow is called a negative discrete flow.

10

1 Quadratic Nonlinear Discrete Systems

(ii) Under a condition of D ¼ B2  4AC [ 0;

ð1:33Þ

a standard form of the 1-dimensional discrete system in Eq. (1.19) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ:

ð1:34Þ

(ii1) For a0 ðpÞ [ 0; the following cases exist. (ii1a) The fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is • monotonically stable (a monotonic sink) if df =dxk jx ¼a1 2 k ð1; 0Þ; • invariantly stable (an invariant sink) if df =dxk jxk ¼a1 ¼ 1; • oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼a1 2 ð2; 1Þ; • flipped if df =dxk jxk ¼a1 ¼ 2 (an oscillatory upper-saddle of the second order for d 2 f =dx2k jxk ¼a2 ¼ a0 [ 0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a1 2 ð1; 2Þ: (ii1b) The fixed-point of xk ¼ a2 ðpÞ is monotonically unstable (a monotonic source) if df =dxk jxk ¼a2 2 ð0; 1Þ: (ii2)

For a0 ðpÞ\0, the following cases exist. (ii2a) The fixed-point of xk ¼ xk þ 1 ¼ a2 ðpÞ is • monotonically stable (a monotonic sink) if df =dxk jx ¼a2 2 k ð1; 0Þ; • invariantly stable (an invariant sink) if df =dxk jx ¼a2 ¼ 1; k • oscillatorilly stable (an oscillatory sink) if df =dxjx ¼a2 2 k ð2; 1Þ; • flipped if df =dxk jxk ¼a2 ¼ 2 (an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼a2 ¼ a0 \0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a2 2 ð1; 2Þ: (ii2b) The fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is monotonically unstable (a monotonic source) if df =dxk jx ¼a1 2 ð0; 1Þ: k

(iii) Under a condition of D ¼ B2  4AC ¼ 0;

ð1:35Þ

a standard form of the 1-dimensional discrete system in Eq. (1.19) is

1.2 Forward Quadratic Discrete Systems

11

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ2 :

ð1:36Þ

(iii1) If a0 ðpÞ [ 0, then the fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is a monotonic upper-saddle of the second-order if d 2 f =dx2k jxk ¼a1 [ 0: The bifurcation at xk ¼ xk þ 1 ¼ a1 ðpÞ for the appearance or vanishing of two simple fixed-points is called a monotonic upper-saddle-node appearing bifurcation of the second-order. (iii2) If a0 ðpÞ\0, then the fixed-point of xk ¼ xk þ 1 ¼ a1 ðpÞ is a monotonic lower-saddle of the second order with d 2 f =dx2k jxk ¼a1 \0: The bifurcation at xk ¼ xk þ 1 ¼ a1 ðpÞ for the appearance or vanishing of two simple fixed-points is called a monotonic lower-saddle-node appearing bifurcation of the second-order. Proof (i) Consider D ¼ B2  4AC\0: (i1) If a0 [ 0, we have xk þ 1  xk ¼ a0 ½ðxk 

B 2 1 Þ þ 2 ðDÞ [ 0: 4A 2A

Thus, such a non-fixed-point discrete flow is called a positive discrete flow. (i2) If a0 \0, we have xk þ 1  xk ¼ a0 ½ðxk 

B 2 1 Þ þ 2 ðDÞ\0: 2A 4A

Thus, such a non-fixed-point discrete flow is called a negative discrete flow. (ii) Let DxkðiÞ ¼ xk  ai ði ¼ 1; 2Þ and xk þ 1ðiÞ ¼ Dxk þ 1ðiÞ . Equation (1.34) becomes Dxk þ 1ðiÞ ¼ ½1 þ a0 ðai  aj ÞDxkðiÞ þ a0 Dx2kðiÞ ði; j 2 f1; 2g; j 6¼ iÞ: Because Dxi is arbitrary small, we have Dxk þ 1ðiÞ ¼ ki DxkðiÞ for ki  1 þ df =dxk jxk ¼ai ¼ 1 þ a0 ðai  aj Þ: The corresponding solution is

12

1 Quadratic Nonlinear Discrete Systems

DxkðiÞ ¼ ðki Þk Dx0ðiÞ where Dx0ðiÞ ¼ x0ðiÞ  ai is an initial condition. (iia) For ki 2 ð0; 1Þ (or df =dxk jxk ¼ai 2 ð1; 0Þ), we have lim ðxkðiÞ  ai Þ ¼ lim DxkðiÞ ¼ lim ðki Þk Dx0ðiÞ ¼ 0 ) lim xkðiÞ ¼ ai :

k!1

k!1

k!1

k!1

Consider ki ¼ 1 þ a0 ðai  aj Þ 2 ð0; 1Þ ) a0 ðai  aj Þ 2 ð1; 0Þ: (iia1) For a0 [ 0, we have ai \aj ) xk ¼ a1 : (iia2) For a0 \0, we have ai [ aj ) xk ¼ a2 : Thus, the fixed-point of xk ¼ ai is monotonically stable. (iib) For ki ¼ 0 (or df =dxk jxk ¼ai ¼ 1), we have ðxkðiÞ  ai Þ ¼ DxkðiÞ ¼ ðki Þk Dx0ðiÞ ¼ 0 ) xkðiÞ ¼ ai ði ¼ 1; 2Þ: Consider ki ¼ 1 þ a0 ðai  aj Þ ¼ 0 ) a0 ðai  aj Þ ¼ 1: (iib1) For a0 [ 0, we have ai \aj ) xk ¼ a1 : (iib2) For a0 \0, we have ai [ aj ) xk ¼ a2 : Thus, the fixed-point of xk ¼ ai is invariantly stable. (iic) For ki 2 ð1; 0Þ (or df =dxk jxk ¼ai 2 ð2; 1Þ), we have

1.2 Forward Quadratic Discrete Systems

13

lim ðxkðiÞ  ai Þ ¼ lim DxkðiÞ ¼ lim kki Dx0ðiÞ k!1 k!1
 0 k k ¼ lim ðjki jÞ ð1Þ Dx0ðiÞ ¼ k!1 0þ

k!1

for k ¼ 2l  1; for k ¼ 2l:

Thus
lim xkðiÞ ¼

k!1

a i aiþ

for k ¼ 2l  1; for k ¼ 2l:

Consider ki ¼ 1 þ a0 ðai  aj Þ 2 ð0; 1Þ ) a0 ðai  aj Þ 2 ð1; 0Þ: (iic1) For a0 [ 0, we have ai \aj ) xk ¼ a1 : (iic2) For a0 \0, we have ai [ aj ) xk ¼ a2 : Thus, the fixed-point of xk ¼ ai is monotonically stable. (iid) For ki ¼ 1 (or df =dxk jxk ¼ai ¼ 2), we have lim ðxkðiÞ  ai Þ ¼ lim DxkðiÞ ¼ lim ð1Þk Dx0ðiÞ ;

k!1

k!1

k!1

so
lim xkðiÞ ¼

k!1

ai þ Dx0ðiÞ ai þ Dx0ðiÞ ai þ Dx0ðiÞ

for k ¼ 2l  1; for k ¼ 2l:

Consider ki ¼ 1 þ a0 ðai  aj Þ ¼ 1 ) a0 ðai  aj Þ ¼ 2: (iid1) For a0 [ 0, we have ai \aj ) xk ¼ a1 : For the fixed-point of xk ¼ a1 , we have

14

1 Quadratic Nonlinear Discrete Systems

Dxk þ 1 ¼ ð1 þ a0 Dxk ÞDxk : Therefore, the fixed-point of xk ¼ a1 is an oscillatory lowersaddle of the second-order. (iid2) For a0 \0, we have ai [ aj ) xk ¼ a2 : For the fixed-point of xk ¼ a2 , we have Dxk þ 1 ¼ ð1 þ a0 Dxk ÞDxk : Thus, the fixed-point of xk ¼ a2 is an oscillatory upper-saddle of the second-order. (iie) For ki \  1 (or df =dxk jxk ¼ai 2 ð1; 2Þ), we have lim ðxkðiÞ  ai Þ ¼ lim DxkðiÞ ¼ lim kki Dx0ðiÞ k!1 k!1
1 for k ¼ 2l  1; k k ¼ lim ðjki jÞ ð1Þ Dx0ðiÞ ¼ k!1 þ 1 for k ¼ 2l:

k!1

Thus
lim xkðiÞ ¼

k!1

ai  1 for k ¼ 2l  1; ai þ 1 for k ¼ 2l:

and ki ¼ 1 þ a0 ðai  aj Þ\  1 ) a0 ðai  aj Þ\  2: (iie1) For a0 [ 0, we have ai \aj ) xk ¼ a1 : (iie2) For a0 \0, we have ai [ aj ) xk ¼ a2 : Thus, the fixed-point of xk ¼ ai is oscillatorilly stable. (iif) If ki [ 1 (or df =dxk jxk ¼ai 2 ð0; 1Þ), we have the following cases.

1.2 Forward Quadratic Discrete Systems

15

lim ðxkðiÞ  ai Þ ¼ lim DxkðiÞ ¼ lim ðki Þk Dx0ðiÞ ¼ 1 ) lim xkðiÞ ¼ 1

k!1

k!1

k!1

k!1

and ki ¼ 1 þ a0 ðai  aj Þ 2 ð1; 1Þ ) a0 ðai  aj Þ 2 ð0; 1Þ: (iif1) For a0 [ 0, we have ai [ aj ) xk ¼ a2 : (iif2) For a0 \0, we have ai \aj ) xk ¼ a1 : Thus, the fixed-point of xk ¼ ai is monotonically unstable. (iii) If a1 ðpÞ ¼ a2 ðpÞ, ki ¼ 1 (or df =dxk jxk ¼ai ¼ 0) and we have Dxk þ 1 ¼ Dxk þ a0 Dx2k ¼ ð1 þ a0 Dxk ÞDxk

and Dxk ¼ xk  xk :

(iii1) For a0 [ 0, Dxk þ 1 [ Dxk [ 0 if Dxk [ 0 and 0 [ Dxk þ 1 [ Dxk if Dxk \0: So a flow of xk reaches to xk ¼ a1 from the initial point of xk0 \a1 and it goes to the positive infinity from xk0 [ a1 . Such a fixed-point is monotonically unstable of the second order, which is called a monotonic upper-saddle of the second-order. (iii2) Similarly, for a0 \0, 0\Dxk þ 1 \Dxk if Dxk [ 0 and Dxk þ 1 \Dxk \0 if Dxk \0: So a flow of xk reaches to xk ¼ a1 from the initial point of xk0 [ a1 and it goes to the negative infinity from xk0 \a1 . Such a fixed-point is monotonically unstable of the second order, which is called a monotonic lower-saddle of the second-order. The theorem is proved.

■

The stability and bifurcation of fixed-points for the quadratic nonlinear discrete system in Eq. (1.19) are illustrated in Fig. 1.2. The stable and unstable fixed-points varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation point of fixed-points occurs at the double-repeated fixed-points at p0 2 @X12 . In Fig. 1.2i, for a0 [ 0, the fixed-point of xk ¼ a2 for D [ 0 is monotonically unstable, and the fixed-point of xk ¼ a1 in a small neighborhood of D ¼ 0 þ is monotonically stable. The fixed point of xk ¼ a1 can be a monotonic sink, a zero-invariant sink, an oscillatory sink, a flipped invariance and an oscillatory source. The monotonic bifurcation of two simple fixed-points also occurs at D ¼ 0: The discrete flow of xk is a forward upper-branch discrete flow for a0 [ 0, and the fixed-point xk ¼ Bðp0 Þ=2Aðp0 Þ at D ¼ 0 is termed a monotonic upper-saddle of the second-order. Such a bifurcation is termed a monotonic upper-

16

1 Quadratic Nonlinear Discrete Systems xk∗ = a2

Δ0

P-2 mUS

mSI

xk∗ Non-fixed point

|| p ||

oSI iSI

Δ0

Δ=0

xk

(i)

(ii) xk∗ = a2

|| p 0 ||

oLS iSI

oSI

xk∗

a2

xk +1 = xk Δ>0

mSO

xk∗ = a1 Fixed point

Non-fixed point

Δ0

Δ=0

xk +1 − xk

Δ 0

Fig. 1.3 Stability and bifurcation of a repeated fixed-point of the second order in the quadratic forward discrete system. Unstable fixed-points is represented by a dashed curve. The stability switching from the monotonic lower-saddle to monotonic upper-saddle is labelled by a circular symbol. (mLS: monotonic lower-saddle; mUS: monotonic upper-saddle.)

þ ð4AD2 Þ\0: The corresponding phase portrait is presented in Fig. 1.2iv. The period-2 fixed-points based on the quadratic forward discrete map are also presented through red curves, labelled by P-2. To illustrate the stability and bifurcation of fixed-points with singularity in a 1-dimensional, quadratic nonlinear discrete system, the fixed-point of xk þ 1  xk ¼ a0 ðxk  a1 Þ2 is presented in Fig. 1.3. The monotonic upper-saddle and lower-saddle fixed-points of xk ¼ a1 with the second-order are unstable, which are depicted by dashed curves. At a0 ¼ 0, the monotonic upper-saddle and lower-saddle fixedpoints will be switched, which is marked by a circular symbol. Consider a symmetric case for the appearing bifurcations in forward quadratic discrete systems. Such a special case can help one further understand the appearing bifurcation and the corresponding stability. Definition 1.4 If BðpÞ ¼ 0 in Eq. (1.19), a 1-dimensional quadratic discrete system is xk þ 1 ¼ xk þ AðpÞx2k þ CðpÞ:

ð1:37Þ

(i) For AðpÞ  CðpÞ [ 0, the discrete system does not have any fixed-points. (i1) The non-fixed-point discrete flow of the discrete system is called a positive discrete flow if AðpÞ [ 0: (i2) The non-fixed-point discrete flow of the discrete system is called a negative discrete flow if AðpÞ\0: (ii) For AðpÞ  CðpÞ\0, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk þ aÞðxk  aÞ with two symmetric fixed-points

ð1:38Þ

18

1 Quadratic Nonlinear Discrete Systems

xk ¼ a and xk ¼ a; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðp0 Þ : with a0 ¼ Aðp0 Þ and a ¼  Aðp0 Þ

ð1:39Þ

(iii) For Cðp0 Þ ¼ 0, the corresponding standard form with D ¼ 0 is xk þ 1 ¼ xk þ a0 x2k

ð1:40Þ

xk ¼ a ¼ 0 and xk ¼ a ¼ þ 0:

ð1:41Þ

with two fixed-points of

Such a fixed-point of xk ¼ 0 is called a monotonic saddle of the secondorder. (iii1) If a0 [ 0, the fixed-point is a monotonic upper-saddle of the secondorder. (iii2) If a0 \0, the fixed-point is a monotonic lower-saddle of the secondorder. (iv) The fixed-point of xk ¼ 0 for two fixed-points appearance or vanishing is called a monotonic saddle-node appearing bifurcation of the second-order at a point p ¼ p0 2 @X12 , and the appearing bifurcation condition is Cðp0 Þ ¼ 0:

ð1:42Þ

AðpÞ  CðpÞ [ 0;

ð1:43Þ

Theorem 1.4 (i) Under a condition of

a standard form of the 1-dimensional forward, quadratic discrete system in Eq. (1.37) is C A

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ a0 ðx2k þ Þ:

ð1:44Þ

(i1) For AðpÞ [ 0, the non-fixed-point flow of the discrete system is a positive discrete flow. (i2) For AðpÞ\0, the non-fixed-point flow of the discrete system is a negative discrete flow.

1.2 Forward Quadratic Discrete Systems

19

(ii) Under a condition of AðpÞ  CðpÞ\0;

ð1:45Þ

a standard form of the 1-dimensional forward, quadratic nonlinear discrete system in Eq. (1.37) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ a0 ðxk þ aÞðxk  aÞ:

ð1:46Þ

(ii1) For a0 ðpÞ [ 0, there are two cases: (ii1a) The fixed-point of xk ¼ a is: • monotonically stable (a monotonic sink) if df =dxk jxk ¼a 2 ð1; 0Þ; • invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; • oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼a 2 ð2; 1Þ; • flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory upper-saddle of the second-order under d 2 f =dx2k jxk ¼a ¼ a0 [ 0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a 2 ð1; 2Þ: (ii1b) The fixed-point of xk ¼ a is monotonically unstable (a monotonic source) if df =dxk jx ¼a [ 0: k (ii2) For a0 ðpÞ\0, there are two cases: (ii2a) The fixed-point of xk ¼ a is monotonically unstable (a monotonic source) if df =dxk jxk ¼a [ 0: (ii2b) The fixed-point of xk ¼ a is: • monotonically stable (a monotonic sink) if df =dxk jxk ¼a 2 ð1; 0Þ; • invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; • oscillatorilly stable (an oscillatory source) if df =dxk jxk ¼a 2 ð2; 1Þ; • flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory lower-saddle of the second-order with d 2 f =dx2k jxk ¼a ¼ a0 \0); • oscillatorilly unstable (an oscillator source) if df =dxk jxk ¼a 2 ð1; 2Þ: (iii) Under a condition of CðpÞ ¼ 0;

ð1:47Þ

20

1 Quadratic Nonlinear Discrete Systems

a standard form of the 1-dimensional discrete system in Eq. (1.37) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 x2k :

ð1:48Þ

(iii1) If a0 ðpÞ [ 0, then the fixed-point of x ¼ 0 is a monotonic upper-saddle of the second-order for d 2 f =dx2 jx ¼0 [ 0: Such a bifurcation for two fixed-points appearance or vanishing is a monotonic upper-saddlenode appearing bifurcation of the second-order. (iii2) If a0 ðpÞ\0, then the fixed-point x ¼ 0 is a monotonic lower-saddle of the second-order for d 2 f =dx2 jx ¼0 \0: Such a bifurcation for two fixed-points appearance or vanishing is a monotonic lower-saddlenode appearing bifurcation of the second-order. Proof The proof is similar to Theorem 1.3. The theorem is proved.

■

The stability and bifurcation of fixed-points for the quadratic nonlinear system in Eq. (1.37) are illustrated in Fig. 1.4 as a special case of the discrete system in Eq. (1.19) with BðpÞ ¼ 0: The stable and unstable fixed-points varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation of fixed-point occurs at the double-repeated fixed-point at the boundary of p0 2 @X12 . In Fig. 1.4i, for D ¼ 4AC [ 0 and a0 ¼ A [ 0, the fixed-point of xk ¼ a [ 0 for C\0 is unstable, and the fixed-point of xk ¼ a\0 for C\0 is from monotonically stable to oscillatorilly unstable. The bifurcation of fixed-point also occurs at C ¼ 0: The discrete flow of xk is a forward upper-branch discrete flow for a0 [ 0, and the fixed-point of xk ¼ 0 at C ¼ 0 is termed an monotonic upper-saddle of the second-order. Such a bifurcation is termed a monotonic uppersaddle-node bifurcation of the second-order. For D ¼ 4AC\0 and a0 ¼ A [ 0, we have C [ 0: Thus, no any fixed-point exists because of xk þ 1  xk ¼ Ax2k þ C [ 0: Such a 1-dimensional discrete system is termed a non-fixed-point discrete system. For a0 ¼ A [ 0 and C [ 0, the discrete flow of xk is always toward the positive direction. In Fig. 1.4(ii), for D ¼ 4AC [ 0 and a0 ¼ A\0, the fixed-point of xk ¼ a for C [ 0 is unstable, and the fixed-point of xk ¼ a for C [ 0 is from monotonically stable to oscillatorilly unstable. The bifurcation of fixed-point also occurs at C ¼ 0: The discrete flow of xk for the bifurcation point is a forward monotonic lower-branch discrete flow for a0 ¼ A\0, and the bifurcation point of the fixed-point at xk ¼ 0 for C ¼ 0 is termed a monotonic lower-saddle of the second-order. Such a bifurcation is termed a monotonic lower-saddle-node bifurcation of the second-order. For D ¼ 4AC\0 and a0 ¼ A\0, we have C\0: For a0 ¼ A\0 and C\0, the discrete flow of xk is always toward the negative direction without any fixed-points because of xk þ 1  xk ¼ Ax2k þ C\0:

1.2 Forward Quadratic Discrete Systems mSO

xk∗ = a

|| p 0 ||

21

B=0

|| p 0 ||

B=0

xk∗ = a oSO

mSI P-2

xk∗

∗ k

x = −a

mLS

oSI oSO Fixed point

C0

mSI

P-2

mSO

xk∗ Non-fixed point

|| p ||

C0

(ii)

Fig. 1.4 Stability and bifurcation of two fixed-points in the quadratic forward discrete system: (i) a monotonic upper-saddle-node bifurcation ða0 [ 0Þ, (ii) a monotonic lower-saddle-node bifurcation ða0 \0Þ. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; iSI: invariant sink.)

1.2.2

Period-1 Switching Bifurcations

Definition 1.5 Consider a 1-dimensional discrete system in Eq. (1.19) as xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ ¼ xk þ a0 ðpÞðxk  aðpÞÞðxk  bðpÞÞ:

ð1:49Þ

(i) For a\b, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞðxk  bÞ

ð1:50Þ

with two fixed-points xk ¼ a1 ¼ a and xk ¼ a2 ¼ b with D ¼ a20 ða  bÞ2 [ 0:

ð1:51Þ

(ii) For a [ b, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  bÞðxk  aÞ

ð1:52Þ

with two fixed-points of xk ¼ a1 ¼ b and xk ¼ a2 ¼ a with D ¼ a20 ða  bÞ2 [ 0:

ð1:53Þ

22

1 Quadratic Nonlinear Discrete Systems

(iii) For a ¼ b, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞ2

ð1:54Þ

with a repeated fixed-point of xk ¼ a: Such a fixed-point is called a monotonic saddle of the second-order. (iii1) If a0 [ 0, the fixed-point is a monotonic upper-saddle of the secondorder. (iii2) If a0 \0, the fixed-point is a monotonic lower-saddle of the second-order. (iv) The fixed-point of xk ¼ a for two fixed-points switching is called a monotonic saddle-node switching bifurcation of fixed-points at a point p ¼ p0 2 @X12 , and the bifurcation condition is D ¼ a20 ða  bÞ2 ¼ 0 or a ¼ b:

ð1:55Þ

a\b and D ¼ a20 ða  bÞ2 [ 0

ð1:56Þ

Theorem 1.5 (i) Under a condition of

a standard form of the 1-dimensional discrete system in Eq. (1.49) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  aÞðxk  bÞ:

ð1:57Þ

(i1) For a0 ðpÞ [ 0, there are two cases: (i1a) The fixed-point of xk ¼ a is: monotonically stable (a monotonic sink) if df =dxk jxk ¼a 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; oscillatorilly stable (an oscillatory source) if df =dxk jxk ¼a 2 ð2; 1Þ; flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory lower-saddle of the second-order with d 2 f =dx2k jxk ¼a ¼ a0 \0); • oscillatorilly unstable (an oscillator source) if df =dxk jxk ¼a 2 ð1; 2Þ:

• • • •

(i1b) The fixed-point of xk ¼ b is monotonically unstable (a monotonic source) if df =dxk jxk ¼b [ 0: (i2) For a0 ðpÞ\0, there are two cases: (i2a) The fixed-point of xk ¼ a is monotonically unstable (a monotonic source) if df =dxk jxk ¼a [ 0.

1.2 Forward Quadratic Discrete Systems

23

(i2b) The fixed-point of xk ¼ b is: • monotonically stable (a monotonic sink) if df =dxk jxk ¼b 2 ð1; 0Þ; • invariantly stable (an invariant sink) if df =dxk jxk ¼b ¼ 1; • oscillatorilly stable (an oscillatory source) if df =dxk jxk ¼b 2 ð2; 1Þ; • flipped if df =dxk jxk ¼b ¼ 2 (an oscillatory lower-saddle of the second-order with d 2 f =dx2k jxk ¼b ¼ a0 \0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼b 2 ð1; 2Þ: (ii) Under a condition of a [ b and D ¼ a20 ða  bÞ2 [ 0

ð1:58Þ

a standard form of the 1-dimensional discrete system in Eq. (1.49) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  bÞðxk  aÞ:

ð1:59Þ

(ii1) For a0 ðpÞ [ 0, there are two cases: (ii1a) The fixed-point of xk ¼ a is monotonically unstable with df =dxk jxk ¼a [ 0. (ii1b) The fixed-point of xk ¼ b is: monotonically stable (a monotonic sink) if df =dxk jxk ¼b 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼b ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼b 2 ð2; 1Þ; flipped if df =dxk jxk ¼b ¼ 2 (an oscillatory upper-saddle of the second-order with d 2 f =dx2k jxk ¼b ¼ a0 [ 0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼b 2 ð1; 2Þ:

• • • •

(ii2) For a0 ðpÞ\0, there are two cases: (ii2a) The fixed-point of xk ¼ a is: monotonically stable (a monotonic sink) if df =dxk jxk ¼a 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼a 2 ð2; 1Þ; flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory lower-saddle of the second-order with d 2 f =dx2k jxk ¼a ¼ a0 \0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a 2 ð1; 2Þ:

• • • •

(ii2b) The fixed-point of xk ¼ b is monotonically unstable (a monotonic source) if df =dxk jxk ¼b [ 0.

24

1 Quadratic Nonlinear Discrete Systems

(iii) For a ¼ b, the corresponding standard form with D ¼ 0 is xk þ 1 ¼ xk þ f ðx; pÞ ¼ xk þ a0 ðx  aÞ2

ð1:60Þ

(iii1) If a0 ðpÞ [ 0, then the fixed-point x ¼ a is a monotonically uppersaddle of the second order with d 2 f =dx2 jx ¼a [ 0. The fixed-point xk ¼ a for two fixed-points switching is a monotonic upper-saddle-node switching bifurcation of the second order. (iii2) If a0 ðpÞ\0, then the fixed-point xk ¼ a is a monotonic lower-saddle of the second-order with d 2 f =dx2 jx ¼a \0. The fixed-point x ¼ a for two fixed-points switching is a monotonic lower-saddle-node switching bifurcation of the second-order. Proof The theorem can be proved as for Theorem 1.3.

■

Definition 1.6 If CðpÞ ¼ 0 in Eq. (1.19), a 1-dimensional quadratic discrete system is xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk :

ð1:61Þ

(i) For AðpÞ  BðpÞ\0, the corresponding standard form is xk þ 1 ¼ xk þ a0 xk ðxk  aÞ

ð1:62Þ

with two fixed-points of xk ¼ a1 ¼ 0 and xk ¼ a2 ¼ a [ 0 with a0 ¼ AðpÞ and a ¼ 

BðpÞ : AðpÞ

ð1:63Þ

(ii) For AðpÞ  BðpÞ [ 0, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞxk

ð1:64Þ

xk ¼ a1 ¼ a\0 and xk ¼ a2 ¼ 0:

ð1:65Þ

with two fixed-points of

(iii) For BðpÞ ¼ 0, the corresponding standard form is xk þ 1 ¼ xk þ a0 x2k

ð1:66Þ

with a double fixed-point of xk ¼ 0. Such a fixed-point is called a monotonic saddle of the second order. (iii1) If a0 [ 0, the fixed-point is a monotonic upper-saddle of the second order.

1.2 Forward Quadratic Discrete Systems

25

(iii2) If a0 \0, the fixed-point is a monotonic lower-saddle of the second order. (iv) The bifurcation of x ¼ 0 for two fixed-points switching is called a monotonic saddle-node switching bifurcation at a point p ¼ p0 2 @X12 , and the bifurcation condition is Bðp0 Þ ¼ 0:

ð1:67Þ

AðpÞ  BðpÞ\0;

ð1:68Þ

Theorem 1.6 (i) Under a condition of

a standard form of the 1-dimensional discrete system in Eq. (1.61) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 xk ðxk  aÞ:

ð1:69Þ

(i1) For a0 ðpÞ [ 0 there are two cases: (i1a) The fixed-point of xk ¼ 0 is: monotonically stable (a monotonic sink) if df =dxk jxk ¼0 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼0 ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼0 2 ð2; 1Þ; flipped if df =dxk jxk ¼0 ¼ 2 (an oscillatory upper-saddle of the second-order with d 2 f =dx2k jxk ¼a ¼ a0 [ 0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼0 2 ð1; 2Þ:

• • • •

(i1b) The fixed-point of xk ¼ a is monotonically unstable with df =dxk jxk ¼a [ 0. (i2) For a0 ðpÞ\0, there are two cases: (i2a) The fixed-point of xk ¼ 0 is monotonically unstable with df =dxk jxk ¼0 [ 0. (i2b) The fixed-point of xk ¼ a is: • • • •

monotonically stable (a monotonic sink) if df =dxk jxk ¼a 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼a 2 ð2; 1Þ; flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory lower-saddle of the second kind with d 2 f =dx2k jxk ¼a ¼ a0 \0) • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a 2 ð1; 2Þ:

26

1 Quadratic Nonlinear Discrete Systems

(ii) Under a condition of AðpÞ  BðpÞ [ 0;

ð1:70Þ

a standard form of the 1-dimensional quadratic system in Eq. (1.61) is xk þ 1 ¼ xk þ a0 ðxk  aÞxk :

ð1:71Þ

(ii1) For a0 ðpÞ [ 0, there are two cases: (ii1a) The fixed-point of xk ¼ 0 is monotonically unstable with df =dxk jxk ¼0 [ 0. (ii1b) The fixed-point of xk ¼ a is: • • • •

monotonically stable (a monotonic sink) if df =dxk j xk ¼a 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼a ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk j xk ¼a 2 ð2; 1Þ; flipped if df =dxk jxk ¼a ¼ 2 (an oscillatory upper-saddle of the second-order with d 2 f =dx2k jxk ¼a ¼ a0 [ 0) • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼a 2 ð1; 2Þ: (ii2) For a0 ðpÞ\0, there are two cases: (ii2a) The fixed-point of xk ¼ 0 is: monotonically stable (a monotonic sink) if df =dxk jxk ¼0 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jxk ¼0 ¼ 1; oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼0 2 ð2; 1Þ; flipped if df =dxk jxk ¼0 ¼ 2 (an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼0 \0); • oscillatorilly unstable (an oscillatory source) if df =dxk jxk ¼0 2 ð1; 2Þ:

• • • •

(ii2b) The fixed-point of xk ¼ a is monotonically unstable if df =dxk jxk ¼a [ 0. (iii) For BðpÞ ¼ 0, the corresponding standard form with D ¼ 0 is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 x2k :

ð1:72Þ

(iii1) If a0 ðpÞ [ 0, then the fixed-point of xk ¼ 0 is a monotonic upper-saddle of the second-order with d 2 f =dx2k jxk ¼0 [ 0. The fixed-point of xk ¼ 0 for two fixed-point switching is a monotonic upper-saddle-node switching bifurcation of the second-order. (iii2) If a0 ðpÞ\0, then the fixed-point of xk ¼ 0 is a monotonic lower-saddle of the second-order with d 2 f =dx2k jxk ¼0 \0. The fixed-point of xk ¼ 0 for two fixed-points switching is a monotonic lower-saddle-node switching bifurcation of the second-order.

1.2 Forward Quadratic Discrete Systems

27

Proof The theorem can be proved as for Theorem 1.3.

■

The stability and bifurcation of two fixed-points for the 1-dimensional forward discrete system in Eq. (1.49) with D ¼ B2  4AC ¼ a20 ða  bÞ2 0 are presented in Fig. 1.5. The stable and unstable fixed-points varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation point of fixed-points occurs at the double-repeated fixed-point at the boundary of p0 2 @X12 . With varying parameters, the two fixed-points of xk ¼ a; b equal each other (i.e., xk ¼ a ¼ b). Such a fixed-point is a switching bifurcation point at xk ¼ a ¼ b for D ¼ 0. The fixed-points of xk ¼ a; b with D 0 are presented in Fig. 1.5i and ii for a0 [ 0 and a0 \0, respectively. The quadratic discrete system in Eq. (1.61) is as a special case of the discrete system in Eq. (1.19) with CðpÞ ¼ 0. Thus D ¼ B2  4AC ¼ B2 0. The fixed-points exist in the entire domain. In Fig. 1.4iii, for a0 [ 0 and B\0, the fixed-point of xk ¼ 0 is unstable, and the fixed-point of xk ¼ a is from monotonically stable to oscillatorilly unstable. However, for a0 [ 0 and B [ 0, the fixed-point of xk ¼ a is stable, and the fixed-point of xk ¼ 0 is from monotonically stable to oscillatorilly unstable. The

Δ = a02 (a − b) 2

mUS

mSO

∗ k

x =b P-2

xk∗

|| p 0 ||

oSI

mSO

mSI

mSI

oSI

oSI

Δ=0

oSO

mSO

Δ>0 || p ||

a0 B>0

xk∗ = a

mUS

mSO

xk∗ = b oSO P-2

xk∗ = a

Δ>0 a>b

(ii)

|| p 0 ||

xk∗ = 0

oSI

mSO

xk∗

(i) C =0

mSI mLS

Δ>0 a>b

a=b

mSI

P-2

P-2

Δ>0 a0 B0 B>0

Δ=0 B=0

oSO

Δ>0 B0

Δ=0

Δ 0

Fig. 1.7 Stability and bifurcation of a double fixed-point of the second order in the quadratic backward discrete system. Unstable fixed-points is represented by a dashed curve. The stability switching from the lower-saddle to upper-saddle is labelled by a circular symbol. (mUS: monotonic upper-saddle; mLS: monotonic lower-saddle.)

xk is always toward the positive direction due to xk þ 1  xk ¼ a0 ½ðxk þ 1 þ 2AB Þ2 þ ð4AD2 Þ [ 0: The corresponding phase portrait is presented in Fig. 1.6iv. To illustrate the stability and bifurcation of fixed-points with singularity in a 1-dimensional, backward system, the fixed-point of xk þ 1  xk ¼ a0 ðxk þ 1  a1 Þ2 is presented in Fig. 1.7. The fixed-point of xk þ 1 ¼ a1 for a0 \0 and a0 [ 0 is the monotonic upper-saddle and lower-saddle of the second-order, respectively. The monotonic upper-saddle and lower-saddle fixed-points of xk þ 1 ¼ a1 with the second-order are unstable, which are depicted by dashed curves. At a0 ¼ 0, the monotonic upper-saddle and lower-saddle fixed-points will be switched, which is marked by a circular symbol.

1.3.2

Backward Period-1 Switching Bifurcations

Definition 1.8 Consider a 1-dimensional, backward discrete system in Eq. (1.73) as xk ¼ xk þ 1 þ AðpÞx2k þ 1 þ BðpÞxk þ 1 þ CðpÞ ¼ xk þ 1 þ a0 ðpÞðxk þ 1  aðpÞÞðxk þ 1  bðpÞÞ:

ð1:91Þ

(i) For a\b, the corresponding standard form is xk ¼ xk þ 1 þ a0 ðxk þ 1  aÞðxk þ 1  bÞ with two fixed-points

ð1:92Þ

40

1 Quadratic Nonlinear Discrete Systems

xk ¼ xk þ 1 ¼ a1 ¼ a and xk ¼ xk þ 1 ¼ a2 ¼ b with D ¼ a20 ða  bÞ2 [ 0:

ð1:93Þ

(ii) For a [ b, the corresponding standard form is xk ¼ xk þ 1 þ a0 ðxk þ 1  bÞðxk þ 1  aÞ

ð1:94Þ

with two fixed-points of xk ¼ xk þ 1 ¼ a1 ¼ b and xk ¼ xk þ 1 ¼ a2 ¼ a with D ¼ a20 ða  bÞ2 [ 0:

ð1:95Þ

(iii) For a ¼ b, the corresponding standard form is xk ¼ xk þ 1 þ a0 ðxk þ 1  aÞ2

ð1:96Þ

with a repeated fixed-point of xk ¼ xk þ 1 ¼ a. Such a fixed-point is called a monotonic saddle of the second order. (iii1) If a0 [ 0, the fixed-point is a monotonic lower-saddle of the second order. (iii2) If a0 \0, the fixed-point is a monotonic upper-saddle of the second order. (iv) The fixed-point of xk ¼ xk þ 1 ¼ a for two fixed-points switching is called a saddle-node switching bifurcation point of fixed-point at a point p ¼ p0 2 @X12 , and the bifurcation condition is D ¼ a20 ða  bÞ2 ¼ 0 or a ¼ b:

ð1:97Þ

a\b and D ¼ a20 ða  bÞ2 [ 0

ð1:98Þ

Theorem 1.8 (i) Under a condition of

a standard form of the 1-dimensional, backward discrete system in Eq. (1.91) is xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ a0 ðxk þ 1  aÞðxk þ 1  bÞ:

ð1:99Þ

1.3 Backward Quadratic Discrete Systems

41

(i1) For a0 ðpÞ [ 0, there are two cases: (i1a) The fixed-point of xk ¼ xk þ 1 ¼ a is: • monotonically unstable (a monotonic source) if df =dxk þ 1 jxk þ 1 ¼a 2 ð1; 0Þ; • infinitely unstable if df =dxk þ 1 jxk þ 1 ¼a ¼ 1, which is – monotonically unstable with a positive infinity eigenvalue with df =dxk þ 1 jxk þ 1 ¼a ¼ 1 þ ; – oscillatorilly unstable with a negative infinity eigenvalue with df =dxk þ 1 jxk þ 1 ¼a ¼ 1 ; • oscillatorilly unstable (an oscillatory source) if df =dxk þ 1 jxk þ 1 ¼a 2 ð2; 1Þ; • flipped if df =dxk þ 1 jxk þ 1 ¼a ¼ 2 with an oscillatory lower-saddle of the second-order for d 2 f =dx2k þ 1 jxk þ 1 ¼a ¼ a0 [ 0; • oscillatorilly stable (an oscillatory sink) if df =dxk þ 1 jxk þ 1 ¼a 2 ð1; 2Þ: (i1b) The fixed-point of xk ¼ xk þ 1 ¼ b is monotonically stable (a monotonic sink) if df =dxk þ 1 jxk þ 1 ¼b 2 ð0; 1Þ. (i2) For a0 ðpÞ\0, there are two cases: (i2a) The fixed-point of xk ¼ xk þ 1 ¼ a is monotonically stable (a monotonic sink) if df =dxk þ 1 jxk þ 1 ¼a 2 ð0; 1Þ. (i2b) The fixed-point of xk ¼ xk þ 1 ¼ b is: • monotonically unstable (a monotonic source) if df =dxk þ 1 jxk þ 1 ¼b 2 ð1; 0Þ; • infinitely unstable if df =dxk þ 1 jxk þ 1 ¼b ¼ 1; which is – monotonically unstable with a positive infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼b ¼ 1 þ ; – oscillatorilly unstable with a negative infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼b ¼ 1 ; • oscillatorilly unstable (an oscillatory source) if df =dxk þ 1 jxk þ 1 ¼b 2 ð2; 1Þ; • flipped if df =dxk þ 1 jxk þ 1 ¼b ¼ 2 with an oscillatory upper-saddle of the second-order with d 2 f =dx2k þ 1 j xk þ 1 ¼b ¼ a0 \0; • oscillatorilly stable (an oscillatory sink) if df =dxk þ 1 jxk þ 1 ¼b 2 ð1; 2Þ:

42

1 Quadratic Nonlinear Discrete Systems

(ii) Under a condition of a [ b and D ¼ a20 ða  bÞ2 [ 0

ð1:100Þ

a standard form of the 1-dimensional, backward discrete system in Eq. (1.91) is xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ a0 ðxk þ 1  bÞðxk þ 1  aÞ

ð1:101Þ

(ii1) For a0 ðpÞ [ 0, there are two cases: (ii1a) The fixed-point of xk ¼ xk þ 1 ¼ a is monotonically stable (a monotonic sink) if df =dxk þ 1 jxk þ 1 ¼a [ 0. (ii1b) The fixed-point of xk ¼ xk þ 1 ¼ b is: • monotonically unstable (a monotonic source) if df =dxk þ 1 jxk þ 1 ¼b 2 ð1; 0Þ; • infinitely unstable if df =dxk þ 1 jxk þ 1 ¼b ¼ 1; which is: – monotonically unstable with a positive infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼b ¼ 1 þ ; – oscillatorilly unstable with a positive infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼b ¼ 1 ; • oscillatorilly unstable (an oscillatory source) if df =dxk þ 1 jxk þ 1 ¼b 2 ð2; 1Þ; • flipped if df =dxk þ 1 jxk þ 1 ¼b ¼ 2 with an oscillatory lowersaddle of the second-order with d 2 f =dx2k þ 1 jxk þ 1 ¼b ¼ a0 [ 0; • oscillatorilly stable (an oscillatory sink) if df =dxk þ 1 jxk þ 1 ¼b 2 ð1; 2Þ: (ii2) For a0 ðpÞ\0, there are two cases: (ii2a) The fixed-point of xk ¼ xk þ 1 ¼ a is: • monotonically unstable (a monotonic source) if df =dxk þ 1 jxk þ 1‘ ¼a 2 ð1; 0Þ; • infinitely unstable if df =dxk þ 1 jxk þ 1 ¼a ¼ 1; – monotonically unstable with a positive infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼a ¼ 1 þ ; – oscillatorilly unstable with a negative infinity eigenvalue if df =dxk þ 1 jxk þ 1 ¼a ¼ 1 ;

1.3 Backward Quadratic Discrete Systems

43

• oscillatorilly stable (an oscillatory sink) if df =dxk þ 1 jxk þ 1 ¼a 2 ð2; 1Þ; • flipped if df =dxk þ 1 jxk þ 1 ¼a ¼ 2 with an oscillatory uppersaddle of the second-order for d 2 f =dx2k þ 1 j xk þ 1 ¼b ¼ a0 \0; • oscillatorilly stable with df =dxk þ 1 jxk þ 1 ¼a 2 ð1; 2Þ: (ii2b) The fixed-point of xk ¼ xk þ 1 ¼ b is monotonically stable (a monotonic sink) if df =dxk þ 1 jxk þ 1 ¼b [ 0. (iii) For a ¼ b; the corresponding standard form with D ¼ 0 is xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ a0 ðxk þ 1  aÞ2 :

ð1:102Þ

(iii1) If a0 ðpÞ [ 0, then the fixed-point of xk ¼ xk þ 1 ¼ a is an monotonic lower-saddle of the second-order with d 2 f =dx2k þ 1 jxk þ 1 ¼a [ 0. The fixed-point of xk ¼ xk þ 1 ¼ a for the switching of two fixed-points is a monotonic lower-saddle-node switching bifurcation of the second-order. (iii2) If a0 ðpÞ\0, then the fixed-point of xk ¼ xk þ 1 ¼ a is a monotonic uppersaddle of the second order with d 2 f =dx2k þ 1 jxk þ 1 ¼a \0. The fixed-point of xk ¼ a for the switching of two fixed-points is a monotonic upper-saddle-node switching bifurcation of the second-order. Proof The theorem can be proved as for Theorem 1.7.

■

The stability and bifurcation of two fixed-points for the 1-dimensional backward discrete system in Eq. (1.91) with D ¼ B2  4AC ¼ a20 ða  bÞ2 0 are presented in Fig. 1.8. The stable and unstable fixed-points varying with the vector parameter are depicted by solid and dashed curves, respectively. The bifurcation point of fixed-points occurs at the repeated fixed-point at the boundary of p0 2 @X12 . With Δ = a02 (a − b) 2 mSI

∗ k

x =b P-2

xk∗

|| p 0 ||

oSO

xk∗ = a

mSO

mSO

oSO oSI

oSO

oSI

a0 || p ||

a0 a>b

mSO

P-2

P-2

Δ=0

|| p 0 ||

Δ = a02 (a − b) 2

mLS

Δ>0 || p ||

mSI

Δ=0 a=b

Δ>0 a>b

(ii)

Fig. 1.8 Stability and bifurcation of two fixed-points in the quadratic backward discrete system: (i) a monotonic lower-saddle-node switching bifurcation ða0 [ 0Þ, (ii) a monotonic upper-saddle-node switching bifurcation ða0 \0Þ. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; iSI: invariant sink.)

44

1 Quadratic Nonlinear Discrete Systems

varying parameters, the two fixed-points of xk þ 1 ¼ a; b equal each other (i.e., xk ¼ a ¼ b). Such a fixed-point is a bifurcation point at xk þ 1 ¼ a ¼ b for D ¼ 0. The fixed-points of xk ¼ a; b with D 0 are presented in Fig. 1.8i and ii for a0 [ 0 and a0 \0, respectively. In Fig. 1.8i the switching bifurcation is a monotonic lower-saddle bifurcation. In Fig. 1.8ii the switching bifurcation is a monotonic upper-saddle bifurcation. The stable fixed-point is a monotonic sink, but the unstable fixed point is from a monotonic source, monotonic source with a positive infinity eigenvalue to oscillatory source with a negative infinity eigenvalue, flipped invariance with the oscillatory lower- or upper-saddle to the oscillatory sink. The period-2 fixed-point are unstable, which are generated through the oscillatory lower- or upper-saddle bifurcations.

1.4

Forward Bifurcation Trees

In this section, the analytical bifurcation scenario will be discussed. The period-doubling bifurcation scenario will be discussed first through nonlinear renormalization techniques, and the bifurcation scenario based on the saddle-node bifurcation will be discussed, which is independent of period-1 fixed-points.

1.4.1

Period-2 Appearing Bifurcations

After the period-doubling bifurcation of a period-1 fixed-point, the period-doubled fixed-points can be obtained. Consider the period-doubling solutions for a forward discrete quadratic nonlinear system. Theorem 1.9 Consider a 1-dimensional quadratic nonlinear discrete system xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ

ð1:103Þ

where three scalar constants AðpÞ 6¼ 0, BðpÞ and CðpÞ are determined by a vector parameter p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:104Þ

D ¼ B2  4AC [ 0;

ð1:105Þ

Under a condition of

there is a standard form of the 1-dimensional discrete system in Eq. (1.103) as

1.4 Forward Bifurcation Trees

45

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ:

ð1:106Þ

where a0 ¼ AðpÞ; a1;2 ¼

pffiffiffiffi BðpÞ  D with a1 \a2 ; 2AðpÞ

ð1:107Þ

If the fixed-point of xk ¼ xk þ 1 ¼ ai ðpÞ with df =dxk jxk ¼ai ¼ 2 is period-doubled, there is a form for period-2 fixed-points as xk þ 2 ¼ xk þ a0 ðxk  a1 Þðxk  a2 ÞðA1 x2k þ B1 xk þ C1 Þ

ð1:108Þ

A1 ¼ a20 ; B1 ¼ a0 ½2  a0 ða2 þ a1 Þ; C1 ¼ 2  a0 ða2 þ a1 Þ þ a20 a1 a2 :

ð1:109Þ

where

(i) For a0 ðpÞ [ 0, there are two cases: (i1) Under D1 ¼ B21  4A1 C1 [ 0;

ð1:110Þ

there is a standard form as xk þ 2 ¼ xk þ a30 ðxk  a1 Þðxk  a2 Þðxk  b1 Þðxk  b2 Þ 1

ð2Þ

¼ xk þ a30 *22 i¼1 ðxk  ai Þ

ð1:111Þ

where ð2Þ

fai ; i ¼ 1; 2; . . .; 4g ¼ sort fa1 ; a2 ; b1 ; b2 g; 1 1 pffiffiffiffiffiffi b1;2 ¼  2 B1  2 D1 2a0 2a0 1 1 pffiffiffiffiffiffi ¼ ½2  a0 ða1 þ a2 Þ  2 D1 ; 2a0 2a0

ð1:112Þ

D1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ: (i2) Under an oscillatory upper-saddle-node bifurcation of dxk þ 1 jx ¼a ¼ 1 þ a0 ða1  a2 Þ ¼ 1 dxk k 1 ) 2 þ a0 ða1  a2 Þ ¼ 0; d xk þ 1 jxk ¼a1 ¼ a0 [ 0; dx2k 2

ð1:113Þ

46

1 Quadratic Nonlinear Discrete Systems

the second quadratics of the period-2 fixed points has D1 ¼ B21  4A1 C1 b1;2

¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ ¼ 0; 1 ¼ ½2  a0 ða1 þ a2 Þ 2a0 1 ¼ ½2 þ a0 ða1  a2 Þ þ a1 ¼ a1 2a0

ð1:114Þ

and the corresponding standard form becomes xk þ 2 ¼ xk þ a30 ðxk  a1 Þ3 ðxk  a2 Þ:

ð1:115Þ

(ii) For a0 ðpÞ\0, there are two cases: (ii1) Under a condition of D1 ¼ B21  4A1 C1 [ 0;

ð1:116Þ

there is a standard form as xk þ 2 ¼ xk þ a30 ðxk  a1 Þðxk  a2 Þðxk  b1 Þðxk  b2 Þ 2

ð2Þ

¼ xk þ a30 *2i¼1 ðxk  ai Þ

ð1:117Þ

where ð2Þ

fai ; i ¼ 1; 2; . . .; 4g ¼ sort fa1 ; a2 ; b1 ; b2 g; 1 1 pffiffiffiffiffiffi b1;2 ¼  2 B1  2 D1 2a0 2a0 1 1 pffiffiffiffiffiffi ¼ ½2  a0 ða2 þ a1 Þ  2 D1 ; 2a0 2a0

ð1:118Þ

D1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ: (ii2) Under an oscillatory lower-saddle-node bifurcation of dxk þ 1 jx ¼a ¼ 1 þ a0 ða2  a1 Þ ¼ 1 dxk k 2 ) 2 þ a0 ða2  a1 Þ ¼ 0; d 2 xk þ 1 jxk ¼a1 ¼ a0 \0; dx2k

ð1:119Þ

1.4 Forward Bifurcation Trees

47

the second quadratics of the period-2 fixed-points has D1 ¼ B21  4A1 C1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ ¼ 0; 1 b1;2 ¼  ½2  a0 ða1 þ a2 Þ 2a0 1 ¼ ½2 þ a0 ða2  a1 Þ þ a2 ¼ a2 2a0

ð1:120Þ

and the corresponding standard form becomes xk þ 2 ¼ xk þ a30 ðxk  a1 Þðxk  a2 Þ3 :

ð1:121Þ

Proof The proof is straightforward through the simple algebraic manipulation. Consider Ax2k þ Bxk þ C ¼ 0: D ¼ B2  4AC 0; we have a0 ¼ AðpÞ; a1;2

pffiffiffiffi BðpÞ  D with a1 \a2 : ¼ 2AðpÞ

Thus, we have Ax2k þ Bxk þ C ¼ ðxk  a1 Þðxk  a2 Þ: Therefore, xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ: For xk þ 1 ¼ xk ¼ ai (i 2 f1; 2g, if dxk þ 1 d 2 xk þ 1 jx ¼ai ¼ 1 þ a0 ðai  aj Þ ¼ 1; j  ¼ a0 6¼ 0; dxk k dx2k xk ¼ai then period-2 fixed-points exists for the quadratic discrete system. Thus, consider the corresponding second iteration gives ð1Þ

xk þ 2 ¼ xk þ 1 þ a0 *2i1 ¼1 ðxk þ 1  ai Þ: The period-2 discrete system of the quadratic discrete system is

48

1 Quadratic Nonlinear Discrete Systems ð1Þ

xk þ 2 ¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i3 ¼1;i2 6¼i2 ðxk 2

2

ð1Þ

 ai2 Þg

ð1Þ

¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þ½A1 x2k þ B1 xk þ C1 Þ where A1 ¼ a20 ; B1 ¼ a0 ½2  a0 ða2 þ a1 Þ; C1 ¼ 2  a0 ða2 þ a1 Þ þ a20 a1 a2 : If A1 x2k þ B1 xk þ C1 ¼ 0; we have D1 ¼ B21  4A1 C1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ 0; 1 1 pffiffiffiffiffiffi b1;2 ¼  ½2  a0 ða1 þ a2 Þ  2 D1 : 2a0 2a0 Thus A1 x2k þ B1 xk þ C1 ¼ a20 *2j2 ¼1 ðxk  bj2 Þ; and xk þ 2 ¼ xk þ a30 ðxk  a1 Þðxk  a2 Þðxk  b1 Þðxk  b2 Þ 2

ð2Þ

¼ xk þ a30 *2i¼1 ðxk  ai Þ where ð2Þ

fai ; i ¼ 1; 2; . . .; 4g ¼ sortfa1 ; a2 ; b1 ; b2 g: For the period-1 discrete systems, xk þ 1 ¼ xk þ a0 *2i¼1 ðxk  ai Þ: (I) For a0 [ 0, the fixed-point of xk ¼ a2 is monotonically unstable due to dxk þ 1 jx ¼a ¼ 1 þ a0 ða2  a1 Þ 2 ð1; 1Þ; dxk k 2 and the fixed-point of xk ¼ a1 is from monotonically stable to oscillatorilly unstable due to dxk þ 1 jx ¼a ¼ 1 þ a0 ða1  a2 Þ 2 ð1; 1Þ: dxk k 1

1.4 Forward Bifurcation Trees

49

Under dxk þ 1 jx ¼a ¼ 1 þ a0 ða1  a2 Þ ¼ 1 dxk k 1 ) 2 þ a0 ða1  a2 Þ ¼ 0 or a0 ða1  a2 Þ ¼ 2; d 2 xk þ 1 jxk ¼a1 ¼ a0 [ 0; dx2k there is a flipped discrete system of the oscillatory upper-saddle of the second order. Thus, for the period-2 discrete system, D1 ¼ B21  4A1 C1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ ¼ 0; 1 1 b1;2 ¼  ½2  a0 ða1 þ a2 Þ ¼  ½2 þ a0 ða1  a2 Þ þ a1 ¼ a1 2a0 2a0 and the corresponding standard form of the period-2 discrete system becomes xk þ 2 ¼ xk þ a30 ðxk  a1 Þ3 ðxk  a2 Þ with dxk þ 2 jx ¼a ¼ 1 þ 3a30 ðxk  a1 Þ2 ðxk  a2 Þ þ a30 ðxk  a1 Þ3 jxk ¼a1 ¼ 1; dxk k 1 d 2 xk þ 2 jxk ¼a1 ¼ 6a30 ðxk  a1 Þðxk  a2 Þ þ 6a30 ðxk  a1 Þ2 jxk ¼a1 ¼ 0; dx2k d 3 xk þ 2 jxk ¼a1 ¼ 6a30 ðxk  a2 Þ þ 18a30 ðxk  a1 Þjxk ¼a1 ¼ 6a30 ða1  a2 Þ\0: dx3k Therefore, xk ¼ a1 for the period-2 discrete system is a monotonic sink of the third-order. (II) Similarly, for a0 ðpÞ\0, the fixed-point of xk ¼ a1 is monotonically unstable due to dxk þ 1 jx ¼a ¼ 1 þ a0 ða1  a2 Þ 2 ð1; 1Þ; dxk k 1 and the fixed-point of xk ¼ a2 is for monotonically stable to oscillatorilly unstable due to

50

1 Quadratic Nonlinear Discrete Systems

dxk þ 1 jx ¼a ¼ 1 þ a0 ða2  a1 Þ 2 ð1; 1Þ: dxk k 2 Under dxk þ 1 jx ¼a ¼ 1 þ a0 ða2  a1 Þ ¼ 1 dxk k 2 ) 2 þ a0 ða2  a1 Þ ¼ 0 or a0 ða2  a1 Þ ¼ 2; d 2 xk þ 1 jxk ¼a2 ¼ a0 \0; dx2k there is a flipped discrete system of the oscillatory lower-saddle of the second order. Thus, for the period-2 discrete system, D1 ¼ B21  4A1 C1 ¼ a20 ½2 þ a0 ða1  a2 Þ½2 þ a0 ða2  a1 Þ ¼ 0; 1 1 b1;2 ¼  ½2  a0 ða1 þ a2 Þ ¼  ½2 þ a0 ða2  a1 Þ þ a2 ¼ a2 2a0 2a0 and the corresponding standard form of the period-2 discrete system becomes xk þ 2 ¼ xk þ a30 ðxk  a1 Þðxk  a2 Þ3 with dxk þ 2 jx ¼a ¼ 1 þ 3a30 ðxk  a1 Þðxk  a2 Þ2 þ a30 ðxk  a2 Þ3 jxk ¼a2 ¼ 1; dxk k 2 d 2 xk þ 2 jxk ¼a2 ¼ 6a30 ðxk  a1 Þðxk  a2 Þ þ 6a30 ðxk  a2 Þ2 jxk ¼a2 ¼ 0; dx2k d 3 xk þ 2 jxk ¼a2 ¼ 6a30 ðxk  a1 Þ þ 18a30 ðxk  a2 Þjxk ¼a2 ¼ 6a30 ða2  a1 Þ\0: dx3k Thus, xk ¼ a2 for the period-2 discrete system is a monotonic sink of the third-order. ■

This theorem is proved. Based on the standardization, the above theorem can be stated as follows.

Theorem 1.10 Consider a 1-dimensional quadratic nonlinear discrete system as xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ

ð1:122Þ

where three scalar constants AðpÞ 6¼ 0, BðpÞ and CðpÞ are determined by a vector parameter

1.4 Forward Bifurcation Trees

51

p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:123Þ

D ¼ B2  4AC [ 0;

ð1:124Þ

Under a condition of

there is a standard form as ð1Þ

ð1Þ

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðx2k þ B1 xk þ C1 Þ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ ¼

xk þ a0 *2i¼1 ðxk



ð1:125Þ

ð1Þ ai Þ

where B ð1Þ C ð1Þ a0 ¼ AðpÞ; B1 ¼ ; C1 ¼ ; A A 1 ð1Þ pffiffiffiffiffiffiffiffi 1 ð1Þ pffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ b1 ¼  ðB1 þ D Þ; b2 ¼  ðB1  Dð1Þ Þ; 2 2 ð1Þ 2 ð1Þ ð1Þ D ¼ ðB1 Þ  4C1 0; ð1Þ

ð1Þ

ð1Þ

ð1:126Þ

ð1Þ

02i¼1 fai g ¼ sortf02i¼1 fbi gg; ai ai þ 1 for i ¼ 1; 2: (i) Consider a forward period-2 discrete system of Eq. (1.122) as ð1Þ

xk þ 2 ¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk 2

ð1Þ

2

ð2Þ

ð1Þ

 ai2 Þg

ð2Þ

¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þ½a20 ðx2k þ B1 xk þ C1 Þ ð1Þ

ð2Þ

2 ¼ xk þ ½a0 *2j1 ¼1 ðxk  ai1 Þ½a20 *j22 ¼1 ðxk  bj2 Þ 2

ð2Þ

¼ xk þ a10 þ 2 *4i¼1 ðxk  ai Þ ð1:127Þ where 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ b1;2 ¼  ðB1 þ Dð2Þ Þ; b2 ¼  ðB1  Dð2Þ Þ; 2 2 ð2Þ 2 ð2Þ ð2Þ D ¼ ðB1 Þ  4C1 0; with fixed-points

ð1:128Þ

52

1 Quadratic Nonlinear Discrete Systems ð2Þ

xk þ 2 ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 4Þ ð2Þ

ð1Þ

ð2Þ

04i¼1 fai g ¼ sortf02j1 ¼1 faj1 g; 02j2 ¼1 fbj2 gg ð2Þ

ð1:129Þ

ð2Þ

with ai \ai þ 1 : ð1Þ

(ii) For a fixed-point of xk þ 1 ¼ xk ¼ ai1 ði1 2 f1; 2gÞ, if dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1

ð1:130Þ

with • an oscillatory upper-saddle-node bifurcation ðd 2 xk þ 1 =dx2k jxk ¼a1 ¼ a0 [ 0Þ, • an oscillatory lower-saddle-node bifurcation ðd 2 xk þ 1 =dx2k jxk ¼a1 ¼ a0 \0Þ, then the following relations satisfy ð1Þ

1 ð2Þ 2

ð2Þ

ð2Þ

ð2Þ

ai1 ¼  Bi1 ; Di1 ¼ ðB1 Þ2  4C1 ¼ 0;

ð1:131Þ

and there is a period-2 discrete system of the quadratic discrete system in Eq. (1.122) as ð1Þ

ð2Þ

xk þ 2 ¼ xk þ a30 ðxk  ai1 Þ3 ðxk  ai2 Þ

ð1:132Þ

For i1 ; i2 2 f1; 2g; i1 6¼ i2 with dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i1 dx2k xk ¼ai1 d 3 xk þ 2 ð1Þ ð2Þ j  ð1Þ ¼ 6a30 ðai1  ai2 Þ ¼ 12a20 \0: dx3k xk ¼ai1

ð1:133Þ

ð1Þ

Thus, xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-order, and the corresponding bifurcations is a monotonic sink bifurcation for the period-2 discrete system. (ii1) The period-2 fixed-points are trivial and unstable if ð1Þ

xk þ 2 ¼ xk ¼ ai1 for i1 ¼ 1; 2:

ð1:134Þ

(ii2) The period-2 fixed-points are non-trivial and stable if ð2Þ

xk þ 2 ¼ xk ¼ bi1 for i1 ¼ 1; 2: Proof See the proof of theorem 1.9. This theorem is proved.

ð1:135Þ ■

1.4 Forward Bifurcation Trees

1.4.2

53

Period-Doubling Renormalization

The generalized cases of period-doublization of quadratic discrete systems are presented through the following theorem. The analytical period-doubling trees can be developed for quadratic discrete systems. Theorem 1.11 Consider a 1-dimensional quadratic nonlinear discrete system xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ

ð1:136Þ

ð1Þ

¼ xk þ a0 *2i¼1 ðxk  ai Þ:

(i) After l-times period-doubling bifurcations, a period-2l ðl ¼ 1; 2; . . .Þ discrete system for the quadratic discrete system in Eq. (1.136) is given through the nonlinear renormalization as ð2l1 Þ

22

xk þ 2l ¼ xk þ ½a0  f1 þ

2l1

ð2

2

l1

2l1

Þ

Þ

ð2l1 Þ

¼ xk þ ½a0

Þ

*j1 ¼1

22

l1

l1

ð2l Þ

Þ

22

l

l1

Þ

22

l1 1

l

*i¼1 ðxk

ð2l1 Þ

 ai 2

Þg

Þ ð2l Þ

ð2l Þ

ðx2k þ Bj2 xk þ Cj2 Þ

ð2l1 Þ

l

*i¼1 ðxk

2

 ai1

22 1 22 l1

*i2 ¼1;i2 6¼i1 ðxk

2l1 1

*i2 ¼1

ð2l1 Þ 1 þ 22

¼ xk þ ða0

2

*i1 ¼1; ðxk

ð2l1 Þ 22

 ½ða0

2

2l 1

Þ

Þ

2l1

 ai 1

2

2l1

l1

ð2

*i1 ¼1 ðxk

ð2l1 Þ 2

 ½ða0

¼ xk þ a0

ð2l1 Þ

 ai 1

*i1 ¼1 ½1 þ a0

ð2

¼ xk þ ½a0

l1

*i1 ¼1 ðxk

Þ ð2l Þ

ð1:137Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ ð2l Þ

 ai Þ

ð2l Þ

 ai Þ

with l dxk þ 2l ð2l Þ X22l ð2l Þ 22 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l d 2 xk þ 2 l ð2l Þ X22l X22l ð2l Þ 22 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk ð1:138Þ .. . l d r xk þ 2l X22l ð2l Þ X22l ð2l Þ 22 ¼ a0 i1 ¼1 . . . ir ¼1;ir 6¼i1 ;i2 ;...;ir1 *ir þ 1 ¼1;i3 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk l

for r 22 ; where

54

1 Quadratic Nonlinear Discrete Systems ð2l Þ

ð2Þ

ð2l1 Þ 1 þ 22

a0 ¼ ða0 Þ1 þ 2 ; a0 2l

¼ ða0 2l1

l

ð2 Þ

Þ

l1

; l ¼ 1; 2; 3; . . .;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

ð2l Þ

2

ai þ 1 ; 02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg; ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

2

ð2l Þ Di

¼

l1 Iqð21 Þ

2

ð2l Þ ðBi Þ2



ð2l Þ 4Ci

0 for i 2

l1 0Nq11¼1 Iqð21 Þ

00Nq ¼1 Iqð2 2 2

l1

2

Þ

;

¼ flðq1 1Þ2l1 þ 1 ; lðq1 1Þ2l1 þ 2 ; . . .; lq1 2l1 g f1; 2; . . .; M1 g0f∅g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  2l1 ;

ð1:139Þ

l

Iqð22 Þ ¼ flðq2 1Þ2l þ 1 ; lðq2 1Þ2l þ 2 ; . . .; lq2 2l g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ 22 1  22 1 ; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l; lN2 2l ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g with fixed-points ð2l Þ

l

xk þ 2l ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 22 Þ 2l

2l1

ð2l Þ

ð2l1 Þ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 l

ð2 Þ

ð2l Þ

ð2l Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð1:140Þ

l

ð2 Þ

with ai \ai þ 1 : ð2l1 Þ

(ii) For a fixed-point of xk þ 2l1 ¼ xk ¼ ai1

ð2l1 Þ

ði1 2 Iq

 f1; 2; . . .; 2ð2

l1

Þ

gÞ, if

dxk þ 2l1 ð2l1 Þ 22l1 ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 1 þ a0 Pi2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; x ¼a dxk i1 k d 2 xk þ 2l1 j  ð2l1 Þ [ 0 for the second order oscillatory upper saddle; xk ¼ai dx2k 1 d 2 xk þ 2l1 j  ð2l1 Þ \0 for the second order oscillatory lower saddle; xk ¼ai dx2k 1 ð1:141Þ then there is a period-2l fixed-point discrete system

1.4 Forward Bifurcation Trees ð2l Þ

x k þ 2 l ¼ x k þ a0

*

55

ð2l1 Þ i1 2Iq

ð2l1 Þ 3

ðxk  ai1

Þ

22

l

*j2 ¼1 ðxk

ð2l Þ

 aj2 Þð1dði1 ;j2 ÞÞ ð1:142Þ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð1:143Þ

with dxk þ 2l d 2 xk þ 2 l j  ð2l1 Þ ¼ 1; j ð2l1 Þ ¼ 0; dxk xk ¼ai1 dx2k xk ¼ai1 d 3 x k þ 2l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai2 Þ3 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k 2l

 *2j2 ¼1 ðað2 i1 ði1 2 Iqð2

l1

Þ

l1

Þ

ð1:144Þ

l

ð2 Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

; q 2 f1; 2; . . .; N1 gÞ ð2l1 Þ

Thus, xk þ 2l at xk ¼ ai1

is

• a monotonic sink of the third-order if d 3 xk þ 2l =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order if d 3 xk þ 2l =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0.

1

(ii1) The period-2l fixed-points are trivial if ð2l1 Þ

xk þ 2l ¼ xk ¼ ai1

l1

for i1 ¼ 1; 2; . . .; 22 ;

ð1:145Þ

(ii2) The period-2l fixed-points are non-trivial if ð2l Þ

ð2l Þ

xk þ 2l ¼ xk ¼ bi1 ;1 ; bi1 ;2

i1 2 f1; 2; . . .; M2 g0f∅g:

ð1:146Þ

Such a period-2l fixed-point is • monotonically unstable if dxk þ 2l =dxk j

• monotonically invariant if dxk þ 2l =dxk j – a – a

ð2l Þ

xk ¼ai

1 ð2l Þ

2 ð1; 1Þ; ¼ 1, which is

xk ¼ai 1 1  monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jxk [ 0; th 2l1 2l1 monotonic lower-saddle the ð2l1 Þ order for d xk þ 2l =dxk jxk \ 0; 1 þ1 monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dx2l jxk k

– a [ 0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 j xk \ 0;

56

1 Quadratic Nonlinear Discrete Systems

• monotonically unstable if dxk þ 2l =dxk j • invariantly zero-stable if dxk þ 2l =dxk j • oscillatorilly stable if dxk þ 2l =dxk j • flipped if dxk þ 2l =dxk j

ð2l1 Þ

xk ¼ai

ð2l Þ

xk ¼ai

1

ð2l Þ

xk ¼ai

¼ 0;

1

2 ð1; 0Þ;

ð2l Þ

xk ¼ai

2 ð0; 1Þ;

1

¼ 1, which is

1

– an oscillatory upper-saddle of the ð2l1 Þth order if d 2l1 xk þ 2l =dx2lk 1 jxk [ 0; – an oscillatory lower-saddle the ð2l1 Þth order with d 2l1 xk þ 2l =dx2lk 1 jxk \ 0; – an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk þ 2l =dx2lk 1 þ 1 jxk \ 0; – an oscillatory sink the ð2l1 þ 1Þth order with d 2l1 þ 1 xk þ 2l =dx2lk 1 þ 1 jxk [ 0; • oscillatorilly unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, this theorem can be proved. (I) For a quadratic system, if the period-1 fixed-points exists, there is a following expression. ð1Þ

xk þ 1 ¼ xk þ a0 *2i1 ¼1 ðxk  ai1 Þ ð1Þ

For xk þ 1 ¼ xk ¼ ai1 (i1 2 f1; 2g, if dxk þ 1 ð1Þ ð2Þ ð2Þ j  ð1Þ ¼ 1 þ a0 ðai1  aj2 Þ ¼ 1; dxk xk ¼ai1 d 2 xk þ 1 ð1Þ j  ð2Þ ¼ a0 ¼ a0 6¼ 0 dx2k xk ¼ai1 then period-2 fixed-points exists for the quadratic discrete system. Thus, consider the corresponding second iteration gives ð1Þ

xk þ 2 ¼ xk þ 1 þ a0 *2i1 ¼1 ðxk þ 1  ai1 Þ: The period-2 discrete system of the quadratic discrete system is ð1Þ

xk þ 2 ¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk 2

ð1Þ

2

ð2Þ

ð2Þ

¼ xk þ ½a0 *2i1 ¼1 ðxk  ai1 Þ½a20 ðx2k þ B1 xk þ C1 Þ: If ð2Þ

ð2Þ

x2k þ B1 xk þ C1 ¼ 0;

ð1Þ

 ai2 Þg

1.4 Forward Bifurcation Trees

57

we have 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ b1 ¼  ðB1 þ Dð2Þ Þ; b2 ¼  ðB1  Dð2Þ Þ 2 2 ð2Þ 2 ð2Þ ð2Þ D ¼ ðB1 Þ  4C1 0: Thus ð2Þ

ð2Þ

ð2Þ

x2k þ B1 xk þ C1 ¼ *2j2 ¼1 ðxk  bj2 Þ; and ð1Þ

ð2Þ

2 ðxk  bj2 Þ xk þ 2 ¼ xk þ ½a0 *2j1 ¼1 ðxk  ai1 Þ½a20 *j22 ¼1 ð2Þ

¼ x k þ a0

*i¼1 ðxk 4

2

ð2Þ

 ai Þ;

ð2Þ

ð2Þ

where a0 ¼ a10 þ 2 . For a fixed-point of xk þ 2 ¼ xk ¼ ai1 ði1 2 f1; 2; . . .; 4gÞ, if dxk þ 2 ð2Þ ð2Þ ð2Þ j  ð2Þ ¼ 1 þ a0 *4i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1 d 2 xk þ 2 ð2Þ X ð2Þ ð2Þ j  ð2Þ ¼ a0 4i2 ¼1;i2 6¼i1 *4i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; dx2k xk ¼ai1 then, a period-2 discrete system of a quadratic discrete system has a perioddoubling bifurcation. (II) Such a period-2 discrete system can be renormalized nonlinearly. For k ¼ k1 þ 2, the previous period-2 discrete system becomes ð2Þ

xk1 þ 2 þ 2 ¼ xk1 þ 2 þ a0

*i¼1 ðxk1 þ 2 4

ð2Þ

 ai Þ:

Because k1 is an index for iteration, it can be replaced by k. Thus, an equivalent form for the foregoing equations becomes ð21 Þ

xk þ 22 ¼ xk þ 21 þ a0 with

22

1

*i¼1 ðxk þ 21

ð21 Þ

 ai

Þ;

58

1 Quadratic Nonlinear Discrete Systems ð21 Þ

x k þ 2 1 ¼ x k þ a0

22

1

*i¼1 ðxk

ð21 Þ

 ai

Þ

xk þ 22 can be expressed as ð21 Þ

xk þ 22 ¼ xk þ a0

1

22

*i1 ¼1 ðxk

ð2Þ

21

ð21 Þ

 ai1 Þf1 þ 1

¼ xk þ ða0 Þ1 þ 2

22

*i1 ¼1 ðxk

1

ð21 Þ

22

*i1 ¼1 ½1 þ a0 a0 2

ð21 Þ

1

ð22 22 Þ=2

 ai1 Þ *i2 ¼1

1

 ai2 Þg

ð22 Þ

ð22 Þ

22

*i2 ¼1;i2 6¼i1 ðxk

ð21 Þ

½ðx2k þ Bi2 xk þ Ci2 Þ:

If ð22 Þ

ð22 Þ

x2k þ Bi2 xk þ Ci2 ð221 Þ

for i2 2 0Nq11¼1 Iq1 Iqð2

21

Þ

¼0

00Nq ¼1 Iqð2 Þ with 2

2 2

2

¼ flðq1Þ21 þ 1 ; lðq1Þ21 þ 2 gf1; 2; . . .; M1 g;

for q 2 f1; 2; . . .; N1 g; M1 ¼ N1  2211 ; N1 ¼ 1 Iqð2 Þ ¼ flðq1Þ22 þ 1 ; lðq1Þ22 þ 2 ; . . .; lq22 gfM1 þ 1; M1 þ 2; . . .; Mg; 2

2

1

for q 2 f1; 2; . . .; N2 g; M2 ¼ ð22  22 Þ=2; then we have ffi ffi 1 ð22 Þ pffiffiffiffiffiffiffiffiffi 1 ð22 Þ pffiffiffiffiffiffiffiffiffi 2 2 ð22 Þ ð22 Þ bi2 ;1 ¼  ðB1 þ Dð2 Þ Þ; bi2 ;2 ¼  ðB1  Dð2 Þ Þ 2 2 ð22 Þ

ð22 Þ

Dð2 Þ ¼ ðB1 Þ2  4C1 2

0:

Thus ð22 Þ

ð22 Þ

x2k þ Bi2 xk þ Ci2

ð22 Þ

ð22 Þ

¼ ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ;

and 21

ð21 Þ

xk þ 22 ¼ xk þ ða0 Þ1 þ 2

22

1

*j1 ¼1 ðxk

ð21 Þ

 aj1 Þ *Nq¼1 *j

ð22 Þ

ð22 Þ

 *j3 2J ð22 Þ ðxk  bj3 ;1 Þðxk  bj3 ;2 Þ ð4Þ

22

ð22 Þ

¼ xk þ ða0 Þ *2i¼1 ðxk  ai where

Þ

ð22 Þ 2 2Iq

ð22 Þ

ð22 Þ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ

1.4 Forward Bifurcation Trees 22

ð22 Þ

02i¼1 fai ð22 Þ

59 21

ð2Þ

g ¼ sortf02i1 ¼1 fai1 g; 0Nq¼1 0

ð22 Þ

ð22 Þ

ð22 Þ

ð22 Þ

i2 2Iq

ð22 Þ

ð22 Þ

fbi2 ;1 ; bi2 ;2 gg;

ð22 Þ

ai þ 1 ; Di2 ¼ ðBi2 Þ2  4Ci2 0; qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 1 ð22 Þ 1 ð22 Þ ð22 Þ ð22 Þ ð22 Þ ð22 Þ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ 2 2 21 2 i2 2 0Nq11¼1 Iqð21 Þ 00Nq22¼1 Iqð22 Þ ; ai

Iqð21

21

Þ

¼ flðq1 1Þ21 þ 1 ; lðq1 1Þ21 þ 2 gf1; 2; . . .; M1 g;

for q 2 f1; 2; . . .; N1 g; M1 ¼ N1  21 ; N1 ¼ 1; 2 Iqð22 Þ ¼ flðq2 1Þ22 þ 1 ; lðq2 1Þ22 þ 2 ; . . .; lq2 22 g fM1 þ 1; M1 þ 2; . . .; M2 g; for q2 2 f1; 2; . . .; N2 g; M2 ¼ 22 1  22 1 ; pffiffiffiffiffiffiffi ð22 Þ ð22 Þ ð22 Þ Di2 ¼ ðBi2 Þ2  4Ci2 \0; i ¼ 1; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð22 Þ 1 ð22 Þ ð22 Þ ð22 Þ ð22 Þ ð22 Þ bi2 ;1 ¼  ðBi2 þ i jDi2 jÞ; bi2 ;2 ¼  ðBi2  i jDi2 jÞ; 2 2 2 i2 2 J ð2 Þ ¼ flN2 22 þ 1 ; lN2 22 þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g: 2

1

Nontrivial period-22 fixed-points are ð4Þ

xk þ 22 ¼ xk ¼ ai

ð22 Þ

ð22 Þ

2 2 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 g;

2

i 2 f1; 2; . . .; 22 g0f∅g; and trivial period-22 fixed-points are ð4Þ

xk þ 22 ¼ xk ¼ ai

21

ð21 Þ

2 02i1 ¼1 fai1 g;

2

i 2 f1; 2; . . .; 22 g: Similarly, the period-2l discrete system ðl ¼ 1; 2; . . .Þ of the quadratic discrete system in Eq. (1.136) can be developed through the above nonlinear renormalization and the corresponding fixed-points can be obtained. (III) Consider a period-2l1 discrete system as ð2l1 Þ

xk þ 2l1 ¼ xk þ a0

22

l1

*i¼1

ð2l1 Þ

ðxk  ai

Þ:

From the nonlinear renormalizations, let k1 ¼ k þ 2l1 , we have

60

1 Quadratic Nonlinear Discrete Systems ð2l1 Þ

22

xk1 þ 2l1 þ 2l1 ¼ xk1 þ 2l1 þ a0

l1

*i¼1

ð2l1 Þ

ðxk1 þ 2l1  ai

Þ:

Because k1 is an index for iteration, it can be replaced by k. Thus, an equivalent form for the foregoing equations becomes ð2l1 Þ

xk þ 2l ¼ xk þ 2l1 þ a0

22

l1

*i¼1

ð2l1 Þ

ðxk þ 2l1  ai

Þ:

With the period-2l1 discrete system, the foregoing equations becomes ð2l1 Þ

22

xk þ 2l ¼ xk þ ½a0  f1 þ

ð2l1 Þ

2l1

 ai1

ð2

*i1 ¼1 ½1 þ a0 2

ð2

¼ xk þ ½a0

l1

*i1 ¼1 ðxk

l1

Þ

2

2l1

Þ

2l 1

2

*i2 ¼1

Þ

Þ

2l1

*i2 ¼1;i2 6¼i1 ðxk 2

ð2

*i1 ¼1 ðxk

ð2l1 Þ 2

 ½ða0

2l1

l1

 ai1

l1

2l1 1

2

Þ

ð2l1 Þ

 ai2

Þg

Þ ð2l Þ

ð2l Þ

ðx2k þ Bi2 xk þ Ci2 Þ:

If ð2l Þ

ð2l Þ

x2k þ Bi2 xk þ Ci2 ¼ 0 ð2l1 Þ

for i2 2 0Nq11¼1 Iq1 Iqð21

l1

Þ

00Nq ¼1 Iqð2 2 2

l1

Þ

2

with

¼ flðq1Þ2l1 þ 1 ; lðq1Þ2l1 þ 2 ; . . .; lq2l1 g f1; 2; . . .; M1 g0f∅g;

for q1 ¼ 1; 2; . . .; N1 ; M1 ¼ N1  2l1 ; l

Iqð22 Þ ¼ flðq1Þ2l þ 1 ; lðq1Þ2l þ 2 ; . . .; lq2l g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 ¼ 1; 2; . . .; N2 ; M2 ¼ 22 1  22

1

;

then we have 1 ð2l Þ ð2l Þ bi2 ;1 ¼  ðBi2 þ 2 ð2l Þ

ð2l Þ

qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ 2 ð2l Þ

Di2 ¼ ðBi2 Þ2  4Ci2 0: Thus

1.4 Forward Bifurcation Trees

61 ð2l Þ

ð2l Þ

ð2l Þ

ð2l Þ

x2k þ Bi2 xk þ Ci2 ¼ ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ: Therefore ð2l1 Þ

22

xk þ 2l ¼ xk þ ½a0 ð2

 ½ða0

l1

2l1

2l1

2l

ð2l Þ

¼ xk þ a0

Þ

22

l1

l

22

Þ ð2l Þ

2 2

*i2 ¼1

ð2l1 Þ 1 þ 22

¼ xk þ ða0

ð2l1 Þ

 ai1

Þ 2

Þ

l1

*i1 ¼1; ðxk

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ

l

ð2l Þ

*i¼1 ðxk

 ai Þ

ð2l Þ

*i¼1 ðxk

 ai Þ

where ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1 þ 2 ; a0 2l

ð2l1 Þ 1 þ 22

¼ ða0 2l1

l

ð2 Þ

Þ

l1

; l ¼ 1; 2; 3; . . .;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

ð2l Þ

2

ai þ 1 ; 02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ; 2 2

ð2l Þ

Di

Iqð21

l1

ð2l Þ

ð2l Þ

¼ ðBi Þ2  4Ci Þ

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

00Nq22¼1 Iqð22

l1

Þ

;

¼ flðq1 1Þ2l1 þ 1 ; lðq1 1Þ2l1 þ 2 ; . . .; lq1 2l1 g f1; 2; . . .; M1 g0f∅g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  2l1 ; l

Iqð22 Þ ¼ flðq2 1Þ2l þ 1 ; lðq2 1Þ2l þ 2 ; . . .; lq2 2l g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ 22 1  22 1 ; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ 2 ð2l Þ Di ¼ ðBi Þ  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l þ 1 ; lN2 2l þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g: For the period-2l discrete system, we have ð2l Þ

x k þ 2 l ¼ x k þ a0

22

l

*i¼1 ðxk

ð2l Þ

and the local stability and bifurcation at xk ¼ ai determined by

ð2l Þ

 ai Þ l

ði 2 f1; 2; . . .; 22 gÞ can be

62

1 Quadratic Nonlinear Discrete Systems l dxk þ 2l ð2l Þ X22l ð2l Þ 22 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l d 2 xk þ 2l ð2l Þ X22l X22l ð2l Þ 22 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . l d r x k þ 2l X22l ð2l Þ X22l ð2l Þ 22 ¼ a0 i1 ¼1 . . . ir ¼1;ir 6¼i1 ;i2 ;...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk l

for r 22 ; and the period-doubling bifurcations are determined by l dxk þ 2l ð2l Þ X22l ð2l Þ 22 j  ð2l Þ ¼ 1 þ a0 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þj  ð2l Þ ¼ 1; i ¼1 1 xk ¼ai dxk xk ¼ai l l X 2l X 2l d 2 xk þ 2 l ð2 Þ 2 ð2l Þ 2 22 j  ð2l Þ ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þjx ¼að2l Þ 6¼ 0: 2 x ¼a dxk i i k k

Non-trivial period-2l fixed-points are ð2l Þ

xk þ 22 ¼ xk ¼ ai

ð2l Þ

ð2l Þ

2 2 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 g;

l

i 2 f1; 2; . . .; 22 g; and trivial period-2l fixed-points are ð2l Þ

xk þ 22 ¼ xk ¼ ai

2l1

ð2l1 Þ

2 02i1 ¼1 fai1

g;

2l

i 2 f1; 2; . . .; 2 g: This theorem can be easily proved.

1.4.3

■

Period-n Appearing and Period-Doublization

The period-n discrete system for the quadratic nonlinear discrete systems will be discussed, and the period-doublization of the period-n quadratic discrete system is discussed through the nonlinear renormalization. Theorem 1.12 Consider a 1-dimensional quadratic nonlinear discrete system

1.4 Forward Bifurcation Trees

63

xk þ 1 ¼ xk þ AðpÞx2k þ BðpÞxk þ CðpÞ

ð1:147Þ

ð1Þ

¼ xk þ a0 *2i¼1 ðxk  ai Þ:

(i) After n-times iterations, a period-n discrete system for the quadratic discrete system in Eq. (1.147) is xk þ n ¼ xk þ a0 *2i1 ¼1 ðxk  ai2 Þf1 þ n

Xn

j¼1 Qj g ðnÞ

ðnÞ

¼ xk þ ða0 Þ2 1 *2i1 ¼1 ðxk  ai1 Þ½*j22 ¼11 ðx2k þ Bj2 xk þ Cj2 Þ ¼

n ðnÞ xk þ a0 *2i¼1 ðxk

n1



ð1:148Þ

ðnÞ ai Þ

with dxk þ n n ðnÞ X n ðnÞ ¼ 1 þ a0 2i1 ¼1 *2i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk d 2 xk þ n n ðnÞ X n X n ðnÞ ¼ a0 2i1 ¼1 2i2 ¼1;i2 6¼i1 *2i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; dx2k .. . d r xk þ n n X n ðnÞ X n ðnÞ ¼ a0 2i1 ¼1 . . . 2ir ¼1;ir 6¼i1 ;i2 ...;ir1 *2ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk for r 2n ;

ð1:149Þ where n

ðnÞ

ð1Þ

a0 ¼ ða0 Þ2 1 ; Q1 ¼ 0; Q2 ¼ *2i2 ¼1 ½1 þ a0 *2i1 ¼1;i1 6¼i2 ðxk  ai1 Þ; ð1Þ

Qn ¼ *2in ¼1 ½1 þ a0 ð1 þ Qn1 Þ *2in1 ¼1;in1 6¼in ðxk  ain1 Þ; n ¼ 3; 4; . . .; n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2 2 ðnÞ ðnÞ ðnÞ Di2 ¼ ðBi2 Þ2  4Ci2 0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þn þ 1 ; lðq1Þn þ 2 ; . . .; lqn gf1; 2; . . .; Mg0f∅g; for q 2 f1; 2; . . .; Ng; M ¼ 2n1  1; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flNn þ 1 ; lNn þ 2 ; . . .; lM g  f1; 2; . . .; Mg0f∅g; ð1:150Þ

64

1 Quadratic Nonlinear Discrete Systems

with fixed-points ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 2n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ðnÞ

ð1:151Þ

ðnÞ

with ai \ai þ 1 : ðnÞ

ðnÞ

(ii) For a fixed-point of xk þ n ¼ xk ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), if dxk þ n n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 1 þ a0 *2i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1

ð1:152Þ

then there is a new discrete system for onset of the qth -set of period-n fixedpoints based on the second-order monotonic saddle-node bifurcation as ðnÞ

xk þ n ¼ x k þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *2i2 ¼1 ðxk  ai2 Þð1dði1 ;j2 ÞÞ

ð1:153Þ

ðnÞ

ð1:154Þ

n

where ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 : (ii1) If dxk þ n j  ðnÞ ¼ 1 ði1 2 IqðnÞ Þ; dxk xk ¼ai1 d 2 xk þ n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 2a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ2 1 q 2 1 dx2k xk ¼ai1 ðnÞ

ð1:155Þ

ðnÞ

 *2i3 ¼1 ðai1  ai3 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk þ n at xk ¼ ai1 is • a monotonic lower-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ k

i1

k

i1

\0; • a monotonic upper-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0. (ii2) The period-n fixed-points ðn ¼ 2n1  mÞ are trivial if

1.4 Forward Bifurcation Trees

65 ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 2 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

gg

)

for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 2n g0f∅g for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 2 f02i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 2n g0f∅g

)

ð1:156Þ

for n ¼ 2n2 : (ii3) The period-n fixed-points ðn ¼ 2n1  mÞ are non-trivial if ) 2n1 1 m ðnÞ ð1Þ ð2n1 1 mÞ gg xk þ n ¼ xk ¼ aj1 62 f02ii ¼1 fai1 g; 02i2 ¼1 fai2 for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 2n g0f∅g

for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 62 f02i2 ¼1 fai2

gg

)

ð1:157Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 2n g0f∅g for n ¼ 2n2 :

Such a period-n fixed-point is • monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is i1

k

 1  – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k xk [ 0;  1  – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k xk \0;

1 þ1 – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l j xk k [ 0;  – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 xk \0;

• monotonically stable if dxk þ n =dxk jx ¼aðnÞ 2 ð0; 1Þ; i1

k

• invariantly zero-stable if dxk þ n =dxk jx ¼aðnÞ ¼ 0; k

i1

• oscillatorilly stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 0Þ; k

i1

• flipped if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is k

i1

 1  – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k xk [ 0;  1  – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k xk \0;

66

1 Quadratic Nonlinear Discrete Systems 1 þ1 – an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l jxk k \0;  – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l1 þ 1 x [ 0;

k

k

• oscillatorilly unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ. k

i1

ðnÞ

ðnÞ

(iii) For a fixed-point of xk þ n ¼ xk ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), there is a period-doubling of the qth -set of period-n fixed-points if dxk þ n n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 1 þ a0 *2j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1 dxk xk ¼ai1

ð1:158Þ

with • an oscillatory upper-saddle for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0; k

i1

• an oscillatory lower saddle for d 2 xk þ n =dx2k jx ¼aðnÞ \0. k

i1

The corresponding period-2  n discrete system of the quadratic discrete system in Eq. (1.147) is ð2nÞ

xk þ 2n ¼ xk þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

2n

ð2nÞ ð1dði1 ;i2 ÞÞ

 ai1 Þ3 *2i2 ¼1 ðxk  ai2

Þ

ð1:159Þ with dxk þ 2n d 2 xk þ 2n jx ¼aðnÞ ¼ 1; jx ¼aðnÞ ¼ 0; i1 i1 k k dxk dx2k d 3 xk þ 2n ð2nÞ ðnÞ ðnÞ 3 jx ¼aðnÞ ¼ 6a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 2 q 2 1 i1 k dx3k 2n

ðnÞ

ð2nÞ ð1dði1 ;i3 ÞÞ

 *2i3 ¼1 ðai1  ai3 ðnÞ

Þ

ð1:160Þ 6¼ 0:

ðnÞ

Thus, xk þ 2n at xk ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic sink of the third-order if d 3 xk þ 2n =dx3k jx ¼aðnÞ \0, k

i1

• a monotonic source of the third-order if d 3 xk þ 2n =dx3k jx ¼aðnÞ [ 0. k

i1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l  n discrete system of the quadratic discrete system in Eq. (1.147) is

1.4 Forward Bifurcation Trees

67

ð2l1 nÞ

22

xk þ 2l n ¼ xk þ ½a0

2l1 n

 f1 þ ¼

2

*i1 ¼1

ð2l1 nÞ

ðxk  ai1

ð2l1 nÞ 22 Þ ½ða0

l1 n

ð2

 ½ða0

nÞ 2

Þ

l1

ð2

¼ xk þ ða0

2

*j1 ¼1

l

Þ

2

2

*j2 ¼1

2l n

*i¼1

2

*i¼1

l1 n1

l1 n1

ð2 nÞ

ðxk  ai

ð2l nÞ

xk þ Cj2

ð2l nÞ

Þ

ðx2k þ Bj2

ðxk  bj2 l

ð2 nÞ

ðxk  ai l

Þg

ð2l nÞ

Þ

ð2l1 nÞ ai1 Þ

22

2l n

ð2l1 nÞ

 ai 2

ð2l1 nÞ ai1 Þ

22



2l n1

2l1 n

nÞ 2

ð2 nÞ

¼ x k þ a0

2l1 n



2l n1

ð2l1 nÞ 22l1 n xk þ ½a0 *i1 ¼1 ðxk l1

Þ

ð2l1 nÞ 22l1 n ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk

ð2l1 nÞ 22l1 n xk þ ½a0 *i1 ¼1 ðxk

 ¼

l1 n

*i1 ¼1

Þ

Þ ð1:161Þ

with dxk þ 2l n ð2l nÞ X22l n 22l n ð2l nÞ ¼ 1 þ a0 Þ; i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 dxk l d 2 xk þ 2l n ð2l nÞ X22l n X22l n ð2l nÞ 22 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 2 dxk .. . l d r xk þ 2l n X22l n ð2l nÞ X22l n ð2l nÞ 22 n ¼ a0 i1 ¼1 . . . ir ¼1;ir 6¼i1 ;i2 ...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk ð1:162Þ

l

for r 22 n , where ð2nÞ

a0

ð2l nÞ ð2l nÞ g 02i¼1 fai

ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l nÞ

Di

n

ðnÞ

ð2l nÞ

¼ ða0 Þ1 þ 2 ; a0 ¼

ð2l1 nÞ 1 þ 22

¼ ða0

l1 n

; l ¼ 1; 2; 3; . . .;

2l1 n ð2l1 nÞ ð2l nÞ ð2l nÞ 2 sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; 2 ð2l nÞ 2

¼ ðBi

Þ

ð2l nÞ

Þ  4Ci

0

68

1 Quadratic Nonlinear Discrete Systems

for i 2 0Nq11¼1 Iqð21 Iqð21

l1

nÞ

l1

nÞ

00Nq ¼1 Iqð2 nÞ 2 2

l

2

¼ flðq1 1Þð2l1 nÞ þ 1 ; lðq1 1Þð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f∅g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  ð2l1  nÞ; l

Iqð22 nÞ ¼ flðq2 1Þð2l nÞ þ 1 ; lðq2 1Þð2l nÞ þ 2 ; . . .; lq2 ð2l nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ ð22 n1  22 n1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi ð2l nÞ ð2l nÞ 2 ð2l nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di i 2 flN2 ð2l nÞ þ 1 ; lN2 ð2l nÞ þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g

ð1:163Þ

with fixed-points ð2l nÞ

xk þ 2l n ¼ xk ¼ ai 2l n

ð2l nÞ

02i¼1 fai l

ð2 nÞ

with ai

l

; ði ¼ 1; 2; . . .; 22 n Þ 2l1 n

g ¼ sortf02i¼1

ð2l1 nÞ

fai

ð2l nÞ

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð1:164Þ

l

ð2 nÞ

\ai þ 1 : ð2l1 nÞ

(v) For a fixed-point of xk þ ð2l nÞ ¼ xk ¼ ai1

ð2l1 nÞ

ði1 2 Iq

f1; 2; . . .; M1 gÞ,

l

there is a period-ð2  nÞ discrete system if dxk þ 2l1 n ð2l1 nÞ 22l1 n ð2l1 nÞ ð2l1 nÞ j  ð2l1 nÞ ¼ 1 þ a0  ai2 Þ ¼ 1 *i2 ¼1;i2 6¼i1 ðai1 xk ¼ai dxk 1 ð1:165Þ with  • an oscillatory upper-saddle for d 2 xk þ 2l1 n =dx2k   ð2l1 nÞ [ 0; x ¼a  k i1 2 2 • an oscillatory lower-saddle for d xk þ 2l1 n =dxk  ð2l1 nÞ \0. xk ¼ai

1

l

The corresponding period-ð2  nÞ discrete system is

1.4 Forward Bifurcation Trees

69 ð2l nÞ

xk þ 2l n ¼ xk þ a0

*

2l n

ð2l1 nÞ

i1 2Iq1

ð2l1 nÞ 3

ðxk  ai1

ð2l nÞ ð1dði1 ;j2 ÞÞ

 *2j2 ¼1 ðxk  aj2

Þ

Þ

ð1:166Þ

;

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai1

ð2l nÞ

ð2l1 nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

6¼ ai1

ð1:167Þ

with dxk þ 2l n d 2 xk þ 2l n j  ð2l1 nÞ ¼ 1; j  ð2l1 nÞ ¼ 0; xk ¼ai xk ¼ai dxk dx2k 1 1 d 3 xk þ 2l n ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j  ð2l1 Þ ¼ 6a0 ðai1  ai2 Þ * ð2l1 nÞ 3 xk ¼ai i2 2Iq1 ;i2 6¼i1 dxk 1 2l n

ð2l1 nÞ

 *2j2 ¼1 ðai1 ði1 2

l1 Iqð2 nÞ ; q

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

ð1:168Þ

6¼ 0

2 f1; 2; . . .; N1 gÞ: ð2l1 nÞ

Thus, xk þ 2l n at xk ¼ ai1

is

• a monotonic sink of the third-order if d 3 xk þ 2l n =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order if d 3 xk þ 2l n =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0.

1

(v1) The period-2l  n fixed-points are trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

ð1Þ

2l1 n

for n 6¼ 2n1 ; ð2l nÞ l

for j ¼ 1; 2; ; 22 n

9 gg = ;

l

for j ¼ 1; 2; ; 22 n xk þ 2l n ¼ xk ¼ aj

ð2l1 nÞ

2 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

2l1 n

ð2l1 nÞ

2 02i2 ¼1 fai2

9 g= ;

for n ¼ 2n1 : (v2) The period-2l  n fixed-points are non-trivial if

ð1:169Þ

70

1 Quadratic Nonlinear Discrete Systems ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

62 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

9 gg = ;

l

for j ¼ 1; 2; ; 2ð2 nÞ for n 6¼ 2n1 ; ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð2l1 nÞ

62 02i2 ¼1 fai2

9 g=

ð1:170Þ

;

l

for j ¼ 1; 2; ; 22 n for n ¼ 2n1 : Such a period-2l  n fixed-point is • monotonically unstable if dxk þ 2l n =dxk j

2 ð1; 1Þ;

ð2l nÞ

xk ¼ai

• monotonically invariant if dxk þ 2l n =dxk j

1 ð2l nÞ

xk ¼ai

¼ 1, which is

1

1  – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk [ 0 (independent ð2l1 Þ-branch appearance); 1  – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0 (independent ð2l1 Þ-branch appearance); 1 þ1 jxk – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l k [ 0 (dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jxk \0 (dependent ð2l1 þ 1Þ-branch appearance from one branch);

• monotonically stable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• invariantly zero-stable if dxk þ 2l n =dxk j • oscillatorilly stable if dxk þ 2l n =dxk j • flipped if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð0; 1Þ;

1

¼ 0;

ð2l nÞ

xk ¼ai ð2l nÞ

xk ¼ai

1

2 ð1; 0Þ;

1

¼ 1, which is

1

1  – an oscillatory upper-saddle of the ð2l1 Þth order if d 2l1 xk þ 2l n =dx2l k jxk [ 0; 1  – an oscillatory lower-saddle the ð2l1 Þth order with d 2l1 xk þ 2l n =dx2l k jxk \0;

1 þ1 – an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk þ 2l n =dx2l jxk k \0; 1 þ1 jxk – an oscillatory sink the ð2l1 þ 1Þth order with d 2l1 þ 1 xk þ 2l n =dx2l k [ 0;

• oscillatorilly unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

1

2 ð1; 1Þ.

1.4 Forward Bifurcation Trees

71

Proof Through the nonlinear renormalization, the proof of this theorem is similar to the proof of Theorem 1.11. This theorem can be easily proved. ■

1.4.4

Period-n Bifurcation Trees

Consider a period-n discrete system of the quadratic system as ðnÞ

x k þ n ¼ x k þ a0 ðnÞ

2n

*i¼1 ðxk

ðnÞ

 ai Þ

ð1:171Þ

n

where a0 ¼ ða0 Þ2 1 . For n ¼ 1, Eq. (1.171) gives a period-1 discrete system of the quadratic system as ð1Þ ð1Þ xk þ 1 ¼ xk þ a0 *2i¼1 ðxk  ai Þ: ð1:172Þ ð1Þ

• If ai ði ¼ 1; 2Þ are complex, none of fixed-points exists in such a quadratic discrete system. ð1Þ • If ai ði ¼ 1; 2Þ are real, two fixed-points exist in such a quadratic discrete system. For n ¼ 2, Eq. (1.171) gives a period-2 discrete system of the quadratic system as

ð2Þ

xk þ 2 ¼ x k þ a0

22

*i¼1 ðxk

ð2Þ

 ai Þ:

ð1:173Þ

ð2Þ

• If ai ði ¼ 1; 2; . . .; 22 Þ are complex, the period-2 discrete system does not have any fixed-points. ð2Þ • If two of ai ði ¼ 1; 2; . . .; 22 Þ are real, the period-2 discrete system possesses two fixed-points, which are trivial. The two fixed-points are the same as the period-1 fixed-points. ð2Þ • If all of ai ði ¼ 1; 2; . . .; 22 Þ are real, the period-2 discrete system possesses four fixed-points, including two trivial fixed-points for period-1 and two non-trivial fixed-points for period-2. Such two non-trivial fixed points are generated through period-doubling bifurcation, and both of fixed-points are stable for period-2. Thus, the period-2 discrete system has one set of period-2 fixed-points on the period-1 to period-2 period-doubling bifurcation tree. Without any independent period-2 fixed-points exists.

72

1 Quadratic Nonlinear Discrete Systems

For n ¼ 3, Eq. (1.171) gives a period-3 discrete system of the quadratic system as

ð3Þ

x k þ 3 ¼ x k þ a0

23

*i¼1 ðxk

ð3Þ

 ai Þ:

ð1:174Þ

ð3Þ

• If ai ði ¼ 1; 2; . . .; 23 Þ are complex, the period-3 discrete system does not have any fixed-points. ð3Þ • If two of ai ði ¼ 1; 2; . . .; 23 Þ are real, the period-3 discrete system possesses two trivial fixed-points which are the same as the period-1 fixed-points. ð3Þ • If all of ai ði ¼ 1; 2; . . .; 23 Þ are real, the period-3 discrete system possesses eight fixed-points, including two trivial fixed-points for period-1 and six non-trivial fixed-points for period-3. Such non-trivial fixed points are generated through the monotonic upper-saddle or monotonic lower-saddle bifurcations. The period-3 fixed-points are independent of the trivial fixed-points for period-1. Thus, the period-3 discrete system has at most one set of period-3 fixed-points, which is independent of the period-1 fixed-points. For n ¼ 4, Eq. (1.171) gives a period-4 discrete system of the quadratic system as 4 ð4Þ ð4Þ xk þ 4 ¼ xk þ a0 *2i¼1 ðxk  ai Þ: ð1:175Þ ð4Þ

• If ai ði ¼ 1; 2; . . .; 24 Þ are complex, the period-4 discrete system does not have any fixed-points. ð4Þ • If two of ai ði ¼ 1; 2; . . .; 24 Þ are real, the period-4 discrete system possesses two trivial fixed-points which are the same as the period-1 fixed-points. ð4Þ • If four of ai ði ¼ 1; 2; . . .; 24 Þ are real, the period-4 discrete system possesses four trivial fixed-points which are the same as the period-1 and period-2 fixed-points. ð4Þ • If eight of ai ði ¼ 1; 2; . . .; 24 Þ are real, the period-4 discrete system possesses eight fixed-points, including two trivial fixed-points for period-1, two trivial fixed-points for period-2, and four non-trivial fixed-points for period-4. Such non-trivial fixed points are stable, which are generated through the period-doubling bifurcations. All trivial fixed-points for period-4 are unstable. ð4Þ • If all of ai ði ¼ 1; 2; . . .; 24 Þ are real, in addition to the period-4 fixed-points by the period-doubling bifurcation, the period-4 discrete system possesses eight non-trivial fixed-points for period-4, which are generated by the monotonic upper-saddle or lower-saddle bifurcations. The period-4 fixed-points are independent of the trivial fixed-points.

1.4 Forward Bifurcation Trees

73

Thus, the period-4 discrete system has at most two sets of period-4 fixed-points, one is dependent on the period-1 to period-4 period-doubling tree, and one set of period-4 fixed-points is independent of the period-1 to period-4 period-doubling bifurcation tree. For n ¼ 5, Eq. (1.171) gives a period-5 discrete system of the quadratic system as ð5Þ

x k þ 5 ¼ x k þ a0

25

*i¼1 ðxk

ð5Þ

 ai Þ:

ð1:176Þ

ð5Þ

• If ai ði ¼ 1; 2; . . .; 25 Þ are complex, the period-5 discrete system does not have any fixed-points. ð5Þ • If two of ai ði ¼ 1; 2; . . .; 25 Þ are real, the period-5 discrete system possesses two trivial fixed-points which are the same as the period-1 fixed-points. ð5Þ • If twelve (12) of ai ði ¼ 1; 2; . . .; 25 Þ are real, the period-5 discrete system possesses 12 fixed-points, including two trivial fixed-points for period-1 and ten (10) non-trivial fixed-points for one set of period-5. Such non-trivial fixed points are generated through the monotonic upper-saddle or monotonic lower-saddle bifurcations. The period-5 fixed-points are independent of the trivial fixed-points for period-5. ð5Þ • If twenty-two (22) of ai ði ¼ 1; 2; . . .; 25 Þ are real, the period-5 discrete system possesses 20 fixed-points, including two (2) trivial fixed-points for period-1 and 20 non-trivial fixed-points for two sets of period-5. ð5Þ • If thirty-two (32) of ai ði ¼ 1; 2; . . .; 25 Þ are real, the period-5 discrete system possesses 32 fixed-points, including two (2) trivial fixed-points for period-1 and 30 non-trivial fixed-points for three sets of period-5. Thus, the period-5 discrete system has at most three (3) sets of period-5 fixed-points independent of the trivial period-5 fixed-points from the period-1 fixed-points. For n ¼ 6, Eq. (1.171) gives a period-6 discrete system of the quadratic system as ð6Þ

xk þ 6 ¼ x k þ a0 ð6Þ

26

*i¼1 ðxk

ð6Þ

 ai Þ:

ð1:177Þ

• If ai ði ¼ 1; 2; . . .; 26 Þ are complex, the period-6 discrete system does not have any fixed-points. ð6Þ • If two of ai ði ¼ 1; 2; . . .; 26 Þ are real, the period-6 discrete system possesses two trivial fixed-points which are the same as the period-1 fixed-points. ð6Þ • If twelve (14) of ai ði ¼ 1; 2; . . .; 26 Þ are real, the period-6 discrete system possesses 14 fixed-points, including two (2) trivial fixed-points for period-1, six (6) trivial fixed-points for period-3, and six (6) non-trivial fixed-points for

74

1 Quadratic Nonlinear Discrete Systems

period-6. Such non-trivial fixed points are generated through the period-3 period-doubling bifurcation. The six trivial fixed-points for period-3 are unstable. The six non-trivial fixed-points for period-6 are stable. ð6Þ • If twenty-two (26) of ai ði ¼ 1; 2; . . .; 26 Þ are real, the period-6 discrete system possesses 26 fixed-points, including two (2) trivial fixed-points for period-1, 12 non-trivial fixed-points on the period-doubling bifurcation trees of period-3, and 12 non-trivial fixed-points for period-6 caused by monotonic upper- and lower-saddle-nodes bifurcations. ð6Þ • If thirty-two (62) of ai ði ¼ 1; 2; . . .; 26 Þ are real, in addition to 14 fixed-points for period-1 and period-3 bifurcation tree, there are 4 sets of period-6 fixedpoints, which are generated through the monotonic upper- and lower-saddlenode bifurcations. Thus, the period-6 discrete system has at most six sets of period-6 fixed-points including five sets of independent period-6 fixed-point, one set of period-6 fixed-points on the period-3 period-doubling bifurcation tree, and the period-1 fixed-pionts. For such a period-6 discrete system, there exist two complex fixed-points. Similarly, other period-n discrete systems can be discussed. From the previous discussion, the period-n fixed-points for a quadratic discrete system are tabulated in Table 1.1. The dependent sets of period-n fixed-points are on the period-doubling bifurcation trees. The independent sets of period-n fixed-points are generated through monotonic saddle-node bifurcations. From analytical expressions, the maximum sets of period-n fixed-points includes dependent and independent sets of period-n fixed-points. In addition to the period-1 trivial fixed-points, other fixed-points on the bifurcation trees relative to period-n fixed points are also trivial. From the period-n discrete systems of a quadratic discrete system, period-1 to period-4 bifurcation trees are sketched through period-n discrete systems, as shown in Fig. 1.9. The solid and dashed curves are for stable and unstable periodn fixed-points, respectively. The red and dark red colors are for period-n fixed points dependent on and independent of period-1 fixed-points on the bifurcation trees. The period-n fixed-points on the other period-doubling bifurcations are said to be dependent. The dependent period-n fixed-points are obtained from period-doubling bifurcation. However, the onsets of period-n fixed-points are not based on period-doubling bifurcations, which are said to be independent. The onsets of such independent period-n fixed-points are based on the monotonic saddle-node bifurcations. The numerical examples can be found from Luo and Guo (2013). Such dependent and independent period-n fixed-points for quadratic systems are presented. Through such a way, one can find all possible period-n fixed points.

1.5 Backward Bifurcation Trees

75

Table 1.1 Period-n fixed-points for a quadratic discrete system P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12

1.5

Dependent sets

Independent sets

N/A (1)P-1 N/A (1)P-1 N/A (1)P-3 N/A (1)P-1 (1)P-4 N/A (3)P-5

1 N/A 1 1 3 4 9 14

1 1 1 2 3 5 9 16

18 48

18 51

93 165

93 170

to P-2 to P-4 to P-6 to P-8 to P-8 to P-10

N/A (1)P-3 to P-12 (4)P-6 to P-12

Maximum sets

Trivial fixed-points N/A (1)P-1 (1)P-1 (1)P-1 to P-2 (1)P-1 (1)P-1, (1)P-3 (1)P-1 (1)P-1 to P-4 (1)P-4 (1)P-1 (1)P-1 (1)P-5 (1)P-1 (1)P-1 (1)P-3 to P-6 (4) P-6

Backward Bifurcation Trees

In this section, the analytical bifurcation scenario for backward quadratic discrete systems will be discussed as in a similar fashion through nonlinear renormalization techniques, and the backward bifurcation scenario based on the monotonic saddle-node bifurcations will be discussed, which is independent of period-1 fixed-points.

1.5.1

Backward Period-2 Quadratic Discrete Systems

After the backward period-doubling bifurcation of a period-1 fixed-point, the backward period-doubled fixed-points can be obtained and the corresponding stability is determined through dxk þ 1 =dxk . Theorem 1.13 Consider a 1-dimensional backward quadratic discrete system as xk ¼ xk þ 1 þ AðpÞx2k þ 1 þ BðpÞxk þ 1 þ CðpÞ

ð1:178Þ

where three scalar constants AðpÞ 6¼ 0, BðpÞ and CðpÞ are determined by a vector parameter

76

1 Quadratic Nonlinear Discrete Systems a0 > 0

P-1

a0 < 0

mSO

P-1 mSI-oSO

mUSN

mLSN mSI-oSO

mSO

∗ k

x

P-1

|| p ||

xk∗ P-1

|| p ||

(i)

(ii)

a0 > 0

P-1

P-2

a0 < 0

PD

mSO mUSN

P-1

mSI-oSO mSI-oSO

P-2

xk∗

P-1

PD

P-2

|| p ||

mSO

xk∗

P-1

|| p ||

(iii)

(iv)

a0 > 0

P-1 mSO

P-2

mUSN

mLSN

a0 < 0

P-3

mLSN

P-1

P-3

mUSN

mSI-oSO mSI-oSO

mLSN

P-3

xk∗

P-1 P-3

mUSN

|| p ||

P-1

|| p ||

(v)

(vi)

a0 > 0

P-1

mSO mUSN

PD

mSI-oSO

mLSN

P-4

mLSN

P-4

a0 < 0

mLSN

mLSN

P-2 PD

P-4

mSI-oSO

P-2 mUSN

mSO

xk∗

P-4

mUSN

P-4 P-2 P-4

mUSN

P-4

mUSN

P-4 P-1

|| p ||

(vii)

PD

mLSN

P-4

PD

P-1

PD

P-1

xk∗

P-4 P-4

P-4 P-2

PD

|| p ||

P-3

mUSN

mSO

xk∗

P-3

mUSN

mLSN

(vii)

Fig. 1.9 Sketched bifurcation trees based on period-doubling and monotonic saddle-node bifurcations. (i)-(viii) period-1 to period-4 bifurcation trees, based on period-n discrete systems. mUSN: monotonic upper-saddle-node, mLSN: monotonic lower-saddle-node. PD: period-doubling bifurcation. The solid and dashed curves are for stable and unstable fixed-points. The red and dark red colors are for dependent and independent period-n fixed points. mSO: monotonic source, mSI-oSO: monotonic sink to oscillatory source

1.5 Backward Bifurcation Trees

77

p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð1:179Þ

D ¼ B2  4AC [ 0;

ð1:180Þ

Under a condition of

there is a standard form for the backward discrete system as ð1Þ

ð1Þ

xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ a0 ðx2k þ 1 þ B1 xk þ 1 þ C1 Þ ¼ xk þ 1 þ a0 ðxk þ 1  a1 Þðxk þ 1  a2 Þ ¼

xk þ 1 þ a0 *2i¼1 ðxk þ 1



ð1:181Þ

ð1Þ ai Þ

where B ð1Þ C ;C ¼ ; A 1 A 1 ð1Þ pffiffiffiffiffiffiffiffi 1 ð1Þ pffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ b1 ¼  ðB1 þ D Þ; b2 ¼  ðB1  Dð1Þ Þ; 2 2 ð1Þ 2 ð1Þ ð1Þ D ¼ ðB1 Þ  4C1 0; ð1Þ

a0 ¼ AðpÞ; B1 ¼

ð1Þ

ð1Þ

ð1Þ

ð1:182Þ

ð1Þ

02i¼1 fai g ¼ sortf02i¼1 fbi gg; ai ai þ 1 for i ¼ 1; 2: (i) Consider a backward period-2 discrete system of Eq. (1.178) as ð1Þ

xk ¼ xk þ 2 þ ½a0 *2i1 ¼1 ðxk þ 2  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2 2

ð1Þ

2

ð2Þ

ð1Þ

 ai2 Þg

ð2Þ

¼ xk þ 2 þ ½a0 *2i1 ¼1 ðxk þ 2  ai1 Þ½a20 ðx2k þ 2 þ B1 xk þ 2 þ C1 Þ ð1Þ

ð2Þ

2 ¼ xk þ 2 þ ½a0 *2j1 ¼1 ðxk þ 2  ai1 Þ½a20 *2j2 ¼1 ðxk þ 2  bj2 Þ 2

ð2Þ

¼ xk þ 2 þ a10 þ 2 *4i¼1 ðxk þ 2  ai Þ ð1:183Þ where 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ b1;2 ¼  ðB1 þ Dð2Þ Þ; b2 ¼  ðB1  Dð2Þ Þ; 2 2 ð2Þ 2 ð2Þ ð2Þ D ¼ ðB1 Þ  4C1 0; with fixed-points

ð1:184Þ

78

1 Quadratic Nonlinear Discrete Systems ð2Þ

xk ¼ xk þ 2 ¼ ai ; ði ¼ 1; 2; . . .; 4Þ ð2Þ

ð1Þ

ð2Þ

04i¼1 fai g ¼ sortf02j1 ¼1 faj1 g; 02j2 ¼1 fbj2 gg

ð1:185Þ

ð2Þ ð2Þ with ai \ai þ 1 : ð1Þ

(ii) For a fixed-point of xk ¼ xk þ 1 ¼ ai1 ði1 2 f1; 2gÞ, if dxk ð1Þ ð1Þ j  ai2 Þ ¼ 1; ð1Þ ¼ 1 þ a0 ðai 1 dxk þ 1 xk þ 1 ¼ai1

ð1:186Þ

with • an oscillatory lower-saddle-node bifurcation ðd 2 xk =dx2k jxk þ 1 ¼a1 ¼ a0 [ 0Þ, • an oscillatory upper-saddle-node bifurcation ðd 2 xk =dx2k jxk þ 1 ¼a1 ¼ a0 \0Þ, then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi1 ; Di1 ¼ ðB1 Þ2  4C1 ¼ 0; 2

ð1:187Þ

and there is a backward period-2 discrete system of the quadratic discrete system in Eq. (1.178) as ð1Þ

ð2Þ

xk ¼ xk þ 2 þ a30 ðxk þ 2  ai1 Þ3 ðxk þ 2  ai2 Þ

ð1:188Þ

for i1 ; i2 2 f1; 2g; i1 6¼ i2 with dxk d 2 xk jx ¼að1Þ ¼ 1; 2 jx ¼að1Þ ¼ 0; dxk þ 2 k þ 2 i1 dxk þ 2 k þ 2 i1 d 3 xk ð2Þ 3 ð1Þ j  ai2 Þ ¼ 12a20 \0: ð1Þ ¼ 6a ðai 0 1 dx3k þ 2 xk þ 2 ¼ai1

ð1:189Þ

ð1Þ

Thus, xk at xk þ 2 ¼ ai1 is a monotonic source of the third-order, and the corresponding bifurcations is a monotonic source bifurcation for the period-2 discrete system. (ii1) The backward period-2 fixed-points are trivial and unstable if ð1Þ

xk ¼ xk þ 2 ¼ ai1 for i1 ¼ 1; 2: (ii2) The backward period-2 fixed-points are non-trivial and stable if

ð1:190Þ

1.5 Backward Bifurcation Trees

79 ð2Þ

xk ¼ xk þ 2 ¼ bi1 for i1 ¼ 1; 2:

ð1:191Þ

Proof Following the corresponding proof for the forward quadratic discrete system. This theorem can be proved. ■

1.5.2

Backward Period-Doubling Renormalization

The generalized cases of period-doublization of backward quadratic discrete systems are presented through the following theorem. The analytical period-doubling trees can be developed for backward quadratic discrete systems. Theorem 1.14 Consider a 1-dimensional backward quadratic discrete system as xk ¼ xk þ 1 þ AðpÞx2k þ 1 þ BðpÞxk þ 1 þ CðpÞ

ð1:192Þ

ð1Þ

¼ xk þ 1 þ a0 *2i¼1 ðxk þ 1  ai Þ:

(i) After l-times period-doubling bifurcations, a period-2l ðl ¼ 1; 2; . . .Þ discrete system for the quadratic discrete system in Eq. (1.192) is produced through the nonlinear renormalization as ð2l1 Þ

xk ¼ xk þ 2l þ ½a0  f1 þ

2

l1

l1

Þ 2

Þ

Þ

2l1

ð2l1 Þ

¼ xk þ 2l þ ½a0

ð2l1 Þ 22

 ½ða0

Þ

ð2l Þ

with

ð2l1 Þ

2

2

2l 1

*j1 ¼1 22

2

2

2l1 1

l1

*i1 ¼1; ðxk þ 2l l

22 1 22

*i2 ¼1

Þ

22

l

l1

 ai1

l1 2

22

ð2

 ai1

l1

Þ

ð2l1 Þ

 ai2

Þg

Þ ð2l Þ

ð2l Þ

ðx2k þ 2l þ Bj2 xk þ 2l þ Cj2 Þ ð2l1 Þ

 ai1

Þ ð2l Þ

ð2l Þ

ðxk þ 2l  bi2 ;1 Þðxk þ 2l  bi2 ;2 Þ

l

*i¼1 ðxk þ 2l

*i¼1 ðxk þ 2l

Þ

2l1

*i2 ¼1;i2 6¼i1 ðxk þ 2l

2l1

ð2l1 Þ 1 þ 22

¼ xk þ 2 l þ a0

Þ

*i1 ¼1 ðxk þ 2l

l1

¼ xk þ 2l þ ða0

l1

ð2

*i1 ¼1 ½1 þ a0 ð2

ð2

l1

2l1

¼ xk þ 2l þ ½a0  ½ða0

22

*i1 ¼1 ðxk þ 2l

ð2l Þ

 ai Þ

ð2l Þ

 ai Þ

ð1:193Þ

80

1 Quadratic Nonlinear Discrete Systems l dxk ð2l Þ X22l ð2l Þ 22 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk þ 2l  ai2 Þ; dxk þ 2l l d 2 xk ð2l Þ X22l X22l ð2l Þ 22 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l  ai3 Þ; 2 dxk þ 2l

.. .

l d r xk X22l ð2l Þ X22l ð2l Þ 22 ¼ a0 i1 ¼1 . . . ir ¼1;ir þ 1 6¼i1 ;i2 ...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk þ 2l  air þ 1 Þ r dxk þ 2l l

for r 22

ð1:194Þ where ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1 þ 2 ; a0 2l

ð2l1 Þ 1 þ ð2l1 Þ2

¼ ða0 2l1

l

ð2 Þ

Þ

; l ¼ 2; 3; . . .;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

ð2l Þ

2

ai þ 1 ; 02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg; ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ; 2 2

ð2l Þ

Di

l1

Iqð21

ð2l Þ

ð2l Þ

¼ ðBi Þ2  4Ci Þ

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

00Nq ¼1 Iqð2 Þ ; 2 2

l

2

¼ flðq1 1Þ2l1 þ 1 ; lðq1 1Þ2l1 þ 2 ; . . .; lq1 2l1 g f1; 2; . . .; M1 g0f∅g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  2l1 ;

ð1:195Þ

l

Iqð22 Þ ¼ flðq2 1Þ2l þ 1 ; lðq2 1Þ2l þ 2 ; . . .; lq2 2l1 g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ 22 1  22 1 ; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l þ 1 ; lN2 2l þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g with fixed-points ð2l Þ

xk ¼ xk þ 2l ¼ ai ; ði ¼ 1; 2; . . .; 22l Þ 2l

2l1

ð2l Þ

ð2l1 Þ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 l

ð2 Þ

l

ð2 Þ

with ai \ai þ 1 :

ð2l1 Þ

ð2l1 Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð1:196Þ

1.5 Backward Bifurcation Trees

81 ð2l1 Þ

(ii) For a fixed-point of xk ¼ xk þ 2l1 ¼ ai1 dxk

dx

j

ð2l1 Þ x ¼a k þ 2l1 k þ 2l1 i1

ð2l1 Þ

¼ 1 þ a0

22

ð2l Þ

l

ði1 2 Iq

l1

ð2l1 Þ

*i2 ¼1;i2 6¼i1 ðai1

 f1; 2; . . .; 22 gÞ, if ð2l1 Þ

 ai 2

Þ ¼ 1 ð1:197Þ

then there is a backward period-2l fixed-point discrete system ð2l Þ

x k ¼ x k þ 2 l þ a0

*

ð2l1 Þ

i1 2Iq

ð2l1 Þ 3

ðxk þ 2l  ai1

Þ

22

l

*j2 ¼1 ðxk þ 2l

ð2l Þ

 aj2 Þð1dði1 ;j2 ÞÞ ð1:198Þ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð1:199Þ

with dxk d 2 xk j  ð2l1 Þ ¼ 1; 2 j ð2l1 Þ ¼ 0; x ¼a dxk þ 2l k i1 dxk þ 2l xk ¼ai1 d 3 xk ð2l Þ ð2l1 Þ ð2l1 Þ j ð2l1 Þ ¼ 6a0 * ð2l1 Þ ða  ai2 Þ3 i2 2Iq ;i2 6¼i1 i1 dx3k þ 2l xk ¼ai1 2l

ð2l1 Þ

 *2j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

ð1:200Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

; q 2 f1; 2; . . .; N1 gÞ: ð2l1 Þ

Thus, xk at xk þ 2l ¼ ai1

is

 • a monotonic source of the third-order if d 3 xk =dx3k þ 2l   ð2l1 Þ \0; x l ¼ai kþ2 1  • a monotonic sink of the third-order if d 3 xk =dx3k þ 2l   ð2l1 Þ [ 0. x

k þ 2l

¼ai

1

(ii1) The backward period-2l fixed-points are trivial if ð2l1 Þ

xk ¼ xk þ 2l ¼ ai1

l1

ð1:201Þ

:

ð1:202Þ

for i1 ¼ 1; 2; . . .; 22 ;

(ii2) The backward period-2l fixed-points are non-trivial if ð2l Þ

ð2l Þ

xk ¼ xk þ 2l ¼ bj1 ;1 ; bj1 ;2 l

l

j1 2 0Nq¼2 Iqð2 Þ f1; 2; . . .; 22 g0f∅g Such a backward period-2l fixed-point is

82

1 Quadratic Nonlinear Discrete Systems

• monotonically stable if dxk =dxk þ 2l j

2 ð1; 1Þ;

ð2l Þ

x

k þ 2l

• monotonically invariant if dxk =dxk þ 2l j

¼ai

1

¼ 1, which is

ð2l Þ

x

k þ 2l

¼ai

1

1 j – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ 2l x

k þ 2l

[ 0; 1 j – a monotonic upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ 2l x

k þ 2l

\0;

1 þ1 – a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2l j k þ 2l x

k þ 2l

[ 0;

– a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

k þ 2l

• monotonically unstable if dxk =dxk þ 2l j

ð2l Þ

x

k þ 2l

¼ai

2 ð0; 1Þ;

1

• monotonically unstable with infinite eigenvalue if dxk =dxk þ 2l j ¼0 ; • oscillatorilly unstable with infinite eigenvalue if dxk =dxk þ 2l j

x

• flipped if dxk =dxk þ 2l j

ð2l1 Þ x l ¼ai kþ2 1

x

ð2l Þ

kþ2

¼ai l

2 ð1; 0Þ;

k þ 2l

ð2l Þ

x

¼ai

ð2l Þ

¼ 0 ;

þ

• oscillatorilly unstable if dxk =dxk þ 2l j

\0;

k þ 2l

¼ai

1

1

1

¼ 1, which is

1 – an oscillatory lower-saddle of the ð2l1 Þth order if d 2l1 xk =dx2l j k þ 2l x

k þ 2l

[ 0; 1 j – an oscillatory upper-saddle the ð2l1 Þth order with d 2l1 xk =dx2l k þ 2l x

k þ 2l

\0; 1 þ1 j – an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk =dx2l k þ 2l x [ 0; – an oscillatory sink the ð2l1 þ 1Þth order with d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

k þ 2l

• oscillatorilly stable if dxk =dxk þ 2l j

x

k þ 2l

ð2l Þ

¼ai

k þ 2l

\0;

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, following the forward case, this theorem can be proved. ■

1.5.3

Backward Period-n Appearing and Period-Doublization

The period-n discrete system for backward quadratic nonlinear discrete systems will be discussed, and the period-doublization of a backward period-n discrete system is discussed through the nonlinear renormalization. Theorem 1.15 Consider a 1-dimensional backward quadratic discrete system as

1.5 Backward Bifurcation Trees

83

xk ¼ xk þ 1 þ AðpÞx2k þ 1 þ BðpÞxk þ 1 þ CðpÞ

ð1:203Þ

ð1Þ

¼ xk þ 1 þ a0 *2i¼1 ðxk þ 1  ai Þ:

(i) After n-times iterations, a period-n discrete system for the quadratic discrete system in Eq. (1.203) is ð1Þ

xk ¼ xk þ n þ a0 *2i1 ¼1 ðxk þ n  ai1 Þf1 þ ¼

n xk þ a02 1 *2i1 ¼1 ðxk þ n

ðnÞ

¼ xk þ a0

2n

*i¼1 ðxk þ n



Xn

i¼1 Qj g ð1Þ ðnÞ 2n1 1 2 ai1 Þ½*j2 ¼1 ðxk þ n þ Bj2 xk þ n

ðnÞ

þ Cj2 Þ

ðnÞ

 ai Þ ð1:204Þ

with

dxk n ðnÞ X n ðnÞ ¼ 1 þ a0 2i1 ¼1 *2i2 ¼1;i2 6¼i1 ðxk þ n  ai2 Þ; dxk þ n d 2 xk n ðnÞ X n X n ðnÞ ¼ a0 2i¼1 2i2 ¼1;i2 6¼i1 *2i3 ¼1;i3 6¼i1 ;i2 ðxk þ n  ai3 Þ; dx2k þ n .. . d r xk n X n ðnÞ X n ðnÞ ¼ a0 2i1 ¼1 . . . 2ir ¼1;ir 6¼1;i2 ;...;ir1 *2ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk þ n  air þ 1 Þ dxrk þ n for r 2n ;

ð1:205Þ where ðnÞ

a0 ¼ ða0 Þ2

n

1

ð1Þ

; Q1 ¼ 0; Q2 ¼ *2i2 ¼1 ½1 þ a0 *2i1 ¼1;i1 6¼i2 ðxk þ n  ai1 Þ; ð1Þ

Qn ¼ *2in ¼1 ½1 þ a0 ð1 þ Qn1 *2in1 ¼1;in1 6¼in ðxk þ n  ain1 ÞÞ; n ¼ 3; 4; . . .; n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i2 2¼1 fbi2 ;1 ; bi2 ;2 gg; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2

ðnÞ

ðnÞ

2

ðnÞ

Di2 ¼ ðBi2 Þ2  4Ci2 0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þn þ 1 ; lðq1Þn þ 2 ; . . .; lqn gf1; 2; . . .; Mg0f∅g; for q 2 f1; 2; . . .; Ng; M ¼ 2n1  1; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flNn þ 1 ; lNn þ 2 ; . . .; lM g  f1; 2; . . .; Mg0f∅g; ð1:206Þ

84

1 Quadratic Nonlinear Discrete Systems

with fixed-points ðnÞ

xk ¼ xk þ n ¼ ai ; ði ¼ 1; 2; . . .; 2n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

02i¼1 fai g ¼ sortf02i1 ¼1 fai1 g; 0M i1 ¼1 fbi2 ;1 ; bi2 ;2 gg ðnÞ

ð1:207Þ

ðnÞ

with ai \ai þ 1 : ðnÞ

ðnÞ

(ii) For a backward fixed-point of xk ¼ xk þ n ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), if dxk ðnÞ 2n ðnÞ ðnÞ j ðnÞ ¼ 1 þ a 0 *i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk þ n xk þ n ¼ai1

ð1:208Þ

then there is a new discrete system for onset of the qth - set of period-n fixed-points based on the second-order monotonic saddle-node bifurcation as ðnÞ

x k ¼ x k þ n þ a0

*i 2I ðnÞ ðxk þ n 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *2j2 ¼1 ðxk þ n  aj2 Þð1dði1 ;j2 ÞÞ ð1:209Þ n

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð1:210Þ

(ii1) If dxk ðnÞ j ðnÞ ¼ 1 ði1 2 I q Þ; dxk þ n xk þ n ¼ai1 d 2 xk ðnÞ ðnÞ ðnÞ 2 j ðnÞ ¼ 2a 0 *i2 2IqðnÞ ;i2 6¼i1 ðai1  ai2 Þ dx2k þ n xk þ n ¼ai1 ðnÞ

ð1:211Þ

ðnÞ

 *2j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk at xk þ n ¼ ai1 is ðnÞ

• a monotonic upper-saddle of the second-order for d 2 xk =dx2k þ n jxk þ n ¼ ai1 \0; ðnÞ • a monotonic lower-saddle of the second-order for d 2 xk =dx2k þ n jxk þ n ¼ ai1 [ 0. (ii2) The backward period-n fixed-points ðn ¼ 2n1  mÞ are trivial

1.5 Backward Bifurcation Trees

85

ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f02ii ¼1 fai1 g; 02i2 ¼1 fai2 gg n for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 2 g0f∅g for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f02i2 ¼1 fai2

)

)

gg

ð1:212Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 2n g0f∅g for n ¼ 2n2 :

(ii3) The period-n fixed-points ðn ¼ 2n1  mÞ are non-trivial if ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 62 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

gg

)

for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 2n g0f∅g for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 62 f02i2 ¼1 fai2

)

gg

ð1:213Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 2n g0f∅g

for n ¼ 2n2 : Such a backward period-n fixed-point is • monotonically stable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

• monotonically invariant if dxk =dxk þ n jx

2 ð1; 1Þ;

1 ðnÞ

kþn

¼ai

¼ 1, which is

1

1  – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jxk þ n [ 0; 1  – a monotonic upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jxk þ n \0;

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jxk þ n \0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jxk þ n [ 0;

• monotonically unstable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

2 ð0; 1Þ;

1

• monotonically infinity-unstable if dxk =dxk þ n jx • oscillatorilly infinity-unstable if dxk =dxk þ n jx • oscillatorilly unstable if dxk =dxk þ n jx • flipped if dxk =dxk þ n jx

kþn

kþn

ðnÞ

¼ai

kþn

ðnÞ

¼ai

ðnÞ

kþn

¼ai ðnÞ

¼ai

¼ 0þ ;

1

¼ 0 ;

1

2 ð1; 0Þ;

1

¼ 1, which is

1

1  – an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jxk þ n [ 0; 1  – an oscillatory upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jxk þ n \0;

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jxk þ n [ 0; – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jxk þ n \0;

86

1 Quadratic Nonlinear Discrete Systems

• oscillatorilly stable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

2 ð1; 1Þ.

1

ðnÞ

ðnÞ

(iii) For a fixed-point of xk ¼ xk þ n ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), there is a backward period-doubling of the qth -set of period-n fixed-points if dxk ðnÞ 2n ðnÞ ðnÞ j ðnÞ ¼ 1 þ a 0 *j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1 dxk þ n xk þ n ¼ai1

ð1:214Þ

with • an oscillatory lower-saddle for d 2 xk =dx2k þ n jx ¼aðnÞ [ 0; k

• an oscillatory upper-saddle for d 2 xk =dx2k þ n jx

kþn

i1

ðnÞ

¼ai

\0.

1

The corresponding period-2  n discrete system of the quadratic discrete system in Eq. (1.203) is ð2nÞ

xk ¼ xk þ 2n þ a0

*i1 2I n ðxk þ 2n q

ðnÞ

 ai 1 Þ 3

ð1:215Þ

ð2nÞ ð1dði1 ;i2 ÞÞ

2n

 *2i2 ¼1 ðxk þ 2n  ai2

Þ

with dxk dxk þ 2n

j x

ðnÞ ¼ai k þ 2n 1

¼ 1;

d 2 xk j ðnÞ ¼ 0; dx2k þ 2n xk þ 2n ¼ai1

d 3 xk ð2nÞ ðnÞ ðnÞ 3 j ðnÞ ¼ 6a *i2 2I n ;i2 6¼i1 ðai1  ai2 Þ 0 q dx3k þ 2n xk þ 2n ¼ai1 2n

ðnÞ

ð2nÞ ð1dði1 ;i3 ÞÞ

 *2i3 ¼1 ðai1  ai3 ðnÞ

Þ

ð1:216Þ 6¼ 0:

ðnÞ

Thus, xk at xk þ n ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic source of the third-order if d 3 xk =dx3k þ 2n jx

ðnÞ

¼ai

k þ 2n

• a monotonic sink of the third-order if d 3 xk =dx3k þ 2n jx

k þ 2n

ðnÞ

¼ai

\0,

1

[ 0.

1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a backward period-2l  n discrete system of the backward quadratic discrete system in Eq. (1.203) is

1.5 Backward Bifurcation Trees

87

ð2l1 nÞ

xk ¼ xk þ 2l n þ ½a0  f1 þ

2

2l1 n

*i1 ¼1

22

l1 n

*i1 ¼1

ð2l1 nÞ

ðxk þ 2l n  ai1

ð2l1 nÞ 22l1 n xk þ 2l n þ ½a0 *i1 ¼1 ðxk þ 2l n

¼



ð2l1 nÞ 22 Þ ½ða0

l1 n

2

2l n1

*j1 ¼1

2

2l1 n1

ð2l1 nÞ 22l1 n xk þ 2l n þ ½a0 *i1 ¼1 ðxk2l n

¼

l1

ð2

 ½ða0

nÞ 2

Þ

ð2l1 nÞ

ð2

¼ xk þ 2l n þ ða0

l

l1

2

2l n1

*j2 ¼1

nÞ 2

ð2 nÞ

¼ xk þ 2l n þ a0

Þ

ð2l1 nÞ 22l1 n ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2l n

Þ

2

2l1 n

ð2l nÞ

*i¼1

2 2

ð2l nÞ



*i¼1

Þg

ð2l1 nÞ ai1 Þ

ðx2k þ 2l n þ Bj2

2l1 nÞ1

2l n



ð2l1 nÞ

 ai2

ð2l nÞ

xk þ 2l n þ Cj2

Þ

ð2l1 nÞ ai1 Þ ð2l nÞ

ðxk þ 2l n  bj2 l

ð2 nÞ

ðxk þ 2l n  ai l

ð2 nÞ

ðxk þ 2l n  ai

Þ

Þ

Þ ð1:217Þ

with dxk

ð2l nÞ X22l n 22l n i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk þ 2l n

¼ 1 þ a0

ð2l nÞ

 ai 2

Þ; dxk þ 2l n l d 2 xk ð2l nÞ X22l n X22l n ð2l nÞ 22 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l n  ai3 2 dxk þ 2l n .. .

l d r xk X22l n ð2l nÞ X22l n ð2l nÞ 22 n ¼ a0 i1 ¼1 . . . ir ¼1;ir 6¼i1 ;i2 ...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk þ 2l n  air þ 1 Þ r dxk þ 2l n l

for r 22 n ; ð1:218Þ where ð2nÞ

a0

ðnÞ

2l n

ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l nÞ

2n

ð2l1 nÞ 1 þ 22

¼ ða0

ð2l1 nÞ

l

ð2 nÞ

02i¼1 fai

Di

ð2l nÞ

¼ ða0 Þ1 þ 2 ; a0

2

ð2l nÞ 2

ð2l nÞ

Þ  4Ci

l1

ð2

g ¼ sortf02i1 ¼1 fai1 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; ¼ ðBi

Þ

0

nÞ

l1 n

; l ¼ 1; 2; 3; . . .; ð2l nÞ

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

88

1 Quadratic Nonlinear Discrete Systems

for i 2 0Nq11¼1 Iqð21 Iqð21

l1

nÞ

l1

nÞ

00Nq ¼1 Iqð2 nÞ ; l

2 2

2

¼ flðq1 1Þð2l1 nÞ þ 1 ; lðq1 1Þð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f∅g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  ð2l1  nÞ; l

Iqð22 nÞ ¼ flðq2 1Þð2l nÞ þ 1 ; lðq2 1Þð2l nÞ þ 2 ; . . .; lq2 ð2l nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ 22 1  22 1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi l l l ð2 nÞ ð2 nÞ 2 ð2 nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di i 2 flNð2l nÞ þ 1 ; lNð2l nÞ þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g

ð1:219Þ

with fixed-points ð2l nÞ

xk ¼ xk þ 2l n ¼ ai 2l n

ð2l nÞ

02i¼1 fai

ð2l nÞ

with ai

l

; ði ¼ 1; 2; . . .; 22 n Þ 2l1 n

ð2l1 nÞ

g ¼ sortf02i1 ¼1 fai1

ð2l nÞ

ð2l nÞ

\ai þ 1 :

ð1:220Þ ð2l1 nÞ

(v) For a fixed-point of xk ¼ xk þ 2l n ¼ ai1 there is a period-2l  n discrete system if dxk dxk þ 2l1 n

j

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð2l1 nÞ x l1 ¼ai kþ2 n 1

ð2l1 nÞ

¼ 1 þ a0

22

ð2l1 nÞ

ði1 2 Iq

l1 n

ð2l1 nÞ

*i2 ¼1;i2 6¼i1 ðai1

f1; 2; . . .; M1 gÞ,

ð2l1 nÞ

 ai2

Þ ¼ 1 ð1:221Þ

with • an oscillatory lower-saddle for d 2 xk =dx2k þ 2l1 n jx

¼ai

• an oscillatory upper-saddle for d 2 xk =dx2k þ 2l1 n jx

¼ai

k þ 2l1 n

k þ 2l1 n

The corresponding period-ð2l  nÞ discrete system is

ðnÞ

[ 0;

1

ðnÞ 1

\0.

1.5 Backward Bifurcation Trees

89 ð2l nÞ

xk ¼ xk þ 2l n þ a0

*

ð2l1 nÞ

i1 2Iq

2l n

ð2l1 nÞ 3

ðxk þ 2l n  ai1

Þ

ð1:222Þ

ð2l nÞ ð1dði1 ;i2 ÞÞ

 *2i2 ¼1 ðxk þ 2l n  ai2

Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai1

ð2l nÞ

ð2l1 nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

6¼ ai1

ð1:223Þ

with dxk dxk þ 2l n

j

ð2l1 nÞ x l ¼ai k þ 2 n 1

¼ 1;

d 2 xk þ 2l n j ð2l1 nÞ ¼ 0; x l ¼ai dx2k k þ 2 n 1

d 3 xk ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j ðai1  ai2 Þ * ð2l1 nÞ ¼ 6a0 ð2l1 nÞ 3 x ¼a i 2I ;i ¼ 6 i 2 q 2 1 dxk þ 2l n k þ 2l n i1 2l n

ð2l1 nÞ

 *2i3 ¼1 ðai1 ði1 2

l1 Iqð2 nÞ ; q

Thus, xk at

xk þ 2l n

ð2l nÞ ð1dði2 ;i3 ÞÞ

 ai3

Þ

6¼ 0

2 f1; 2; . . .; N1 gÞ ¼

ð2l1 nÞ ai1

ð1:224Þ is

• a monotonic source of the third-order if d 3 xk =dx3k þ 2l n j • a monotonic sink of the third-order if d 3 xk =dx3k þ 2l n j

ð2l1 nÞ

x

k þ 2l n

¼ai

ð2l1 nÞ

x

k þ 2l n

¼ai

\0;

1

[ 0.

1

(v1) The backward period-2l  n fixed-points are trivial if ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

ð1Þ

2l1 n

ð2l1 nÞ

2 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 2ð2 nÞ for n 6¼ 2n1 ; ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

2l1 n

ð2l1 nÞ

2 02i2 ¼1 fai2

g

)

l

for j ¼ 1; 2; . . .; 22 n for n ¼ 2n1 : (v2) The backward period-2l  n fixed-points are non-trivial if

ð1:225Þ

90

1 Quadratic Nonlinear Discrete Systems ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

62 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 2ð2 nÞ for n 6¼ 2n1 ; ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

2l1 n

ð2l1 nÞ

62 f02i2 ¼1 fai2

g

ð1:226Þ

)

l

for j ¼ 1; 2; . . .; 22 n for n ¼ 2n1 : Such a backward period-2l  n fixed-point is • monotonically stable if dxk =dxk þ 2l n j

2 ð1; 1Þ;

ð2l nÞ

x

k þ 2l n

• monotonically invariant if dxk =dxk þ 2l n j

¼ai

1 ð2l nÞ

x

k þ 2l n

¼ai

¼ 1, which is

1

– a monotonic lower-saddle of the ð2l1 Þth order for dxk =dxk þ 2l n jx

k þ 2l n

[0

(independent ð2l1 Þ-branch appearance); – a monotonic upper-saddle the ð2l1 Þth order for dxk =dxk þ 2l n jx

\0

(independent ð2l1 Þ-branch appearance) – a monotonic sink of the ð2l1 þ 1Þth order for dxk =dxk þ 2l n jx

[0

(dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic source the ð2l1 þ 1Þth order for dxk =dxk þ 2l n jx

k þ 2l n

dent ð2l1 þ 1Þ-branch appearance from one branch); • monotonically unstable if dxk =dxk þ 2l n j

ð2l nÞ

x

k þ 2l n

¼ai

• monotonically infinity-unstable if dxk =dxk þ 2l n j • monotonically infinity-unstable if dxk =dxk þ 2l n j • oscillatorilly unstable if dxk =dxk þ 2l n j • flipped if dxk =dxk þ 2l n j

ð2l nÞ x l ¼ai k þ 2 n 1

x

k þ 2l n

k þ 2l n

\0 (depen-

2 ð0; 1Þ;

1 ð2l nÞ

x

¼ai

x

¼ai

k þ 2l n

1

ð2l nÞ

k þ 2l n

ð2l nÞ

¼ai

k þ 2l n

¼ 0þ ; ¼ 0þ ;

1

2 ð1; 0Þ;

1

¼ 1, which is

1 – an oscillatory lower-saddle of the ð2l1 Þth order if d 2l1 xk =dx2l j k þ 2l n x

k þ 2l n

1 – an oscillatory upper-saddle the ð2l1 Þth order with d 2l1 xk =dx2l j k þ 2l n x

th

– an oscillatory source of the ð2l1 þ 1Þ order if – an oscillatory sink the ð2l1 þ 1Þth order with • oscillatorilly stable if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

1

[ 0;

k þ 2l n

\0;

d xk =dxk2lþ1 þ2l1n jx l [ 0; k þ 2 n 1 þ1  d 2l1 þ 1 xk =dx2l j \0; k þ 2l n x l 2l1 þ 1

2 ð1; 1Þ:

k þ 2 n

1.5 Backward Bifurcation Trees a0 > 0

91 P-1

a0 < 0

mSI

P-1 mSO-oSI

mLSN

mUSN mSO-oSI

mSI

xk∗+1

P-1

|| p ||

xk∗+1 P-1

|| p ||

(i)

(ii) P-2

P-1

a0 > 0

UPD

mSI mLSN

P-1

mSO-oSI mSO-oSI

P-2

xk∗+ 2

P-1

UPD

P-2

|| p ||

mSI

xk∗+ 2

P-1

|| p ||

(iii)

(iv) P-1

a0 > 0 mSI

P-2

mUSN

mLSN

a0 < 0

P-3

mUSN

P-1

P-3

mLSN

mSO-oSI mSO-oSI

mLSN

P-3

xk∗+3

P-1

P-1

|| p ||

(v)

(vi)

a0 > 0

P-1

mSI mUSN

UPD

mSO-oSI

a0 < 0 mUSN

mUSN

P-4

mUSN

P-4

mUSN

P-2 UPD

mSO-oSI

P-2 mLSN

xk∗+ 4

mSI

P-4

mLSN

P-4 P-2 P-4

mLSN

P-4

mLSN

P-4 P-1

|| p ||

(vii)

UPD

mUSN

P-4

UPD

xk∗+ 4

P-1

UPD

P-4 P-1

P-4 P-4

P-4 P-2

UPD

|| p ||

P-3

mLSN

mSI

P-3

mUSN

|| p ||

xk∗+3

P-3

mLSN

mUSN

(vii)

Fig. 1.10 Sketched backward bifurcation trees based on period-doubling and monotonic saddle-node bifurcations for backward period-n discrete systems of quadratic discrete systems. (i)–(viii) period-1 to period-4 bifurcation trees, based on period-n discrete systems. mUSN: monotonic upper-saddle-node, mLSN: monotonic lower-saddle-node. UPD: unstable period-doubling bifurcation. The solid and dashed curves are for stable and unstable fixed-points. The red and dark red colors are for dependent and independent period-n fixed points. mSI: monotonic sink, mSO-oSI: monotonic source to oscillatory sink

92

1 Quadratic Nonlinear Discrete Systems

Proof Through the nonlinear renormalization, the proof of this theorem can follow the proof for the forward discrete system. This theorem can be easily proved. ■

1.5.4

Backward Period-n Bifurcation Trees

Similarly, from the period-n discrete systems of a backward quadratic discrete system, period-1 to period-4 bifurcation trees are sketched through period-n discrete systems, as shown in Fig. 1.10. As for the forward quadratic discrete system, solid and dashed curves are for stable and unstable period-n fixed-points. The red and dark red colors are for period-n fixed points dependent on and independent of period-1 fixed-points on the bifurcation trees. The backward period-n fixed-points on the other period-doubling bifurcations are also said to be dependent. The dependent period-n fixed-points are obtained from unstable period-doubling bifurcation. However, the onsets of period-n fixed-points are based on unstable period-doubling bifurcations, which are also said to be independent as well. The onsets of such independent backward period-n fixed-points are based on the monotonic saddle-node bifurcations.

References Luo ACJ (2010) A Ying-Yang theory for nonlinear discrete dynamical systems. International Journal of Bifurcation and Chaos 20(4):1085–1098 Luo ACJ (2012) Regularity and Complexity in Dynamical Systems. Springer, New York Luo ACJ (2019) The stability and bifurcation of fixed-points in low-degree polynomial systems. Journal of Vibration Testing and System Dynamics 3(4):403–451 Luo ACJ, Guo Y (2013) Vibro-Impact Dynamics. Wiley, New York

Chapter 2

Cubic Nonlinear Discrete Systems

In this Chapter, the stability and stability switching of fixed-points in cubic polynomial discrete systems are discussed. As in Luo (2019), the monotonic uppersaddle-node and lower-saddle-node appearing and switching bifurcations are discussed and the third-order monotonic sink and source switching bifurcations are discussed as well. The third-order monotonic sink and source flower-bundle switching bifurcations for simple fixed-points are presented. The third-order monotonic sink and source switching bifurcations for monotonic saddle and nodes are discovered. Graphical illustrations of global stability and bifurcations are presented. The bifurcation trees for cubic nonlinear discrete systems are discussed through period-doublization and monotonic saddle-node bifurcations.

2.1

Period-1 Cubic Discrete Systems

In this section, period-1 fixed-points in cubic nonlinear discrete systems will be discussed, and the stability and bifurcation conditions will be developed. Definition 2.1 Consider a cubic nonlinear discrete system xk þ 1 ¼ xk þ AðpÞx3k þ BðpÞx2k þ CðpÞxk þ DðpÞ  xk þ a0 ðpÞðxk  aðpÞÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ

ð2:1Þ

where four scalar constants AðpÞ 6¼ 0; BðpÞ; CðpÞ and DðpÞ are determined by A ¼ a0 ; B ¼ ða þ B1 Þa0 ; C ¼ ðaB1 þ C1 Þa0 ; D ¼ aa0 C1 ; p ¼ ðp1 ; p2 ; . . .; pm ÞT :

© Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3_2

ð2:2Þ

93

94

2 Cubic Nonlinear Discrete Systems

(i) If D1 ¼ B21  4C1 \0 for p 2 X1  Rm

ð2:3Þ

then the cubic nonlinear discrete system has a simple fixed-point only as xk ¼ a for p 2 X1  Rm

ð2:4Þ

and the standard form of such a 1-dimensional system is xk þ 1 ¼ xk þ a0 ðxk  aÞðx2k þ B1 xk þ C1 Þ:

ð2:5Þ

D1 ¼ B21  4C1 [ 0 for p 2 X2  Rm

ð2:6Þ

(ii) If

then there are three fixed-points with 1 2

a0 ¼ AðpÞ; b1;2 ¼  ðB1 ðpÞ 

pffiffiffiffiffiffi D1 Þ with b1 [ b2 ;

a1 ¼ minfa; b1 ; b2 g; a3 ¼ maxfa; b1 ; b2 g; a2 2 fa; b1 ; b2 g 6¼ fa1 ; a3 g;

ð2:7Þ

2

Dij ¼ ðai  aj Þ [ 0 for i; j 2 f1; 2; 3g but i 6¼ j:

(ii1) If ai 6¼ aj with Dij ¼ ðai  aj Þ2 [ 0 for i; j 2 f1; 2; 3g but i 6¼ j:

ð2:8Þ

the cubic forward discrete system has three different, simple fixed-points as xk ¼ a1 ; xk ¼ a2 ; xk ¼ a3

ð2:9Þ

and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ:

ð2:10Þ

(ii2) If at p ¼ p1 a1 ¼ b2 ; a2 ¼ a; a3 ¼ b1 ; D12 ¼ ða1  a2 Þ ¼ ða  b2 Þ2 ¼ 0;

ð2:11Þ

the cubic nonlinear discrete system has a double-repeated fixed-point and a simple fixed-point as xk ¼ a1 ; xk ¼ a1 ; xk ¼ a2

ð2:12Þ

2.1 Period-1 Cubic Discrete Systems

95

and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ:

ð2:13Þ

Such a discrete flow at the fixed-point of xk ¼ a1 is called a monotonicsaddle discrete flow of the second-order. The fixed-point of xk ¼ a1 for two different fixed-points switching is called a switching bifurcation point of fixed-point at p ¼ p1 with the second-order multiplicity, and the switching bifurcation condition is pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 a ¼ b1 ¼ minf ðB1 ðpÞ þ D1 Þ;  ðB1 ðpÞ  D1 Þg ð2:14Þ 2

2

(ii3) If at p ¼ p2 , a2 ¼ b1 ; a3 ¼ a; a1 ¼ b2 ; D23 ¼ ða2  a3 Þ ¼ ða  b1 Þ2 ¼ 0;

ð2:15Þ

the cubic forward discrete system has three fixed-points as xk ¼ a1 ; xk ¼ a2 ; xk ¼ a2

ð2:16Þ

and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ2 :

ð2:17Þ

Such a discrete flow at the fixed-point of xk ¼ a2 is called a monotonic saddle discrete flow of the second-order. The fixed-point of xk ¼ a2 for two different fixed-points switching is called a switching bifurcation point of fixed-point at a point p ¼ p1 with the second-order multiplicity, and the switching bifurcation condition is pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 a ¼ b2 ¼ maxf ðB1 ðpÞ þ D1 Þ;  ðB1 ðpÞ  D1 Þg ð2:18Þ 2

2

(ii3) If at p ¼ p3 , a1 ¼ b2 ; a2 ¼ a; a3 ¼ b1 ; D12 ¼ ða1  a2 Þ2 ¼ ða  b2 Þ2 ¼ 0; D23 ¼ ða2  a3 Þ2 ¼ ða  b1 Þ2 ¼ 0; D13 ¼ ða1  a3 Þ2 ¼ ðb2  b1 Þ2 ¼ 0; the cubic nonlinear system has three repeated fixed-point as

ð2:19Þ

96

2 Cubic Nonlinear Discrete Systems

xk ¼ a1 ; xk ¼ a2 and xk ¼ a3

ð2:20Þ

and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞ3 :

ð2:21Þ

Such a discrete flow at the fixed-point of xk ¼ a1 is called a monotonic sink or source discrete flow of the third-order. The fixed-point of xk ¼ a1 at a point p ¼ p3 for three different fixed-points switching is called a switching bifurcation point of fixed-point with the third-order multiplicity, and the switching bifurcation condition is 1 2

a ¼ b ¼  B1 ðpÞ:

ð2:22Þ

(iii) If D1 ¼ B21  4A1 C1 ¼ 0 for p ¼ p0 2 @X12  Rm1 ;

ð2:23Þ

then there exist 1 2

a0 ¼ Aðp0 Þ; and b1 ¼ b2 ¼ b ¼  B1 ðp0 Þ: (iii1)

ð2:24Þ

For a\b; the cubic nonlinear system has a double-repeated fixedpoint plus a monotonic lower-branch simple fixed-point xk ¼ a1 ¼ a; xk ¼ a2 ¼ b and xk ¼ a2 ¼ b

ð2:25Þ

with the corresponding standard form of xk þ ! ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ2 :

ð2:26Þ

Such a discrete flow at the fixed-point of x ¼ a2 is called a monotonic saddle discrete flow of the second-order. The fixed-point of xk ¼ a2 for two different fixed-points appearing is called an appearing bifurcation point of fixed-points at a point p ¼ p0 2 @X12 with the second-order multiplicity, and the appearing bifurcation condition is D1 ¼ B21  4C1 ¼ 0 with a\b:

ð2:27Þ

2.1 Period-1 Cubic Discrete Systems

(iii2)

97

For a [ b; the cubic nonlinear system has a double-repeated fixed-point plus a simple fixed-point xk ¼ a1 ¼ b and xk ¼ a1 ¼ b; xk ¼ a2 ¼ a

ð2:28Þ

with the corresponding standard form of xk þ 1 ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ:

ð2:29Þ

Such a discrete flow at the fixed-point of xk ¼ a1 is called a monotonic saddle discrete flow of the second order. The fixed-point of xk ¼ a1 ¼ b for two different fixed-point appearing is called a bifurcation point of fixed-point at a point p ¼ p0 2 @X12 with the lower-branch second-order multiplicity, and the bifurcation appearing condition is also D1 ¼ B21  4C1 ¼ 0 with a [ b: (iii3)

ð2:30Þ

For a ¼ b; the cubic forward discrete system has a triple-repeated fixed-point as xk ¼ a1 ¼ a and xk ¼ a1 ¼ a; xk ¼ a2 ¼ a

ð2:31Þ

with the corresponding standard form of xk þ 1 ¼ xk þ a0 ðxk  a1 Þ3 :

ð2:32Þ

Such a discrete flow at the fixed-point of x ¼ a1 is called a monotonic source or sink discrete flow of the third-order. The fixed-point of x ¼ a1 ¼ a for three fixed-points switching or two fixed-points switching is called a switching bifurcation of fixed-point at a point p ¼ p0 2 @X12 with the third-order multiplicity, and the switching bifurcation condition is D1 ¼ B21  4C1 ¼ 0 with a ¼ b:

ð2:33Þ

From the previous definitions, the conditions of stability and bifurcation in forward cubic nonlinear discrete systems are stated in the following theorem.

98

2 Cubic Nonlinear Discrete Systems

Theorem 2.1 (i) Under a condition of D1 ¼ B21  4C1 \0

ð2:34Þ

a standard form of the 1-dimensional discrete system in Eq. (2.1) is 1 2

1 4

xk þ 1 ¼ xk þ f ðxk þ 1 ; pÞ ¼ xk þ a0 ðxk  a1 Þ½ðxk þ B1 Þ2 þ ðD1 Þ: ð2:35Þ (i1) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic source) with df =dxk jx ¼a1 2 ð0; 1Þ: k (i2) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is • monotonically stable with df =dxk jx ¼a1 2 ð1; 0Þ (a monotonic sink); k • invariantly stable with df =dxk jx ¼a1 ¼ 1 (an invariant sink); k • oscillatorilly stable with df =dxk jx ¼a1 2 ð2; 1Þ (an oscillatory k sink); • flipped with df =dxk jx ¼a1 ¼ 2; which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a1 k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a1 k \ 0; – an oscillatory source of the third-order for d 2 f =dx2k jx ¼a1 ¼ 0 and k

d 3 f =dx3k jx ¼a1 \0; k

• oscillatorilly unstable with df =dxk jx ¼a1 2 ð1; 2Þ (an oscillatory k source). (i3) If a0 ðpÞ ¼ 0; then the fixed-point of xk ¼ a1 is stability switching. (ii) Under the conditions of D1 ¼ B21  4C1 [ 0; a1 ; a2 ; a3 ¼ sortfb2 ; a; b1 g; ai 6¼ aj ; ai \ai þ 1 ;

ð2:36Þ

2

Dij ¼ ðai  aj Þ 6¼ 0 for i; j 2 f1; 2; 3g; a standard form of the 1-dimensional forward discrete system in Eq. (2.31) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ:

ð2:37Þ

2.1 Period-1 Cubic Discrete Systems

99

(ii1a) if a0 ðpÞ [ 0; then the fixed-points of xk ¼ a1 is monotonically unstable with df =dxk jx ¼a1 2 ð0; 1Þ (a monotonic source). k (ii1b) If a0 ðpÞ [ 0; then the fixed-points of xk ¼ a2 is • monotonically stable with df =dxk jx ¼a2 2 ð1; 0Þ (a monotonic sink); k • invariantly stable with df =dxk jx ¼a2 ¼ 1 (an invariant sink); k • oscillatorilly stable with df =dxk jx ¼a2 2 ð2; 1Þ (an oscillatory k sink). • flipped with df =dxk jx ¼a2 ¼ 2; which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a2 k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a2 k \0; – an oscillatory sink of the third-order for d 2 f =dx2k jx ¼a2 ¼ 0 and k

d 3 f =dx3k jx ¼a2 [ 0; k

• oscillatorilly unstable with df =dxk jx ¼a2 2 ð1; 2Þ (an oscillatory k source). (ii1c) If a0 ðpÞ [ 0; then the fixed-points of xk ¼ a3 ; is monotonically unstable with df =dxk jx ¼a3 2 ð0; 1Þ (a monotonic source). k (ii2a) If a0 ðpÞ\0; then the fixed-points of xk ¼ a1 is • monotonically stable with df =dxk jx ¼a1 2 ð1; 0Þ (a monotonic sink); k • invariantly stable with df =dxk jx ¼a1 ¼ 1 (an invariant sink); k • oscillatorilly stable with df =dxk jx ¼a1 2 ð2; 1Þ (an oscillatory k sink); • flipped with df =dxk jx ¼a1 ¼ 2 (an oscillatory upper-saddle of the k

second-order if d 2 f =dx2k jx ¼a1 [ 0Þ; k • oscillatorilly unstable with df =dxk jx ¼a2 2 ð1; 2Þ (an oscillatory k source). (ii2b) If a0 ðpÞ\0; then the fixed-points of xk ¼ a2 is monotonically unstable with df =dxk jx ¼a2 2 ð0; 1Þ (a monotonic source). k (ii2c) If a0 ðpÞ\0; then the fixed-points of xk ¼ a3 is • monotonically stable with df =dxk jx ¼a3 2 ð1; 0Þ (a monotonic k sink); • invariantly stable with df =dxk jx ¼a3 ¼ 1 (an invariant sink); k

100

2 Cubic Nonlinear Discrete Systems

• oscillatorilly stable with df =dxk jx ¼a3 2 ð2; 1Þ (an oscillatory k source); • flipped with df =dxk jx ¼a3 ¼ 2 (an oscillatory lower-saddle of the k

second-order for d 2 f =dx2k jx ¼a3 \0Þ; k • oscillatorilly unstable with df =dxk jx ¼a3 2 ð1; 2Þ (an oscillatory k source). (iii) Under a condition of D1 ¼ B21  4C1 [ 0; a1 ; a2 ; a3 ¼ sortfb2 ; a; b1 g; ai 6¼ aj ; ai  ai þ 1

ð2:38Þ

2

D12 ¼ ða1  a2 Þ ¼ 0; for i; j 2 f1; 2; 3g a standard form of the 1-dimensional forward discrete system in Eq. (2.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ:

ð2:39Þ

(iii1a) If a0 ðpÞ [ 0; then the fixed-points of xk ¼ a1 is monotonically unstable with d 2 f =dx2k jx ¼a1 \0 (a monotonic lower-saddle of the second-order). k (iii1b) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a2 is monotonically unstable with df =dxk jx ¼a2 2 ð0; 1Þ (a monotonic source). k (iii1c) The bifurcation of fixed-point at xk ¼ a1 for the two different fixed-points switching is a monotonic lower-saddle-node switching bifurcation of the second-order at a point p ¼ p1 . (iii2a) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is monotonically unstable with d 2 f =dx2k jx ¼a1 [ 0 (a monotonic upper-saddle of the secondk order). (iii2b) If a0 ðpÞ\0; then the fixed-point of xk ¼ a2 is • monotonically stable with df =dxk jx ¼a2 2 ð1; 0Þ (a monotonic k sink); • invariantly stable with df =dxk jx ¼a2 ¼ 1 (an invariant sink); k • oscillatorilly stable with df =dxk jx ¼a2 2 ð2; 1Þ (an oscillatory k sink); • flipped with df =dxk jx ¼a2 ¼ 2 (an oscillatory lower-saddle of the k

second-order if d 2 f =dx2k jx ¼a2 \0Þ; k • oscillatorilly unstable with df =dxk jx ¼a2 2 ð1; 2Þ (an oscillak tory source).

2.1 Period-1 Cubic Discrete Systems

101

(iii2c) The bifurcation of fixed-point at xk ¼ a1 for the two different fixed-point switching is a monotonic upper-saddle-node switching bifurcation of the second order at a point p ¼ p1 . (iv) For D1 ¼ B21  4C1 [ 0; a1 ; a2 ; a3 ¼ sortfb2 ; a; b1 g; ai 6¼ aj ; ai  ai þ 1

ð2:40Þ

2

D23 ¼ ða2  a3 Þ ¼ 0; for i; j 2 f1; 2; 3g a standard form of the 1-dimensional discrete system in Eq. (2.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ2 :

ð2:41Þ

(iv1a) If a0 ðpÞ [ 0; then the fixed-points of xk ¼ a2 is monotonically unstable with d 2 f =dx2k jx ¼a2 [ 0 (a monotonic upper-saddle of the secondk order). (iv1b) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable with df =dxk jx ¼a1 2 ð0; 1Þ (a monotonic source). k (iv1c) The bifurcation of fixed-point at xk ¼ a2 for the two different fixed-points switching is a monotonic upper-saddle-node switching bifurcation of the second order at a point p ¼ p1 . (iv2a) If a0 ðpÞ\0; then the fixed-point of xk ¼ a2 is monotonically unstable with d 2 f =dx2k jx ¼a2 \0 (a monotonic lower-saddle of the second-order). k (iv2b) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is • monotonically stable with df =dxk jx ¼a1 2 ð1; 0Þ (a monotonic k sink); • invariantly stable with df =dxk jx ¼a1 ¼ 1 (an invariant sink); • oscillatorilly stable with df =dxk jx ¼a1 2 ð2; 1Þ (an oscillatory sink); • flipped with df =dxk jx ¼a1 ¼ 2 (an oscillatory upper-saddle of the k

second-order for d 2 f =dx2k jx ¼a1 [ 0Þ; k • oscillatorilly unstable with df =dxk jx ¼a1 2 ð1; 2Þ (an oscillak tory source). (iv2c) The bifurcation of fixed-point at xk ¼ a2 for the two different fixed-point switching is a monotonic lower-saddle-node bifurcation of the second-order at a point p ¼ p1 .

102

2 Cubic Nonlinear Discrete Systems

(v) For D1 ¼ B21  4C1 0; a1 ; a2 ; a3 ¼ sortfb2 ; a; b1 g; ai 6¼ aj ; ai  ai þ 1 ;

ð2:42Þ

2

Dij ¼ ðai  aj Þ ¼ 0 for i; j ¼ 1; 2; 3 but i 6¼ j; a standard form of the 1-dimensional forward discrete system in Eq. (2.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ3 :

ð2:43Þ

(v1a) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable (a third-order monotonic source, d 3 f =dx3k jx ¼a1 [ 0Þ. k (v1b) The bifurcation of fixed-point at xk ¼ a1 for three different fixed-points switching is a monotonic source switching bifurcation of the third-order at a point p ¼ p1 . (v2a) If a0 ðpÞ\0; then the fixed-point of x ¼ a1 is monotonically stable (a third-order monotonic sink, d 3 f =dx3k jx ¼a1 \0Þ. k (v2b) The bifurcation of fixed-point at x ¼ a2 for three fixed-points switching is a monotonic sink switching bifurcation of the third-order at a point p ¼ p1 . (vi) For D1 ¼ B21  4A1 C1 ¼ 0; a\b a1 ¼ a; a2 ¼ b; D12 ¼ ða1  a2 Þ2 6¼ 0

ð2:44Þ

at p ¼ p0 2 @X12  Rm1 , a standard form of the 1-dimensional discrete system is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ2 :

ð2:45Þ

(vi1a) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic source, df =dxk jx ¼a1 [ 0Þ. k (vi1b) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a2 is monotonically unstable (a monotonic upper-saddle of the second-order, d 2 f =dx2k jx ¼a2 [ 0Þ. k (vi1c) The bifurcation of fixed-point at x ¼ a2 for two different fixed-point vanishing or appearance is a monotonic upper-saddle-node appearing bifurcation of the second-order at a point p ¼ p0 2 @X12 . (vi2a) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is

2.1 Period-1 Cubic Discrete Systems

103

• monotonically stable with df =dxk jx ¼a1 2 ð1; 0Þ (a monotonic k sink); • invariantly stable with df =dxk jx ¼a1 ¼ 1 (an invariant sink); • oscillatorilly stable with df =dxk jx ¼a1 2 ð2; 1Þ (an oscillatory sink); • flipped with df =dxk jx ¼a1 ¼ 2 (an oscillatory upper-saddle of the k

second-order for d 2 f =dx2k jx ¼a1 [ 0Þ; k • oscillatorilly unstable with df =dxk jx ¼a1 2 ð1; 2Þ (an oscillak tory source). (vi2b) If a0 ðpÞ\0; then the fixed-point of xk ¼ a2 is monotonically unstable (a monotonic lower-saddle of the second-order, d 2 f =dx2k jx ¼a2 \0Þ. k (vi2c) The bifurcation of fixed-point at x ¼ a2 for two different fixed-points vanishing or appearance is a monotonic lower-saddle-node appearing bifurcation of the second order at a point p ¼ p0 2 @X12 . (vii) For D1 ¼ B21  4A1 C1 ¼ 0; a [ b a1 ¼ b; a2 ¼ a; D12 ¼ ða1  a2 Þ2 6¼ 0

ð2:46Þ

at p ¼ p0 2 @X12  Rm1 , a standard form of the 1-dimensional, forward discrete system is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ:

ð2:47Þ

(vii1a) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic lower-saddle of the second-order, d 2 f =dx2k jx ¼a1 \0Þ. k (vii1b) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a2 is monotonically unstable (a monotonic source, df =dxk jx ¼a2 [ 0Þ. k (vii1c) The bifurcation of fixed-point at x ¼ a1 for two different fixed-points vanishing or appearance is a monotonic lower-saddle-node appearing bifurcation of the second order at a point p ¼ p0 2 @X12 . (vii2a) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is monotonically unstable (a monotonically upper-saddle of the second-order, d 2 f =dx2k jx ¼a1 [ 0Þ. k (vii2b) If a0 ðpÞ\0; then the fixed-point of xk ¼ a2 is • monotonically stable with df =dxk jx ¼a2 2 ð1; 0Þ (a monotonic k sink); • invariantly stable with df =dxk jx ¼a2 ¼ 1 (an invariant sink); • oscillatorilly stable with df =dxk jx ¼a2 2 ð2; 1Þ (an oscillatory sink);

104

2 Cubic Nonlinear Discrete Systems

• flipped with df =dxk jx ¼a2 ¼ 2 (an oscillatory lower-saddle of the k

second-order if d 2 f =dx2k jx ¼a2 \0Þ; k • oscillatorilly unstable with df =dxk jx ¼a2 2 ð1; 2Þ (an oscillak tory source). (vii2c) The bifurcation of fixed-point at x ¼ a1 for two different fixed-points vanishing or appearance is a monotonically upper-saddle-node appearing bifurcation of the second-order at a point p ¼ p0 2 @X12 . (viii) For D1 ¼ B21  4A1 C1 ¼ 0; a ¼ b a2 ¼ a; a2 ¼ a3 ¼ b;

ð2:48Þ

2

D12 ¼ ða1  a2 Þ ¼ 0 at p ¼ p0 2 @X12  Rm1 , a standard form of the 1-dimensional, forward discrete system is xk þ 1 ¼ xk þ 1 þ f ðxk ; pÞ ¼ a0 ðxk  a1 Þ3 :

ð2:49Þ

(viii1a) If a0 ðpÞ [ 0; then the fixed-point of xk ¼ a1 is monotonically unstable (a third-order monotonic source, d 3 f =dx3k jx ¼a1 [ 0Þ. k (viii1b) The bifurcation of fixed-point at xk ¼ a1 for one fixed-point to three different three fixed-point switching is a monotonic source switching bifurcation of the third order at a point p ¼ p0 2 @X12 . (viii2a) If a0 ðpÞ\0; then the fixed-point of xk ¼ a1 is monotonically stable (a third-order monotonic sink, d 3 f =dx3k jx ¼a1 \0Þ. k (viii2b) The bifurcation of fixed-point at xk ¼ a1 for one fixed-point to three different three fixed-point switching is a monotonic sink switching bifurcation of the third order at a point p ¼ p0 2 @X12 . Proof The proof is similar to Theorem 1.2.

2.2

■

Period-1 to Period-2 Bifurcation Trees

In this section, period-1 stability and bifurcation of cubic nonlinear discrete systems are discussed graphically and period-2 fixed-points are also presented for a better understanding of complex bifurcations. The 1-dimensional cubic nonlinear discrete system can be expressed by a factor of ðxk  aÞ and a quadratic form of a0 ðx2k þ B1 xk þ C1 Þ as in Eq. (2.1). Three period-1 fixed-points do not have any intersections. Thus, only one bifurcation

2.2 Period-1 to Period-2 Bifurcation Trees

105

occurs at D1 ¼ B21  4C1 ¼ 0: The bifurcation of fixed-points occurs at the double or triple repeated fixed-point at the boundary of p0 2 @X12 . For D1 ¼ B21  4C1 [ 0; x2k þ B1 xk þ C1 ¼ 0 gives two fixed-points of xk ¼ b1 ; b2 . For a0 [ 0; if a [ maxfb1 ; b2 g, then the fixed-point of xk ¼ a3 ¼ a is monotonically unstable, and the fixed-point of xk ¼ a2 ¼ maxfb1 ; b2 g is from monotonically stable to oscillatorilly unstable, and the fixed-points of xk ¼ a1 ¼ minfb1 ; b2 g is monotonically unstable. For D1 ¼ B21  4C1 \0; x2k þ B1 xk þ C1 ¼ 0 does not have any real solutions. For D1 ¼ B21  4C1 ¼ 0; x2k þ B1 xk þ C1 ¼ 0 has a double repeated fixed-point of x ¼ b ¼ 12B1 . The condition of D1 ¼ B21  4C1 ¼ 0 gives B21 ¼ 4C1 :

ð2:50Þ

From Eq. (2.2), one obtains B1 ¼ a þ

B C B and C1 ¼ þ aða þ Þ: A A A

ð2:51Þ

Thus, equation (2.50) gives a¼

ffi B 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2  3AC : 3A 3A

ð2:52Þ

Further, the double repeated fixed-point of xk ¼ b ¼ 12B1 is given by xk ¼ b ¼ 

ffi B 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  B2  3AC : 3A 3A

ð2:53Þ

If B2 [ 3AC; such a double repeated fixed-point exists. If B2 \3AC; such a double repeated fixed-point does not exist. From Eq. (2.51), another fixed-point is xk ¼ a; which is different from xk ¼ b: If B2 ¼ 3AC; such a double repeated fixed-point with fixed-point of xk ¼ a has an intersected point at xk ¼ 3AB : The bifurcation diagram for a [ maxfb1 ; b2 g and a0 [ 0 is presented in Fig. 2.1 (i). The stable and unstable fixed-points varying with the vector parameter are presented by solid and dashed curves, respectively. Such a fixed-point of xk ¼ b is a monotonic lower-saddle-node (mLSN) appearing or vanishing bifurcation. • The fixed-point of xk ¼ a is a monotonic source (mSO), which is monotonically unstable. • The fixed-point of xk ¼ maxfb1 ; b2 g is – – – – –

a monotonic sink (mSI), an invariant sink (iSI), an oscillatory sink (oSI), an oscillatory saddle bifurcation (oUS or oLS), an oscillatory source (oSO).

106

2 Cubic Nonlinear Discrete Systems a0 > 0

a3

mSO

a0 < 0 iSI

oSI

mSI

oSO

oSO

oSI

mSI

a2

a3

P-2

mSO

a2

iSI

mLSN

mUSN

P-2

mSI

P-2

iSI

mSO

oSI

xk∗

oSO

xk∗

a1

a1

P-2

|| p ||

Δ1 < 0

Δ1 = 0

Δ1 > 0

|| p ||

Δ1 < 0

Δ1 = 0

(i)

Δ1 > 0

(ii) a3

a0 > 0

a0 < 0

mSO

oSO oSI

iSI

mLSN oSI

oSO

mSO

x

a2 mSO mSI

a1

P-2 iSI

oSI

∗ k

x || p ||

Δ1 < 0

Δ1 = 0

a2

iSI

mSI

∗ k

P-2

mSI

P-2 mUSN

a3

a1

P-2

Δ1 > 0

|| p ||

(iii)

oSO

Δ1 < 0

Δ1 = 0

Δ1 > 0

(iv)

Fig. 2.1 Stability and bifurcation of three independent fixed-points in the 1-dimensional, cubic nonlinear discrete system: For a [ fb1 ; b2 g: (i) a mLSN bifurcation (a0 [ 0Þ, (ii) a mUSN bifurcation (a0 \0Þ. For a\fb1 ; b2 g: (iii) a mUSN bifurcation (a0 [ 0Þ, (iv) a mLSN bifurcation (a0 \0Þ. mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; iSI: invariant sink). The period-2 fixed-points are presented on the period-1 bifurcation trees through red curves

• The fixed-point of xk ¼ minfb1 ; b2 g is a monotonic source (mSO), which is monotonically unstable. However, the bifurcation diagram for a [ maxfb1 ; b2 g and a0 \0 is presented in Fig. 2.1(ii). The fixed-point of xk ¼ b is a monotonic upper-saddle-node (mUSN) appearing or vanishing bifurcation. • The fixed-point of xk ¼ a is – a monotonic sink (mSI) first, – an invariant sink (iSI), – an oscillatory sink (oSI),

2.2 Period-1 to Period-2 Bifurcation Trees

107

– an oscillatory saddle bifurcation (oUS or oLS), – an oscillatory source (oSO). • The fixed-point of xk ¼ maxfb1 ; b2 g is a monotonic source (mSO). • The fixed-point of xk ¼ minfb1 ; b2 g is – – – – –

a monotonic sink (mSI) first, an invariant sink (iSI), an oscillatory sink (oSI), an oscillatory saddle bifurcation (oUS, oLS), an oscillatory source (oSO).

The bifurcation diagram for a\minfb1 ; b2 g and a0 [ 0 is presented in Fig. 2.1(iii). The fixed-point of xk ¼ b is a monotonic upper-saddle-node (mUSN) appearing or vanishing bifurcation. • The fixed-point of xk ¼ maxfb1 ; b2 g is a monotonic source (mSO). • The fixed-point of xk ¼ minfb1 ; b2 g is – – – – –

a monotonic sink (mSI) first, an invariant sink (iSI), an oscillatory sink (oSO), an oscillatory saddle bifurcation (oUS, oLS), an oscillatory source (oSO).

• The fixed-point of xk ¼ a is a monotonic source (mSO). The bifurcation diagram for a\minfb1 ; b2 g and a0 \0 is presented in Fig. 2.1(iv). The fixed-point of xk ¼ b is a monotonic lower-saddle-node (mLSN) appearing or vanishing bifurcation. • The fixed-point of xk ¼ maxfb1 ; b2 g is – – – – –

a monotonic sink (mSI) first, an invariant sink (iSI), an oscillatory sink (oSI), an oscillatory saddle bifurcation (oUS, oLS), an oscillatory source (oSO).

• The fixed-point of xk ¼ minfb1 ; b2 g is a monotonic source (mSO). • The fixed-point of xk ¼ a is – – – – –

a monotonic sink (mSI) first, an invariant sink (iSI), an oscillatory sink (oSI), an oscillatory saddle bifurcation (oUS, oLS), an oscillatory source (oSO).

108

2 Cubic Nonlinear Discrete Systems

Table 2.1 Stability and bifurcation of a 1-dimensional cubic nonlinear discrete system (xk þ 1 ¼ xk þ a0 ðxk  aÞ½x2k þ B1 ðpÞxk þ C1 ðpÞÞ Figure 2.1

a2

a1

a3

Bifurcation

(i) a0 [ 0 (ii) a0 \0 (iii) a0 \0 (iv) a0 \0

a [ maxfb1 ; b2 g mSO mSI-oSO mSO 2nd mLSN a [ maxfb1 ; b2 g mSI-oSO mSO mSI-oSO 2nd mUSN a\minfb1 ; b2 g mSO mSI-oSO mSO 2nd mUSN a\minfb1 ; b2 g mSI-oSO mSO mSI-oSO 2nd mLSN ffiffiffiffiffi ffi p 2 Notice that b1;2 ¼ 12ðB1 D1 Þ; D1 ¼ B1  4C1 . Bifurcation condition: D1 ¼ 0: (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; mSI-oSO: for monotonic sink to oscillatory source via the oscillatory sink)

The stability and bifurcations of fixed-points of the 1-dimensional cubic nonlinear, forward discrete system are summarized in Table 2.1. The period-2 fixed-points are also presented as well through the period-doubling. For D1 ¼ B21  4C1 0; the 1-dimensional cubic nonlinear, forward discrete system in Eq. (2.1) have three fixed-points. Three fixed-points are xk ¼ a; b1 ; b2 : Assume ai  ai þ 1 for i ¼ 1; 2 with a1;2;3 ¼ sortða; b1 ; b2 Þ. With varying parameters, two of three fixed-points (i.e., ai ¼ aj for i; j 2 f1; 2; 3g but i 6¼ jÞ will be intersected each other with the corresponding discriminant of Dij ¼ ðai  aj Þ2 ¼ 0; and in the vicinity of the intersection point, Dij ¼ ðai  aj Þ2 [ 0: The two intersected points of a ¼ b1;2 gives a¼

B1 1

2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B21  4C1 ;

ð2:54Þ

or a2 þ aB1 ¼ C1 :

ð2:55Þ

With Eqs. (2.2) or (2.51), the foregoing equation gives a ¼ b1;2 ¼ 

ffi B 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2  3AC : 3A 3A

ð2:56Þ

If B2 [ 3AC; such a intersected point of x ¼ a and x ¼ b1 or x ¼ b2 exists. If B \3AC; such an intersected point does not exist. Such the intersection point is for the two fixed-point switching, which is called the monotonic saddle-node bifurcation. The stability and bifurcation diagrams for a0 [ 0 and a0 \0 are presented in Fig. 2.2(i) and (ii), respectively. Three fixed-points are intersected at a point with Dij ¼ ðai  aj Þ2 ¼ 0 and a1 ¼ a2 ¼ a3 ¼ 3AB ; and in the vicinity of the intersection point, Dij ¼ ðai  aj Þ2 [ 0 for i; j ¼ 1; 2; 3 but i 6¼ j: The intersection points for a0 [ 0 and a0 \0 are called the monotonic source and sink bifurcations of the third 2

2.2 Period-1 to Period-2 Bifurcation Trees a0 > 0

109 a0 < 0

mSO

a3

mSO

oSI

mUSN

oSO oSO

P-2

mSI

mSO mSO mUSN

a3 mLSN

P-2

oSI mSI mLSN

P-2

P-2 mSO

P-2

xk∗

oSI

a2

mSO

mSI

mUSN mSO

mLSN

xk∗

mSO

P-2

Δ 23 = 0

Δ12 = 0 Δ13 = 0

oSO

P-2

mSI

a1 oSO Δ 23 = 0

Δ12 = 0 Δ13 = 0

|| p ||

(i) a0 > 0

oSO

P-1

a2

a1

|| p ||

oSI

oSI

mSI oSI

(ii) mSO

oSO

a0 < 0

oSO

P-2 mSO

3rd mSO mSO mSI

a3

P-2 P-2

3rd mSI

a3

mSO

mSI mSI

a2

a2 a1

|| p ||

oSI

oSI

P-2

xk∗

mSI

xk∗

mSO

Δ12 = 0 Δ 23 = 0 Δ13 = 0

(iii)

P-2

a1

|| p ||

oSI

oSO P-2

oSI

Δ12 = 0 Δ 23 = 0 Δ13 = 0

(iv)

Fig. 2.2 Stability and bifurcation of fixed-points switching in the 1-dimensional, cubic nonlinear discrete system. For two fixed-points switching: (i) a0 [ 0; (ii) a0 \0: For three fixed-point switching: (iii) 3rd order monotonic source bifurcation (a0 [ 0Þ, (iv) 3rd order monotonic sink bifurcation (a0 \0Þ. mLSN: mono lower-saddle-node, mUSN: monotonic upper-saddle-node. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; iSI: invariant sink). The period-2 fixed-points are presented on the period-1 bifurcation trees through red curves

order, respectively. The corresponding stability and bifurcation diagrams for three fixed-points switching are presented in Fig. 2.2(iii) and (iv). The period-2 fixed-points are also presented as well through the period-doubling. In the 1-dimensional cubic nonlinear, forward discrete system of Eq. (2.1), x2k þ B1 xk þ C1 ¼ 0 gives two fixed-points of xk ¼ b1 ; b2 for D1 ¼ B21  4C1 [ 0: One of the two fixed-points has one intersection with xk ¼ a and there are three different fixed-points for a ¼ a2 2 ðminfb1 ; b2 g; maxfb1 ; b2 gÞ: For this case, the intersection point occurs at a ¼ minfb1 ; b2 g for p1 2 @X23 or a ¼ maxfb1 ; b2 g for p2 2 @X23 . The bifurcation of fixed-point occurs at the double repeated fixed-point at D1 ¼ B21  4C1 ¼ 0 for p0 2 @X12 . Such a bifurcation is a monotonic lower- or upper-saddle-node bifurcation. For a ¼ 12B1 with D1 ¼ B21  4C1 ¼ 0; three fixed-points are repeated with three multiplicity. The intersected point of a ¼ 12B1 with Eq. (2.51) gives

110

2 Cubic Nonlinear Discrete Systems 1 2

B A

a ¼  ða þ Þ:

ð2:57Þ

Thus a¼

B : 3A

ð2:58Þ

Such a bifurcation at the intersection point is also a third-order monotonic source or sink bifurcation. The bifurcation diagrams for six cases of three fixed-points with one intersection are presented in Fig. 2.3(i)–(vi) and the stability and bifurcations are listed in Table 2.2. The corresponding period-2 fixed points are sketched as well. The 1-dimensional cubic nonlinear system is expressed by a factor of ðxk  aÞ and a quadratic form of a0 ðx2k þ B1 xk þ C1 Þ as in Eq. (2.1). For D1 ¼ B21  4C1 [ 0; x2k þ B1 xk þ C1 ¼ 0 gives two fixed-points of xk ¼ b1 ; b2 . The two fixed-points do not have any intersections with xk ¼ a: For D1 ¼ B21  4C1 ¼ 0; there are two parameters of p1 2 @X12 and p2 2 @X12 , and the two double repeated fixed-points are at xk ðpi Þ ¼ 12B1 ðpi Þ (i ¼ 1; 2Þ. With the two repeated fixed-points, the two fixed-points of xk ¼ b1 ; b2 formed a closed path in the bifurcation diagram. The bifurcation points of fixed-point occur at the two double repeated fixed-points of D1 ¼ B21  4C1 ¼ 0 for pi 2 @X12 (xk ¼ b1 ; b2 ). Such a bifurcation at the intersection point is also a monotonic lower or upper-saddle-node bifurcation. The stable and unstable fixed-points varying with the vector parameter are also represented by solid and dashed curves, respectively. The bifurcation diagrams for four cases of three fixed-points are presented in Fig. 2.4(i)–(vi), and the stability and bifurcations are summarized in Table 2.3. If the two repeated fixed-points have two intersections with xk ¼ aðpi Þ (i ¼ 1; 2Þ, i.e., aðpi Þ ¼ 12B1 ðpi Þ: The two triple repeated fixed-points at xk ¼ aðpi Þ (i ¼ 1; 2Þ are the third-order, monotonic sink or source bifurcations. The stability and bifurcation diagrams of fixed-points are formed by the fixed-point of xk ¼ aðpÞ and the closed loop of fixed-points of xk ¼ b1 ; b2 , as shown in Fig. 2.4(v) and (vi) for a0 [ 0 and a0 \0; respectively. The stability and bifurcations are also summarized in Table 2.3, and the corresponding period-2 fixed-points are sketched as well. In the 1-dimensional cubic nonlinear system in Eq. (2.1), x2k þ B1 xk þ C1 ¼ 0 for D1 ¼ B21  4C1 [ 0 gives two fixed-points of xk ¼ b1 ; b2 , which have an intersection with xk ¼ a: The intersected point are at a ¼ b1 or a ¼ b2 with Eq. (2.55). The double repeated fixed-point requires D1 ¼ B21  4C1 ¼ 0 and the two fixed-points of xk ¼ a; b1 under D1 ¼ B21  4C1 [ 0 and xk ¼ b2 for D1 ¼ B21  4C1 \0: Similarly, the two fixed-points of xk ¼ a; b2 under D1 ¼ B21  4C1 [ 0 and xk ¼ b2 for D1 ¼ B21  4C1 \0: Such a bifurcation for two fixed-points appearance and vanishing is called a monotonic lower or upper-saddle-node appearing bifurcation. The stable and unstable fixed-points varying with the vector

2.2 Period-1 to Period-2 Bifurcation Trees a0 > 0

a0 < 0

mUSN

mSO

mSI oSI

a oSI mSI

oSO

xk∗

mUSN mSO

min(b1 , b2 )

Δ12 > 0

Δ1 < 0 Δ1 = 0 Δ1 > 0 Δ12 = 0 Δ12 > 0

xk∗

oSO

Δ12 > 0

(ii) max(b1 , b2 )

a0 > 0

mSO

oSO

a0 < 0

P-1

oSI

P-2

P-2

P-2 mLSN

mSO

mSI

xk∗

Δ 23 > 0 Δ1 < 0

Δ1 = 0 Δ1 > 0 Δ 23 = 0 Δ 23 > 0

a

oSO

mUSN

P-2

Δ 23 > 0

P-2

Δ1 = 0 Δ1 > 0 Δ 23 = 0 Δ 23 > 0

Δ1 < 0

|| p ||

mSO

oSI oSO

(iii)

(iv) max(b1 , b2 )

a0 > 0

mSO

P-1

3rd mSO

mSI

oSI

oSO

oSI

min(b1 , b2 ) xk∗

Δ1 = 0

Δ1 > 0

a=b

P-1

a min(b1 , b2 )

mSI

oSI

oSO P-2

P-2

P-2

|| p ||

(v)

mSO

3rd mSI mSI

oSO

xk∗ Δ1 < 0

P-2

mSI oSI

mSO

oSO

a

mSO

a=b

max(b1 , b2 )

a0 < 0

P-2

|| p ||

i∈{1,2}

mSO

a

|| p ||

P-1

a = min bi

mLSN

mSO

max(b1 , b2

mSI

oSO

mSI oSI mSI

x

P-2

Δ1 = 0 Δ1 > 0 Δ12 = 0 Δ12 > 0

Δ1 < 0

(i)

∗ k

min(b1 , b2 )

oSI

|| p ||

mUSN

P-2

mSI

P-1

mSO

P-1

mLSN mSO

a

P-2

P-2

mSI

oSI

oSO

P-2

mLSN

|| p ||

111

Δ1 < 0

Δ1 = 0

Δ1 > 0

(vi)

Fig. 2.3 Stability and bifurcation of fixed-points in the 1-dimensional, cubic nonlinear discrete system before the oscillatory saddle bifurcation for sink branches: (i) the mLSN (D1 ¼ 0Þ and mUSN (a ¼ maxfb1 ; b2 g) bifurcations (a0 [ 0Þ, (ii) the mUSN (D1 ¼ 0Þ and mLSN (a ¼ maxfb1 ; b2 g) bifurcations (a0 \0Þ. (iii) the mUSN (D1 ¼ 0Þ and mLSN (a ¼ minfb1 ; b2 g) bifurcations (a0 [ 0Þ, (iv) the mLSN (D1 ¼ 0Þ and mUSN (a ¼ minfb1 ; b2 g) bifurcations (a0 \0Þ, (v) the third order mSO bifurcation (D1 ¼ 0 and a ¼ bÞ (a0 [ 0Þ, (vi) the third-order mSI bifurcation (D1 ¼ 0 and a ¼ bÞ (a0 \0Þ. The bifurcation points are marked by circular symbols. (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle; oUS: oscillatory upper-saddle; oLS: oscillatory lower-saddle; iSI: invariant sink). The period-2 fixed-points are presented on the period-1 bifurcation trees through the red curves

a1 a2

a3 nd

B-I nd

B-II

B-III

mSO mSI-oSO-mSI mSO 2 mLSN 2 mUSN a ¼ maxfb1 ; b2 g mSI-oSO-mSI mSO mSI-oSO-mSI 2nd mUSN 2nd mLSN a ¼ maxfb1 ; b2 g mSO mSI-oSO-mSI mSO 2nd mUSN 2nd mUSN a ¼ minfb1 ; b2 g mSI-oSO-mSI mSO mSI-oSO-mSI 2nd mLSN 2nd mUSN a ¼ minfb1 ; b2 g mSO mSI-oSO-mSI mSO D1 ¼ 0 a ¼ 12B1 3rd order SO 1 mSI-oSO-mSI mSO mSI-oSO-mSI D1 ¼ 0 a ¼ 2B1 3rd order SI ffiffiffiffiffi ffi p Notice that b1;2 ¼ 12ðB1 D1 Þ; D1 ¼ B21  4C1 . Bifurcation-I (B-I): D1 ¼ 0: Bifurcation-II (B-II): a ¼ maxfb1 ; b2 g or a ¼ minfb1 ; b2 g. Bifurcation-III (B-III): D1 ¼ 0 and a ¼ 2AB11 : mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node. mSO: monotonic source, mSI: monotonic sink. mSI-oSO-mSI: from monotonic sink to oscillatory source then to monotonic sink

(i) a0 [ 0 (ii) a0 \0 (iii) a0 [ 0 (iv) a0 \0 (v) a0 [ 0 (vi) a0 \0

Figure 2.3

Table 2.2 Stability and bifurcation of a 1-dimensional cubic nonlinear system (_x ¼ a0 ðx  aÞ½x2 þ B1 ðpÞx þ C1 ðpÞ; a 2 ðminfb1 ; b2 g; maxfb1 ; b2 gÞÞ

112 2 Cubic Nonlinear Discrete Systems

2.2 Period-1 to Period-2 Bifurcation Trees

113

(i)

(ii)

(iii)

( iv )

(v )

(v i)

Fig. 2.4 Stability and bifurcation of three fixed-points in the 1-dimensional, cubic nonlinear discrete system: For a\fb1 ; b2 g: (i) two mUSN bifurcations (a0 [ 0Þ, (ii) two mLSN bifurcations (a0 \0Þ. For a [ fb1 ; b2 g: (iii) two mLSN bifurcations (a0 [ 0Þ, (iv) two mUSN bifurcations (a0 \0Þ. (v) two 3rd order mSO bifurcations (a0 [ 0Þ, (vi) two 3rd mSI bifurcations (a0 \0Þ.Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. (mLSN: monotonic lower-saddle-node; mUSN: monotonic upper-saddle-node; (mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle). The period-2 fixed-points are presented on the period-1 bifurcation trees through red curves

114

2 Cubic Nonlinear Discrete Systems

Table 2.3 Stability and bifurcation of a 1-dimensional cubic nonlinear system (xk þ 1 ¼ xk þ a0 ðxk  aÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ; a 2 ðminfb1 ; b2 g; maxfb1 ; b2 gÞÞ Figure 2.4

a

b1

b2

B-I

B-II

B-III

(i) a0 [ 0 (ii) a0 \0 (iii) a0 [ 0 (iv) a0 \0 (v) a0 \0 (iv) a0 \0

mSO mSI-oSO-mSI mSO mUSN mUSN – mSI-oSO mSO mSI-oSO mLSN mLSN – mSO mSI-oSO-mSI mSO mLSN mLSN – mSI-oSO mSO mSI-oSO mUSN mUSN – mSO mSI-oSO-mSI mSO – – 3rdmSO mSI-oSO-mSI mSO mSI-oSO-mSI – – 3rd mSI ffiffiffiffiffi ffi p Notice that b1;2 ¼ 12ðB1 D1 Þ; D1 ¼ B21  4C1 . Bifurcation-I (B-I): D1 ¼ 0: Bifurcation-II (B-II): a ¼ maxfb1 ; b2 g. Bifurcation-III(B-III): a ¼ minfb1 ; b2 g. (mLSN: monotonic lower-saddle-node; mUSN: monotonic upper-saddle-node; mSI-oSO: monotonic sink to oscillatory source via oscillatory sink.)

parameter are also represented by solid and dashed curves, respectively. The bifurcation diagrams for four cases of three fixed-points are presented in Fig. 2.5 (i)–(vi). If the double repeated fixed-point has an intersection with xk ¼ aðp0 Þ ¼ 12B1 ¼ 3AB : The two triple repeated fixed-points of xk ¼ aðp0 Þ for a0 [ 0 and a0 \0 are the third-order monotonic sink and source bifurcations, respectively. The stability and bifurcation diagrams of fixed-points are shown in Fig. 2.5(v) and (vi). The period-2 fixed-points are sketched through the red curves.

2.3

Higher-Order Period-1 Switching Bifurcations

Consider a 1-dimensional, cubic nonlinear, forward discrete system with a double repeated fixed-point and one simple fixed-point. (i) For b\a; the discrete system is xk þ 1 ¼ xk þ a0 ðpÞðxk  bðpÞÞ2 ðxk  aðpÞÞ;

ð2:59Þ

For such a system, if a0 [ 0; the double repeated fixed-point of xk ¼ b is a monotonic lower-saddle, which is unstable, and the simple fixed-point of xk ¼ a is a monotonic source, which is monotonically unstable. If a0 \0; the double repeated fixed-point of xk ¼ b is a monotonic upper-saddle, which is monotonically unstable, and the simple fixed-point of xk ¼ a is from a monotonic sink to the oscillatory source. Such a fixed-point is from monotonically stable to oscillatorilly unstable. (ii) For b [ a; the 1-dimensional cubic nonlinear, forward discrete system is xk þ 1 ¼ xk þ a0 ðpÞðxk  aðpÞÞðxk  bðpÞÞ2 :

ð2:60Þ

For such a system, if a0 [ 0; the double-repeated fixed-point of xk ¼ b is a monotonic upper-saddle, which is monotonically unstable, and the simple

2.3 Higher-Order Period-1 Switching Bifurcations

115

(i)

(ii)

(iii)

( iv )

(v)

(v i)

Fig. 2.5 Stability and bifurcation of fixed-points in the 1-dimensi, cubic nonlinear discrete system: (i) the LSN (D1 ¼ 0Þ and mUSN (a ¼ maxfb1 ; b2 g) bifurcations (a0 [ 0Þ, (ii) the mUSN (D1 ¼ 0Þ and mLSN (a ¼ maxfb1 ; b2 g) bifurcations (a0 \0Þ. (iii) the USN (D1 ¼ 0Þ and mLSN (a ¼ minfb1 ; b2 g) bifurcations (a0 [ 0Þ, (iv) the mLSN (D1 ¼ 0Þ and USN (a ¼ minfb1 ; b2 g) bifurcations (a0 \0Þ. (v) the third order mSO bifurcation (D1 ¼ 0 and a ¼ bÞ (a0 [ 0Þ, (vi) the third order mSI bifurcation (D1 ¼ 0 and a ¼ bÞ (a0 \0Þ. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. (mLSN: monotonic lower-saddle-node; mUSN: monotonic upper-saddle-node; mSO: monotonic source; mSI: monotonic sink; oSO: oscillatory source; oSI: oscillatory sink; mLS: monotonic lower-saddle; mUS: monotonic upper-saddle). The period-2 fixed-points are presented on the period-1 bifurcation trees through red curves

116

2 Cubic Nonlinear Discrete Systems

fixed-point of xk ¼ a is a monotonic source, which is monotonically unstable. If a0 \0; the double fixed-point of xk ¼ b is a monotonic lower-saddle, which is monotonically unstable, and the simple fixed-point of xk ¼ a is from a monotonic sink to oscillatory source. Such a fixed-point is from monotonically stable to oscillatorilly unstable. (iii) For b ¼ a; the discrete system on the boundary is xk þ 1 ¼ xk þ a0 ðpÞðxk  bðpÞÞ3 :

ð2:61Þ

For such a system, if a0 [ 0; the triple fixed-point of xk ¼ b with the third multiplicity is a source switching bifurcation of the third-order for the (mUS: mSO) to (mSO:mLS) fixed-point. If a0 \0; the triple fixed-point of xk ¼ b with the third multiplicity is a sink switching bifurcation of the third- order for the (mLS:mSI-oSO) to (mSO:mUS) fixed-point. With parameter changes, the bifurcation diagram for the cubic nonlinear system is presented in Fig. 2.6. The acronyms mLSN, mUSN, mSI-oSO, and mSO are for monotonic lower-saddle-node, monotonic upper-saddle-node, monotonic sink to oscillatory source, and monotonic source, respectively. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation point is marked by a circular symbol. To illustrate the stability and bifurcation of fixed-point with singularity in a 1-dimensional, cubic nonlinear system, the fixed-point of xk þ 1 ¼ xk þ a0 ðxk  a1 Þ3 is presented in Fig. 2.7. The third order monotonic sink and source of fixed-points of x ¼ a1 with the third order multiplicity are stable and unstable, respectively. The stable and unstable fixed-points are depicted by solid and dashed curves, respectively. At a0 ¼ 0; the fixed-points with the 3rd order monotonic sink and source are switched, which is marked by a circular symbol.

a0 > 0

xk∗ = a

|| p1 ||

xk∗ = b

a0 < 0

x =b

mSO

mUS

xk∗ || p ||

|| p1 || P-1

∗ k

mLS ∗ k

x =b

∗ k

x =a ab

xk∗ || p ||

P-2 mSI

xk∗ = a oSO

xk∗ = a

mSI

mLS

mSO

oSO oSI

mUS

oSI P-1 P-2

ab

(ii)

Fig. 2.6 Stability and bifurcation of a triple fixed-point with a simple fixed-point in a 1-dimensional, cubic nonlinear discrete system: (i) a 3rd order source switching bifurcation for (mUS:mSO) to (mSO:mLS) switching (a0 [ 0Þ, (ii) a 3rd order monotonic sink switching bifurcation (a0 \0Þ for (mLS:mSI-oSO) to (mSI-oSO:mUS) switching. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The period-2 fixed-points are depicted through red curves

2.4 Direct Cubic Polynomial Discrete Systems Fig. 2.7 Stability of a triple fixed-point in the 1-dimensional, cubic nonlinear, forward discrete system: Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The stability switching is labelled by a circular symbol

|| p 0 ||

xk∗ = a1

xk∗ 3rd order mSI

|| p ||

2.4

117

3rd order mSO

a0 = 0

a0 < 0

a0 > 0

Direct Cubic Polynomial Discrete Systems

For the 1-dimensional, cubic nonlinear, forward discrete systems, the stability and bifurcation of fixed-points can be described through an alternative way as follows. Definition 2.2 Consider a 1-dimensional, cubic nonlinear discrete system xk þ 1 ¼ xk þ AðpÞx3k þ BðpÞx2k þ CðpÞxk þ DðpÞ  a0 ðpÞ½ðxk þ

B 3 Þ þ pðpÞðxk 3A

þ

ð2:62Þ

B Þ þ qðpÞ 3A

where four scalar constants AðpÞ 6¼ 0;BðpÞ;CðpÞ and DðpÞ satisfy A ¼ a0 ; p ¼

C A



B2 ;q 3A2

¼

D BC 2B3  2þ 3 A 3A 27A

ð2:63Þ

p ¼ ðp1 ; p2 ; . . .; pm ÞT : (i) If D¼

q2 p3 þ [ 0; 4 27

ð2:64Þ

the cubic nonlinear discrete system has one fixed-point as pffiffiffiffi q pffiffiffiffi q B xk ¼ a  ð þ DÞ1=3 þ ð  DÞ1=3  2

2

3A

ð2:65Þ

and the corresponding standard form is 1 2

1 4

xk þ 1 ¼ xk þ a0 ðxk  aÞ½ðxk þ B1 Þ2 þ ðD1 Þ ¼ a0 ðxk  aÞ½x2k þ B1 xk þ C1 Þ

ð2:66Þ

118

2 Cubic Nonlinear Discrete Systems

where A ¼ a0 ; B ¼ ða þ B1 Þa0 ; C ¼ ðaB1 þ C1 Þa0 ; D ¼ aa0 C1 :

ð2:67Þ

(ii) If D¼

q2 p3 þ \0 4 27

ð2:68Þ

the cubic nonlinear discrete system has three fixed-points as pffiffiffiffi q pffiffiffiffi q B xk ¼ a ¼ ð þ DÞ1=3 þ ð  DÞ1=3  ; 2 2 3A pffiffiffiffi q pffiffiffiffi q B xk ¼ b1 ¼ xð þ DÞ1=3 þ x2 ð  DÞ1=3  ; 2 2 3A pffiffiffiffi q pffiffiffiffi q B xk ¼ b2 ¼ x2 ð þ DÞ1=3 þ xð  DÞ1=3  ; 2 2 3A pffiffiffi pffiffiffi pffiffiffiffiffiffiffi 1 þ i 3 2 1  i 3 x¼ ;x ¼ ; i ¼ 1: 2

ð2:69Þ

2

The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ

ð2:70Þ

a1 ¼ minðb1 ; b2 ; aÞ; a3 ¼ maxðb1 ; b2 ; aÞ; a2 2 fb1 ; b2 ; ag; a2 6¼ fa1 ; a3 g

ð2:71Þ

where

(iii) If D¼

q2 p3 q2 p3 þ ¼ 0; ¼  6¼ 0 4 27 4 27

ð2:72Þ

the 1-dimensional, cubic nonlinear discrete system has the a double repeated fixed-point with the second multiplicity plus a simple fixed-point as q 2

xk ¼ a ¼ 2ð Þ1=3 

B ; 3A

xk ¼ b1 ¼ b; xk ¼ b2 ¼ b; q 2 pffiffiffi 1 þ i 3 2 ;x 2

q B ; 2 3A pffiffiffi pffiffiffiffiffiffiffi 1  i 3 ; i ¼ 1: 2

b ¼ xð Þ1=3 þ x2 ð Þ1=3  x¼

¼

ð2:73Þ

2.4 Direct Cubic Polynomial Discrete Systems

119

(iii1) The corresponding standard form for a\b is xk þ 1 ¼ xk þ a0 ðxk  aÞðxk  bÞ2 :

ð2:74Þ

Such a discrete flow with the fixed-point of xk ¼ b is called – a monotonic upper-saddle discrete flow of the second-order at a point p ¼ p1 2 @X12 for a0 [ 0; – a monotonic lower-saddle discrete flow of the second-order at a point p ¼ p1 2 @X12 for a0 \0. The bifurcation of fixed-point at xk ¼ b for two different fixed-points appearance or vanishing is called – a monotonic upper-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 for a0 [ 0; – a monotonic lower-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 for a0 \0. The corresponding upper (or lower)-saddle-node appearing bifurcation condition is D¼

q2 p3 q2 p3 þ ¼ 0; ¼  6¼ 0; a\b: 4 27 4 27

ð2:75Þ

(iii2) The corresponding standard form for a [ b is xk þ 1 ¼ xk þ a0 ðxk  bÞ2 ðxk  aÞ:

ð2:76Þ

Such a discrete flow with the fixed-point of xk ¼ b is called – a monotonic point p ¼ p1 – a monotonic point p ¼ p1

lower-saddle discrete flow of the second-order at a 2 @X12 for a0 [ 0; upper-saddle discrete flow of the second-order at a 2 @X12 for a0 \0:

The bifurcation of fixed-point at x ¼ b for two fixed-points appearing or vanishing is called – a monotonic lower-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 for a0 [ 0; – a monotonic upper-saddle-node appearing bifurcation of the second order at a point p ¼ p1 2 @X12 for a0 \0: and the corresponding lower (or upper)-saddle-node appearing bifurcation condition is 1 4

1 27

1 4

1 27

D ¼ q2 þ p3 ¼ 0; q2 ¼  p3 6¼ 0; a [ b:

ð2:77Þ

120

2 Cubic Nonlinear Discrete Systems

(iv) If 1 4

1 27

1 4

1 27

D ¼ q2 þ p3 ¼ 0; q2 ¼  p3 ¼ 0;

ð2:78Þ

the 1-dimensional discrete system has a triple fixed-point as xk ¼ a ¼ 

B  B B ; x ¼ b1 ¼  ; xk ¼ b2 ¼  : 3A k 3A 3A

ð2:79Þ

The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞ3 :

ð2:80Þ

Such a discrete flow at the fixed-point of xk ¼ a is called – a monotonic source discrete flow of the third-order for a0 [ 0; – a monotonic sink discrete flow of the third-order for a0 \0: The bifurcation of fixed-point at xk ¼ 3AB for one fixed-point to three fixed-points is called – a monotonic source switching bifurcation of the third-order at p ¼ p1 2 @X12 for a0 [ 0; – a monotonic sink switching bifurcation of the third-order at p ¼ p1 2 @X12 for a0 [ 0. The corresponding switching bifurcation condition of the third-order source and sink is 1 4

1 27

1 4

1 27

D ¼ q2 þ p3 ¼ 0; q2 ¼  p3 ¼ 0:

ð2:81Þ

From the afore-described stability and bifurcation of the 1-dimensional, cubic nonlinear forward discrete systems, the stability and bifurcations of fixed-point in Eq. (2.62) are similar to Theorem 2.1. The 1-dimensional cubic nonlinear, forward discrete system has the following four cases: (i) One real solution of simple fixed-point of xk ¼ a requires D ¼ ðq2Þ2 þ ðp3Þ3 [ 0 for Eq. (2.62), equivalent to D1 ¼ B21  4C1 \0 for Eq. (2.1). Aa3 þ Ba2 þ Ca þ D ¼ 0:

ð2:82Þ

(ii) Three different solutions of simple fixed-points of xk ¼ a; b1 ; b2 require D\0 for Eq. (2.62), equivalent to D1 ¼ B21  4C1 [ 0 for Eq. (2.1).

2.4 Direct Cubic Polynomial Discrete Systems

121

1 Aa3 þ Ba2 þ Ca þ D ¼ 0 and b1;2 ¼ ðB1

2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B21  4C1 Þ

ð2:83Þ

(iii) The double repeated fixed-point requires D ¼ 0 and ðq2Þ2 ¼ ðp3Þ3 6¼ 0 for Eq. (2.62), equivalent to D1 ¼ B21  4C1 ¼ 0 or a ¼ b1;2 for Eq. (2.1). a ¼ b1;2 ¼ 

ffi B 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2  3AC with B2 [ 3AC: 3A 3A

ð2:84Þ

(iv) The triple repeated fixed-point requires D ¼ 0 and ðq2Þ2 ¼ ðp3Þ3 ¼ 0 for Eq. (2.62), equivalent to D1 ¼ B21  4C1 ¼ 0 and a ¼ b1;2 for Eq. (2.1). a ¼ b1;2 ¼ 

2.5

B with B2 ¼ 3AC: 3A

ð2:85Þ

Forward Cubic Discrete Systems

In this section, the analytical bifurcation scenario will be discussed. The period-doubling bifurcation scenario will be discussed first through nonlinear renormalization techniques, and the bifurcation scenario based on the saddle-node bifurcation will be discussed, which is independent of period-1 fixed-points.

2.5.1

Period-Doubled Cubic Discrete Systems

After the period-doubling bifurcation of a period-1 fixed-point, the period-doubled fixed-points in the cubic discrete system can be obtained. Consider the period-doubling solutions for a forward cubic nonlinear discrete system first. Theorem 2.2 Consider a 1-dimensional cubic nonlinear discrete system as xk þ 1 ¼ xk þ AðpÞx3k þ BðpÞx2k þ CðpÞxk þ DðpÞ ¼ xk þ a0 ðpÞðxk  aðpÞÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ

ð2:86Þ

where four scalar constants AðpÞ 6¼ 0;BðpÞ;CðpÞ and DðpÞ are determined by A ¼ a0 ; B ¼ ða þ B1 Þa0 ; C ¼ ðaB1 þ C1 Þa0 ; D ¼ aa0 C1 ; p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð2:87Þ

122

2 Cubic Nonlinear Discrete Systems

Under D1 ¼ B21  4C1 \0;

ð2:88Þ

the standard form of such a 1-dimensional forward discrete system is xk þ 1 ¼ xk þ a0 ðxk  aÞðx2k þ B1 xk þ C1 Þ:

ð2:89Þ

D1 ¼ B21  4C1 [ 0;

ð2:90Þ

Under

the standard form of such a 1-dimensional forward discrete system is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ:

ð2:91Þ

Thus, a general standard form of such a 1-dimensional cubic discrete system is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ Ax3k þ Bx2k þ Cxk þ D  xk þ a0 ðxk  aÞ½x2k þ B1 xk þ C1  ¼

xk þ a0 *3i¼1 ðxk



ð2:92Þ

ð1Þ ai Þ

where ð1Þ

1 2

a0 ¼ AðpÞ; b1;2 ¼  ðB1 ðpÞ  ð1Þ

pffiffiffiffiffiffiffiffi Dð1Þ Þ for Dð1Þ [ 0;

ð1Þ

ð2Þ

a1 ¼ minfa; b1 ; b2 g; a3 ¼ maxfa; b1 ; b2 g; a1 2 fa; b1 ; b2 g 6¼ fa1 ; a3 g; qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 ð1Þ b1;2 ¼  ðB1 ðpÞ  i jDð1Þ jÞ; i ¼ 1 for Dð1Þ \0 2

ð1Þ a1

ð1Þ

ð1Þ

ð1Þ

ð1Þ

¼ a; a2 ¼ b1 ; a3 ¼ b2 : ð2:93Þ

(i) Consider a forward period-2 discrete system of Eq. (2.86) as ð1Þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk

ð1Þ

ð32 3Þ=2

ðx2k þ Bi2 xk þ Ci2 Þ

ð1Þ

ð32 3Þ=2

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ

xk þ 2 ¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þf1 þ

3

¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þ½a30 *i2 ¼1 ¼ xk þ ½a0 *3j1 ¼1 ðxk  ai1 Þ½a30 *j2 ¼1

3

ð2Þ

ð2Þ

ð1Þ

 ai2 Þg

ð2Þ

ð2Þ

ð2Þ

¼ xk þ a10 þ 3 *3i¼1 ðxk  ai Þ 2

ð2:94Þ

2.5 Forward Cubic Discrete Systems

123

where ð2Þ

ð2Þ

1 2

bi;1 ¼  ðBi þ ð2Þ

Di

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð2Þ ð2Þ ð2Þ ð2Þ Di Þ; bi;2 ¼  ðBi  Di Þ; 2

ð2Þ

ð2Þ

¼ ðBi Þ2  4Ci 0; i 2 Iqð2 Þ ; 0

Iqð2 Þ ¼ flðq1Þ20 m1 þ 1 ; lðq1Þ20 m1 þ 2 ; ; lq20 m1 g; 0

ð2:95Þ

m1 2 f1; 2g; q 2 f1; 2g; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i jDð2Þ jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0;

with fixed-points ð2Þ

xk þ 2 ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 32 Þ 2

ð2Þ

ð1Þ

ð2Þ

ð2Þ

ð2:96Þ

03i¼1 fai g ¼ sortf03j1 ¼1 faj1 g; 03j2 ¼1 fbj2 ;1 ; bj2 ;2 gg ð2Þ

ð2Þ

with ai \ai þ 1 : ð1Þ

(ii) For a fixed-point of xk þ 1 ¼ xk ¼ ai1 (i1 2 f1; 2; 3g), if dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1

ð2:97Þ

with • an oscillatory upper-saddle-node bifurcation (d 2 xk þ 1 =dx2k jx ¼að1Þ [ 0Þ, i1

k

• an oscillatory lower-saddle-node bifurcation (d 2 xk þ 1 =dx2k jx ¼að1Þ \0Þ, i1

k

• a third-order oscillatory sink bifurcation (d 3 xk þ 1 =dx3k jx ¼að1Þ [ 0Þ, k

i1

• a third-order oscillatory source bifurcation (d 3 xk þ 1 =dx3k jx ¼að1Þ \0Þ, k

i1

then the following relations satisfy ð1Þ

1 ð2Þ 2

ð2Þ

ð2Þ

ð2Þ

ai1 ¼  Bi1 ; Di1 ¼ ðBi1 Þ2  4Ci1 ¼ 0;

ð2:98Þ

and there is a period-2 discrete system of the cubic discrete system in Eq. (2.86), as xk þ 2 ¼ xk þ a40 *i

ð2 Þ 1 2Iq

ð1Þ

ð2Þ

ðxk  ai1 Þ3 *3i2 ¼1 ðxk  ai2 Þð1dði1 ;i2 ÞÞ 2

ð2:99Þ

124

2 Cubic Nonlinear Discrete Systems

for i1 2 f1; 2; 3g; i1 6¼ i2 with dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i1 dx2k xk ¼ai1

ð2:100Þ

ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-orderif d 3 xk þ 2 ð1Þ j  ð1Þ ¼ 6a40 P ð20 Þ ðx  ai1 Þ3 i2 2Iq ;i2 6¼i1 k dx3k xk ¼ai1 

2 ð1Þ P3i3 ¼1 ðai1



ð2:101Þ

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ \0;

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the period-2 discrete system; ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a40 dx3k xk ¼ai1

*

ð20 Þ

i2 2Iq

2 *3i3 ¼1 ðað1Þ i1

ð1Þ

;i2 6¼i1



ðxk  ai1 Þ3

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ

ð2:102Þ [ 0;

and the corresponding bifurcations is a third-order monotonic source bifurcation for the period-2 discrete system. (ii1) The period-2 fixed-points are trivial and unstable if ð1Þ

xk þ 2 ¼ xk ¼ ai1 for i1 ¼ 1; 2; 3:

ð2:103Þ

(ii2) The period-2 fixed-points are non-trivial and stable if ð2Þ

ð2Þ

xk þ 2 ¼ xk ¼ bi1 ;1 ; bi1 ;2 for i1 ¼ 1; 2; 3:

ð2:104Þ

Proof The proof is straightforward through the simple algebraic manipulation. Consider Ax2k þ Bxk þ C ¼ 0: Under D ¼ B2  4AC 0; we have a0 ¼ AðpÞ; b1;2

pffiffiffiffi BðpÞ D with b1 \b2 : ¼ 2AðpÞ

2.5 Forward Cubic Discrete Systems

125

Under D ¼ B2  4AC\0; we have a0 ¼ AðpÞ; b1;2

pffiffiffiffiffiffi pffiffiffiffiffiffiffi BðpÞ i jDj ; i ¼ 1: ¼ 2AðpÞ

Thus, we have Ax2k þ Bxk þ C ¼ ðxk  ai2 Þðxk  ai3 Þ: Therefore, xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ: where fa1 ; a2 ; a3 g ¼ sortfa; b1 ; b2 g for real b1;2 ; a1 ¼ a; a2;3 ¼ b1;2 ; and dxk þ 1 jx ¼a ¼ 1 þ a0 *3i1 ¼1;i1 6¼i ðai  ai1 Þ; dxk k i d 2 xk þ 1 X jxk ¼ai ¼ a0 3i1 ¼1;i1 6¼i *3i2 ¼1;i2 6¼i1 ðai  ai2 Þ; dx2k d 3 xk þ 1 jxk ¼ai ¼ a0 : dx3k For real xk þ 1 ¼ xk ¼ ai (i 2 f1; 2; 3g, if dxk þ 1 jx ¼a ¼ 1 þ a0 *3i1 ¼1;i1 6¼i ðai  ai1 Þ ¼ 1; dxk k i with • an oscillatory upper-saddle-saddle bifurcation (d 2 xk þ 1 =dx2k jx ¼ai [ 0Þ, k

• an oscillatory lower-saddle-node bifurcation (d 2 xk þ 1 =dx2k jx ¼ai \0Þ, k • a third-order oscillatory sink bifurcation (d 2 xk þ 1 =dx2k jx ¼ai ¼ 0 and d 3 xk þ 1 = dx3k jx ¼ai [ 0Þ, k k • a third-order oscillatory source bifurcation (d 2 xk þ 1 =dx2k jx ¼ai ¼ 0 and d 3 xk þ 1 = dx3k jx ¼ai \0Þ, k

k

then period-2 fixed-points exists for the cubic discrete system. The period-2 discrete system of the cubic discrete system is

126

2 Cubic Nonlinear Discrete Systems ð1Þ

xk þ 2 ¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk 3

ð1Þ

3

ð32 3Þ=2

¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þ½ða0 Þ3 *i2 ¼1

ð2Þ

ð1Þ

 ai2 Þg ð2Þ

ðx2k þ Bi2 xk þ Ci2 Þ

If ð2Þ

ð2Þ

x2k þ Bi xk þ Ci

¼ 0;

we have bi;1 ¼  ðBi þ

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð2Þ ð2Þ ð2Þ ð2Þ Di Þ; bi;2 ¼  ðBi  Di Þ;

ð2Þ Di

ð2Þ 4Ci

ð2Þ

1 2

¼

ð2Þ

ð2Þ ðBi Þ2



2

0; i 2

0 Iqð2 Þ ;

Iqð2 Þ ¼ flðq1Þ20 m1 þ 1 ; lðq1Þ20 m1 þ 2 ; ; lq20 m1 g; 0

m1 2 f1; 2g; q 2 f1; 2g; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i jDð2Þ jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0; Thus ð2Þ

ð2Þ

x2k þ Bi xk þ Ci

ð2Þ

¼ *2j¼1 ðxk  bi;j Þ;

and ð1Þ

ð32 3Þ=2

xk þ 2 ¼ xk þ a40 *3i1 ¼1 ðxk  ai1 Þ *i2 ¼1 21

ð2Þ

ð2Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ

ð2Þ

¼ xk þ a40 *3i¼1 ðxk  ai Þ where ð2Þ

ð1Þ

ð2Þ

ð2Þ

fai ; i ¼ 1; 2; . . .; 32 g ¼ sortf03i1 ¼1 fai1 g; 03i2 ¼1 fbi2 ;1 ; bi2 ;2 gg: For the period-1 cubic discrete systems, ð1Þ

xk þ 1 ¼ xk þ a0 *3i¼1 ðxk  ai Þ: ð1Þ

The fixed-point of xk ¼ ai1 (i1 2 f1; 2; 3g) is monotonically unstable due to dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ 2 ð1; 1Þ; dxk xk ¼ai1

2.5 Forward Cubic Discrete Systems

127

ð1Þ

and the fixed-point of xk ¼ ai1 (i1 2 f1; 2; 3g) is from monotonically stable to oscillatorilly unstable due to dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ 2 ð1; 1Þ: dxk xk ¼ai1 Under dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1 dxk xk ¼ai1 ð1Þ

ð1Þ

) 2 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 0; • there is a flipped discrete system of the oscillatory upper-saddle of the second order if d 2 xk þ 1 X ð1Þ ð1Þ j  ð1Þ ¼ a0 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ [ 0; dx2k xk ¼ai1 • there is a flipped discrete system of the oscillatory lower-saddle of the second order if d 2 xk þ 1 X ð1Þ ð1Þ j  ð1Þ ¼ a0 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ\0; dx2k xk ¼ai1 • there is a flipped discrete system of the oscillatory sink of the third order if d 2 xk þ 1 d 3 xk þ 1 j j  ð1Þ ¼ a0 [ 0; ð1Þ ¼ 0;  dx2k xk ¼ai1 dx3k xk ¼ai1 • there is a flipped discrete system of the oscillatory source of the third order if d 2 xk þ 1 d 3 xk þ 1 j j  ð1Þ ¼ a0 \0: ð1Þ ¼ 0;  dx2k xk ¼ai1 dx3k xk ¼ai1 The corresponding standard form of the period-2 discrete system becomes xk þ 2 ¼ xk þ a40

*

ð20 Þ

i1 2Iq

ð1Þ

ðxk  ai1 Þ3

32

*i ¼1 ðxk 2

ð2Þ

 ai2 Þð1dði1 ;i2 ÞÞ

with dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i dx2k xk ¼ai ð1Þ

• xk ¼ ai if

for the period-2 discrete system is a monotonic sink of the third-order

128

2 Cubic Nonlinear Discrete Systems

d 3 xk þ 2 j  ð1Þ ¼ 6a40 dx3k xk ¼ai1

*

ð20 Þ

i2 2Iq

ð1Þ

;i2 6¼i1

32

ðxk  ai1 Þ3

ð1Þ

*i ¼1 ðai 1 3

ð2Þ

 ai3 Þð1dði2 ;i3 ÞÞ \0;

ð1Þ

• xk ¼ ai for the period-2 discrete system is a monotonic source of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a40 dx3k xk ¼ai1

*

ð20 Þ

i2 2Iq

ð1Þ

;i2 6¼i1

ðxk  ai1 Þ3

ð1Þ

32

*i ¼1 ðai 1 3

ð2Þ

 ai3 Þð1dði2 ;i3 ÞÞ [ 0: ■

This theorem is proved.

2.5.2

Period-Doubling Renormalization

The generalized cases of period-doublization of cubic discrete systems are presented through the following theorem. The analytical period-doubling trees can be developed for cubic discrete systems. Theorem 2.3 Consider a 1-dimensional cubic nonlinear discrete system as xk þ 1 ¼ xk þ Ax3k þ Bx2k þ Cxk þ D

ð2:105Þ

ð1Þ

¼ xk þ a0 *3i¼1 ðxk  ai Þ:

(i) After l-times period-doubling bifurcations, a period- 2l discrete system ( l ¼ 1; 2; . . .) for the cubic discrete system in Eq. (2.105) is given through the nonlinear renormalization as ð2l1 Þ

32

xk þ 2l ¼ xk þ ½a0  f1 þ ¼

l1

*i1 ¼1 ðxk

ð2

ð2l1 Þ 32l1 3 *i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk

l1

2l1

Þ 3

Þ

ð2l1 Þ

¼ xk þ ½a0

Þ

ð3 32

l1

l1

l

ð2l Þ

Þ

32

l1

l

*i¼1 ðxk

32

Þg

ð2l Þ

Þ=2

l

ð2l Þ

Þ ð2l Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ

*i¼1 ðxk

 ai Þ

ð2l Þ

ðx2k þ Bj2 xk þ Cj2 Þ

ð2l1 Þ

ð32 32 l1

Þ=2

 ai1

*i2 ¼1

ð2l1 Þ 1 þ 32

¼ xk þ ða0

l1

*j1 ¼1

32

ð2l1 Þ

 ai 2

ð2l1 Þ ai1 Þ



2l

*i1 ¼1; ðxk

ð2l1 Þ 32

 ½ða0

¼ x k þ a0

Þ

2l1

ð2l1 Þ 32l1 xk þ ½a0 *i1 ¼1 ðxk

 ½ða0

ð2l1 Þ

 ai1

ð2l Þ

 ai Þ

ð2:106Þ

2.5 Forward Cubic Discrete Systems

129

with l dxk þ 2l ð2l Þ X32l ð2l Þ 32 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l d 2 xk þ 2 l ð2l Þ X32l X32l ð2l Þ 32 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . l d r xk þ 2 l X32l ð2l Þ X32l ð2l Þ 32 ¼ a0 i1 ¼1 ir ¼1;i3 6¼i1 ;i2 ...ir1 *ir þ 1 ¼1;i3 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk l

for r  32 : ð2:107Þ

where ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1 þ 3 ; a0 2l ð2l Þ 03i¼1 fai g

ð2l Þ

1 2

¼

Iqð21

l1

¼ Þ

Þ

l1

; l ¼ 1; 2; 3; ;

2l1 ð2l Þ ð2l Þ ð2l Þ 2 sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð2l Þ

bi;1 ¼  ðBi ð2l Þ Di

ð2l1 Þ 1 þ 32

¼ ða0

ð2l Þ ðBi Þ2

þ

qffiffiffiffiffiffiffiffiffi ð2l Þ ð2l Þ Di Þ; bi;2



ð2l Þ 4Ci

ð2l Þ

,ai qffiffiffiffiffiffiffiffiffi 1 ð2l Þ ð2l Þ ¼  ðBi  Di Þ;

ð2l Þ

 ai þ 1 ;

2

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

l

00Nq22¼1 Iqð22 Þ

¼ flðq1 1Þ2l1 m1 þ 1 ; lðq1 1Þ2l1 m1 þ 2 ; ; lq1 2l1 m1 g f1; 2; ; M1 g0f∅g;

for q1 2 f1; 2; ; N1 g; M1 ¼ N1  2l1 with m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ2l m1 þ 1 ; lðq2 1Þ2l m1 þ 2 ; ; lq2 2l m1 g fM1 þ 1; M1 þ 2; ; M2 g0f∅g; l

l1

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ; ; lM2 g  fM1 þ 1; M1 þ 2; ; M2 g0f∅g

ð2:108Þ

with fixed-points ð2l Þ

l

xk þ 2l ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 32 Þ 2l

2l1

ð2l Þ

ð2l1 Þ

03i¼1 fai g ¼ sortf03i1 ¼1 fai1 ð2l Þ

ð2l Þ

with ai \ai þ 1 :

ð2l Þ

ð2l Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð2:109Þ

130

2 Cubic Nonlinear Discrete Systems ð2l1 Þ

(ii) For a fixed-point of xk þ 2l1 ¼ xk ¼ ai1

ð2l1 Þ

( i1 2 Iq

 f1; 2; . . .; 3ð2

dxk þ 2l1 ð2l1 Þ 32l1 ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 1 þ a0  ai1 Þ ¼ 1; *i2 ¼1;i2 6¼j1 ðai1 x ¼a dxk i1 k d s xk þ 2l1 j ð2l1 Þ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk ¼ai1 d r xk þ 2l1 l1 j  ð2l1 Þ 6¼ 0 for 1\r  32 ; xk ¼ai dxrk 1

l1

Þ

g), if

ð2:110Þ

with • a r th -order oscillatory sink for d r xk þ 2l1 =dxrk j th

• a r -order oscillatory source for d

r

ð2l1 Þ

xk ¼ai

[ 0 and r ¼ 2l1 þ 1;

1

xk þ 2l1 =dxrk j  ð2l1 Þ \0 x ¼a k

i1

2l1 þ 1; • a r th -order oscillatory upper-saddle for d r xk þ 2l1 =dxrk j

r ¼ 2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ 2l1 =dxrk j r ¼ 2l1 ;

ð2l1 Þ

xk ¼ai

and r ¼ [ 0 and

1

ð2l1 Þ

xk ¼ai

\0 and

1

then there is a period- 2l fixed-point discrete system ð2l Þ

x k þ 2 l ¼ x k þ a0

*

ð2l1 Þ i1 2Iq

ð2l1 Þ 3

ðxk  ai1

Þ

32

l

*j2 ¼1 ðxk

ð2l Þ

 aj2 Þð1dði1 ;j2 ÞÞ ð2:111Þ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð2:112Þ

dxk þ 2l d 2 xk þ 2 l j  ð2l1 Þ ¼ 1; j ð2l1 Þ ¼ 0: dxk xk ¼ai1 dx2k xk ¼ai1

ð2:113Þ

and

ð2l1 Þ

• xk þ 2l at xk ¼ ai1

is a monotonic sink of the third-order if

d xk þ 2 l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai 2 Þ 3 3 x ¼a i 2I ;i ¼ 6 i 2 q 2 1 dxk i1 k 3

2l

ð2l1 Þ

 *3j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

; q 2 f1; 2; . . .; N1 gÞ;

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

ð2:114Þ

2.5 Forward Cubic Discrete Systems

131 ð2l1 Þ

and such a bifurcation at xk ¼ ai1 is a third-order monotonic sink bifurcation. ð2l1 Þ • xk þ 2l at xk ¼ ai1 is a monotonic source of the third-order if d 3 x k þ 2l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai2 Þ3 3 x ¼a i 2I ;i ¼ 6 i 2 q 2 1 dxk i1 k 2l

ð2l1 Þ

 *3j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ [ 0

ð2:115Þ

; q 2 f1; 2; . . .; N1 gÞ ð2l1 Þ

and such a bifurcation at xk ¼ ai1 bifurcation.

is a third-order monotonic source

(ii1) The period- 2l fixed-points are trivial if ð2l1 Þ

xk þ 2l ¼ xk ¼ ai1

l1

for i1 ¼ 1; 2; . . .; 32 ;

ð2:116Þ

(ii2) The period- 2l fixed-points are non-trivial if ð2l Þ

ð2l Þ

2 xk þ 2l ¼ xk ¼ 0M j1 ¼1 fbj1 ;1 ; bj1 ;2 g:

ð2:117Þ

Such a period- 2l fixed-point is • monotonically unstable if dxk þ 2l =dxk j

• monotonically invariant if dxk þ 2l =dxk j

2 ð1; 1Þ;

ð2l Þ

xk ¼ai

1 ð2l Þ

xk ¼ai

¼ 1; which is

1

– a monotonic upper-saddle of the ð2l1 Þth order for 1 d 2l1 xk þ 2l =dx2l k jx [ 0; k

1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \0; k

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx [ 0; k

– a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \0; k

• monotonically stable if dxk þ 2l =dxk j

• invariantly zero-stable if dxk þ 2l =dxk j • oscillatorilly stable if dxk þ 2l =dxk j • flipped if dxk þ 2l =dxk j

ð2l1 Þ

xk ¼ai

2 ð0; 1Þ;

ð2l Þ

xk ¼ai

1 ð2l Þ

xk ¼ai ð2l Þ

xk ¼ai

¼ 0;

1

2 ð1; 0Þ;

1

¼ 1; which is

1

– an oscillatory upper-saddle of the ð2l1 Þth order for 1 d 2l1 xk þ 2l =dx2l k jx [ 0; k

– an oscillatory lower-saddle of the ð2l1 Þth order for 1 d 2l1 xk þ 2l =dx2l k jx \0; k

132

2 Cubic Nonlinear Discrete Systems

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx\0; k

– an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, this theorem can be proved. (I) For a cubic discrete system, if the period-1 fixed-points exists, there is a following expression. ð1Þ

xk þ 1 ¼ xk þ a0 *3i1 ¼1 ðxk  ai1 Þ ð1Þ

For xk þ 1 ¼ xk ¼ ai1 (i1 2 f1; 2; 3g, if dxk þ 1 ð1Þ 3 ð1Þ ð1Þ j ð1Þ ¼ 1 þ a 0 *i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk þ 1 ¼ai1 d 2 xkþ1 ð1Þ X3 ð1Þ ð1Þ 3 j ð2Þ ¼ a 0 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðai1  ai4 Þ 6¼ 0 dx2k xk þ 1 ¼ai1 with d s xk þ 1 j ð2Þ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk þ 1 ¼ai1 d r xk þ 1 j ð2Þ 6¼ 0 for 1\r  3; dxrk xk þ 1 ¼ai1 then period-2 fixed-points exists for the cubic discrete system. Thus, consider the corresponding second iteration gives ð1Þ

xk þ 2 ¼ xk þ 1 þ a0 *3i1 ¼1 ðxk þ 1  ai1 Þ: The forward period-2 discrete system of the cubic discrete system is ð1Þ

xk þ 2 ¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þf1 þ ð1Þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk 3

3

ð2Þ

ð2Þ

3 ðx2k þ Bi2 xk þ Ci2 Þ: ¼ xk þ ½a0 *3i1 ¼1 ðxk  ai1 Þ½a30 *3i2 ¼1 2

1

If ð2Þ

ð2Þ

x2k þ Bi2 xk þ Ci2 ¼ 0; we have

ð1Þ

 ai2 Þg

2.5 Forward Cubic Discrete Systems

133

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ 2 2 0 ð2Þ ð2Þ Dð2Þ ¼ ðBi Þ2  4Ci 0; i 2 Iqð2 Þ ;

ð2Þ bi;1

Iqð2 Þ ¼ flðq1Þ20 m1 þ 1 ; lðq1Þ20 m1 þ 2 ; ; lq20 m1 g; 0

m1 2 f1; 2g; q 2 f1; 2g: Thus ð2Þ

ð2Þ

ð2Þ

ð2Þ

x2k þ Bi2 xk þ Ci2 ¼ ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ; and xk þ 2 ¼ xk þ ½a0

*i ¼1 ðxk 1

¼ xk þ a10 þ 3

3

32

ð1Þ

 ai1 Þ½a30

*i¼1 ðxk

ð32 31 Þ=2

*i ¼1 2

ð2Þ

ð2Þ

ðxk  bi;1 Þðxk  bi;2 Þ

ð2Þ

 ai Þ: ð2Þ

For a fixed-point of xk þ 2 ¼ xk ¼ ai1 (i1 2 f1; 2; . . .; 32 g), if dxk þ 2 2 ð2Þ ð2Þ ð2Þ j  ð2Þ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1 with d s xk þ 2 j  ð2Þ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk ¼ai1 d r xk þ 2 j  ð2Þ 6¼ 0 for 1\r  32 ; dxrk xk ¼ai1 then, the forward period-2 discrete system of a cubic discrete system has a period-doubling bifurcation. (II) Such a period-2 discrete system can be renormalized nonlinearly. For k ¼ k1 þ 2; the previous period-2 discrete system becomes ð21 Þ

xk 1 þ 2 þ 2 ¼ x k 1 þ 2 þ a0

32

*i¼1 ðxk1 þ 2

ð21 Þ

 ai

Þ

Because k1 is index for iteration, it can be replaced by k: Thus, an equivalent form for the foregoing equations becomes ð21 Þ

xk þ 22 ¼ xk þ 21 þ a0 with

32

1

*i¼1 ðxk þ 21

ð21 Þ

 ai

Þ:

134

2 Cubic Nonlinear Discrete Systems ð21 Þ

x k þ 2 1 ¼ x k þ a0

ð21 Þ

32

*i¼1 ðxk

 ai

Þ

xk þ 22 can be expressed as ð21 Þ

xk þ 22 ¼ xk þ a0

32

1

*i1 ¼1 ðxk

ð21 Þ

¼ xk þ ða0 Þ1 þ 3

21

ð21 Þ

 ai1 Þf1 þ 32

1

*i1 ¼1 ðxk

32

1

ð21 Þ

*i1 ¼1 ½1 þ a0 2

ð21 Þ

1

ð32 32 Þ=2

 ai1 Þ *i2 ¼1

1

32

*i2 ¼1;i2 6¼i1 ðxk ð22 Þ

ð22 Þ

ð22 Þ

ð221 Þ

for i2 2 0Nq11¼1 Iq1 Iqð21

Þ

21

¼0

00Nq ¼1 Iqð2 Þ with 2 2

2

2

¼ flðq1 1Þ221 m1 þ 1 ; lðq1 1Þ221 m1 þ 2 ; ; lq1 221 m1 g f1; 2; ; M1 g0f∅g;

for q1 2 f1; 2; ; N1 g; M1 ¼ N1  221  m1 ; m1 2 f1; 2g; Iqð22 Þ ¼ flðq2 1Þ22 m1 þ 1 ; lðq2 1Þ22 m1 þ 2 ; ; lq2 22 m1 g 2

fM1 þ 1; M1 þ 2; ; M2 g0f∅g; 2

21

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; then we have 1 ð22 Þ ð22 Þ bi2 ;1 ¼  ðBi2 þ 2 ð22 Þ

Di2

ð22 Þ

qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 1 ð22 Þ ð22 Þ ð22 Þ ð22 Þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ 2 ð22 Þ

¼ ðBi2 Þ2  4Ci2 0

Thus ð22 Þ

ð22 Þ

x2k þ Bi2 xk þ Ci2

ð22 Þ

ð22 Þ

¼ ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ:

and ð2Þ

xk þ 22 ¼ xk þ ða0 Þ1 þ 3

21

32

1

*j1 ¼1 ðxk

 *Nq11¼1 *j

ð221 Þ 2 2Iq1

 *Nq22¼1 *j

ð22 Þ 3 2Iq2

ð21 Þ

 aj 1 Þ ð22 Þ

ð22 Þ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ ð22 Þ

ð22 Þ

ðxk  bj3 ;1 Þðxk  bj3 ;2 Þ ð22 Þ

ð22 Þ

 *j4 2J ð22 Þ ðxk  bj4 ;1 Þðxk  bj4 ;2 Þ ð22 Þ

22

ð22 Þ

¼ xk þ ða0 Þ *3i¼1 ðxk  ai where

Þ

ð22 Þ

½ðx2k þ Bi2 xk þ Ci2 Þ:

If x2k þ Bi2 xk þ Ci2

ð21 Þ

 ai2 Þg

2.5 Forward Cubic Discrete Systems ð2Þ

ð22 Þ

a0 ¼ ða0 Þ1 þ 3 ; a0 22

135 ð221 Þ 1 þ 32

¼ ða0 221

ð2 Þ 2

Þ

21

;

ð2 Þ

ð22 Þ

2

ð22 Þ

ð22 Þ

2 g ¼ sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ,ai qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 1 ð22 Þ 1 ð22 Þ ð22 Þ ð22 Þ ð22 Þ ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

03i¼1 fai ð22 Þ

bi;1

2

ð22 Þ Di

¼

ð22 Þ

 ai þ 1 ;

2

ð22 Þ ðBi Þ2



ð22 Þ 4Ci

0 for i 2 0Nq11¼1 Iqð21 Þ 00Nq22¼1 Iqð22 1

2

Þ

Iqð21 Þ ¼ flðq1 1Þ21 m1 þ 1 ; lðq1 1Þ21 m1 þ 2 ; ; lq1 21 m1 g 1

f1; 2; ; M1 g0f∅g; for q1 2 f1; 2; ; N1 g; M1 ¼ N1  21  m1 ; m1 2 f1; 2g; Iqð22 Þ ¼ flðq2 1Þ22 m1 þ 1 ; lðq2 1Þ22 m1 þ 2 ; ; lq2 22 m1 g 2

fM1 þ 1; M1 þ 2; ; M2 g0f∅g; 2

21

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð22 Þ 1 ð22 Þ ð22 Þ ð22 Þ ð22 Þ ð22 Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð22 Þ ð22 Þ ð22 Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; i 2 J ð2 Þ ¼ flN2 22 m1 þ 1 ; lN2 22 m1 þ 2 ; ; lM2 g 2

 fM1 þ 1; M1 þ 2; ; M2 g0f∅g: Non-trivial period-22 fixed-points are ð22 Þ

xk þ 22 ¼ xk ¼ ai

ð21 Þ

ð21 Þ

2 2 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 g;

1

i 2 f1; 2; . . .; 32 g: and trivial period-22 fixed-points are ð22 Þ

xk þ 22 ¼ xk ¼ ai

221

ð221 Þ

2 03i1 ¼1 fai1

g;

1

i 2 f1; 2; . . .; 32 g: Similarly, the period-2l discrete systems (l ¼ 1; 2; . . .) of the cubic discrete system in Eq. (2.105) can be developed through the above nonlinear renormalization and the corresponding fixed-points can be obtained. (III) Consider a period-2l1 discrete system as ð2l1 Þ

xk þ 2l1 ¼ xk þ a0

32

l1

*i¼1

ð2l1 Þ

ðxk  ai

Þ:

From the nonlinear renormalizations, let k1 ¼ k þ 2l1 , we have

136

2 Cubic Nonlinear Discrete Systems ð2l1 Þ

32

xk1 þ 2l1 þ 2l1 ¼ xk1 þ 2l1 þ a0

l1

*i¼1

ð2l1 Þ

ðxk1 þ 2l1  ai

Þ:

Because k1 is index for iteration, it can be replaced by k: Thus, an equivalent form for the foregoing equations becomes ð2l1 Þ

xk þ 2l ¼ xk þ 2l1 þ a0

32

l1

*i¼1

ð2l1 Þ

ðxk þ 2l1  ai

Þ:

With the period-2l1 discrete system, the foregoing equations becomes ð2l1 Þ

xk þ 2l ¼ xk þ ½a0

32

l1

*i1 ¼1 ðxk

2l1

 f1 þ

ð2

*i1 ¼1 ½1 þ a0 3

ð2

¼ xk þ ½a0

l1

Þ

ð2l1 Þ 3

 ½ða0

Þ

ð2l1 Þ

 ai1

2l1

*i1 ¼1 ðxk 3

2l1

2l

l1

Þ

*i2 ¼1;i2 6¼i1 ðxk 3

ð2

 ai1

ð3 3

*i2 ¼1

Þ

2l1

2l1

l1

Þ

ð2l1 Þ

 ai 2

Þ ð2l Þ

Þ=2

Þg

ð2l Þ

ðx2k þ Bi2 xk þ Ci2 Þ:

If ð2l Þ

ð2l Þ

x2k þ Bi2 xk þ Ci2 ¼ 0 ð2l1 Þ

for i2 2 0Nq11¼1 Iq1 Iqð21

l1

Þ

00Nq ¼1 Iqð2 Þ with 2 2

l

2

¼ flðq1 1Þ2l1 m1 þ 1 ; lðq1 1Þ2l1 m1 þ 2 ; ; lq1 2l1 m1 g f1; 2; ; M1 g0f∅g;

for q1 2 f1; 2; ; N1 g; M1 ¼ N1  2l1  m1 ; m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ2l m1 þ 1 ; lðq2 1Þ2l m1 þ 2 ; ; lq2 2l m1 g fM1 þ 1; M1 þ 2; ; M2 g0f∅g; l

l1

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; then we have 1 ð2l Þ ð2l Þ bi2 ;1 ¼  ðBi2 þ 2 ð2l Þ

ð2l Þ

qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ 2 ð2l Þ

Di2 ¼ ðBi2 Þ2  4Ci2 0: Thus ð2l Þ

ð2l Þ

ð2l Þ

ð2l Þ

x2k þ Bi2 xk þ Ci2 ¼ ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ:

2.5 Forward Cubic Discrete Systems

137

Therefore ð2l1 Þ

32

xk þ 2l ¼ xk þ ½a0

ð2l1 Þ 3

Þ

 ½ða0

l1

*i1 ¼1; ðxk 2l1

ð2l Þ

¼ xk þ a0

Þ

32

2l

ð3 3

*i2 ¼1

ð2l1 Þ 1 þ 32

¼ xk þ ða0

ð2l1 Þ

 ai1

l1

l

2l1

32

Þ=2

l

Þ ð2l Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ

*i¼1 ðxk

ð2l Þ

 ai Þ

ð2l Þ

*i¼1 ðxk

 ai Þ

where ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1þ3 ; a0 2l

ð2l1 Þ 1 þ 32

¼ ða0 2l1

l

ð2 Þ

Þ

l1

; l ¼ 1; 2; 3; ;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

2 03i¼1 fai g ¼ sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ,ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

2

ð2l Þ Di l1

Iqð21

¼ Þ

ð2l Þ ðBi Þ2

ð2l Þ

 ai þ 1 ;

2



ð2l Þ 4Ci

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

00Nq ¼1 Iqð2 Þ ; 2 2

l

2

¼ flðq1 1Þ2l1 m1 þ 1 ; lðq1 1Þ2l1 m1 þ 2 ; ; lq1 2l1 m1 g f1; 2; ; M1 g0f∅g;

for q1 2 f1; 2; ; N1 g; M1 ¼ N1  2l1  m1 ; m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ2l m1 þ 1 ; lðq2 1Þ2l m1 þ 2 ; ; lq2 2l m1 g fM1 þ 1; M1 þ 2; ; M2 g0f∅g; l

l1

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ; ; lM2 g  fM1 þ 1; M1 þ 2; ; M2 g0f∅g: For the period-2l discrete system, we have ð2l Þ

x k þ 2 l ¼ x k þ a0

32

l

*i¼1 ðxk

ð2l Þ

and the local stability and bifurcation at xk ¼ ai determined by

ð2l Þ

 ai Þ l

(i 2 f1; 2; . . .; 32 g) can be

138

2 Cubic Nonlinear Discrete Systems l dxk þ 2l ð2l Þ X32l ð2l Þ 32 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l d 2 xk þ 2 l ð2l Þ X32l X32l ð2l Þ 32 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . l d r xk þ 2 l X32l ð2l Þ X32l ð2l Þ 32 ¼ a0 i1 ¼1 ir ¼1;ir 6¼i1 ;i2 ;...ir1 *ir þ 1 ¼1;i3 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk l

for r  32 ; and the period-doubling bifurcations are determined by dxk þ 2l j ð2l Þ ¼ 1; dxk xk ¼ai d s xk þ 2 l j ð2l Þ ¼ 0; for s ¼ 2; 3; ; r  1; dxsk xk ¼ai d r xk þ 2l l j  ð2l Þ 6¼ 0; for r  32 : r x ¼a dxk i k Nontrivial period-2l fixed-points are ð2l Þ

xk þ 22 ¼ xk ¼ ai

ð2l Þ

ð2l Þ

2 2 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 g;

l

i 2 f1; 2; . . .; 32 g; and trivial period-2l fixed-points are ð2l Þ

xk þ 22 ¼ xk ¼ ai

2l1

ð2l1 Þ

2 03i1 ¼1 fai1

g;

l

i 2 f1; 2; . . .; 32 g: This theorem is proved.

2.5.3

■

Period-n Appearing and Period-Doublization

A period-n discrete system for a cubic nonlinear discrete system will be discussed, and the period-doublization of period-n discrete systems is discussed through the nonlinear renormalization.

2.5 Forward Cubic Discrete Systems

139

Theorem 2.4 Consider a 1-dimensional, forward cubic discrete system as xk þ 1 ¼ xk þ Ax3k þ Bx2k þ Cxk þ D

ð2:118Þ

ð1Þ

¼ xk þ a0 *3i¼1 ðxk  ai Þ:

(i) After n-times iterations, a period-n discrete system for the cubic discrete system in Eq. (2.118) is xk þ n ¼ xk þ a0 *3i1 ¼1 ðxk  ai2 Þf1 þ ¼

ð3n 1Þ=2 3 x k þ a0 *i1 ¼1 ðxk ðnÞ

¼ x k þ a0

3n

*i¼1 ðxk



Xn

j¼1

Qj g

ð3n 3Þ=2 ai1 Þ½*j2 ¼1 ðx2k

ð2l Þ

ð2l Þ

þ Bj2 xk þ Cj2 Þ

ðnÞ

 ai Þ ð2:119Þ

with dxk þ n n ðnÞ X n ðnÞ ¼ 1 þ a0 3i1 ¼1 *3i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk d 2 xk þ n n ðnÞ X n X n ðnÞ ¼ a0 3i1 ¼1 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; dx2k .. . d r xk þ n n X n ðnÞ X n ðnÞ ¼ a0 3i1 ¼1 3ir ¼1;ir 6¼i1 ;i2 ...;ir1 *3ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk

for r  3n ;

ð2:120Þ where n

ðnÞ

a0 ¼ ða0 Þð3 1Þ=2 ; Q1 ¼ 0; Q2 ¼ *3i2 ¼1 ½1 þ a0 Qn ¼

*i ¼1 ½1 þ a0 ð1 þ Qn1 Þ n 3

ðnÞ

n

3 ðxk in1 ¼1;in1 6¼in

ð1Þ

ðnÞ

*i ¼1;i 6¼i ðxk 1 1 2 3

ð1Þ

 ai1 Þ;

ð1Þ

 ain1 Þ; n ¼ 3; 4; ; ðnÞ

03i¼1 fai g ¼ sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2

ðnÞ Di2

¼

ðnÞ ðBi2 Þ2

2



ðnÞ 4Ci2

0 for i2 2 0Nq¼1 IqðnÞ ;

IqðnÞ ¼ flðq1Þn þ 1 ; lðq1Þn þ 2 ; ; lqn gf1; 2; ; Mg0f∅g; for q 2 f1; 2; ; Ng; M ¼ ð3n  3Þ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flNn þ 1 ; lNn þ 2 ; ; lM g  f1; 2; ; Mg0f∅g; with fixed-points

ð2:121Þ

140

2 Cubic Nonlinear Discrete Systems ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 3n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

2 03i¼1 fai g ¼ sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ðnÞ

ð2:122Þ

ðnÞ

with ai \ai þ 1 : ðnÞ

ðnÞ

(ii) For a fixed-point of xk þ n ¼ xk ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), if dxk þ n n ðnÞ ðnÞ ð2l Þ jx ¼aðnÞ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk k i1

ð2:123Þ

d 2 xk þ n n ðnÞ X n ðnÞ ð2l Þ j  ðnÞ ¼ a0 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; dx2k xk ¼ai1

ð2:124Þ

with

then there is a new discrete system for onset of the qth -set of period-n fixed-points based on the second-order monotonic saddle-node bifurcation as ðnÞ

x k þ n ¼ x k þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *3j2 ¼1 ðxk  aj2 Þð1dði1 ;j2 ÞÞ

ð2:125Þ

ðnÞ

ð2:126Þ

n

where ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 : (ii1) If dxk þ n j  ðnÞ ¼ 1 ði1 2 IqðnÞ Þ; dxk xk ¼ai1 d 2 xk þ n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 2a0 *i 2I ðnÞ ;i 6¼i ðai1  ai1 Þ2 1 q 2 1 dx2k xk ¼ai1 ðnÞ

ð2:127Þ

ðnÞ

 *3j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk þ n at xk ¼ ai1 is • a monotonic lower-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ \0; k

i1

• a monotonic upper-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0: k

(ii2) The period-n fixed-points (n ¼ 2n1  mÞ are trivial if

i1

2.5 Forward Cubic Discrete Systems

141

ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f03ii ¼1 fai1 g; 03i2 ¼1 fai2 gg n for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 3 g0f£g for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f03i2 ¼1 fai2

)

)

gg

ð2:128Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 3n g0f£g for n ¼ 2n2 :

(ii3) The period-n fixed-points (n ¼ 2n1  m) are non-trivial if ) 2n1 1 m ðnÞ ð1Þ ð2n1 1 mÞ xk ¼ xk þ n ¼ aj1 62 f03ii ¼1 fai1 g; 03i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 3n g0f£g

for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

gg xk ¼ xk þ n ¼ aj1 62 f03i2 ¼1 fai2 for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 3n g0f£g

)

ð2:129Þ

for n ¼ 2n2 : Such a period-n fixed-point is • monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk þ n =dxk jx ¼aðnÞ ¼ 1; which is i1

k

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jxk [ 0; 1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

1 þ1 – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l jx k k [ 0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx \0; k

• monotonically stable if dxk þ n =dxk jx ¼aðnÞ 2 ð0; 1Þ; i1

k

• invariantly zero-stable if dxk þ n =dxk jx ¼aðnÞ ¼ 0; k

i1

• oscillatorilly stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 0Þ; k

i1

• flipped if dxk þ n =dxk jx ¼aðnÞ ¼ 1; which is k

i1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jxk [ 0; 1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

142

2 Cubic Nonlinear Discrete Systems

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n = dxk2l1 þ 1 jx \0; k

– an oscillatory sink the ð2l1 þ 1Þth order for 1 þ1 d 2l1 þ 1 xk þ n =dx2l jx [ 0; k k

• oscillatorilly unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ: i1

k

ðnÞ

ðnÞ

(iii) For a fixed-point of xk þ n ¼ xk ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), there is a period-doubling of the qth -set of period-n fixed-points if dxk þ n n ðnÞ ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *3j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; dxk k i1 d s xk þ n j  ðnÞ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk ¼ai1 d r xk þ n j  ðnÞ 6¼ 0 for 1\r  3n dxrk xk ¼ai1

ð2:130Þ

with • a r th -order oscillatory source for d r xk þ n =dxrk jx ¼aðnÞ \ 0 and r ¼ 2l1 þ 1; i1

k

• a r th -order oscillatory sink for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ 2l1 þ 1; i1

k

• a r th -order oscillatory upper-saddle for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ i1

k

2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ 2l1 . k

i1

The corresponding period- 2  n discrete system of the cubic discrete system in Eq. (2.118) is ð2nÞ

xk þ 2n ¼ xk þ a0

*i ¼I ðnÞ ðxk 1 q

ðnÞ

2n

ð2nÞ ð1dði1 ;j2 ÞÞ

 ai1 Þ3 *3j2 ¼1 ðxk  aj2

Þ

ð2:131Þ with dxk þ 2n d 2 xk þ 2n jx ¼aðnÞ ¼ 1; jx ¼aðnÞ ¼ 0; i1 i1 k k dxk dx2k d 3 xk þ 2n ð2nÞ ðnÞ ðnÞ 3 jx ¼aðnÞ ¼ 6a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 2 q 2 1 i1 k dx3k 2n

ðnÞ

ð2nÞ ð1dði1 ;j2 ÞÞ

 *3j2 ¼1 ðai1  aj2

Þ

:

ð2:132Þ

2.5 Forward Cubic Discrete Systems

143

ðnÞ

ðnÞ

Thus, xk þ 2n at xk ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic sink of the third-order if d 3 xk þ 2n =dx3k jx ¼aðnÞ \0; k

i1

• a monotonic source of the third-order if d 3 xk þ 2n =dx3k jx ¼aðnÞ [ 0: k

i1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l  n discrete system of the cubic discrete system in Eq. (1.118) is ð2l1 nÞ

xk þ 2l n ¼ xk þ ½a0  f1 þ

ð2l1 nÞ

ð2l1 nÞ 3

¼

ð2l1 nÞ

ðxk  ai1

Þ

3

2l1 n

*i1 ¼1 2l1 n

ð2l1 nÞ

ðxk  ai1

l l1 ð32 n 32 n Þ=2

*j1 ¼1

ð2l1 nÞ 32l1 n xk þ ½a0 *i1 ¼1 ðxk ð2l1 nÞ 3

Þ

 ½ða0

2l1 n

ð3

*j2 ¼1

¼

Þ

l1 n

ð2l nÞ 32l n xk þ a0 *i¼1 ðxk



ð2l1 nÞ

 ai2

Þg

Þ ð2l nÞ

ðx2k þ Bj2

ð2l nÞ

x k þ Cj 2

Þ

ð2l1 nÞ ai1 Þ



2l n

ð2l1 nÞ 1 þ 32

¼ xk þ ða0

Þ

l1 ð2l1 Þ 32l1 n 32 n *i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk

¼ xk þ ½a0  ½ða0

l1 n

32

*i1 ¼1

l1 n

32 l

32 n

*i¼1

Þ=2

ð2l nÞ

ð2l nÞ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ ð2l nÞ

ðxk  ai

Þ

ð2l nÞ ai Þ

ð2:133Þ with dxk þ 2l n ð2l nÞ X32l n 32l n ð2l nÞ ¼ 1 þ a0 Þ; i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 dxk l d 2 xk þ 2l n ð2l nÞ X32l n X32l n ð2l nÞ 32 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 2 dxk .. . l d r xk þ 2l n X32l n ð2l nÞ X32l n ð2l nÞ 32 n ¼ a0 i1 ¼1 ir ¼1;ir 6¼i1 ;i2 ...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk l

for r  32 n ; ð2:134Þ where

144

2 Cubic Nonlinear Discrete Systems ð2nÞ

ðnÞ

2l n

2n

ð2 nÞ

ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l nÞ

l1

ð2

Þ

l1

g ¼ sortf03i1 ¼1 fai1 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; 2 ð2l nÞ 2

¼ ðBi

nÞ

ð2l nÞ

Þ  4Ci

for i 2 0Nq11¼1 Iqð21 Iqð21

ð2l1 nÞ 1 þ 32

¼ ða0

2l1 n

l

03i¼1 fai

Di

ð2l nÞ

¼ ða0 Þ1 þ 3 ; a0

a0

l1

nÞ

nÞ

l1 n

; l ¼ 1; 2; 3; . . .; ð2l nÞ

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

0

00Nq ¼1 Iqð2 nÞ ; l

2 2

2

¼ flðq1 1Þð2l1 nÞ þ 1 ; lðq1 1Þð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f£g;

ð2:135Þ

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  n; l

Iqð22 nÞ ¼ flðq2 1Þð2l nÞ þ 1 ; lðq2 1Þð2l nÞ þ 2 ; . . .; lq2 ð2l nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f£g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ ð32 n  32 n Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi l l l ð2 nÞ ð2 nÞ 2 ð2 nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di i 2 flNð2l1 nÞ þ 1 ; lNð2l1 nÞ þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f£g with fixed-points ð2l nÞ

xk þ 2l n ¼ xk ¼ ai 2l n

ð2l nÞ

03i¼1 fai l

ð2 nÞ

with ai

l

; ði ¼ 1; 2; . . .; 32 n Þ 2l1 n

g ¼ sortf03i¼1

ð2l1 nÞ

fai

ð2l1 nÞ

2 g; 0M i¼1 fbi;1

ð2l1 nÞ

; bi;2

gg

l

ð2 nÞ

\ai þ 1 : ð2:136Þ l1

(v) For a fixed-point of xk þ ð2l nÞ ¼ xk ¼ aið21

nÞ

there is a period- ð2l  nÞ discrete system if

(i1 2 Iqð2

l1

nÞ

; q 2 f1; 2; . . .; N1 g),

2.5 Forward Cubic Discrete Systems

145

dxk þ 2l1 n ð2l1 nÞ 32l1 n ð2l1 nÞ ð2l1 nÞ j  ð2l1 nÞ ¼ 1 þ a0  ai2 Þ ¼ 1; *i2 ¼1;i2 6¼i1 ðai1 xk ¼ai dxk 1 d s xk þ 2l1 n j  ð2l1 nÞ ¼ 0; for s ¼ 2; . . .; r  1; xk ¼ai dxsk 1 r d xk þ 2l1 n l1 j  ð2l1 nÞ 6¼ 0 for 1\r  32 n r x ¼a dxk i1 k ð2:137Þ with • a r th -order oscillatory sink for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

2l1 þ 1; • a r th -order oscillatory source for d r xk þ 2l n =dxrk j

[ 0 and r ¼

1

ð2l nÞ

xk ¼ai

\0 and r ¼

1

2l1 þ 1; • a r th -order oscillatory upper-saddle for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

r ¼ 2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ 2l n =dxrk j r ¼ 2l1 ;

[ 0 and

1

ð2l nÞ

xk ¼ai

\0 and

1

The corresponding period- 2l  n discrete system is ð2l nÞ

xk þ 2l n ¼ xk þ a0 

l

32 n *j2 ¼1 ðxk

*

ð2l1 nÞ i1 2Iq



ð2l1 nÞ 3

ðxk  ai1

Þ

ð2:138Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ aj 2 Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð2:139Þ with dxk þ 2l n d 2 xk þ 2l n j  ð2l1 nÞ ¼ 1; j  ð2l1 nÞ ¼ 0; xk ¼ai xk ¼ai dxk dx2k 1 1 d 3 xk þ 2l n ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j  ð2l1 Þ ¼ 6a0 ðai1  ai 2 Þ * ð2l1 nÞ 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k 2l n

ð2l1 nÞ

 *3j2 ¼1 ðai1 ði1 2 Iqð2

l1

nÞ

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

6¼ 0

; q 2 f1; 2; . . .; N1 gÞ: ð2:140Þ

146

2 Cubic Nonlinear Discrete Systems ð2l1 nÞ

xk þ 2l n at xk ¼ ai1

is

• a monotonic sink of the third-order for d 3 xk þ 2l n =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order for d 3 xk þ 2l n =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0:

1

(v1) The period- 2l  n fixed-points are trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

2 f03ii ¼1 fai1 g; 03i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 2ð2 nÞ for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð2l1 nÞ

2 03i2 ¼1 fai2

g

)

l

for j ¼ 1; 2; . . .; 32 n for n ¼ 2n1 : ð2:141Þ l

(v2) The period- 2  n fixed-points are non-trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

62 f03ii ¼1 fai1 g; 03i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 32 n for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð2l1 nÞ

62 03i2 ¼1 fai2

g

)

l

for j ¼ 1; 2; . . .; 32 n for n ¼ 2n1 : ð2:142Þ Such a period- 2l  n fixed-point is • monotonically unstable if dxk þ 2l n =dxk j  ð2l nÞ 2 ð0; 1Þ; x ¼ai • monotonically invariant if dxk þ 2l n =dxk j k  1ð2l nÞ ¼ 1; which is xk ¼ai th

1

1 – a monotonic upper-saddle of the ð2l1 Þ order for d 2l1 xk þ 2l n = dx2l k jxk [ 0 (independent ð2l1 Þ-branch appearance); 1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0 (independent ð2l1 Þ-branch appearance) 1 þ1 jx – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l k k [ 0 (dependent ð2l1 þ 1Þ-branch appearance from one branch); 1 þ1 jx – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n = dx2l k k \0 (dependent ð2l1 þ 1Þ-branch appearance from one branch);

2.5 Forward Cubic Discrete Systems

147

• monotonically stable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• invariantly zero-stable if dxk þ 2l n =dxk j • oscillatorilly stable if dxk þ 2l n =dxk j • flipped if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð0; 1Þ;

1 ð2l1 nÞ

xk ¼ai ð2l nÞ

xk ¼ai

1

¼ 0;

2 ð1; 0Þ;

1

¼ 1; which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk [ 0; 1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0; – an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n = dxk2l1 þ 1 jx \0; k

1 þ1 – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l jx k k [ 0;

• oscillatorilly unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, the proof of this theorem is similar to the proof of Theorem 2.3. This theorem can be easily proved. ■

2.5.4

Sampled Period-n Appearing Bifurcations

Consider a period-n discrete system of the cubic discrete system as ðnÞ

x k þ n ¼ x k þ a0

3n

*i¼1 ðxk

ðnÞ

 ai Þ

ð2:143Þ

n

ðnÞ

where a0 ¼ ða0 Þð3 1Þ=2 . For n ¼ 1; Eq. (2.143) gives a period-1 discrete system of the cubic system as ð1Þ

xk þ 1 ¼ x k þ a0

*i¼1 ðxk 3

ð1Þ

 ai Þ:

ð2:144Þ

ð1Þ

• If two of ai (i ¼ 1; 2; 3Þ are complex, only one fixed-point exists in such a cubic discrete system. ð1Þ • If ai (i ¼ 1; 2; 3Þ are real, three fixed-points exist in such a cubic discrete system. For n ¼ 2; equation (2.143) gives a period-2 discrete system of the cubic system as

148

2 Cubic Nonlinear Discrete Systems ð2Þ

x k þ n ¼ x k þ a0

32

*i¼1 ðxk

ð2Þ

 ai Þ:

ð2:145Þ

ð2Þ

• If eight of ai (i ¼ 1; 2; . . .; 32 ) are complex, the period-2 discrete system has only one trivial fixed-point. ð2Þ • If three of ai (i ¼ 1; 2; . . .; 32 ) are real, the period-2 discrete system possesses three fixed-points. One fixed-point is trivial from period-1 and two fixed-points are for period-2, or the three fixed-points are the same as the period-1 fixed-points. Such two non-trivial fixed points are generated through period-doubling bifurcation. ð2Þ • If five of ai (i ¼ 1; 2; . . .; 32 ) are real, the period-2 discrete system possesses five fixed-points, including three trivial fixed-points for period-1 and two non-trivial fixed-points for period-2. Such two non-trivial fixed points are generated through period-doubling bifurcation, and both of fixed-points are stable for period-2. ð2Þ • If nine of ai (i ¼ 1; 2; . . .; 32 ) are real, the period-2 discrete system possesses nine fixed-points, including three trivial fixed-points for period-1 and six non-trivial fixed-points for period-2. Such six non-trivial fixed points are generated through different period-doubling bifurcation. Two sets of period-2 fixed-points are stable, and one set of period-2 fixed-points is unstable. With three unstable trivial period-2 fixed-points, we have five unstable fixed points. Thus, the period-2 discrete system of the cubic nonlinear discrete system has three sets of period-2 fixed-points on the period-1 to period-2 period-doubling bifurcation tree. No any independent period-2 fixed-points exists. For numerical simulations, one set of stable asymmetric period-2 fixed points can be obtained. For n ¼ 3; Eq. (2.143) gives a period-3 discrete system of the cubic nonlinear discrete system as ð3Þ

xk þ n ¼ x k þ a0 ð3Þ

33

*i¼1 ðxk

ð3Þ

 ai Þ:

ð2:146Þ

• If one of ai (i ¼ 1; 2; . . .; 33 ) is real, the period-3 discrete system does have one trivial fixed-point from period-1. ð3Þ • If three of ai (i ¼ 1; 2; . . .; 33 ) is real, the period-3 discrete system does have three trivial fixed-points from period-1. ð3Þ • If nine of ai (i ¼ 1; 2; . . .; 33 ) are real, the period-3 discrete system possesses three trivial fixed-points and one set of six period-3 fixed points. ð3Þ • If all of ai (i ¼ 1; 2; . . .; 33 ) are real, the period-3 discrete system possesses 27 fixed-points, including three trivial fixed-points for period-1 and 4 sets of non-trivial fixed-points. Such non-trivial fixed points are generated through the

2.5 Forward Cubic Discrete Systems

149

monotonic upper-saddle or monotonic lower-saddle bifurcations. The period-3 fixed-points are independent of the trivial fixed-points for period-1. Thus, the period-3 discrete system has at most four sets of period-3 fixed-points, which are independent of the period-1fixed-points. For n ¼ 4; Eq. (2.143) gives a period-4 discrete system of the cubic system as ð4Þ

xk þ 4 ¼ x k þ a0

34

*i¼1 ðxk

ð4Þ

 ai Þ:

ð2:147Þ

ð4Þ

• If one of ai (i ¼ 1; 2; . . .; 34 ) are real, the period-4 discrete system does have one trivial fixed-point from period-1. ð4Þ • If three of ai (i ¼ 1; 2; . . .; 34 ) are real, the period-4 discrete system possesses three trivial fixed-points from period-1 or period-2. ð4Þ • If nine of ai (i ¼ 1; 2; . . .; 34 ) are real, the period-4 discrete system possesses nine trivial fixed-points which are the same as the period-1 and period-2 fixed-points. ð4Þ • If 17 of ai (i ¼ 1; 2; . . .; 34 ) are real, the period-4 discrete system possesses eight fixed-points, including three trivial fixed-points for period-1, six trivial fixed-points for period-2, and eight non-trivial fixed-points for period-4. Such non-trivial fixed points are stable, which are generated through the period-doubling bifurcations from 4 period-2 branches. All trivial fixed-points for period-4 are unstable. ð4Þ • If all of ai (i ¼ 1; 2; . . .; 34 ) are real, in addition to the period-4 fixed-points by the period-doubling bifurcation, the period-4 discrete system possesses eight sets of non-trivial period-4 fixed-points, which are generated by the monotonic upper-saddle or lower-saddle bifurcations. The period-4 fixed-points are independent of the trivial fixed-points. Thus, the period-4 discrete system has at most nine sets of period-4 fixed-points, two sets are dependent on the period-1 to period-4 period-doubling trees, and eights set of period-4 fixed-points are independent of the period-1 to period-4 period-doubling bifurcation trees. Similarly, other period-n discrete systems can be discussed. From the previous discussion, the period-n fixed-points for a cubic discrete system are tabulated in Table 2.4. The dependent sets of period-n fixed-points are on the period-doubling bifurcation trees. The independent sets of period-n fixed-points are generated through monotonic saddle-node bifurcations. From analytical expressions, the maximum sets of period-n fixed-points includes dependent and independent sets of period-n fixed-points. In addition to the period-1 trivial fixed-points, other fixed-points on the bifurcation trees relative to period-n fixed points are also trivial.

150

2 Cubic Nonlinear Discrete Systems

Table 2.4 Period-n fixed-points for a cubic discrete system Dependent sets P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12

2.6

Independent sets

N/A (1)P-1 to P-2 N/A (2)P-1 to P-4 N/A (4)P-3 to P-6 N/A (2)P-1 to P-8 (8)P-4 to P-8 N/A (24)P-5 to P-10 N/A (4)P-3 to P-12 (57)P-6 to P-12

Maximum sets

3 1-3 4 8 24 57 156 401

3 3 4 10 24 61 156 411

1093 2928

1093 2952

8052 22082

8052 22143

Trivial fixed-points N/A (3)P-1 (3)P-1 (2)P-1 to P-2 (3)P-1 (3)P-1, (4)P-3 (3)P-1 (2)P-1 to P-4 (8)P-4 (3)P-1 (3)P-1 (24)P-5 (1)P-1 (3)P-1 (4)P-3 to P-6 (57)P-6

Backward Cubic Nonlinear Discrete Systems

In this section, the analytical bifurcation trees for backward cubic nonlinear discrete systems will be discussed, as in a similar fashion, through nonlinear renormalization techniques, and the bifurcation scenario based on the monotonic saddle-node bifurcations will be discussed, which is independent of period-1 fixed-points.

2.6.1

Backward Period-2 Cubic Discrete Systems

After the period-doubling bifurcation of a period-1 fixed-point in the backward cubic nonlinear discrete system, the period-doubled fixed-points can be obtained. Consider the period-doubling solutions for a backward cubic nonlinear discrete system as follows. Theorem 2.5 Consider a 1-dimensional backward cubic nonlinear discrete system as xk ¼ xk þ 1 þ AðpÞx3k þ 1 þ BðpÞx2k þ 1 þ CðpÞxk þ 1 þ DðpÞ ¼ xk þ 1 þ a0 ðpÞðxk þ 1  aðpÞÞ½x2k þ 1 þ B1 ðpÞxk þ 1 þ C1 ðpÞ

ð2:148Þ

where four scalar constants AðpÞ 6¼ 0;BðpÞ;CðpÞ and DðpÞ are determined by A ¼ a0 ; B ¼ ða þ B1 Þa0 ; C ¼ ðaB1 þ C1 Þa0 ; D ¼ aa0 C1 ; p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð2:149Þ

2.6 Backward Cubic Nonlinear Discrete Systems

151

Under D1 ¼ B21  4C1 \0;

ð2:150Þ

the standard form of such a 1-dimensional backward cubic discrete system is xk ¼ xk þ 1 þ a0 ðxk þ 1  aÞðx2k þ 1 þ B1 xk þ 1 þ C1 Þ:

ð2:151Þ

D1 ¼ B21  4C1 [ 0;

ð2:152Þ

Under

the standard form of such a backward cubic discrete system is xk ¼ xk þ 1 þ a0 ðxk þ 1  a1 Þðxk þ 1  a2 Þðxk þ 1  a3 Þ:

ð2:153Þ

Thus, a general standard form of such a backward cubic discrete system is xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ Ax3k þ 1 þ Bx2k þ 1 þ Cxk þ 1 þ D  xk þ 1 þ a0 ðxk þ 1  aÞ½x2k þ 1 þ B1 xk þ 1 þ C1 Þ ¼ xk þ 1 þ a0 *3i¼1 ðxk þ 1 

ð2:154Þ

ð1Þ ai Þ

where ð1Þ

1 2

a0 ¼ AðpÞ; b1;2 ¼  ðB1 ðpÞ  ð1Þ

pffiffiffiffiffiffiffiffi Dð1Þ Þ for Dð1Þ [ 0;

ð1Þ

ð2Þ

a1 ¼ minfa; b1 ; b2 g; a3 ¼ maxfa; b1 ; b2 g; a1 2 fa; b1 ; b2 g 6¼ fa1 ; a3 g; pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 ð1Þ b1;2 ¼  ðB1 ðpÞ  i Dð1Þ Þ; i ¼ 1 for Dð1Þ \0

ð2:155Þ

2

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

a1 ¼ a; a2 ¼ b1 ; a3 ¼ b2

(i) Consider a backward period-2 discrete system of Eq. (2.148) as ð1Þ

xk ¼ xk þ 2 þ ½a0 *3i1 ¼1 ðxk þ 2  ai1 Þ  f1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2 3

3

ð1Þ

ð1Þ

 ai2 Þg ð2Þ

ð2Þ

¼ xk þ 2 þ ½a0 *3i1 ¼1 ðxk þ 2  ai1 Þ½a30 *3i2 ¼1 ðx2k þ 2 þ Bi2 xk þ 2 þ Ci2 Þ ð1Þ

ð2Þ

3 ¼ xk þ 2 þ ½a0 *3j1 ¼1 ðxk þ 2  ai1 Þ½a30 *j32 ¼1 ðxk þ 2  bj2 Þ 2

ð2Þ

¼ xk þ 2 þ a10 þ 3 *3i¼1 ðxk þ 2  ai Þ 2

ð2:156Þ where

152

2 Cubic Nonlinear Discrete Systems

1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ ð2Þ bi;1 ¼  ðBi þ Dð2Þ Þ; bi;2 ¼  ðBi  2 2 0 ð2Þ ð2Þ ð2Þ Di ¼ ðBi Þ2  4Ci 0; i 2 Iqð2 Þ ;

qffiffiffiffiffiffiffiffi ð2Þ Di Þ;

Iqð2 Þ ¼ flðq1Þ20 m1 þ 1 ; lðq1Þ20 m1 þ 2 ; ; lq20 m1 g; 0

ð2:157Þ

m1 2 f1; 2g; q 2 f1; 2g; qffiffiffiffiffiffiffiffi 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i Dð2Þ Þ; bi;2 ¼  ðBi  i Di Þ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0;

with backward fixed-points ð2Þ

xk ¼ xk þ 2 ¼ ai ; ði ¼ 1; 2; . . .; 32 Þ ð2Þ

2

ð1Þ

ð2Þ

ð2Þ

ð2:158Þ

03i¼1 fai g ¼ sortf03j1 ¼1 faj1 g; 03j2 ¼1 fbj2 ;1 ; bj2 ;2 gg ð2Þ

ð2Þ

with ai \ai þ 1 : ð1Þ

(ii) For a backward fixed-point of xk þ 1 ¼ xk ¼ ai1 (i1 2 f1; 2; 3g), if dxk ð1Þ ð1Þ 3 j ð1Þ ¼ 1 þ a0 * i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk þ 1 xk þ 1 ¼ai1

ð2:159Þ

with • an oscillatory lower-saddle-saddle bifurcation (d 2 xk =dx2k þ 1 jx

ð1Þ

kþ1

• an oscillatory upper-saddle-node bifurcation (d

2

xk =dx2k þ 1 jx

ð1Þ

kþ1

• a third-order oscillatory source bifurcation (d 3 xk =dx3k þ 1 jx • a third-order oscillatory sink bifurcation (d

xk =dx3k þ 1 jx

kþ1

¼ai ð1Þ

kþ1

3

¼ai

¼ai

\0),

1

[ 0),

1

ð1Þ

¼ai

[ 0),

1

\0),

1

then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi1 ; Di1 ¼ ðBi1 Þ2  4Ci1 ¼ 0; 2

ð2:160Þ

and there is a backward period-2 discrete system of the cubic discrete system in Eq. (2.148) as xk ¼ xk þ 2 þ a40

*

ð20 Þ

i1 2Iq

ð1Þ

ðxk þ 2  ai1 Þ3

for i1 2 f1; 2; 3g; i1 6¼ i2 with

33

*i ¼1 ðxk þ 2 2

ð2Þ  ai2 Þð1dði1 ;i2 ÞÞ ð2:161Þ

2.6 Backward Cubic Nonlinear Discrete Systems

153

dxk d 2 xk jx ¼að1Þ ¼ 1; 2 jx ¼að1Þ ¼ 0; dxk þ 2 k þ 2 i1 dxk þ 2 k þ 2 i1

ð2:162Þ

ð1Þ

• xk at xk þ 2 ¼ ai1 is a monotonic source of the third-order if d 3 xk j  ð1Þ ¼ 6a40 dx3k þ 2 xk ¼ai1

*

ð20 Þ

i1 2Iq

ð1Þ

;i2 6¼i1

ð1Þ

ðai1  ai2 Þ3

ð2:163Þ

ð2Þ ð1dði2 ;i3 ÞÞ  *3i3 ¼1 ðað1Þ \0; i1  ai3 Þ 3

and the corresponding bifurcations is a third-order monotonic source bifurcation for the backward period-2 discrete system; ð1Þ • xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-order if d 3 xk j  ð1Þ ¼ 6a40 dx3k þ 2 xk ¼ai1 

*

ð20 Þ

i1 2Iq

ð1Þ

;i2 6¼i1

ð1Þ 33 *i ¼1 ðai1 3



ð1Þ

ðai1  ai2 Þ3

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ

ð2:164Þ [ 0;

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the backward period-2 discrete system. (ii1) The backward period-2 fixed-points are trivial and unstable if ð1Þ

xk ¼ xk þ 2 ¼ ai1 for i1 ¼ 1; 2; 3:

ð2:165Þ

(ii2) The backward period-2 fixed-points are non-trivial and stable if ð2Þ

ð2Þ

xk ¼ xk þ 2 ¼ bi1 ;1 ; bi1 ;2 for i1 ¼ 1; 2; 3:

ð2:166Þ

Proof Following the corresponding proof for the forward cubic discrete system. This theorem can be proved. ■

2.6.2

Backward Period-Doubling Renormalization

The generalized cases of period-doublization of backward cubic discrete systems are presented through the following theorem. The analytical backward period-doubling trees can be developed for backward cubic discrete systems.

154

2 Cubic Nonlinear Discrete Systems

Theorem 2.6 Consider a 1-dimensional backward cubic discrete system as xk ¼ xk þ 1 þ AðpÞx3k þ 1 þ BðpÞx2k þ 1 þ CðpÞxk þ 1 þ DðpÞ

ð2:167Þ

ð1Þ

¼ xk þ 1 þ a0 *3i¼1 ðxk þ 1  ai Þ:

(i) After l-times backward period-doubling bifurcations, a backward period- 2l discrete system (l ¼ 1; 2; . . .) for the backward cubic discrete system in Eq. (2.167) is given through the nonlinear renormalization as ð2l1 Þ

xk ¼ xk þ 2l þ ½a0  f1 þ

3

ð2l1 Þ 2

Þ

l1

Þ

2l1

ð2l1 Þ

¼ xk þ 2l þ ½a0

ð2l1 Þ 22

Þ

ð2l Þ

with for r  22

Þ

3

3

2l

2l1

ð3 3

*j1 ¼1 32

Þ=2

l1

*i1 ¼1; ðxk þ 2l l

ð32 32

l1

*i2 ¼1

Þ

32

l

l1

Þ=2 32

ð2

 ai1

l1

Þ

ð2l1 Þ

 ai 2

Þg

Þ ð2l Þ

ð2l Þ

ðx2k þ 2l þ Bj2 xk þ 2l þ Cj2 Þ ð2l1 Þ

 ai1

Þ ð2l Þ

ð2:168Þ

ð2l Þ

ðxk þ 2l  bi2 ;1 Þðxk þ 2l  bi2 ;2 Þ

l

*i¼1 ðxk þ 2l

*i¼1 ðxk þ 2l

Þ

*i2 ¼1;i2 6¼i1 ðxk þ 2l

*i1 ¼1 ðxk þ 2l

ð2l1 Þ 1 þ 32

¼ xk þ 2 l þ a0

ð2l1 Þ

 ai1 2l1

2l1

l1

¼ xk þ 2l þ ða0

l1

ð2

*i1 ¼1 ½1 þ a0 ð2

 ½ða0

l1

2l1

¼ xk þ 2l þ ½a0  ½ða0

32

*i1 ¼1 ðxk þ 2l

ð2l Þ

 ai Þ

ð2l Þ

 ai Þ

l

l dxk ð2l Þ X32l ð2l Þ 32 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk þ 2l  ai2 Þ; dxk þ 2l l d 2 xk ð2l Þ X32l X32l ð2l Þ 32 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l  ai3 Þ; dx2k þ 2l

.. .

l d r xk X32l ð2l Þ X32l ð2l Þ 32 ¼ a0 i1 ¼1 ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk þ 2l  air þ 1 Þ r dxk þ 2l

ð2:169Þ where

2.6 Backward Cubic Nonlinear Discrete Systems ð2l Þ

ð2Þ

ð2l1 Þ 1 þ 32

a0 ¼ ða0 Þ1 þ 3 ; a0 2l

¼ ða0 2l1

l

ð2 Þ

Þ

l1

155

; l ¼ 2; 3; ;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

ð2l Þ

2 03i¼1 fai g ¼ sortf03i1 ¼1 fai1 g; 0M i2 2¼1 fbi2 ;1 ; bi2 ;2 gg ,ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

 ai þ 1 ;

ð2l Þ Di

l

2

l1

Iqð21

¼ Þ

2

ð2l Þ ðBi Þ2



ð2l Þ 4Ci

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

00Nq22¼1 Iqð22 Þ ;

¼ flðq1 1Þ2l1 m1 þ 1 ; lðq1 1Þ2l1 m1 þ 2 ; ; lq1 2l1 m1 g f1; 2; ; M1 g0f∅g;

for q1 2 f1; 2; ; N1 g; M1 ¼ N1  2l1  m1 ; m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ2l m1 þ 1 ; lðq2 1Þ2l m1 þ 2 ; ; lq2 m1 2l g fM1 þ 1; M1 þ 2; ; M2 g0f∅g; l

l1

for q2 2 f1; 2; ; N2 g; M2 ¼ ð32  32 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ; ; lM2 g  f1; 2; ; M2 g0f∅g

ð2:170Þ

with fixed-points ð2l Þ

l

xk ¼ xk þ 2l ¼ ai ; ði ¼ 1; 2; . . .; 32 Þ 2l

2l1

ð2l Þ

ð2l1 Þ

03i¼1 fai g ¼ sortf02i¼1 fai l

ð2l1 Þ

2 g; 0M i¼1 fbi;1

ð2l1 Þ

; bi;2

gg

ð2:171Þ

l

ð2 Þ

ð2 Þ

with ai \ai þ 1 : ð2l1 Þ

(ii) For a backward fixed-point of xk ¼ xk þ 2l1 ¼ ai1 . . .; N1 g), if dxk dxk þ 2l1

j

x

kþ2

ð2l1 Þ

¼ai l1

ð2l1 Þ

¼ 1 þ a0

32

l1

ð2l1 Þ

*i2 ¼1;i2 6¼i1 ðai1

ð2l1 Þ

(i1 2 Iq ð2l1 Þ

 ai 2

; q 2 f1; 2;

Þ ¼ 1 ð2:172Þ

1

then there is a backward period- 2l fixed-point discrete system ð2l Þ

xkl ¼ xk þ 2l þ a0

*

ð2l1 Þ i1 2Iq

ð2l1 Þ 3

ðxk þ 2l  ai1

Þ

32

l

*j2 ¼1 ðxk þ 2l

ð2l Þ

 aj2 Þð1dði1 ;j2 ÞÞ ð2:173Þ

156

2 Cubic Nonlinear Discrete Systems

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð2:174Þ

with dxk d 2 xk j j ð2l1 Þ ¼ 1; ð2l1 Þ ¼ 0; dxk þ 2l xk þ 2l ¼ai1 dx2k þ 2l xk þ 2l ¼ai1 d 3 xk ð2l Þ ð2l1 Þ ð2l1 Þ j ðai1  ai2 Þ3 * ð2l1 Þ ¼ 6a0 ð2l1 Þ 3 i2 2Iq ;i2 6¼i1 dxk þ 2l xk þ 2l ¼ai1 2l

ð2l1 Þ

 *3j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0

; q 2 f1; 2; . . .; N1 gÞ

ð2l1 Þ

xk at xk þ 2l ¼ ai1

ð2:175Þ

is

• a monotonic sink of the third-order if d 3 xk =dx3k þ 2l j

ð2l1 Þ

x

k þ 2l

• a monotonic source of the third-order if d 3 xk =dx3k þ 2l j

¼ai

x

k þ 2l

[ 0;

1 ð2l1 Þ

¼ai

\0:

1

(ii1) The backward period- 2l fixed-points are trivial if ð2l1 Þ

xk ¼ xk þ 2l ¼ ai1

l1

for i1 ¼ 1; 2; ; 32 ;

ð2:176Þ

(ii2) The backward period- 2l fixed-points are non-trivial if ð2l Þ

ð2l Þ

xk ¼ xk þ 2l ¼ bj1 ;1 ; bj1 ;2

j1 2 1; 2; . . .; M2 g0f£g

:

ð2:177Þ

Such a period- 2l fixed-point is • monotonically stable if dxk =dxk þ 2l j

2 ð0; 1Þ;

ð2l Þ

x

¼ai

k þ 2l

• monotonically invariant if dxk þ 2l =dxk j

1

¼ 1; which is

ð2l Þ

xk ¼ai 1 th

1 j – a monotonic lower-saddle of the ð2l1 Þ order for d 2l1 xk =dx2l k þ 2 l x

k þ 2l

th

– a monotonic upper-saddle the ð2l1 Þ order for

[ 0;

1 d 2l1 xk =dx2l j \0; k þ 2l xk þ 2l

– a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l1 jx

k þ 2l

– a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

[ 0;

k þ 2l

• monotonically unstable if dxk =dxk þ 2l j

x

k þ 2l

2 ð0; 1Þ;

ð2l Þ

¼ai

1

• monotonically infinity-unstable if dxk =dxk þ 2l j

x

k þ 2l

ð2l Þ

¼ai

1

¼ 0þ ;

\0;

2.6 Backward Cubic Nonlinear Discrete Systems

157

 • oscillatorilly infinity-unstable if dxk =dxk þ 2l j  ð2l Þ ¼ 0 ; x l ¼ai kþ2 1 • oscillatorilly unstable if dxk =dxk þ 2l j  ð2l Þ 2 ð1; 0Þ; x

• flipped if dxk =dxk þ 2l j

ð2l1 Þ x l ¼ai kþ2 1

k þ 2l

¼ai

1

¼ 1; which is

1 – an oscillatory lower-saddle of the ð2l1 Þth order if d 2l1 xk =dx2l j k þ 2 l x

k þ 2l

– an oscillatory upper-saddle the ð2l1 Þ

th

order with

[ 0;

1 d 2l1 xk =dx2l j k þ 2l xk þ 2l

\0; – an oscillatory sink of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

k þ 2l

\0;

1 þ1 – an oscillatory source the ð2l1 þ 1Þth order with d 2l1 þ 1 xk =dx2l j k þ 2l x

k þ 2l

[ 0; • oscillatorilly stable if dxk =dxk þ 2l j

ð2l Þ

x

k þ 2l

¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, following the forward case, this theorem can be proved. ■

2.6.3

Backward Period-n Appearing and Period-Doublization

The period-n discrete system for backward cubic nonlinear discrete systems will be discussed, and the backward period-doublization of period-n discrete systems is discussed through the nonlinear renormalization. Theorem 2.7 Consider a 1-dimensional backward cubic nonlinear discrete system xk ¼ xk þ 1 þ Ax3k þ 1 þ Bx2k þ 1 þ Cxk þ 1 þ D ð1Þ

¼ xk þ 1 þ a0 *3i¼1 ðxk þ 1  ai Þ:

ð2:178Þ

(i) After n-times iterations, a backward period-n discrete system for the cubic discrete system in Eq. (2.178) is xk ¼ xk þ n þ a0 *3i1 ¼1 ðxk þ n  ai2 Þf1 þ ¼

n xk þ n þ a03 1 *3i1 ¼1 ðxk þ n n

ð3 3Þ=2

 ½*j2 ¼1

ðnÞ

¼ x k þ n þ a0

Xn

j¼1

 ai1 Þ

l

ð2l Þ

ð2 Þ

ðx2k þ n þ Bj2 xk þ n þ Cj2 Þ 3n

*i¼1 ðxk þ n

ðnÞ

 ai Þ

Qj g ð2:179Þ

158

2 Cubic Nonlinear Discrete Systems

with for r  3n dxk n ðnÞ X n ðnÞ ¼ 1 þ a0 3i1 ¼1 *3i2 ¼1;i2 6¼i1 ðxk þ n  ai2 Þ; dxk þ n d 2 xk n ðnÞ X n X n ðnÞ ¼ a0 3i1 ¼1 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðxk þ n  ai3 Þ; dx2k þ n .. . d r xk n X n ðnÞ X n ðnÞ ¼ a0 3i1 ¼1 3ir ¼1;ir 6¼i1 ;i2 ...;ir1 *3ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk þ n  air þ 1 Þ dxrk þ n ð2:180Þ where ðnÞ

a0 ¼ ða0 Þð3

n

1Þ=2

ð1Þ

; Q1 ¼ 0; Q2 ¼ *3i2 ¼1 ½1 þ a0 *3i1 ¼1;i1 6¼i2 ðxk þ n  ai1 Þ; ð1Þ

Qn ¼ *3in ¼1 ½1 þ a0 ð1 þ Qn1 Þ *3in1 ¼1;in1 6¼in ðxk þ n  ain1 Þ; n ¼ 3; 4; . . .; n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

03i¼1 fai g ¼ sortf03i1 ¼1 fai1 g; 0M i1 ¼1 fbi2 ;1 ; bi2 ;2 gg; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2 2 ðnÞ ðnÞ ðnÞ Di2 ¼ ðBi2 Þ2  4Ci2 0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þn þ 1 ; lðq1Þn þ 2 ; . . .; lqn gf1; 2; . . .; Mg0f£g; for q 2 f1; 2; . . .; Ng; M ¼ ð3n  3Þ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flNn þ 1 ; lNn þ 2 ; . . .; lM g  f1; 2; . . .; Mg0f£g; ð2:181Þ

with backward fixed-points ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 3n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

2 03i¼1 fai g ¼ sortf03i¼1 fai g; 0M i¼1 fbi;1 ; bi;2 gg

ð2:182Þ

ðnÞ ðnÞ with ai \ai þ 1 : ðnÞ

ðnÞ

(ii) For a backward fixed-point of xk ¼ xk þ n ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), if dxk n ðnÞ ðnÞ ð2l Þ jx ¼aðnÞ ¼ 1 þ a0 *3i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk þ n k i1

ð2:183Þ

2.6 Backward Cubic Nonlinear Discrete Systems

159

with d 2 xk n ðnÞ X n ðnÞ ð2l Þ jx ¼aðnÞ ¼ a0 3i2 ¼1;i2 6¼i1 *3i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; 2 dxk þ n k þ n i1

ð2:184Þ

then there is a new discrete system for onset of the qth -set of period-n fixedpoints based on the second-order monotonic saddle-node bifurcation as ðnÞ

xk ¼ x k þ n þ a0

*i 2I ðnÞ ðxk þ n 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *3j2 ¼1 ðxk þ n  aj2 Þð1dði1 ;j2 ÞÞ ð2:185Þ n

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð2:186Þ

(ii1) If dxk j ði1 2 IqðnÞ Þ; ðnÞ ¼ 1 dxk þ n xk þ n ¼ai1 d 2 xk ðnÞ ðnÞ ðnÞ 2 j ðnÞ ¼ 2a 0 *i2 2IqðnÞ ;i2 6¼i1 ðai1  ai2 Þ dx2k þ n xk þ n ¼ai1 ðnÞ

ð2:187Þ

ðnÞ

 *3j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk at xk þ n ¼ ai1 is • a monotonic upper-saddle of the second-order for d 2 xk =dx2k þ n jx

¼ai

kþn

¼ai

\0; • a monotonic lower-saddle of the second-order for d 2 xk =dx2k þ n jx [ 0:

ðnÞ

kþn

1

ðnÞ 1

(ii2) The backward period-n fixed-points ( n ¼ 2n1  m) are trivial ) 2n1 1 m ðnÞ ð1Þ ð2n1 1 mÞ xk þ n ¼ xk ¼ aj1 2 f03ii ¼1 fai1 g; 03i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 3n g0f£g

for n 6¼ 2n2 ; ðnÞ

2n1 1 m

ð2n1 1 mÞ

g xk þ n ¼ xk ¼ aj1 2 03i2 ¼1 fai2 for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 3n g0f£g for n ¼ 2n2 :

)

ð2:188Þ

160

2 Cubic Nonlinear Discrete Systems

(ii3) The backward period-n fixed-points ( n ¼ 2n1  m) are non-trivial if ) 2n1 1 m ðnÞ ð1Þ ð2n1 1 mÞ xk ¼ xk þ n ¼ aj1 62 f03ii ¼1 fai1 g; 03i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 3n g0f£g

for n 6¼ 2n2 ; 2n1 1 m

ðnÞ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 62 03i2 ¼1 fai2

)

g

ð2:189Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 3n g0f£g for n ¼ 2n2 :

Such a backward period-n fixed-point is • monotonically stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk =dxk þ n jx

kþn

ðnÞ

¼ai

¼ 1; which is

1

th

– a monotonic lower-saddle of the ð2l1 Þ

1 order for d 2l1 xk =dx2l k þ n j x

kþn

[ 0; 1 – a monotonic upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jx \0; kþn

– a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx

kþn

[ 0; – a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx

kþn

\0; • monotonically stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonic infinity-unstable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

kþn

¼ai

• oscillatory infinity-unstable if dxk =dxk þ n jx • oscillatorilly unstable if dxk =dxk þ n jx • flipped if dxk =dxk þ n jx

kþn

kþn

ðnÞ

¼ai

ðnÞ

¼ai

1

ðnÞ

¼ 0þ ; ¼ 0 ;

1

2 ð1; 0Þ;

1

¼ 1; which is

1

1 – an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ n j x

kþn

[ 0; 1 – an oscillatory upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ n j x

kþn

\0; – an oscillatory sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx \0;

kþn

2.6 Backward Cubic Nonlinear Discrete Systems

161

– an oscillatory source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx

kþn

[ 0; • oscillatorilly stable if dxk =dxk þ n jx

kþn

ðnÞ

¼ai

2 ð1; 1Þ:

1

ðnÞ

ðnÞ

(iii) For a backward fixed-point of xk ¼ xk þ n ¼ ai1 ( i1 2 Iq , q 2 f1; 2; . . .; Ng), there is a backward period-doubling of the qth -set of period-n fixed-points if dxk n ðnÞ ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *3j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; dxk þ n k þ n i1 d s xk j ðnÞ ¼ 0; for s ¼ 2; . . .; r  1; dxsk þ n xk þ n ¼ai1 d r xk n j ðnÞ 6¼ 0 for 1\r  3 dxrk þ n xk þ n ¼ai1

ð2:190Þ

with • a r th -order oscillatory sink for d r xk =dxrk þ n jx

kþn

ðnÞ

¼ai

\0 and r ¼ 2l1 þ 1;

1

• a r th -order oscillatory source for d r xk =dxrk þ n jx

ðnÞ

kþn

¼ai

[ 0 and r ¼

1

2l1 þ 1; • a r th -order oscillatory lower-saddle for d r xk =dxrk þ n jx

ðnÞ

kþn

r ¼ 2l1 ; • a r th -order oscillatory upper-saddle for d r xk =dxrk þ n jx

¼ai

kþn

r ¼ 2l1 ;

[ 0 and

1

ðnÞ

¼ai

\0 and

1

The corresponding period- 2  n discrete system of the backward cubic discrete system in Eq. (2.178) is ð2nÞ

xk ¼ xk þ 2n þ a0

*i 2I ðnÞ ðxk þ 2n 1 q

ðnÞ

 ai 1 Þ 3

ð2nÞ ð1dði1 ;j2 ÞÞ

2n

 *3j2 ¼1 ðxk þ 2n  aj2

Þ

ð2:191Þ

with dxk dxk þ 2n

j x

ðnÞ

¼ai k þ 2n

1

¼ 1;

d 2 xk þ 2n j ðnÞ ¼ 0; dx2k þ 2n xk þ 2n ¼ai1

d xk ð2nÞ ðnÞ ðnÞ 3 j ðnÞ ¼ 6a *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 0 2 q 2 1 dx3k þ 2n xk þ 2n ¼ai1 3

2n

ðnÞ

ð2nÞ ð1dði1 ;j2 ÞÞ

 *3j2 ¼1 ðai1  aj2

Þ

ð2:192Þ 6¼ 0:

162

2 Cubic Nonlinear Discrete Systems ðnÞ

ðnÞ

Thus, xk at xk þ 2n ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic source of the third-order if d 3 xk =dx3k þ 2n jx

ðnÞ

¼ai

k þ 2n

• a monotonic sink of the third-order if d 3 xk =dx3k þ 2n jx

ðnÞ

k þ 2n

¼ai

\0;

1

[ 0:

1

(iv) After l-times backward period-doubling bifurcations of period-n fixed points, a period-2l  n discrete system of the backward cubic discrete system in Eq. (2.178) is ð2l1 nÞ

xk ¼ xk þ 2l n þ ½a0  f1 þ ¼

ð2l1 nÞ

3

*i1 ¼1

32

l1 n

*i1 ¼1

ð2l1 nÞ

ðxk þ 2l n  ai1

ð2l1 nÞ 32l1 n xk þ 2l n þ ½a0 *i1 ¼1 ðxk þ 2l n ð2

 ½ða0

l1

nÞ 3

Þ

ð2l1 nÞ

l1

ð2

¼ xk þ 2l n þ ½a0 ð2

 ½ða0

l1

nÞ 3

Þ

nÞ

ð2l1 nÞ

ð3

ð2 nÞ

¼ xk þ 2l n þ a0

2l n

*j1 ¼1 3

ð3

2l n

*j2 ¼1

Þ

3

l1 n

2l n

*i¼1

3

2l1 n

*i1 ¼1

ð2l1 nÞ 32

¼ xk þ 2l n þ ða0

l

Þ

ð2l1 nÞ 32l1 n ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2l n

2l1 n

Þ=2



ð2l nÞ

ðx2k þ 2l n þ Bj2 ð2

3

l

32 n

*i¼1

Þ=2

Þg

ð2l1 nÞ ai1 Þ

ðxk þ 2l n  ai1 2l1 n

ð2l1 nÞ

 ai 2

l1

nÞ

ð2l nÞ

xk þ 2l n þ Cj2

Þ

Þ ð2l nÞ

ð2l nÞ

ðxk þ 2l n  bj2 ;1 Þðxk þ 2l n  bj2 ;2 Þ ð2l nÞ

ðxk þ 2l n  ai l

ð2 nÞ

ðxk þ 2l n  ai

Þ

Þ

l

ð2:193Þ

with for r  32 n dxk ð2l nÞ X32l n 32l n ð2l nÞ ¼ 1 þ a0 Þ; i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk þ 2l n  ai2 dxk þ 2l n l d 2 xk ð2l nÞ X32l n X32l n ð2l nÞ 32 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l n  ai3 2 dxk þ 2l n

.. .

l d r xk X32l n ð2l nÞ X32l n ð2l nÞ 32 n ¼ a0 i1 ¼1 ir ¼1;ir 6¼i1 ;i2 ...;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk þ 2l n  air þ 1 Þ r dxk þ 2l n

ð2:194Þ where

2.6 Backward Cubic Nonlinear Discrete Systems ð2nÞ

a0

ðnÞ

ð2l nÞ

¼ ða0 Þ1 þ 3 ; a0 2n

ð2l nÞ ð2l nÞ g 03i¼1 fai

ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l1 nÞ 1 þ 32

¼ ða0

l1 n

; l ¼ 1; 2; 3; . . .;

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; 2

ð2l nÞ

ð2l nÞ 2

ð2l nÞ

¼ ðBi

for i 2

l l 0Nq11¼1 Iqð21 nÞ 00Nq22¼1 Iqð22 nÞ ;

l1

Þ

2l1 n ð2l1 nÞ ð2l nÞ ð2l nÞ 2 sortf03i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

Di

Iqð21

163

nÞ

Þ  4Ci

0

¼ flðq1 1Þð2l1 nÞ þ 1 ; lðq1 1Þð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f£g;

ð2:195Þ

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1  ð2l1  nÞ; l

Iqð22 nÞ ¼ flðq2 1Þð2l nÞ þ 1 ; lðq2 1Þð2l nÞ þ 2 ; . . .; lq1 ð2l nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f£g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ ð32 n  32 n Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi  i jDi jÞ; bi;2 2 pffiffiffiffiffiffiffi ð2l nÞ ð2l nÞ 2 ð2l nÞ Di ¼ ðBi Þ  4Ci \0; i ¼ 1; i 2 flNð2l1 nÞ þ 1 ; lNð2l1 nÞ þ 2 ; . . .; lM2 g  fM1 þ 1; M1 þ 2; . . .; M2 g0f£g

with fixed-points ð2l nÞ

xk ¼ xk þ 2l n ¼ ai 2l n

ð2l nÞ

03i¼1 fai l

ð2 nÞ

with ai

l

; ði ¼ 1; 2; . . .; 32 n Þ 2l1 n

g ¼ sortf03i¼1

ð2l1 nÞ

fai

ð2l1 nÞ

2 g; 0M i¼1 fbi;1

ð2l1 nÞ

; bi;2

gg

l

ð2 nÞ

\ai þ 1 : ð2:196Þ

(v)

ð2

l1

nÞ

l1

ð2

nÞ

For a fixed-point of xk ¼ xk þ 2l n ¼ ai1 (i1 2 Iq ; q 2 f1; 2; . . .; l N1 g), there is a backward period-ð2  nÞ discrete system if

with

dxk j ð2l1 nÞ ¼ 1; dxk þ 2l1 n xk þ 2l1 n ¼ai1 d s xk j ð2l1 nÞ ¼ 0; for s ¼ 2; . . .; r  1; s dxk þ 2l1 n xk þ 2l1 n ¼ai1 d r xk 2l1 n j ð2l1 nÞ 6¼ 0 for 1\r  3 r dxk þ 2l1 n xk þ 2l1 n ¼ai1

ð2:197Þ

164

2 Cubic Nonlinear Discrete Systems

• a r th -order oscillatory source for d r xk =dxrk þ 2l n j

r ¼ 2l1 þ 1; • a r th -order oscillatory sink for d r xk =dxrk þ 2l n j

[ 0 and

ð2l nÞ

x

k þ 2l1 n

¼ai

1

ð2l nÞ

x

k þ 2l1 n

r ¼ 2l1 þ 1; • a r th -order oscillatory upper-saddle for d r xk =dxrk þ 2l n j and r ¼ 2l1 ; • a r th -order oscillatory lower-saddle for d r xk =dxrk þ 2l n j

¼ai

ð2l nÞ

x

k þ 2l1 n

¼ai

k þ 2l1 n

¼ai

\0

1

ð2l nÞ

x

and r ¼ 2l1 .

\0 and

1

[0

1

The corresponding backward period- ð2l  nÞ discrete system is ð2l nÞ

xk ¼ xk þ 2l n þ a0 l

32 n *j2 ¼1 ðxk þ 2l n



*

ð2l1 nÞ i1 2Iq



ð2l1 nÞ 3

ðxk þ 2l n  ai1

Þ

ð2:198Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ aj2 Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai 1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð2:199Þ

with dxk dxk þ 2l n

j

ð2l1 nÞ x l ¼ai k þ 2 n 1

¼ 1;

d 2 xk j ð2l1 nÞ ¼ 0; dx2k þ 2l n xk þ 2l n ¼ai1

d 3 xk ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j ða  ai 2 Þ * ð2l1 nÞ ¼ 6a0 ð2l1 nÞ i2 2Iq ;i2 6¼i1 i1 dx3k þ 2l n xk þ 2l n ¼ai1 2l n

ð2l1 nÞ

 *3j2 ¼1 ðai1 ði1 2 Iqð2

l1

nÞ

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

6¼ 0

; q 2 f1; 2; . . .; N1 gÞ ð2:200Þ ð2l1 nÞ

xk at xk þ 2l n ¼ ai1

is

• a monotonic source of the third-order if d 3 xk =dx3k þ 2l n j • a monotonic sink of the third-order if d 3 xk =dx3k þ 2l n j

x

k þ 2l n

k þ 2l n

(v1) The backward period- 2l  n fixed-points are trivial if

ð2l1 nÞ

x

¼ai

ð2l1 nÞ

¼ai

1

\0;

1

[ 0:

2.6 Backward Cubic Nonlinear Discrete Systems ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

165 2l1 n

ð1Þ

ð2l1 nÞ

2 f03ii ¼1 fai1 g; 03i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 32 n for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð2l1 nÞ

2 03i2 ¼1 fai2

g

ð2:201Þ

)

l

for j ¼ 1; 2; . . .; 32 n for n ¼ 2n1 : (v2) The backward period- 2l  n fixed-points are non-trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

62 f03ii ¼1 fai1 g; 03i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 32 n for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð2l1 nÞ

62 03i2 ¼1 fai2

g

ð2:202Þ

)

l

for j ¼ 1; 2; . . .; 32 n for n ¼ 2n1 : Such a backward period- 2l  n fixed-point is • monotonically stable if dxk =dxk þ 2l n j

ð2l nÞ

x

k þ 2l n

• monotonically invariant if dxk =dxk þ 2l n j

¼ai

2 ð1; 1Þ;

1 ð2l nÞ

x

k þ 2l n

¼ai

¼ 1; which is

1

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l j k þ 2l n x

[ 0 (independent ð2l1 Þ-branch appearance); 1 j – a monotonic upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ 2l n x

k þ 2l n

k þ 2l n

\0

(independent ð2l1 Þ-branch appearance) 1 þ1 j – a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2l k þ 2l n x [ 0 (dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l1n jx (dependent ð2l1 þ 1Þ-branch appearance from one branch); • monotonically unstable if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

• monotonically infinity-unstable if dxk =dxk þ 2l n j • oscillatorilly infinity-unstable if dxk =dxk þ 2l n j

x

x

ð2l nÞ

k þ 2l n

k þ 2l n

k þ 2l n

2 ð0; 1Þ;

1

¼ai

1

ð2l nÞ

¼ai

1

¼ 0þ ;

¼ 0 ;

k þ 2l n

\0

166

2 Cubic Nonlinear Discrete Systems

• oscillatorilly unstable if dxk =dxk þ 2l n j • flipped if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

x

ð2l nÞ

k þ 2l n

¼ai

2 ð1; 0Þ;

1

¼ 1; which is

1

1 – an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l j k þ 2l n x

k þ 2l n

[ 0; 1 j – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ 2l n x

k þ 2l n

\0; 1 þ1 j – an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2l k þ 2l n x

k þ 2l n

[ 0; 1 þ1 j – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk = dx2l k þ 2l n x

k þ 2l n

\ 0; • oscillatorilly stable if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, the proof of this theorem can follow the proof for the forward cubic discrete system. This theorem can be easily proved. ■

Reference Luo ACJ (2019) The stability and bifurcation of fixed-points in low-degree polynomial systems. J Vib Test Syst Dyn 3(4):403–451

Chapter 3

Quartic Nonlinear Discrete Systems

In this Chapter, the stability and bifurcation of the quartic nonlinear discrete systems will be presented, which is similar to Luo (2019). The fourth-order monotonic upper-saddle and monotonic lower-saddle appearing bifurcations of two second-order monotonic upper-saddles or monotonic lower-saddles will be presented. The 3rd order monotonic sink and source switching bifurcations of monotonic lower-saddle with monotonic sink and monotonic upper-saddle with monotonic source will be discussed. Graphical illustrations of global stability and bifurcations are presented. The bifurcation trees for quartic nonlinear discrete systems are discussed through period-doublization and monotonic saddle-node bifurcations.

3.1

Period-1 Appearing Bifurcations

In this section, period-1 fixed-points in quartic nonlinear discrete systems will be discussed, and the stability and appearing and switching bifurcation conditions for period-1 fixed points will be developed. Definition 3.1 Consider a 1-dimensional, forward, quartic nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ ¼

xk þ a0 ðpÞ½x2k

þ B1 ðpÞxk þ C1 ðpÞ½x2k

ð3:1Þ

þ B2 ðpÞxk þ C2 ðpÞ

where AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

© Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3_3

ð3:2Þ

167

168

3 Quartic Nonlinear Discrete Systems

(i) If Di ¼ B2i  4Ci \0 for i ¼ 1; 2

ð3:3Þ

the quartic nonlinear discrete system does not have any fixed-point, and the corresponding standard form is 1 2

1 4

1 2

1 4

xk þ 1 ¼ xk þ a0 ½ðxk þ B1 Þ2 þ ðD1 Þ½ðxk þ B2 Þ2 þ ðD2 Þ:

ð3:4Þ

The discrete flow of such a discrete system without fixed-points is called a non-fixed-points discrete flow. (i1) If a0 [ 0, the non-fixed-point discrete flow is called the positive discrete flow. (i2) If a0 \0, the non-fixed-point discrete flow is called the negative discrete flow. (ii) If Di ¼ B2i  4Ci [ 0 and Dj ¼ B2j  4Cj \0 for i; j 2 f1; 2g; i 6¼ j

ð3:5Þ

the quartic polynomial discrete system has two simple fixed-points, i.e., ðiÞ

1 2

xk ¼ b1 ¼  ðBi þ

pffiffiffiffiffi pffiffiffiffiffi 1 ðiÞ Di Þ; xk ¼ b2 ¼  ðBi  Di Þ: 2

ð3:6Þ

The corresponding standard form is 1 2

1 4

xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ½ðxk þ Bj Þ2 þ ðDj Þ

ð3:7Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

a1 ¼ minðb1 ; b2 Þ and a2 ¼ maxðb1 ; b2 Þ

ð3:8Þ

Such a discrete flow of fixed-points is called a discrete flow of two simple fixed-points. (iii) If Di ¼ B2i  4Ci ¼ 0 and Dj ¼ B2j  4Cj \0 for i; j 2 f1; 2g; i 6¼ j

ð3:9Þ

the quartic polynomial, forward discrete system has a double repeated fixedpoint, i.e., ðiÞ

1 2

ðiÞ

1 2

xk ¼ b1 ¼  Bi ; xk ¼ b2 ¼  Bi :

ð3:10Þ

3.1 Period-1 Appearing Bifurcations

169

The corresponding standard form is 1 2

1 4

xk þ 1 ¼ xk þ a0 ðxk  a1 Þ2 ½ðxk þ Bj Þ2 þ ðDj Þ

ð3:11Þ

ðiÞ

ð3:12Þ

where ðiÞ

a1 ¼ b1 ¼ b2 : Such a discrete flow of the fixed-point of xk ¼ a1 is called

• a monotonic upper-saddle discrete flow of the second-order for a0 [ 0; • a monotonic lower-saddle discrete flow of the second-order for a0 \0. The fixed-point of x ¼ a1 for two fixed-points appearance or vanishing is called • a monotonic upper-saddle two fixed-points at p ¼ p1 • a monotonic lower-saddle two fixed-points at p ¼ p1

appearing bifurcation of the second-order for 2 @X12 for a0 [ 0; appearing bifurcation of the second-order for 2 @X12 for a0 \0.

The appearing bifurcation condition of the upper or lower-saddle is 1 2

Di ¼ B2i  4Ci ¼ 0ði 2 f1; 2gÞ and a1 ¼  Bi : (iv) If

Di ¼ B2i  4Ci  0 for i ¼ 1; 2;

ð3:13Þ

ð3:14Þ

the quartic nonlinear discrete system has four fixed-points, i.e., ðiÞ

1 2

xk ¼ b1 ¼  ðBi þ

pffiffiffiffiffi pffiffiffiffiffi 1 ðiÞ Di Þ; xk ¼ b2 ¼  ðBi  Di Þ for i ¼ 1; 2: 2

ð3:15Þ

(iv1) A standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þðxk  a4 Þ

ð3:16Þ

where Di ¼ B2i  4Ci [ 0; i ¼ 1; 2; ð1Þ

ð2Þ

bk 6¼ bl

for k; l 2 f1; 2g;

a1;2;3;4 2

ð1Þ ð1Þ ð2Þ ð2Þ fb1 ; b2 ; b1 b2 g

ð3:17Þ with am \am þ 1 :

Such a discrete flow of fixed-points is called a discrete flow of four simple fixed-points.

170

3 Quartic Nonlinear Discrete Systems

(iv2) The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ

ð3:18Þ

where Di ¼ B2i  4Ci [ 0; Dj ¼ B2j  4Cj [ 0 for i; j ¼ 1; 2; ðiÞ

ðjÞ

ai1 ¼ bk ¼ bl ; ði; kÞ 6¼ ðj; lÞ; i; j; k; l 2 f1; 2g;

ð3:19Þ

ai1 62 fai2 ; ai3 g for ia 2 f1; 2; 3; 4g and a 2 f1; 2; 3; 4g: (iv2a) Such a discrete flow of fixed-point xk ¼ ai1 ðai1 \minfai2 ; ai3 g or ai1 [ maxfai2 ; ai3 gÞ is called • a monotonic upper-saddle discrete flow of the second-order for a0 [ 0; • a monotonic lower-saddle discrete flow of the second-order for a0 \0. The fixed-point of xk ¼ ai1 ðai1 \minfai2 ; ai3 g or ai1 [ maxfai2 ; ai3 gÞ for two fixed-points switching is called • a monotonic upper-saddle switching bifurcation of order for fixed-points at a point p ¼ p1 2 @X12 for • a monotonic lower-saddle switching bifurcation of order for fixed-points at a point p ¼ p1 2 @X12 for

the seconda0 [ 0; the seconda0 \0.

(iv2b) Such a discrete flow of fixed-point xk ¼ ai1 ðminfai2 ; ai3 g\ai1 \maxfai2 ; ai3 gÞ is called • a monotonic lower-saddle discrete flow of the second order for a0 [ 0; • a monotonic upper-saddle discrete flow of the second order for a0 \0. The fixed-point of xk ¼ ai1 ðminfai2 ; ai3 g\ai1 \maxfai2 ; ai3 gÞ for two fixed-points switching is called • a monotonic lower-saddle switching bifurcation of the secondorder for fixed-points at a point p ¼ p1 2 @X12 for a0 [ 0; • a monotonic upper-saddle switching bifurcation of the secondorder for fixed-points at a point p ¼ p1 2 @X12 for a0 \0. (iv2c) The corresponding monotonic supper- or lower-saddle switching bifurcation condition for switching of two fixed-points is

3.1 Period-1 Appearing Bifurcations

171

Di ¼ B2i  4Ci [ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj [ 0 ðj 2 f1; 2gÞ; ðiÞ bk

¼

ðjÞ bl ; ði; kÞ

ð3:20Þ

6¼ ðj; lÞ; ði; j; k; l 2 f1; 2gÞ:

(iv2d) The corresponding monotonic upper-or lower-saddle appearing bifurcation condition for appearance or vanishing of two fixed-points is Di ¼ B2i  4Ci ¼ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj [ 0 ðj 2 f1; 2g; j 6¼ iÞ; ðiÞ bk

¼

ðiÞ bl ; ði; kÞ

ð3:21Þ

6¼ ðj; lÞ; ði; j; k; l 2 f1; 2gÞ:

(iv3) The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai2 Þ

ð3:22Þ

where Di ¼ B2i  4Ci [ 0; Dj ¼ B2j  4Cj ¼ 0 for i; j ¼ 1; 2; 1 2

ðiÞ

ðiÞ

ai1 ¼  Bj ¼ bl ; ai2 ¼ bk ; k 6¼ l; k; l 2 f1; 2g;

ð3:23Þ

ai1 ¼ ai3 for ia 2 f1; 2; 3; 4g and a 2 f1; 2; 3; 4g: (iv3a) Such a discrete flow of the fixed-point of xk ¼ ai1 ðai1 \ ai2 Þ is called • a monotonic sink discrete flow of the third-order for a0 [ 0; • a monotonic source discrete flow of the third-order for a0 \0. The fixed-point of x ¼ ai1 for one fixed-point to three fixed-points is called • a monotonic sink switching bifurcation of the third-order for fixed-point at point p ¼ p1 2 @X12 for a0 [ 0; • a monotonic source switching bifurcation of the third-order for fixed-points at point p ¼ p1 2 @X12 for a0 \0. (iv3b) Such a discrete flow of the fixed-point of xk ¼ ai1 ðai1 [ ai2 Þ is called • a monotonic source discrete flow of the third-order for a0 [ 0; • a monotonic sink discrete flow of the third-order for a0 \0. The fixed-point of x ¼ ai1 for one fixed-point to three fixed-points is called • a monotonic source switching bifurcation of the third-order at point p ¼ p1 2 @X12 for a0 [ 0; • a monotonic sink switching bifurcation of the third-order at point p ¼ p1 2 @X12 for a0 \0.

172

3 Quartic Nonlinear Discrete Systems

(iv3c) The corresponding monotonic sink or source switching bifurcation condition of the third-order is Di ¼ B2i  4Ci [ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj ¼ 0 ðj 2 f1; 2gÞ; ðiÞ bk

¼

1 ðiÞ  Bj ; bk 2

6¼

ðiÞ bl ; ðk

:

ð3:24Þ

6¼ l; k; l 2 f1; 2gÞ:

(iv4) The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ2

ð3:25Þ

where Di ¼ B2i  4Ci ¼ 0; i ¼ 1; 2; ð1Þ

ð1Þ

ð2Þ

1 2

ð2Þ

1 2

b1 ¼ b2 ¼  B1 ; b1 ¼ b2 ¼  B2 ; B1 6¼ B2 ; a1 ¼

1 1 minf B1 ;  B2 g; a2 2 2

¼

ð3:26Þ

1 1 maxf B1 ;  B2 g: 2 2

Such a discrete flow with the two fixed-points of xk ¼ a1 and xk ¼ a2 is called • a (2nd mUS:2nd mUS) discrete flow for a0 [ 0; • a (2nd mLS:2nd mLS) discrete flow for a0 \0. The fixed-points of xk ¼ a1 and xk ¼ a2 for two sets of two fixed-points switching or appearing are called two upper- or lower-saddle switching or appearing bifurcations of the second-order at a point p ¼ p1 2 @X12 , and the bifurcation condition is Di ¼ B2i  4Ci ¼ 0; i ¼ 1; 2 ; ð1Þ

ð1Þ

ð2Þ

1 2

ð2Þ

1 2

b1 ¼ b2 ¼  B1 ; b1 ¼ b2 ¼  B2 :

ð3:27Þ

(iv5) The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þ4

ð3:28Þ

where Di ¼ B2i  4Ci ¼ 0; i ¼ 1; 2; ðiÞ

ðiÞ

1 2

b1 ¼ b2 ¼  Bi ; B1 ¼ B2 :

ð3:29Þ

3.1 Period-1 Appearing Bifurcations

173

Such a discrete flow at the fixed-point of xk ¼ a1 is called • a monotonic upper-saddle discrete flow of the fourth-order for a0 [ 0; • a monotonic lower-saddle discrete flow of the fourth-order for a0 \0. The fixed-point of xk ¼ a1 for two double repeated fixed-points switching or four simple fixed-points appearance is called • a monotonic upper-saddle switching or appearing bifurcation of the fourth-order at point p ¼ p1 2 @X12 for a0 [ 0; • a monotonic lower-saddle switching or appearing bifurcation of the fourth-order at point p ¼ p1 2 @X12 for a0 \0. The corresponding upper- or lower-saddle bifurcation condition is ðiÞ

ðiÞ

Di ¼ B2i  4Ci ¼ 0; a1 ¼ b1 ¼ b2 ; i ¼ 1; 2:

ð3:30Þ

Theorem 3.1 (i) Under conditions of Di ¼ B2i  4Ci \0 for i ¼ 1; 2

ð3:31Þ

a standard form of Eq. (3.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ 1 2

1 4

1 2

1 4

¼ xk þ a0 ½ðxk þ B1 Þ2 þ ðD1 Þ½ðxk þ B2 Þ2 þ ðD2 Þ

ð3:32Þ

with a0 ¼ AðpÞ, which has a non-fixed-point discrete flow. (i1) If a0 ðpÞ [ 0, the non-fixed-point discrete flow is called a positive discrete flow. (i2) If a0 ðpÞ [ 0, the non-fixed-point discrete flow is called a negative discrete flow. (ii) Under a condition of Di ¼ B2i  4Ci [ 0 and Dj ¼ B2j  4Cj \0 for i; j 2 f1; 2g; i 6¼ j

ð3:33Þ

a standard form of Eq. (3.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ 1 2

1 4

¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ½ðxk þ Bj Þ2 þ ðDj Þ

ð3:34Þ

174

3 Quartic Nonlinear Discrete Systems

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

a1 ¼ minðb1 ; b2 Þ and a2 ¼ maxðb1 ; b2 Þ; pffiffiffiffiffi ðiÞ pffiffiffiffiffi 1 1 ðiÞ b1 ¼  ðBi þ Di Þ; b2 ¼  ðBi  Di Þ: 2

ð3:35Þ

2

(ii1a) For a0 ðpÞ [ 0, the fixed-point of xk ¼ a1 is • • • •

monotonically stable (a monotonic sink) if df =dxk jx ¼a1 2 ð1; 0Þ; k invariantly stable (an invariant sink) if df =dxk jx ¼a1 ¼ 1; k oscillatorilly stable (an oscillatory sink) if df =dxk jx ¼a1 2 ð2; 1Þ; k slipped if df =dxk jx ¼a1 ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a1 k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a1 k \0; • oscillatorilly unstable (an oscillatory source) if df =dxk jx ¼a1 2 k ð1; 2Þ. (ii1b) For a0 ðpÞ [ 0, the fixed-point of xk ¼ a2 is monotonically unstable (a monotonic source) if df =dxk jx ¼a2 2 ð0; 1ÞÞ. k (ii2a) For a0 ðpÞ\0, the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic source) if df =dxk jx ¼a1 2 ð0; 1Þ. k (ii2b) For a0 ðpÞ\0, the fixed-point of xk ¼ a2 is • • • •

monotonically stable (a monotonic sink) if df =dxk jx ¼a2 2 ð1; 0ÞÞ; k invariantly stable (an invariant sink) if df =dxk jx ¼a2 ¼ 1; k oscillatorilly stable (an oscillatory sink) if df =dxk jx ¼a2 2 ð2; 1ÞÞ; k slipped if df =dxk jx ¼a2 ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a2 k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a2 k \0; • oscillatorilly unstable (an oscillatory source) if df =dxk jx ¼a2 2 k ð1; 2Þ.

3.1 Period-1 Appearing Bifurcations

175

(iii) Under conditions of Di ¼ B2i  4Ci ¼ 0 and Dj ¼ B2j  4Cj \0 for i; j 2 f1; 2g; i 6¼ j

ð3:36Þ

a standard form of Eq. (3.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ 1 2

1 4

¼ a0 ðxk  a1 Þ2 ½ðxk þ Bj Þ2 þ ðDj Þ

ð3:37Þ

where ðiÞ

ðiÞ

1 2

a1 ¼ b1 ¼ b2 ¼  Bi :

ð3:38Þ

(iii1) For a0 ðpÞ [ 0, the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic upper-saddle, d 2 f =dx2k jx ¼a1 [ 0Þ. k

• Such a discrete flow at the fixed-point of xk ¼ a1 is called a monotonic upper-saddle discrete flow of the second-order. • The bifurcation of fixed-point of at xk ¼ a1 for two fixed-points appearance or vanishing is called a monotonic upper-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 . (iii2) For a0 ðpÞ\0, the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic lower-saddle, d 2 f =dx2k jx ¼a1 \0Þ. k

• Such a discrete flow at the fixed-point of xk ¼ a1 is called a monotonic lower-saddle discrete flow of the second-order. • The bifurcation of fixed-point of at xk ¼ a1 for two fixed-points appearance or vanishing is called a monotonic lower-saddle-node bifurcation of the second-order at a point p ¼ p1 2 @X12 . (iv) Under conditions of Di ¼ B2i  4Ci [ 0; i ¼ 1; 2 ð1Þ

ð2Þ

bk 6¼ bl ðiÞ b1

¼

for k; l 2 f1; 2g;

pffiffiffiffiffi ðiÞ 1  ðBi þ Di Þ; b2 2

¼

1  ðBi 2

pffiffiffiffiffi  Di Þ for i ¼ 1; 2

ð3:39Þ

a standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þðxk  a4 Þ where

ðiÞ

ðiÞ

a1;2;3;4 2 02i¼1 fb1 ; b2 g with am \am þ 1 :

ð3:40Þ

ð3:41Þ

176

3 Quartic Nonlinear Discrete Systems

(iv1) For a0 ðpÞ [ 0, the fixed-points of xk ¼ a1 ; a2 ; a3 ; a4 are monotonically stable to oscillatorilly unstable, monotonically unstable, monotonically stable to oscillatorilly unstable, and monotonically unstable, respectively. The discrete flow is called a (mSI-oSO:mSO:mSI-oSO:mSO) discrete flow. (iv2) For a0 ðpÞ\0, the fixed-points of xk ¼ a1 ; a2 ; a3 ; a4 are monotonically unstable, monotonically stable to oscillatorilly unstable, monotonically unstable, and monotonically stable to oscillatorilly unstable, respectively. The discrete flow is called a (mSO:mSI-oSO:mSO:mSI-oSO) discrete flow. (iv3) The fixed-point of xk ¼ ai ði ¼ 1; 2; 3; 4Þ is • • • • •

monotonically unstable (a monotonic source) if df =dxk jxk ¼ai 2 ð0; 1Þ; monotonically stable (a monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ; k invariantly stable (an invariant sink) if df =dxk jx ¼ai ¼ 1; k oscillatorilly stable (an oscillatory sink) if df =dxk jx ¼ai 2 ð2; 1Þ; k flipped if df =dxjx ¼ai ¼ 2, which is – an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼ai k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼ai k \0;

• oscillatorilly unstable (an oscillatory source) if df =dxjxk ¼ai 2 ð1; 2Þ. (v)

Under conditions of Di ¼ B2i  4Ci [ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj [ 0ðj 2 f1; 2gÞ; pffiffiffiffiffiffi ðaÞ pffiffiffiffiffiffi : 1 1 ðaÞ b1 ¼  ðBa þ Da Þ; b2 ¼  ðBa  Da Þ for a ¼ i; j 2

ðiÞ

ð3:42Þ

2

ðjÞ

bk ¼ bl ; ði; kÞ 6¼ ðj; lÞ; ði; j; k; l 2 f1; 2gÞ; a standard form of Eq. (3.1) is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ

ð3:43Þ

where ðiÞ

ðjÞ

ðiÞ

ðiÞ

ai1 ¼ bk ¼ bl 2 02i¼1 fb1 ; b1 g; ði; kÞ 6¼ ðj; lÞ; i; j; k; l 2 f1; 2g; ðiÞ

ðiÞ

ai1 62 fai2 ; ai3 g  02i¼1 fb1 ; b1 g for ia 2 f1; 2; 3g and a 2 f1; 2; 3g:

ð3:44Þ

(v1) The fixed-points of xk ¼ ai2 ; ai3 are • • • •

monotonically unstable (a monotonic source) if df =dxk jxk ¼ai2 ;ai3 2 ð0; 1Þ; monotonically stable (a monotonic sink) if df =dxk jxk ¼ai2 ;ai3 2 ð1; 0Þ; invariantly stable (an invariant sink) if df =dxk jx ¼ai ;ai ¼ 1; k 2 3 oscillatorilly stable (an oscillatory sink) if df =dxk jxk ¼ai2 ;ai3 2 ð2; 1Þ;

3.1 Period-1 Appearing Bifurcations

• flipped if df =dxk jx ¼ai k

177

2

¼ 2, which is

;ai3

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼ai2 ;ai3 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼ai2 ;ai3 \0; • oscillatorilly unstable (an oscillatory source, df =dxk jxk ¼ai2 ;ai3 2 ð1; 2Þ). (v2) The fixed-point of xk ¼ ai1 is • monotonically unstable (a monotonic upper-saddle, d 2 f =dx2k jxk ¼ai1 [ 0Þ, • monotonically unstable (a monotonic lower-saddle, d 2 f =dx2k jxk ¼ai1 \0). (v3) The bifurcation of fixed-point at x ¼ ai1 for two fixed-points switching or appearing is called • a monotonic upper-saddle-node ðd 2 f =dx2k jx ¼ai [ 0Þ switching or k 1 appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 ; • a monotonic lower-saddle-node ðd 2 f =dx2k jx ¼ai \0Þ switching or k 1 appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 . (vi) Under conditions of Di ¼ B2i  4Ci [ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj ¼ 0 ðj 2 f1; 2gÞ; pffiffiffiffiffi ðiÞ pffiffiffiffiffi 1 1 ðiÞ b1 ¼  ðBi þ Di Þ; b2 ¼  ðBi  Di Þ; ðjÞ

b1;2 ¼

2 1 ðiÞ  Bj ; bk 2

ð3:45Þ

2

ðiÞ

¼ bl ; ðk 6¼ l; k; l 2 f1; 2gÞ

a standard form of Eq. (3.1) is where

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai2 Þ ðjÞ

ðiÞ

ð3:46Þ

ðiÞ

ai1 ¼ b1;2 ¼ bl ; ai2 ¼ bk ; a1 \a2 ; for i; j; l 2 f1; 2g; ia 2 f1; 2g and a 2 f1; 2g:

ð3:47Þ

(vi1) The fixed-point of xk ¼ ai2 is • monotonically unstable (a monotonic source) if df =dxk jxk ¼ai2 2 ð0; 1Þ; • monotonically stable (a monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ; k 2 • invariantly stable (an invariant sink) if df =dxk jx ¼ai ¼ 1; k

2

178

3 Quartic Nonlinear Discrete Systems

• oscillatorilly stable (an oscillatory sink if df =dxk jx ¼ai 2 ð2; 1Þ; k 2 • flipped if df =dxk jx ¼ai ¼ 2, which is k

2

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼ai k 2 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼ai k 2 \0; • oscillatorilly unstable (an oscillatory source) if df =dxk jx ¼ai 2 k 2 ð1; 2Þ. (vi2) The fixed-point of xk ¼ ai1 with df =dxk jx ¼ai ¼ 0 and d 2 f =dx2k jx ¼ai k k 1 1 ¼ 0 is • unstable of the third-order monotonic source for d 3 f =dx3k jx ¼ai [ 0; k

1

• stable of the third-order monotonic sink for d 3 f =dx3k jx ¼ai \0. k

(vi3) The bifurcation of fixed-point at d

2

f =dx2k jx ¼ai k 1

xk

1

¼ ai1 with df =dxk jx ¼ai ¼ 0 and k

1

¼ 0 for one fixed-point to three fixed-points is called

• a monotonic source switching bifurcation of the third-order at a point p ¼ p1 2 @X12 ðd 3 f =dx3k jx ¼ai [ 0Þ, k 1 • a monotonic sink switching bifurcation of the third-order at a point p ¼ p1 2 @X12 ðd 3 f =dx3k jx ¼ai \0Þ. 1

(vii) Under conditions of Di ¼ B2i  4Ci ¼ 0ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj ¼ 0ðj 2 f1; 2gÞ; ðaÞ

ðaÞ

ð3:48Þ

1 2

b1 ¼ b2 ¼  Ba for a ¼ i; j; B1 6¼ B2 ; a standard form of Eq. (3.1) is where

xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ2 ðxk  a2 Þ2 1 2

1 2

1 2

1 2

a1 ¼ minf B1 ;  B2 g; a2 ¼ maxf B1 ;  B2 g:

ð3:49Þ

ð3:50Þ

(vii1a) For a0 ðpÞ [ 0, the fixed-points of xk ¼ ai ði ¼ 1; 2Þ are unstable of a monotonic upper-saddle of the second-order if d 2 f =dx2k jx ¼ai [ 0. k (vii1b) The fixed-points of xk ¼ ai ði ¼ 1; 2Þ for two fixed-points vanishing and appearance are called a monotonic upper-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 .

3.1 Period-1 Appearing Bifurcations

179

(vii2a) For a0 ðpÞ\0, the fixed-points of xk ¼ ai ði ¼ 1; 2Þ are unstable of a monotonic lower-saddle of the second-order if d 2 f =dx2k jx ¼ai \0. k (vii2b) The fixed-point of xk ¼ ai ði ¼ 1; 2Þ for two fixed-points vanishing and appearance are called a monotonic lower-saddle-node appearing bifurcation of the second-order at a point p ¼ p1 2 @X12 . (viii) Under conditions of Di ¼ B2i  4Ci ¼ 0 ði 2 f1; 2gÞ and Dj ¼ B2j  4Cj ¼ 0 ðj 2 f1; 2gÞ; ðaÞ

ðaÞ

1 2

b1 ¼ b2 ¼  Ba for a ¼ i; j;

ð3:51Þ

B1 6¼ B2 ; the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  a1 Þ4

ð3:52Þ

where 1 2

1 2

a1 ¼  B1 ¼  B2 :

ð3:53Þ

(viii1a) For a0 ðpÞ [ 0, the fixed-point of xk ¼ a1 is unstable of a monotonic upper-saddle of the fourth-order if d 4 f =dx4k jx ¼a1 [ 0. k (viii1b) The fixed-point of xk ¼ a1 for four fixed-points vanishing and appearance are called a upper-saddle-node appearing bifurcation of the fourth order at a point p ¼ p1 2 @X12 . (viii2a) For a0 ðpÞ\0, the fixed-point of xk ¼ a1 is unstable of a monotonic lower-saddle of the fourth-order if d 4 f =dx4k jx ¼a1 \0. k (viii2b) The fixed-point of xk ¼ a1 for four fixed-points vanishing and appearance are called a monotonic lower-saddle-node appearing bifurcation of the fourth-order at a point p ¼ p1 2 @X12 . Proof As for quadratic discrete systems, the proof is completed.

3.2

■

Period-1 to Period-2 Bifurcation Trees

In this section, period-1 stability and bifurcation of quartic nonlinear discrete systems are discussed graphically and period-2 fixed-points on the period-1 to period-2 bifurcation trees are also presented for a better understanding of complex bifurcations.

180

3 Quartic Nonlinear Discrete Systems

As discussed before, a quartic nonlinear discrete system is expressed by the product of two quadratic polynomials, i.e., xk þ 1 ¼ xk þ f ðxk ; pÞ

ð3:54Þ

¼ xk þ a0 ðpÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ½x2k þ B2 ðpÞxk þ C2 ðpÞ:

Thus, for xk þ 1 ¼ xk , the period-1 fixed-points are determined by the roots of two quadratic polynomial equations, i.e.,  x2 k þ B1 ðpÞxk þ C1 ðpÞ ¼ 0;

ð3:55Þ

 x2 k þ B2 ðpÞxk þ C2 ðpÞ ¼ 0:

 • If x2 k þ Bi ðpÞxk þ Ci ðpÞ 6¼ 0 for i ¼ 1; 2, such a quartic discrete system does not have any period-1 fixed-points.  • If x2 k þ Bi ðpÞxk þ Ci ðpÞ ¼ 0 for i ¼ 1; 2, such a quartic discrete system has four period-1 fixed-points.  2  • If x2 k þ Bi ðpÞxk þ Ci ðpÞ ¼ 0 and xk þ Bj ðpÞxk þ Cj ðpÞ 6¼ 0 for i; j 2 f1; 2g and i 6¼ j, such a quartic discrete system has two period-1 fixed-points.

The roots of such quadratic equations are determined by the corresponding discriminant of the quadratic equations, i.e., Di ¼ B2i  4Ci for i ¼ 1; 2:

ð3:56Þ

If Di \0, the quadratic equation of x2k þ Bi ðpÞxk þ Ci ðpÞ ¼ 0 does not have any roots. If Di [ 0, the quadratic equation of x2k þ Bi ðpÞxk þ Ci ðpÞ ¼ 0 has two roots. If Di ¼ 0, the quadratic equation of x2k þ Bi ðpÞxk þ Ci ðpÞ ¼ 0 has a repeated root. With parameter variation, suppose one of two quadratic polynomial equations has one root intersected with the roots of the other quadratic polynomial equation. ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ There are six cases for a0 [ 0: (i) b2 ¼ b1 ; (ii) b1 ¼ b1 ¼ b2 ¼ 12Bi : ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðjÞ

ðiÞ

ðiÞ

ðiÞ

ðjÞ

(iii) b1 ¼ b1 , (iv) b2 ¼ b2 (v) b2 ¼ b1 ¼ b2 ¼ 12Bi , (vi) b1 ¼ b2 , as presented in Fig. 3.1. The intersected points for simple fixed-roots is a monotonic saddle-node bifurcation of the second-order for the subscritical case. The monotonic lower-saddle-node and monotonic upper-saddle-node bifurcations are shown in Fig. 3.1(i, ii) and (iv, vi), respectively. P-2 is for period-2 fixed-point. Open curves of P-2 are for mSI-oSO. Closed loops of P-2 is for mSI-oSO-mSI. The bifurcation dynamics for the 1-dimensional quartic nonlinear, forward, discrete system is determined by xk þ 1 ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ with ia ; a 2 f1; 2; 3g for four fixed-points or

ð3:57Þ

3.2 Period-1 to Period-2 Bifurcation Trees

181

b1(i )

a0 > 0 mSO

mSO

mSI-oSO

mLSN

b2( j )

mLSN

mUSN

P-2

mSO mUSN

x

b

∗ k

x

Δj >0

|| p ||

(iv)

a0 > 0

b1(i )

mSO

a0 > 0

3rd mSO

mUSN

mSI-oSO

b2(i )

mSO

b2( j ) P-2

|| p ||

P-2

b1(i )

mSI-oSO

mSO 3rd mSI

b2( j )

mSI-oSO

xk∗

b2(i ) P-2

Δj >0

Δj 0

Δj 0

Δj 0

mSO mUSN

b1( j )

a0 > 0

mSO

( j) 1

b

P-1

P-2

mSI-oSO

P-1 P-2

mUSN

b1(i )

mSI-oSO-mSI

mSI-oSO-mSI mUSN

mLSN P-2

P-2

mSO

b2(i )

xk∗

b2( j )

mSI-oSO

Δj 0

b2( j )

mSO mUSN

xk∗

b2(i )

mSI-oSO

P-2

|| p ||

mLSN

P-2

|| p ||

Δj 0

(vi)

Fig. 3.1 Stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ ðiÞ ðjÞ ðiÞ ðjÞ system ða0 [ 0Þ: (i) b2 ¼ b1 , (ii) b1 ¼ b1 ¼ b2 ¼ 12Bi . (iii) b1 ¼ b1 , (iv) b2 ¼ b2 ðjÞ ðiÞ ðiÞ ðiÞ ðjÞ 1 (v) b2 ¼ b1 ¼ b2 ¼ 2Bi , (vi) b1 ¼ b2 : mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node, mSI-oSO: monotonic sink to oscillatory source, mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: Period-2 fixed-point. Open curve for P-2 is for mSI-oSO. Closed loop for P-2 is for mSI-oSO-mSI

182

3 Quartic Nonlinear Discrete Systems 1 2

1 4

xk þ 1 ¼ xk þ a0 ðxk  ai Þ2 ½ðxk þ Bj Þ2  Dj 

ð3:58Þ

with i; j 2 f1; 2g for two fixed-points. If the intersected point occurs at the repeated root, the third-order monotonic source and monotonic sink switching bifurcations are presented in Fig. 3.1(ii) and (iv), respectively. The corresponding bifurcation dynamics for the 1-dimensional quartic, forward discrete system is determined by xk þ 1 ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai2 Þ

ð3:59Þ

with ia ; a 2 f1; 2g. The stable and unstable fixed pints are presented by solid and dashed curves, respectively. The intersected points are marked by circular symbols, which are for bifurcation points. Without losing generality, suppose the two roots of ðiÞ ðiÞ the quadratic polynomial equation have a relation of b1 [ b2 for i ¼ 1; 2. The repeated roots of the two quadratic polynomial equations are also the monotonic upper or lower-saddle-node bifurcations for two fixed-points appearance and vanishing. The period-2 fixed-points are sketched as well. Similarly, the six cases of stability and bifurcation diagrams varying with parameter for a0 \0 are presented in Fig. 3.2. The stability and bifurcation conditions for a0 \0 are opposite to a0 [ 0: If the roots of two quadratic equations do not have any intersections, the open loops for stability and bifurcation diagrams of fixed-points for a0 [ 0 and a0 \0 are presented in Fig. 3.3. There are four cases of open loops for a0 [ 0: (i) Bi \Bj , ðjÞ

ðjÞ

(ii) Bi [ Bj , (iii) b2 \  12Bi \b1 , (vi) Di ¼ Dj ; Bi 6¼ Bj and four cases of open ðjÞ

ðjÞ

loops for a0 \0: (v) Bi \Bj , (vi) Bi [ Bj , (vii) b2 \  12Bi \b1 ; (viii) Di ¼ Dj ; Bi 6¼ Bj . The two bifurcations occur at the same time because the quadratic equations have Di ¼ Dj ; Bi 6¼ Bj . The bifurcation points are only for two fixed-points appearance or vanishing from the discriminants of the quadratic equations. The bifurcation dynamics for the 1-dimensional quartic discrete system is from xk þ 1 ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ

ð3:60Þ

with ia ; a 2 f1; 2; 3g. With varying vector parameter, the open loops of stability and bifurcation diagrams will become closed loops. Thus, the closed loops of stability and bifurcation diagrams of fixed-points for a0 [ 0 and a0 \0 are presented in Fig. 3.4. There are ðjÞ ðjÞ six cases of closed loops: (i) Bi \Bj , (ii) Bi [ Bj , (iii) b2 \  12 Bi \b1 for a0 [ 0; ðjÞ

ðjÞ

(iv) Bi \Bj , (v) Bi [ Bj , (vi) b2 \  12 Bi \b1 for a0 \0. For such a closed loop, the bifurcation points are the upper and lower-saddle bifurcations of the second order at both ends. The bifurcation points are determined from the discriminants of the quadratic equations. The corresponding period-2 fixed-points are sketched as well.

3.2 Period-1 to Period-2 Bifurcation Trees a0 < 0

183

b1(i )

mSI-oSO

b1( j )

a0 < 0

mSI-oSO

P-2

P-2 mSO

mLSN

b1( j )

mUSN

mSO

b2( j )

mSI-oSO-mSI

mUSN

P-2

(i ) 2

b

mSO

x

x

(iv) P-2

b1(i )

mSI-oSO

a0 < 0

b

P-2

mLSN

mSO

b2(i ) b2( j )

Δj >0

mSO

|| p ||

b1(i )

P-2 mLSN

(iii)

b1( j )

mSI-oSO P-2

P-1

mSO

b1(i )

mSO mLSN

Δj 0

Δj 0

a0 < 0 mSO

mSI-oSO-mSI

b1( j )

b1( j )

P-2

P-2

b1(i )

4th mUSN

mSO

x

Δi, j = 0

4th mLSN

4th mLSN

P-2

b2(i )

b

mSI-oSO-mSI

b2( j ) ∗ k

x

mSI-oSO-mSI

|| p ||

mSO

4th mUSN

(i ) 2

P-2 ∗ k

b1(i )

Δi, j > 0

Δi, j = 0

b2( j )

mSO

Δi, j > 0

Δi, j = 0

|| p ||

(i)

Δi , j = 0

(ii)

a0 > 0

P-2

a0 < 0

b1( j )

mSO

b1( j )

mSI-oSO

(i ) 1

b1(i )

b

P-2

mSI-oSO 4th mUSN

mSO 4th

− Bi = − B j 1 2

1 2

− 12 Bi = − 12 B j

mLSN mSI-oSO

mSO

b2(i )

xk∗

mSI-oSO

( j) 2

b

P-2

b2(i )

xk∗

b2( j )

mSO

P-2

|| p ||

Δj >0

Δj 0

Δj >0

Δj 0

Δj 0

Δj 0

a0 > 0 P-1

P-2

LSN

mSI-oSO-mSI

P-2

mLSN mUSN

b2( j )

mSO

mSO

b2(i )

P-2

mUSN

xk∗

b1(i )

mSI-oSO

b1( j )

mUSN

b1( j )

mSO

mSO

b2(i )

mSI-oSO

b2( j )

mSI-oSO-mSI

xk∗

mUSN

P-2

Δj >0

Δj 0

b1( j )

b1( j )

a0 > 0

mUSN

mSO

mSO

b1(i )

P-2

mSI-oSO-mSI

mLSN

x

mLSN P-2 mSO

mSO

mUSN

b2( j )

b2( j )

mUSN

mSI-oSO-mSI

∗ k

Δj >0

Δj 0

a0 > 0

mSO

mSO

mUSN

P-2 (i ) 2

mUSN

b

b1( j ) mUSN

mLSN

P-2 mSO

b1(i )

( j) 1

b

mSI-oSO-mSI mUSN

P-2

mLSN

mSO P-2

mUSN

b2( j )

xk∗

Δj 0

|| p ||

b

xk∗

mSI-oSO-mSI

Δj 0

|| p ||

(iv)

a0 > 0

a0 > 0

b1(i )

3rd mSO mSO

mUSN

b2(i )

P-2 mSO

mSI-oSO-mSI mSO

b2( j ) mUSN

xk∗

P-2

|| p ||

b2(i )

Δj 0

(ii)

(v)

a0 > 0

a0 > 0

mSO

mSO

P-2

b1(i )

mUSN

P-2

mLSN

mSO

(i ) 2

b

( j) 2

b

b2( j ) LSN

mSO P-2

b1( j )

mUSN

b1( j )

mUSN

(i ) 1

b

mUSN

mUSN mSI-oSO-mSI

xk∗

b1(i )

P-2

3rd mSI

Δj 0

mUSN

mUSN

mUSN mSI-oSO-mSI

|| p ||

b1( j )

P-2

P-2

b2( j )

xk∗

mSO

mUSN

mSO

b1( j )

mUSN mSI-oSO-mSI

(i)

mUSN

b2(i )

( j) 2

P-2

(i ) 2

b

xk∗ mSI-oSO-mSI

|| p ||

Δj 0

|| p ||

Δj 0

(iii)

(vi)

Fig. 3.8 Closed loops of stability and bifurcation of fixed-points in the 1-dimeisonal, quartic ðiÞ ðjÞ ðiÞ ðjÞ ðjÞ ðiÞ ðiÞ nonlinear discrete system (a0 [ 0): (i) b2 ¼ b1 andb1 ¼ b2 , (ii) b1 ¼ b1 ¼ b2 ¼ 12 Bi with ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ ðiÞ ðjÞ b1 ¼ b1 and b2 ¼ b2 , (iii) b1 ¼ b1 and b2 ¼ b2 , (iv) b2 ¼ b2 and b1 ¼ b1 , ðjÞ

ðiÞ

ðiÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

ðiÞ

ðjÞ

(v) b2 ¼ b1 ¼ b2 ¼ 12Bi with b1 ¼ b1 and b2 ¼ b2 , (vi) b1 ¼ b2 and b2 ¼ b1 . mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node, mSI-oSO-mSI: monotonic sink to oscillatory source to monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: Period-2 fixed-point. Closed loop of P-2 are for mSI-oSO-mSI

3.3 Higher-Order Period-1 Quartic Discrete Systems

191

b1(i )

a0 < 0

a0 < 0

P-2

mLSN

mSO

mLSN

b1( j )

P-2

mSO

b2(i ) mLSN

mUSN

mLSN

P-2

b1( j ) P-2

mLSN

xk∗ || p ||

( j) 2

b

b2( j )

mSO

b2(i ) mLSN

mSO

xk∗

Δj 0

b1(i )

USN

Δj >0

Δj 0

197 a0 > 0

b1(2) mSO

mUS

P-2

mSO

b1(2)

mUSN

b2(2)

mUSN mSI-oSO

mSI-oSO P-2

xk∗

b2(2)

|| p ||

Δ2 > 0

Δ2 < 0 Δ2 = 0

xk∗

mUS

|| p ||

Δ2 < 0 Δ2 = 0

(i)

b1

Δ2 > 0

(iv)

a0 > 0 3rd

mUS

mSO

mSO

a0 > 0

b1(2)

b1(2)

mSO

b1

b1

mLS 4th

USN

mUSN

mUS

mLS

mSI-oSO

xk∗

P-2

b2(2)

xk∗

mSI-oSO

b2(2)

P-2

|| p ||

Δ2 > 0

Δ2 < 0 Δ2 = 0

|| p ||

Δ2 < 0 Δ2 = 0

(ii)

Δ2 > 0

(v)

a0 > 0 mSO

a0 > 0

b1(2)

b1(1) mUS

4th mUS

mUSN mUS

mLS

b1

3rd SI

∗ k

x

mSI-oSO

b2(2)

xk∗

mUS

b2(1)

P-2

|| p ||

Δ2 < 0 Δ2 = 0

(iii)

Δ2 > 0

|| p ||

Δ1 < 0 Δ1 = 0

Δ1 > 0

(vi)

Fig. 3.10 Stability and bifurcation of three fixed-points with intersection in the 1-dimensional, ð2Þ quartic nonlinear forward discrete system ða0 [ 0Þ: (i) without intersection b1 [ b1 , (ii) an ð2Þ ð2Þ ð2Þ intersection at b1 ¼ b1 , (iii) an intersection at b1 ¼ b2 , (iv) without intersection b1 \b1 , (v) an intersection at b1 ¼ 12B1 , (vi) D1 ¼ 0: mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: Period-2 fixed-point. Open curves of P-2 are for mSI-oSO

198

3 Quartic Nonlinear Discrete Systems b1

a0 < 0

a0 < 0

b1(2) mSI-oSO

mLS

P-2

mSI-oSO

b1(2)

mLSN

b2(2)

mLSN mSO

P-2

mSO

xk∗

b2(2)

|| p ||

Δ2 > 0

Δ2 < 0 Δ2 = 0

xk∗

b1

mLS

Δ2 < 0 Δ2 = 0

|| p ||

(i)

Δ2 > 0

(iv)

a0 < 0

a0 < 0

P-2 3rd

b1(2)

P-2

b1

mLS mSI-oSO

b1(2)

mSI-oSO

mSI mUS 4th

mLSN

mLSN

mLS

mUS

b1

mSO

xk∗

b2(2)

|| p ||

Δ2 > 0

Δ2 < 0 Δ2 = 0

xk∗

mSO

|| p ||

Δ2 < 0 Δ2 = 0

(ii)

b2(2)

Δ2 > 0

(v)

a0 < 0

a0 < 0

b1(2)

mSI-oSO

mLS

b1(1)

P-2

4th mLS

LSN mLS

mUS

x

|| p ||

b1

3rd mSO

∗ k

mSO

Δ2 < 0 Δ2 = 0

(iii)

Δ2 > 0

b2(2)

xk∗ || p ||

mLS

Δ1 < 0 Δ1 = 0

b2(1)

Δ1 > 0

(vi)

Fig. 3.11 Stability and bifurcation of three fixed-points with intersection in the 1-dimensional, ð2Þ quartic nonlinear forward discrete system ða0 \0Þ: (i) without intersection b1 [ b1 , (ii) an ð2Þ ð2Þ ð2Þ intersection at b1 ¼ b1 , (iii) an intersection at b1 ¼ b2 , (iv) without intersection b1 \b1 , (v) an intersection at b1 ¼ 12B1 , (vi) D1 ¼ 0: mLSN: monotonic lower-saddle-node, mUSN: monotonic upper-saddle-node, mSI: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: Period-2 fixed-point. Open curves of P-2 are for mSI-oSO

3.3 Higher-Order Period-1 Quartic Discrete Systems

199

sink bifurcations of the third order, accordingly. In Fig. 3.11(v), the monotonic lower-saddle fixed-point for a0 \0 intersects with a repeated fixed-point with a monotonic lower-saddle. The intersection point is an unstable fixed-point, which is called a 4th order monotonic lower-saddle-node bifurcation. In Fig. 3.11(vi), the two monotonic second-order lower saddle fixed-points are presented for a0 \0. The two monotonic lower-saddle fixed-points appear at the monotonic lower-saddle bifurcation of the fourth-order. Consider a 1-dimensional, quartic nonlinear, forward, discrete system with two double fixed-points. (i) For b 6¼ a, the forward discrete system is xk þ 1 ¼ xk þ a0 ðpÞðxk  bðpÞÞ2 ðxk  aðpÞÞ2 :

ð3:93Þ

For such a system, if a0 [ 0, two repeated fixed-points of xk ¼ a; b are two monotonic upper-saddles, which are monotonically unstable. If a0 \0, two repeated fixed-points of xk ¼ a; b are two monotonic lower-saddles, which are monotonically unstable. (ii) For a ¼ b, the discrete system on the boundary is xk þ 1 ¼ xk þ a0 ðpÞðxk  bðpÞÞ4 :

ð3:94Þ

With parameter changes, the bifurcation diagram for the quartic nonlinear discrete system is presented in Fig. 3.12. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation point is marked by a circular symbol. In Fig. 3.12(i), if a0 [ 0, two repeated fixed-points of xk ¼ a; b

a0 > 0

a0 < 0

a

a

mLS

mUS 4th mUS

4th mUS

b xk∗

|| p ||

b xk∗

mUS

b2 < b1

b1 = b2

(i)

b2 > b1

mLS

|| p ||

b2 < b1

b1 = b2

b2 > b1

(ii)

Fig. 3.12 Stability and bifurcation of two mUS or mLS fixed-points with intersection in the 1-dimensional, quartic nonlinear discrete system: (i) (mUS:mUS)-flow ða0 [ 0Þ, (i) (LS:LS)-flow ða0 \0Þ. 4th mLS: 4th order monotonic lower-saddle bifurcation, 4th mUS- 4th order monotonic upper-saddle bifurcation. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

200

3 Quartic Nonlinear Discrete Systems

|| p 0 ||

xk∗ = a1

xk∗ 4th mLS

|| p ||

a0 < 0

4th mUS

a0 = 0

a0 > 0

Fig. 3.13 Stability of a repeated fixed-point with the fourth multiplicity in the 1-dimensional, quartic nonlinear discrete system: Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The stability switching is labelled by a circular symbol. 4th mLS: fourth-order monotonic lower-saddle bifurcation, 4th mUS: fourth-order monotonic upper-saddle bifurcation

are the monotonic upper-saddles of the second order. The two monotonic uppersaddles intersect at a point of xk ¼ a ¼ b with the fourth multiplicity, which is a monotonic upper-saddle bifurcation of the fourth-order for the (mUS:mUS) to (mUS:mUS) fixed-points. If a0 \0, two repeated fixed-points of xk ¼ a; b are the monotonic lower-saddle of the second order, which are intersected at a point of xk ¼ a ¼ b, as shown in Fig. 3.12(ii). Such a quartically repeated fixed-point is called a monotonic lower-saddle bifurcation of the fourth order for the (mLS:mLS) to (mLS:mLS) fixed-point. To illustrate the stability and bifurcation of fixed-point with singularity in a 1-dimensional, quadratic nonlinear system, the fixed-point of xk þ 1 ¼ xk þ a0 ðxk  a1 Þ4 is presented in Fig. 3.13. The fourth-order, monotonic upper and lower-saddles of fixed-point of xk ¼ a1 with the other order multiplicity are monotonically unstable, and the monotonic upper and lower saddle fixed-points of the fourth-order are invariant. At a0 ¼ 0, the monotonic lower-saddle fixed-point switches to the monotonic upper-saddle fixed-point, which is a switching point marked by a circular symbol.

3.4

Period-1 Switching Bifurcations

For further discussion on the switching bifurcations in the quartic nonlinear system, the following definitions are presented.

3.4 Period-1 Switching Bifurcations

3.4.1

201

Simple Period-1 Switching Bifurcations

Definition 3.4 Consider a 1-dimensional, quartic nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ ¼ xk þ a0 ðpÞðxk  aÞðxk 

bÞ½x2k

ð3:95Þ

þ B2 ðpÞxk þ C2 ðpÞ

where AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð3:96Þ

(i) If D2 ¼ B22  4C2 \0;

ð3:97Þ

fa1 ; a2 g ¼ sortfa; bg; a1  a2 ;

the quartic nonlinear discrete system has any two fixed-points. The corresponding standard form is 1 2

1 4

xk þ 1 ¼ xk þ a0 ðpÞðxk  a1 Þðxk  a2 Þ½ðxk þ B2 Þ2 þ ðD2 Þ:

ð3:98Þ

(i1) For a0 [ 0, the discrete fixed-point flow is a (mSI-oSO:mSO) discrete flow. (i1a) The fixed-point of xk ¼ a1 is • • • •

monotonically stable (monotonic sink) if df =dxk jx ¼a1 2 ð1; 0Þ; k invariantly stable (zero-invariant sink) if df =dxk jx ¼a1 ¼ 1; k oscillatorilly stable (oscillatory sink) if df =dxk jx ¼a1 2 ð2; 1Þ; k flipped if df =dxk jx ¼a1 ¼ 2, where is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a1 [ 0; k

– an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a1 \0; k

• oscillatorilly unstable (oscillatory source) if df =dxk jx ¼a1 2 ð1; 2Þ. k

xk

(i1b) The fixed-point of ¼ a2 is monotonically unstable (monotonic source) if df =dxk jx ¼a2 2 ð0; 1Þ. k

202

3 Quartic Nonlinear Discrete Systems

(i2) For a0 \0, the fixed-point flow is a (mSO:mSI-oSO) flow. (i2a) The fixed-point of xk ¼ a1 is monotonically unstable (monotonic source) if df =dxk jx ¼a1 2 ð0; 1Þ. k (i2b) The fixed-point of xk ¼ a2 is • • • •

monotonically stable (monotonic sink) if df =dxk jx ¼a2 2 ð1; 0Þ; k invariantly stable (zero-invariant sink) if df =dxk jx ¼a2 ¼ 1; k oscillatorilly stable (oscillatory sink) if df =dxk jx ¼a2 2 ð2; 1Þ; k flipped if df =dxk jx ¼a2 ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼a2 [ 0; k

– an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼a2 \0; k

• oscillatorilly unstable (oscillatory source) if df =dxk jx ¼a2 2 ð1; 2Þ. k

(i3) Under D12 ¼ ða1  a2 Þ2 ¼ 0 with a1 ¼ a2

ð3:99Þ

the quartic nonlinear discrete system has a standard form as 1 2

1 4

xk þ 1 ¼ xk þ a0 ðpÞðxk  a1 Þ2 ½ðxk þ B2 Þ2 þ ðD2 Þ:

ð3:100Þ

(i3a) For a0 ðpÞ [ 0, the fixed-point of xk ¼ a1 is monotonically unstable (a monotonic upper-saddle of second-order, d 2 f =dx2k jx ¼a1 [ 0Þ. k

• Such a discrete flow is called a monotonic upper-saddle discrete flow of the second-order. • The bifurcation of fixed-point at xk ¼ a1 for two fixed-points switching of xk ¼ a1 ; a2 is called a monotonic upper-saddle-node switching bifurcation of the second-order at a point p ¼ p1 . (i3b) For a0 ðpÞ\ 0, the fixed-point of xk ¼ a1 is monotonically unstable (an lower-saddle of second-order, d 2 f =dx2k jx ¼a1 \ 0Þ. • Such a discrete flow is called a lower-saddle discrete flow of the second-order. • The bifurcation of fixed-point at xk ¼ a1 for two fixed-points switching of xk ¼ a1 ; a2 is called a monotonic lower-saddle-node switching bifurcation of the second-order at a point p ¼ p1 .

3.4 Period-1 Switching Bifurcations

203

(ii) If D2 ¼ B22  4C2 [ 0;

ð3:101Þ

the 1-dimensional quartic nonlinear discrete system has four fixed-points as ð2Þ

1 2

xk ¼ b1 ¼  ðB2 þ

pffiffiffiffiffiffi pffiffiffiffiffiffi 1 ð2Þ D2 Þ; xk ¼ b2 ¼  ðB2  D2 Þ 2

ð2Þ

ð2Þ

fa1 ; a2 ; a3 :a4 g ¼ sortfa; b; b1 ; b2 g; ai \ai þ 1:

ð3:102Þ

(ii1) The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þðxk  a4 Þ:

ð3:103Þ

(ii1a) For a0 [ 0, the discrete flow is called an (mSI-oSO:mSO: mSI-oSO:mSO) discrete flow. (ii1b) For a0 \ 0, the discrete flow is called an (mSO:mSI-oSOI:mSO: mSI-oSO) discrete flow. (ii2) The fixed-point of xk ¼ ai1 ði1 2 f1; 2; 3; 4gÞ is • • • • •

monotonically unstable (monotonic source) if df =dxk jxk ¼ai1 2 ð0; 1Þ; monotonically stable (monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ; k 1 invariantly stable (zero-invariant sink) if df =dxk jx ¼ai ¼ 1; k 1 oscillatorilly stable (oscillatory sink) if df =dxk jx ¼ai 2 ð2; 1Þ; k 1 flipped if df =dxk jx ¼ai ¼ 2, which is k

1

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼a1 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼a1 \0; • oscillatorilly unstable (oscillatory source) if df =dxk jxk ¼ai1 2 ð1; 2Þ. (ii3) Under Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; ai1 ¼ ai2 ; i1 ; i2 2 f1; 2; 3; 4g; i1 6¼ i2

ð3:104Þ

the standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai3 Þðxk  ai4 Þ ia 2 f1; 2; 3; 4g; a ¼ 1; 3; 4:

ð3:105Þ

(ii3a) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic upper-saddle of second-order, d 2 f =dx2k jx ¼ai [ 0Þ. k

1

204

3 Quartic Nonlinear Discrete Systems

• Such a discrete flow is called an upper-saddle discrete flow at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for two fixed-points switching of xk ¼ ai1 ; ai2 is called a monotonical upper-saddlenode switching bifurcation of the second-order at a point p ¼ p1 . (ii3b) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic lower-saddle of second-order, d 2 f =dx2k jx ¼ai \0Þ. k

1

• Such a discrete flow is called a lower-saddle discrete flow of the second-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for two fixed-points switching of xk ¼ ai1 ; ai2 is called a lower-saddle-node switching bifurcation of the second-order at a point p ¼ p1 . (ii4) Under Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; ai1 ¼ ai2 ¼ a3 ; i1 ; i2 ; i3 2 f1; 2; 3; 4g; i1 6¼ i2 6¼ i3 ;

ð3:106Þ

the standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai4 Þ ia 2 f1; 2; 3; 4g; a ¼ 1; 4:

ð3:107Þ

(ii4a) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic source of the third-order, d 3 f =dx3k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic source flow of the third-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for three simple fixedpoint bundle-switching of xk ¼ ai1 ; ai2 ; ai3 is called a source bundle-switching bifurcation of the third-order at a point p ¼ p1 . (ii4b) The fixed-point of xk ¼ ai1 is monotonically stable (a monotonic sink of the third-order, d 3 f =dx3k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic sink discrete flow of the third-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for three simple fixed-point bundle-switching of xk ¼ ai1 ; ai2 ; ai3 is called a monotonic sink bundle-switching bifurcation of the third-order at a point p ¼ p1 .

3.4 Period-1 Switching Bifurcations

205

(ii5) Under Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; ð3:108Þ

Di3 i4 ¼ ðai4  ai4 Þ2 ¼ 0; ai1 ¼ ai2 ¼ ai3 ¼ ai4 ; i1 ; i2 ; i3 ; i4 2 f1; 2; 3; 4g; i1 6¼ i2 6¼ i3 6¼ i4 the standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ4 :

ð3:109Þ

(ii5a) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic upper-saddle of the fourth-order, d 4 f =dx4k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic upper-saddle flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for four simple fixed-points bundle-switching of xk ¼ a1;2;3;4 is called a monotonic upper-saddle-node bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . (ii5b) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic lower-saddle flow of the third-order, d 4 f =dx4k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic lower-saddle flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for four simple fixed-points bundle-switching of xk ¼ a1;2;3;4 is called a monotonic lower-saddle-node bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . (iii) If D2 ¼ B22  4C2 ¼ 0;

ð3:110Þ

the 1-dimensional quartic nonlinear discrete system has three fixed-point as ð2Þ

ð2Þ

1 2

xk ¼ b1 ¼ b2 ¼  B2 ; ð2Þ

ð2Þ

fa1 ; a2 ; a3 g ¼ sortfa; b; b1 ¼ b2 g; ai \ ai þ 1 ; ð2Þ

ð2Þ

ai1 ;i2 ¼ b1 ¼ b2 ; ai3 ¼ a; ai4 ¼ b; ia 2 f1; 2; 3g; a 2 f1; 2; 3; 4g:

ð3:111Þ

206

3 Quartic Nonlinear Discrete Systems

The corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ:

ð3:112Þ

(iii1) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic upper-saddle, d 2 f =dx2k jx ¼ai [ 0Þ. k

1

• The discrete flow is a monotonic upper-saddle flow of the secondorder at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for the appearing or vanishing of two simple fixed-points is called the monotonic uppersaddle-node appearing bifurcation of the second-order. (iii2) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic lower-saddle, d 2 f =dx2k jx ¼ai \ 0Þ. k

1

• The discrete flow is a monotonic lower-saddle discrete flow at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for the appearing or vanishing of two simple fixed-points is called the monotonic lowersaddle-node appearing bifurcation of the second-order. (iii3) Under Di3 i4 ¼ ðai3  ai4 Þ2 ¼ 0; ai3 ¼ ai4 ; ai1 6¼ ai3 ; ia 2 f1; 2; 3g; a 2 f1; 2; 3; 4g;

ð3:113Þ

the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai3 Þ2 ia 2 f1; 2g; a ¼ 1; 3:

ð3:114Þ

The fixed-point of xk ¼ ai1 ; ai3 is monotonically unstable (a monotonic upper-saddle of the second-order, d 2 f =dx2k jx ¼ai ;ai [ 0Þ and monok 1 3 tonically unstable (a monotonic lower-saddle of the second-order, d 2 f =dx2k jx ¼ai ;ai \ 0Þ. k

1

3

• Such a discrete flow is called a (mUS:mUS) or (mLS:mLS) discrete flow. • The bifurcation of fixed-point at xk ¼ ai1 for two simple fixed-point onset of xk ¼ ai1 ; ai2 and at xk ¼ ai3 for two fixed-point switching of xk ¼ ai3 ; ai4 is called a (mUS:mUS) or (mLS:mLS) switching bifurcation at a point p ¼ p1 .

3.4 Period-1 Switching Bifurcations

207

(iii4) Under Di1 i3 ¼ ðai1  ai3 Þ2 ¼ 0; ai1 ¼ ai2 ; ai1 ¼ ai3 ai1 6¼ ai4 ; ia 2 f1; 2; 3g; a 2 f1; 2; 3; 4g;

ð3:115Þ

the standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai4 Þ ia 2 f1; 2g; a ¼ 1; 4:

ð3:116Þ

(iii4a) The fixed-point of xk ¼ ai1 is monotonically unstable (a thirdorder monotonic source, d 3 f =dx3k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic source pitchfork discrete flow of the third-order. • The bifurcation of fixed-point at xk ¼ ai1 for one simple fixedpoint of xk ¼ ai1 switching to three simple fixed-points of xk ¼ ai1 ;i2 ;i3 is called a monotonic upper-saddle-node pitchfork switching bifurcation of the third-order at a point p ¼ p1 . (iii4b) The fixed-point of xk ¼ ai1 is monotonically stable (a third-order monotonic sink, d 3 f =dx3k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic sink discrete flow of the third-order. • The bifurcation of fixed-point at xk ¼ ai1 for one simple fixed-point of xk ¼ ai1 switching to three simple fixed-points of x ¼ ai1 ;i2 ;i3 is called a monotonic sink pitchfork switching bifurcation of the third-order at a point p ¼ p1 . (iii5) Under Di1 i3 ¼ ðai1  ai3 Þ2 ¼ 0; Di3 i4 ¼ ðai3  ai4 Þ2 ¼ 0; ai1 ¼ ai2 ; ai1 ¼ ai3 ai1 ¼ ai4 ; ia 2 f1; 2; 3g; a 2 f1; 2; 3; 4g;

ð3:117Þ

the standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ4 :

ð3:118Þ

(iii5a) For a0 [ 0, the fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic upper-saddle of the fourth-order, d 4 f =dx4k jxk ¼ai1 [ 0Þ. • Such a discrete flow is called a monotonic upper-saddle discrete flow of the fourth-order.

208

3 Quartic Nonlinear Discrete Systems

• The bifurcation of fixed-point at xk ¼ ai1 for two simple fixed-points switching to four simple fixed-points is called a monotonic upper-saddle-node flower-bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . (iii5b) For a0 [ 0, the fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic lower-saddle of the fourth-order, d 4 f =dx4k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic lower-saddle discrete flow of the fourth-order. • The bifurcation of fixed-point at xk ¼ ai1 for two simple fixed-points switching to four simple fixed-points is called a monotonic lower-saddle-node flower-bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . Based on the previous definition, the stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system ða0 [ 0Þ is presented in Fig. 3.14. In Fig. 3.14(i)–(iii), monotonic upper-saddle-node (mUSN) and monotonic lower-saddle-node (mLSN) switching bifurcations are at two locations for two simple fixed-points, and one monotonic upper-saddle-node (mUSN) appearing bifurcation is for two simple fixed-points. In Fig. 3.14(iv), a third-order monotonic sink (3rd mSI) pitchfork-switching bifurcation for a switching of one monotonic sink fixed-point to three monotonic simple fixed-points is presented, and one monotonic upper-saddle-node (mUSN) switching bifurcation for two monotonic simple fixed-points switching is also presented. In Fig. 3.14(v), a third-order source (3rd mSO) bundle-switching bifurcation for three fixed-point bundle-switching is presented, and a monotonic upper-saddle-node (mUSN) appearing bifurcation for two fixed-point onsets is also presented. In Fig. 3.14(vi), a fourth-order upper-saddle (4th mUS) flower-bundle switching bifurcation for four simple fixed-points are presented. Similarly, the stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system ða0 \ 0Þ is presented in Fig. 3.15. In Fig. 3.15(i)– (iii), monotonic-lower-saddle-node (mLSN) and monotonic-upper-saddle-node (mUSN) switching bifurcations are at two locations for two simple fixed-points, and one monotonic-lower-saddle-node (mLSN) appearing bifurcation is for two simple fixed-points appearing. In Fig. 3.15(iv), a third-order monotonic source (3rd mSO) pitchfork-switching bifurcation for a switching of one monotonic source fixed-point to three monotonic simple fixed-points is presented, and one monotoniclower-saddle-node (mLSN) switching bifurcation for two simple fixed-points switching is also presented. In Fig. 3.15(v), a third-order monotonic sink (3rd mSI) bundle-switching bifurcation for three fixed-point bundle-switching is presented, and a monotonic lower-saddle-node (mLSN) appearing bifurcation for two fixed-point onset is also presented. In Fig. 3.15(vi), a fourth-order order monotonic-lower-saddle (4th mLS) flower-bundle switching bifurcation for four simple fixed-points are presented. The period-2 fixed-points are also sketched for mSI-oSO or mSI-oSO-mSI. For the further discussion on the switching bifurcation, the following definition is given for the 1-dimensional, quartic nonlinear discrete system.

3.4 Period-1 Switching Bifurcations

209 b

a0 > 0

mSO

a0 > 0

P-2

mUSN

b1(1)

mLSN

mSI-oSO

P-2

P-2

a

P-2 P-2

b

mSO mUSN

3rd mSI

mSO

mSO

b1(1) P-2

a

mUSN

b2(1)

mSI-oSO

xk∗

Δ2 < 0 Δ2 = 0

b

xk∗

P-2

|| p ||

(1) 2

Δ2 > 0

P-2

Δ2 < 0

|| p ||

Δ2 = 0 Δ2 > 0

(i)

(iv)

a0 > 0

a0 > 0

b mSO

mUSN

mLSN

P-2

mSO

b1(1) P-2

a

P-2

mSO

b1(1) P-2

b

3rd mSO

mSI-oSO

P-2

a

mSO

mSO mUSN

x

Δ2 < 0 Δ2 = 0

P-2

b2(1)

∗ k

x

P-2

|| p ||

USN

b2(1)

mSI-oSO ∗ k

Δ2 > 0

mSI-oSO

Δ2 < 0 Δ2 = 0

|| p ||

(ii)

(v) b

a0 > 0

a0 > 0

P-2

b1(1)

mLSN

b mSO

a

P-2

4thmUS mSO

mSO

mSI-oSO

xk∗

|| p ||

a

P-2

USN

Δ1 0

b1(1)

mSO

P-2

mSO

mUSN P-2

Δ2 > 0

mSI-oSO

b2(1)

xk∗

|| p ||

mSI-oSO

Δ2 < 0

Δ2 = 0

b2(1)

Δ2 > 0

(vi)

Fig. 3.14 Stability and bifurcations of fixed-points in the 1-dimeisonal, quartic nonlinear discrete system (a0 [ 0): (i)-(iii) Two (mUSN and mLSN) switching and one mUSN appearing bifurcations, (iv) 3rd mSI pitchfork-switching bifurcation, (v) 3rd mSO bundle-switching bifurcation, (vi) 4thm mUS flower-bundle switching bifurcation. mLSN: monotonic-lower-saddle-node, mUSN: monotonic-upper-saddle-node, mSI-oSO: monotonic-sink to oscillatory source, mSI-oSO-mSI: monotonic-sink to oscillatory source to monotonic sink, mSO: monotonic-source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: Period-2 fixed-point. Open curves of P-2 are for mSI-oSO. Closed loop of P-2 are for mSI-oSO-mSI

210

3 Quartic Nonlinear Discrete Systems b

a0 < 0 mLSN

P-2 mSO mUSN

P-2

P-2

LSN

b1(1) P-2

mSO

a b2(1)

mSO

Δ2 < 0 Δ2 = 0

mSO

b2(1)

Δ2 < 0

|| p ||

(i)

mLSN mUSN

Δ2 < 0 Δ2 = 0

a

mSO

P-2 mSI-oSO

mLSN

b2(1) ∗

Δ2 < 0 Δ2 = 0

|| p ||

(ii)

Δ2 > 0

(v) P-2

a0 < 0

b1(1)

a0 < 0

b

mSI-oSO mLSN

P-2

b1(1)

mUSN mSO

4th mLS

a

mSO

Δ2 < 0 Δ2 = 0

(iii)

Δ2 > 0

b

mSO

mSI-oSO

P-2

xk∗

|| p ||

mSI-oSO

P-2

P-2 mLSN

b2(1)

mSO

xk

Δ2 > 0

P-2

b

mSO

P-2

a

mSO-oSO

xk

mSI-oSO 3rd mSI

P-2

P-2

∗

P-2 mSI-oSO

P-2 b (1) 1

mSO

b1(1)

a0 < 0

b

mSI-oSO

|| p ||

Δ2 = 0 Δ2 > 0

(iv)

a0 < 0

mLSN

P-2

xk∗

Δ2 > 0

P-2

b1(1)

mSI-oSO

3rd mSO

P-2

P-2

xk∗

b

mSI-oSO

a mLSN

|| p ||

a0 < 0

mSI-oSO

mSI-oSO

mSO

(1) 2

b

mSO

xk∗

|| p ||

Δ2 < 0

Δ2 = 0

a P-2

b2(1)

Δ2 > 0

(vi)

Fig. 3.15 Stability and bifurcations of fixed-points in the 1-dimeisonal, quartic nonlinear discrete system (a0 \0): (i)-(iii) Two (mLSN and mUSN) switching and one mLSN appearing bifurcations, (iv) 3rd mSO pitchfork switching bifurcation, (v) 3rd mSI bundle-switching bifurcation, (vi) 4th mLS flower-bundle switching bifurcation. mLSN: monotonic-lower-saddle-node, mUSN: monotonic-upper-saddle-node, mSI-oSO: monotonic-sink to oscillatory source, mSI-oSO: monotonic-sink to oscillatory source to monotonic sink, mSO: monotonic-source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. Open curves of P-2 are for mSI-oSO. Closed loop of P-2 are for mSI-oSO-mSI

3.4 Period-1 Switching Bifurcations

211

Definition 3.5 Consider a 1-dimensional, quartic nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ

ð3:119Þ

¼ a0 ðpÞðxk  aÞðxk  bÞðxk  cÞðxk  dÞ where AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð3:120Þ

(i) If fa1 ; a2 ; a3 ; a4 g ¼ sortfa; b; c; dg; ai  ai þ 1 ;

ð3:121Þ

the quartic nonlinear discrete system has any four simple fixed-points. The standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðpÞðxk  a1 Þðxk  a2 Þðxk  a2 Þðxk  a3 Þ:

ð3:122Þ

(i1) For a0 [ 0, the fixed-point flow is a (mSI-oSO:mSO:mSI-oSO:mSO) flow. (i1a) The fixed-point of xk ¼ a1;3 is • • • •

monotonically stable (monotonic sink) if df =dxk jx ¼a1;3 2 ð1; 0Þ, k invariantly stable (invariant sink) if df =dxk jx ¼a1;3 ¼ 1, k oscillatorilly stable (oscillatory sink) if df =dxk jx ¼a1;3 2 ð2; 1Þ, k flipped if df =dxk jx ¼a1;3 ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼a1;3 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼a1;3 \0; • oscillatorilly stable (oscillatory source) if df =dxk jx ¼a1;3 2 ð1; 2Þ. k

(i1b) The fixed-point of x ¼ a2;4 is monotonically unstable (a monotonic source) if df =dxk jx ¼a2;4 2 ð0; 1Þ. k (i2) For a0 \ 0, the fixed-point flow is a (mSO:mSI-oSO:mSO:mSI-oSO) flow. (i2a) The fixed-point of xk ¼ a1;3 is monotonically unstable (a monotonic source) if df =dxk jx ¼a1;3 2 ð0; 1Þ. k (i2b) The fixed-point of xk ¼ a2;4 is

212

3 Quartic Nonlinear Discrete Systems

• • • •

monotonically stable (monotonic sink) if df =dxk jx ¼a2;4 2 ð1; 0Þ, k invariantly stable (invariant sink) if df =dxk jx ¼a2;4 ¼ 1, k oscillatorilly stable (oscillatory sink) if df =dxk jx ¼a2;4 2 ð2; 1Þ, k flipped if df =dxk jx ¼a2;4 ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼a2;4 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼a2;4 \0; • oscillatorilly stable (oscillatory source) if df =dxk jx ¼a2;4 2 ð1; 2Þ. k

(ii) If Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0 with ai1 ¼ ai2 ; i1 ; i2 2 f1; 2; 3; 4g;

ð3:123Þ

the quartic nonlinear discrete system has a standard form as xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai3 Þðxk  ai4 Þ:

ð3:124Þ

(ii1) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic-uppersaddle of second-order, d 2 f =dx2k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic-upper-saddle discrete flow of the second-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for two fixed-points switching of xk ¼ ai1 ; ai2 is called a monotonic-upper-saddle-node switching bifurcation of the second-order at a point p ¼ p1 . (ii2) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic-lowersaddle of second-order, d 2 f =dx2k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a lower-saddle discrete flow of the second-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for two fixed-points switching of xk ¼ ai1 ; ai2 is called a lower-saddle-node switching bifurcation of the second-order at a point p ¼ p1 . (ii3) The fixed-point of xk ¼ aj ðj ¼ i3 ; i4 Þ is • monotonically unstable (monotonic source) if df =dxk jxk ¼aj 2 ð0; 1Þ; • monotonically stable (monotonic sink) if df =dxk jx ¼aj 2 ð1; 0Þ; k • invariantly stable (invariant sink) if df =dxk jx ¼aj ¼ 1; k

3.4 Period-1 Switching Bifurcations

213

• oscillatorilly stable (oscillatory sink) if df =dxk jx ¼aj 2 ð2; 1Þ; k • flipped if df =dxk jx ¼aj ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼aj [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼aj \0; • oscillatorilly stable (oscillatory source) if df =dxk jx ¼aj 2 ð1; 2Þ. k

(iii) If Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; ai1 ¼ ai2 ¼ a3 ; i1 ; i2 ; i3 2 f1; 2; 3; 4g; i1 6¼ i2 6¼ i3 ;

ð3:125Þ

the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai4 Þ ia 2 f1; 2; 3; 4g; a ¼ 1; 4:

ð3:126Þ

(iii1) The fixed-point of xk ¼ ai1 is monotonically unstable (a third-order monotonic source,d 3 f =dx3k jx ¼ai [ 0). k

1

• Such a discrete flow is called a monotonic source discrete flow of the third-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of three simple fixed-points of xk ¼ ai1 ; ai2 ; ai3 is called a third-order monotonic source bundle-switching bifurcation at a point p ¼ p1 . (iii2) The fixed-point of xk ¼ ai1 is monotonically stable (a third-order monotonic sink, d 3 f =dx3k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic sink discrete flow of the third-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of three simple fixed-points of xk ¼ ai1 ; ai2 ; ai3 is called a monotonic sink bundle-switching bifurcation of the third-order at a point p ¼ p1 . (iii3) The fixed-point of xk ¼ ai4 is • monotonically unstable (monotonic source) if df =dxk jxk ¼ai4 2 ð0; 1Þ; • monotonically stable (monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ; k 4 • invariantly stable (invariant sink) if df =dxk jx ¼ai ¼ 1; k

4

214

3 Quartic Nonlinear Discrete Systems

• oscillatorilly stable (oscillatory sink) if df =dxk jx ¼ai 2 ð2; 1Þ; k 4 • flipped if df =dxk jx ¼ai ¼ 2, which is k

4

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jxk ¼ai4 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jxk ¼ai4 \0; • oscillatorilly stable (oscillatory source) if df =dxk jxk ¼ai4 2 ð1; 2Þ. (iv) If Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; Di3 i4 ¼ ðai4  ai4 Þ2 ¼ 0; ai1 ¼ ai2 ¼ ai3 ¼ ai4 ;

ð3:127Þ

i1 ; i2 ; i3 ; i4 2 f1; 2; 3; 4g; i1 6¼ i2 6¼ i3 6¼ i4 ; the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ4 :

ð3:128Þ

(iv1) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonic-uppersaddle of the fourth-order, d 4 f =dx4k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic-upper-saddle discrete flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of four simple fixed-points of xk ¼ a1;2;3;4 is called a monotonicupper-saddle-node bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . (iv2) The fixed-point of xk ¼ ai1 is monotonically unstable (a 4th order monotonic-lower-saddle, d 4 f =dx4k jx ¼ai \ 0Þ. k

1

• Such a flow is called a monotonic-lower-saddle discrete flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of four simple fixed-points of xk ¼ a1;2;3;4 is called a monotonic-lowersaddle bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . From the previous definition, stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system is presented in Fig. 3.16. For a0 [ 0, the bifurcations and stability of fixed-points are presented in Fig. 3.16(i)–(iii). In Fig. 3.16(i), four monotonic-upper-saddle-node (mUSN) and two monotonic-lowersaddle-node (mLSN) switching bifurcation network are presented for all possible switching bifurcation between two simple fixed-points. In Fig. 3.16(ii), a third-order

3.4 Period-1 Switching Bifurcations

215

monotonic-sink (3rd mSI) bundle-switching bifurcation for three simple fixed-point is presented, and there are three possible monotonic-upper-saddle-node (mUSN) and monotonic-lower-saddle-node (mLSN) switching bifurcations for two simple fixed-points. Figure 3.16(iii) a fourth-order monotonic-upper-saddle (4th mUS) bundle-switching bifurcation for four simple fixed-points are presented. Similarly, For a0 \ 0, the bifurcations and stability of fixed-points are presented in Fig. 3.16 (iv)–(vi). In Fig. 3.16(iv), four monotonic-lower-saddle-node (mLSN) and two monotonic-upper-saddle-node (mUSN) switching bifurcation network are presented for all possible switching bifurcation between two simple fixed-points. In Fig. 3.16 (v), a third-order monotonic source (mSO) bundle-switching bifurcation for three simple fixed-point is presented, and there are three possible monotonic-lower-saddle-node (mLSN) and monotonic-upper-saddle-node (mUSN) switching bifurcations for two simple fixed-points. Figure 3.16(vi) a fourth-order monotonic-lower-saddle (4th mLS) bundle-switching bifurcation for four simple fixed-points are presented. The corresponding period-2 fixed points are sketched as well. For the switching bifurcation between the second-order and simple fixed-points, the following definition is given for the 1-dimensional, quartic nonlinear discrete system.

3.4.2

Higher-Order Period-1 Switching Bifurcations

Definition 3.6 Consider a 1-dimensional, quartic nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ

ð3:129Þ

2

¼ xk þ a0 ðpÞðxk  aÞ ðxk  bÞðxk  cÞ where AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð3:130Þ

(i) If fa1 ; a2 ; a3 g ¼ sortfa; b; cg; ai \ai þ 1 i1 ; i2 ; i3 2 f1; 2; 3g

ð3:131Þ

the quartic nonlinear discrete system has a standard form as xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ2 ðxk  ai2 Þðxk  ai3 Þ:

ð3:132Þ

216

3 Quartic Nonlinear Discrete Systems

a0 > 0

P-2

c

mUSN

a0 < 0

P-2

mSO

c

P-2 P-2

mUSN mLSN P-2

mLSN

mLSN

mLSN

P-1

b

mUSN mUSN

b

mSO P-2

mUSN

a

mSO

P-2

mUSN

mLSN

mSI-oSO

mLSN

a

mSI-oSO

xk∗ d

Δ12 > 0

|| p ||

P-2

xk∗

P-2

mSO

d Δ12 = 0 Δ 21 > 0

Δ12 > 0

|| p ||

Δ12 = 0 Δ 21 > 0

(i)

(iv) c

mSO

a0 > 0

mUSN

P-2

P-2 P-2

a0 < 0

P-2

mUSN

mSI-oSO

P-2

mLSN

P-2

a mSI-oSO

d

P-2

b

a0 > 0

d

Δ 23 > 0 Δ 23 = 0 Δ 32 > 0

Δ12 > 0 Δ12 = 0 Δ 21 > 0

|| p ||

(v) a0 < 0

mSO

c 4th mUS P-2

P-2 P-2

mSI-oSO

4th mLS

b

mSO

xk∗ P-2

(iii)

P-2

P-2

a

xk∗ Δ ij = 0

b mSI-oSO

P-2 mSI-oSO

d

c

mSO

P-2

mSO

Δ ij > 0

a

P-2

Δ 23 > 0 Δ 23 = 0 Δ 32 > 0

Δ12 > 0 Δ12 = 0 Δ 21 > 0

3rd mSO mSO

xk∗

(ii)

|| p ||

mSO

c

mSI-oSO

P-2

3rd mSI

|| p ||

mLSN P-2

mUSN

P-2

xk∗

P-2

mLSN

b

mSO

mSI-oSO

Δ ji > 0 i, j = 1,2,3,4; i ≠ j

|| p ||

a

d Δ ij > 0

Δ ij = 0

Δ ji > 0 i, j = 1,2,3,4; i ≠ j

(vi)

Fig. 3.16 Stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system ða0 [ 0Þ: (i) Four USN and two LSN switching bifurcation network, (ii) 3rd order mSI bundle-switching bifurcation, (iii) 4th order mUS bundle-switching bifurcation, ða0 \0Þ: (iv) Four mLSN and two mUSN switching bifurcation network, (v) 3rd order mSO bundle-switching bifurcation, (vi) 4th order mLS bundle-switching bifurcation. mLSN: monotonic lower-saddle-node, mUSN: upper-saddle-node, mSI-oSO: sink, mSO: source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. Open curves of P-2 are for mSI-oSO. Closed loop of P-2 are for mSI-oSO-mSI

3.4 Period-1 Switching Bifurcations

217

(i1a) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotonicupper-saddle of second-order, d 2 f =dx2k jx ¼ai [ 0Þ. Such a discrete flow k 1 is called a monotonic upper-saddle discrete flow of the second-order at xk ¼ ai1 . (i1b) The fixed-point of xk ¼ ai1 is monotonically unstable (a monotoniclower-saddle of second-order, d 2 f =dx2k jx ¼ai \ 0Þ. Such a discrete flow k 1 is called a monotonic lower-saddle discrete flow of the second-order at xk ¼ ai1 . (i1c) The fixed-point of xk ¼ aj ðj ¼ i2 ; i3 Þ is • • • • •

monotonically unstable (monotonic source) if df =dxk jx ¼aj 2 ð0; 1Þ; k monotonically stable (monotonic sink) if df =dxk jx ¼aj 2 ð1; 0Þ; k invariantly stable (invariant sink) if df =dxk jx ¼aj ¼ 1; k oscillatorilly stable (oscillatory sink) if df =dxk jx ¼aj 2 ð2; 1Þ; k flipped if df =dxk jx ¼aj ¼ 2, which is k

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼aj k [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼aj \ 0; k

• oscillatorilly stable (oscillatory source) if df =dxk jx ¼aj 2 ð1; 2Þ. k

(ii) If

a ¼ ai1 ; b ¼ ai2 ; c ¼ ai3 ; ð3:133Þ

Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; ai2 ¼ ai3 ; i1 ; i2 ; i3 2 f1; 2g; i1 6¼ i2 6¼ i3 ; the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðx  ai1 Þ2 ðx  ai2 Þ2 ia 2 f1; 2g; a ¼ 1; 2:

ð3:134Þ

The fixed-points of xk ¼ ai1 ; ai2 are monotonically unstable (a monotonicupper-saddle of the second-order, d 2 f =dx2k jx ¼ai orai [ 0 or a monotonick

lower-saddle of the second-order, d 2 f =dx2k jx ¼ai k

1

2

orai1 \ 2

0Þ.

• Such a discrete flow is called a (mUS:mUS) or (mLS:mLS) discrete flow. • The bifurcation of fixed-point at xk ¼ ai2 for two simple fixed-points switching of xk ¼ b; c is called a monotonic-upper-saddle or monotoniclower-saddle switching bifurcation of the second-order at a point p ¼ p1 .

218

3 Quartic Nonlinear Discrete Systems

(iii) If a ¼ ai1 ; ai3 ; ai2 2 fb; cg; ð3:135Þ

Di1 i3 ¼ ðai1  ai3 Þ2 ¼ 0; ai1 ¼ ai3 ; ai3 6¼ ai2 ; i1 ; i2 ; i3 2 f1; 2; 3g; i1 6¼ i2 6¼ i3 ; the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðx  ai1 Þ3 ðx  ai2 Þ

ð3:136Þ

ia 2 f1; 2g; a ¼ 1; 2:

(iii1) The fixed-point of xk ¼ ai1 is monotonically unstable (a 3rd order monotonic source, d 3 f =dx3k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic source discrete flow of the third-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a switching of secondorder and simple fixed-points of xk ¼ ai1 ; ai2 is called a monotonic source switching bifurcation of the third-order at a point p ¼ p1 . (iii2) The fixed-point of xk ¼ ai1 is monotonically stable (a third-order monotonic sink, d 3 f =dx3k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic sink discrete flow of the third-order at xk ¼ ai1 . • The bifurcation offixed-point at xk ¼ ai1 for a switching of one secondorder and one simple fixed-points of xk ¼ ai1 ; ai2 is called a monotonic sink switching bifurcation of the third-order at a point p ¼ p1 . (iii3) The fixed-point of xk ¼ ai2 is • • • • •

monotonically unstable (monotonic source) if df =dxk jx ¼ai 2 ð0; 1Þ; k 2 monotonically stable (monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ; k 2 invariantly stable (invariant sink) if df =dxk jx ¼ai ¼ 1; k 2 oscillatorilly stable (oscillatory sink) if df =dxk jx ¼ai 2 ð2; 1Þ; k 2 flipped if df =dxk jx ¼ai ¼ 2, which is k

2

– an oscillatory upper-saddle of the second-order for d 2 f =dx2k jx ¼ai k 2 [ 0; – an oscillatory lower-saddle of the second-order for d 2 f =dx2k jx ¼ai k 2 \0; • oscillatorilly stable (oscillatory source) if df =dxk jx ¼ai 2 ð1; 2Þ. k

2

3.4 Period-1 Switching Bifurcations

219

(iv) If Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; Di2 i3 ¼ ðai2  ai3 Þ2 ¼ 0; i1 ; i2 ; i3 2 f1; 2; 3g; i1 6¼ i2 6¼ i3 ;

ð3:137Þ

the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ4 :

ð3:138Þ

(iv1) The fixed-point of xk ¼ ai1 is monotonically unstable (a fourth-order monotonic-upper-saddle, d 4 f =dx4k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a monotonic-upper-saddle discrete flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of one second-order and two simple fixed-points of xk ¼ a1;2;3 is called a upper-saddle bundle-switching bifurcation of the fourthorder at a point p ¼ p1 . (iv2) The fixed-point of xk ¼ ai1 is monotonically unstable (a fourth-order monotonic lower-saddle, d 4 f =dx4k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a monotonic-lower-saddle discrete flow of the fourth-order at xk ¼ ai1 . • The bifurcation of fixed-point at xk ¼ ai1 for a bundle switching of one second-order and two simple fixed-points of xk ¼ a1;2;3 is called a monotonic-lower-saddle bundle-switching bifurcation of the fourth-order at a point p ¼ p1 . From Definition 3.6, stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system is presented in Fig. 3.17. For a0 [ 0, the bifurcations and stability of fixed-points are presented in Fig. 3.17(i)–(iii). In Fig. 3.17(i), there is a switching bifurcation network with two third-order monotonic-source switching bifurcations and one monotonic upper-saddle-node bifurcation. The two third-order monotonic source (3rd mSO) switching bifurcations are for (mLS:mSO) switching to (mUS:mSO) fixed-points and for (mSO:mUS) switching to (mSO:mLS)fixed-points. The upper-saddle-node (mUSN) switching bifurcation is for two simple fixed-points. In Fig. 3.17(ii), a fourth-order monotonic-upper-saddle (4th US) bundle-switching bifurcation for (mSI-oSO:mLS:mSO) switching to (mSO: mLS:mSI-oSO) fixed-points. In Fig. 3.17(iii), a fourth-order monotonic-uppersaddle (4th mUS) bundle-switching bifurcation for (mSI-oSO:mSO:mUS) switching to (mSO:mSI-oSO:mUS) fixed-points. Similarly, for a0 \0, the bifurcations and stability of fixed-points are presented in in Fig. 3.17(iv)–(vi). In Fig. 3.17(iv), the switching bifurcation network consists of two third-order monotonic sink switching

220

3 Quartic Nonlinear Discrete Systems

(i)

(iv)

(ii)

(v)

(iii)

(vi)

Fig. 3.17 Stability and bifurcations of fixed-points in the 1-dimeisonal, quartic nonlinear discrete system (a0 [ 0): (i) two 3rd order mSO and mUSN switching bifurcation network, (ii) 4th mUS bundle switching bifurcation, (iii) 4th order mUS bundle-switching bifurcation, (a0 \0): (iv) two 3rd order mSI and mLSN switching bifurcation network, (v) 4th order mLS bundle-switching bifurcation, (vi) 4th order mLS bundle switching. mLSN: monotonic-lower-saddle-node, mUSN: monotonic-upper-saddle-node, mSI-oSO: monotonic-sink to oscillatory source, mSI-oSO-mSI: monotonic-sink to oscillatory source to monotonic sink, mSO: monotonic-source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: period-2 fixed-point. Open curves of P-2 are for mSI-oSO. Closed loop of P-2 are for mSI-oSO-mSI

3.4 Period-1 Switching Bifurcations

221

bifurcations and one monotonic lower-saddle-node bifurcation. The two 3rd order monotonic sink (3rd mSI) switching bifurcations are for the (mLS:mSO) switching to (mUS:mSO)-fixed-points and for the (mSO:mUS) switching to (mSO:mLS)fixed-points. The monotonic upper-saddle-node (mUSN) switching bifurcation is for two simple fixed-points. In Fig. 3.17(v), a fourth-order monotonic lower-saddle (4th mLS) bundle-switching bifurcation for (mSO:mUS:mSI-oSO) switching to (mSI-oSO:mUS:mSO) fixed-points. In Fig. 3.17(vi), a fourth-order monotonic-lowersaddle (4th mLS) bundle-switching bifurcation for (mSO:mSI-oSO:mLS) switching to (mSI-oSO:mSO:mLS) fixed-points. For the switching bifurcation between the third-order and simple fixed-points, the following definition is given for the 1-dimensional, quartic nonlinear discrete system. Definition 3.7 Consider a 1-dimensional, quartic nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ

ð3:139Þ

3

¼ xk þ a0 ðpÞðxk  aÞ ðxk  bÞ where AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð3:140Þ

(i) If fa1 ; a2 g ¼ sortfa; bg; ai \ai þ 1

ð3:141Þ

i1 ; i2 2 f1; 2g; the quartic nonlinear discrete system has a standard form as xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ3 ðxk  ai2 Þ:

ð3:142Þ

(i1a) The fixed-point of xk ¼ ai1 is monotonically unstable (a third-order monotonic-source, d 3 f =dx3k jx ¼ai [ 0Þ. k 1 (i1b) The fixed-point of xk ¼ ai2 is • • • • •

monotonically stable (monotonic sink) if df =dxk jx ¼ai 2 ð1; 0Þ, k 2 invariantly stable (invariant sink) if df =dxk jx ¼ai ¼ 1, k 2 oscillatorilly stable (oscillatory sink) if df =dxk jx ¼ai 2 ð2; 1Þ, k 2 flipped if df =dxk jx ¼ai ¼ 2, k 2 oscillatorilly stable (oscillatory source) if df =dxk jx ¼ai 2 ð1; 2Þ. k

2

222

3 Quartic Nonlinear Discrete Systems

(i1c) Such a discrete flow is called a (3rd mSO:mSI-oSO) or (mSI-oSO:3rd mSO)-discrete flow. (i2a) The fixed-point of x ¼ ai1 is monotonically stable (a third-order monotonic-sink, d 3 f =dx3k jx ¼ai \ 0Þ. k 1 (i2b) The fixed-point of x ¼ ai2 is monotonically unstable (a monotonic source, df =dxk jx ¼ai 2 ð0; 1ÞÞ. k 2 (i2c) Such a discrete flow is called a (3rd mSI:mSO) or (mSO:3rd mSI)discrete flow. (ii) If

a ¼ ai1 ; b ¼ ai2 ; Di1 i2 ¼ ðai1  ai2 Þ2 ¼ 0; ai1 ¼ ai2 ;

ð3:143Þ

i1 ; i2 2 f1; 2g; i1 6¼ i2 ; the corresponding standard form is xk þ 1 ¼ xk þ f ðxk ; pÞ ¼ xk þ a0 ðxk  ai1 Þ4 :

ð3:144Þ

(ii1) The fixed-point of xk ¼ ai1 is monotonically unstable (a fourth-order monotonically upper-saddle, d 4 f =dx4k jx ¼ai [ 0Þ. k

1

• Such a discrete flow is called a fourth-order monotonic upper-saddle discrete flow. • The bifurcation of fixed-point at xk ¼ ai1 for a switching of one third-order and one simple fixed-points of xk ¼ a1;2 is called a fourth-order upper-saddle switching bifurcation at a point p ¼ p1 . (ii2) The fixed-point of xk ¼ ai1 is monotonically unstable (a fourth-order monotonic-lower-saddle, d 4 f =dx4k jx ¼ai \ 0Þ. k

1

• Such a discrete flow is called a fourth-order monotonic-lower-saddle discrete flow. • The bifurcation of fixed-point at xk ¼ ai1 for a switching of one third-order and one simple fixed-points of xk ¼ a1;2 is called a fourth-order monotonic lower-saddle switching bifurcation at a point p ¼ p1 . From the Definition 3.7, the stability and bifurcations of fixed-points in the 1-dimensional, quartic nonlinear discrete system is presented in Fig. 3.18. In Fig. 3.18(i), the fourth-order monotonic upper-saddle (4th mUS) switching bifurcation for a0 [ 0 is presented for the third-order monotonic sink (3rd mSI) with a

3.4 Period-1 Switching Bifurcations

(i)

223

(ii)

Fig. 3.18 Stability and bifurcations of fixed-points in the 1-dimeisonal, quartic nonlinear discrete system (a0 [ 0): (i) 4th order US switching bifurcation of (3rdmSI: mSO) to (mSI-oSO: 3rd mSO), (a0 \ 0):(ii) 4th order mLS switching bifurcation of (3rd mSO: mSI-oSO) to (mSO: 3rd mSI). mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic-source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: period-2 fixed-point. Open curves of P-2 are for mSI-oSO. Closed loop of P-2 are for mSI-oSO-mSI

simple monotonic source (mSO) fixed-points (i.e., (3rd mSI:mSO)) to the third-order monotonic source (3rd mSO) with a simple sink to oscillatory source (mSI-oSO) fixed-points (i.e., (mSI-oSO:3rd mSO)). Similarly, in Fig. 3.18(ii), the fourth-order lower-saddle (4th mLS) switching bifurcation for a0 \ 0 is presented for the third-order source (3rd mSO) with a simple monotonic sink to oscillatory source (mSI-oSO) fixed-points (i.e., (3rd mSO:mSI-oSO)) to the 3rd order monotonic sink (3rd mSI) with a simple monotonic source (mSO) fixed-points (i.e., (mSO: 3rd mSI)). For the switching bifurcation between the two second-order monotonic fixed-points, the following definition was presented in Definition 3.3, and the corresponding illustrations are presented in Fig. 3.12.

3.5

Forward Quartic Discrete Systems

In this section, the analytical bifurcation scenario of a quartic nonlinear discrete system will be discussed. The period-doubling bifurcation scenario will be discussed first through nonlinear renormalization techniques, and the bifurcation scenario based on the saddle-node bifurcation will be discussed, which is independent of period-1 fixed-points.

224

3 Quartic Nonlinear Discrete Systems

3.5.1

Period-2 Quartic Discrete Systems

After the period-doubling bifurcation of a period-1 fixed-point, the period-doubled fixed-points of a quartic nonlinear discrete system can be obtained. Consider the period-doubling solutions of a forward quartic nonlinear discrete system first. Theorem 3.2 Consider a 1-dimensional, forward, quartic nonlinear discrete system xk þ 1 ¼ xk þ AðpÞx4k þ BðpÞx3k þ CðpÞx2k þ DðpÞxk þ EðpÞ ¼ xk þ a0 ðpÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ½x2k þ B2 ðpÞxk þ C2 ðpÞ

ð3:145Þ

where a0 ðpÞ ¼ AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm ÞT :

ð3:146Þ

Di ¼ B2i  4Ci [ 0; for i ¼ 1; 2 with 1 1 pffiffiffiffiffiffi 1 1 pffiffiffiffiffiffi a1;2 ¼ B1 D1 ; a3;4 ¼ B2 D2 ; 2 2 2 2

ð3:147Þ

Under

the standard form of such a 1-dimensional system is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þðxk  a4 Þ:

ð3:148Þ

Thus, a general standard form of such a 1-dimensional quartic discrete system is xk þ 1 ¼ xk þ Ax4k þ Bx3k þ Cx2k þ Dxk þ E ð1Þ

¼ xk þ a0 *4i¼1 ðxk  ai Þ

ð3:149Þ

where ð1Þ bi;1

¼

1 ð1Þ  ðBi þ 2

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð1Þ ð1Þ ð1Þ ð1Þ Di Þ; bi;2 ¼  ðBi  Di Þ 2

ð1Þ for Di  0; i 2 f1; 2g; ð1Þ ð1Þ ð1Þ ð1Þ 04i¼1 ai ¼ sortf02i¼1 fbi;2 ; bi;2 gg; ai

pffiffiffiffiffiffiffi ð1Þ for Di \0; i 2 f1; 2g; i ¼ 1; ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ðiÞ

 ai þ 1 ; ð1Þ

a1 ¼ b1;1 ; a2 ¼ b1;2 ; a3 ¼ b2;1 ; a4 ¼ b2;2 :

ð3:150Þ

3.5 Forward Quartic Discrete Systems

225

(i) Consider a forward period-2 discrete system of Eq. (3.145) as ð1Þ

xk þ 2 ¼ xk þ ½a0

*i ¼1 ðxk 1

 ai1 Þf1 þ

¼ xk þ ½a0

*i ¼1 ðxk 1

 ai1 Þ½a40

¼ xk þ ½a0

*j ¼1 ðxk 1

 ai1 Þ½a40

4 4 3

¼ xk þ a10 þ 4

42

ð1Þ ð1Þ

*i¼1 ðxk

*i ¼1 ½1 þ a0 *i ¼1;i 6¼i ðxk 1 2 2 1 4

4

ð42 4Þ=2

ðx2k þ Bi2 xk þ Ci2 Þ

42 4

 bj2 Þ

*i ¼1 2

*j ¼1 ðxk 2

ð2Þ

ð1Þ

 ai2 Þg

ð2Þ

ð2Þ

ð2Þ

 ai Þ ð3:151Þ

where bi;1 ¼  ðBi þ

qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ð2Þ ð2Þ ð2Þ Dð2Þ Þ; bi;2 ¼  ðBi  Di Þ;

ð2Þ Di

ð2Þ 4Ci

ð2Þ

ð2Þ

1 2

¼

ð2Þ ðBi Þ2



2

 0; i 2

0 0Nq11¼1 Iqð21 Þ

00Nq ¼1 Iqð2 Þ 2 2

2

2

Iqð21 Þ ¼ flðq1 1Þ 20 m1 þ 1 ; lðq1 1Þ 20 m1 þ 2 ;    ; lq1 20 m1 g 0

f1; 2;    ; M1 g0f∅g; q1 2 f1; 2;    ; N1 g; M1 ¼ N1 20 m1 ; m1 2 f1; 2g; Iqð22 Þ ¼ flðq2 1Þ 21 m1 þ 1 ; lðq2 1Þ 21 m1 þ 2 ;    ; lq2 21 m1 g 1

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g;

ð3:152Þ

q2 2 f1; 2;    ; N2 g; M2 ¼ ð42  4Þ=2; qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i D Þ; bi;2 ¼  ðBi  i Di Þ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0; i 2 J ð2 Þ ¼ flN2 21 m1 þ 1 ; lN2 21 m1 þ 2 ;    ; lM2 g fM1 þ 1; M1 þ 2;    ; M2 g 1

with fixed-points ð2Þ

xk þ 2 ¼ xk ¼ ai ði ¼ 1; 2; . . .; 42 Þ ; ð2Þ

ð1Þ

ð2Þ

ð2Þ

2 04i¼1 fai g ¼ sortf04j1 ¼1 faj1 g; 0M j2 ¼1 fbj2 ;1 ; bj2 ;2 gg 2

with

ð2Þ ð2Þ ai \ai þ 1 ;

ð3:153Þ

M2 ¼ ð42  4Þ=2: ð1Þ

(ii) For a fixed-point of xk þ 1 ¼ xk ¼ ai1 ði1 2 f1; 2; . . .; 4gÞ, if dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *4i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1 with

ð3:154Þ

226

3 Quartic Nonlinear Discrete Systems

• a r th -order oscillatory upper-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ k

i1

k

i1

[ 0; r ¼ 2l1 Þ; • a r th -order oscillatory lower-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ \0; r ¼ 2l1 Þ; • a r th -order oscillatory sink bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ [ 0; r ¼ k

i1

k

i1

2l1 þ 1Þ; • a r th -order oscillatory source bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ [ 0; r ¼ 2l1 þ 1Þ; then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi1 ; Di1 ¼ ðBi1 Þ2  4Ci1 ¼ 0; 2

ð3:155Þ

and there is a period-2 discrete system of the quartic discrete system in Eq. (3.145) as ð1Þ

ð2Þ

ð3:156Þ

dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i1 dx2k xk ¼ai1

ð3:157Þ

xk þ 2 ¼ xk þ a50

*

ð20 Þ

i2 2Iq1

ðxk  ai2 Þ3

42

*i ¼1 ðxk 3

 ai3 Þð1dði2 ;i3 ÞÞ :

For i1 2 f1; 2; . . .; 4g; i1 6¼ i2 with

ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a50 dx3k xk ¼ai1

*

ð1Þ

ð20 Þ

i2 2Iq1 ;i2 6¼i1

ð1Þ

ðai1  ai2 Þ3

ð3:158Þ

ð2Þ ð1dði2 ;i3 ÞÞ

*4i3 ¼1 ðað1Þ \ 0; i1  ai3 Þ 2

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the period-2 discrete system; ð1Þ • xk þ 2 at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a50 dx3k xk ¼ai1

*

ð1Þ

ð20 Þ

i2 2Iq1 ;i2 6¼i1

ð1Þ 42 *i ¼1 ðai 1 3



ð1Þ

ðai1  ai2 Þ3 ð2Þ ai3 Þð1dði2 ;i3 ÞÞ

ð3:159Þ [ 0;

3.5 Forward Quartic Discrete Systems

227

and the corresponding bifurcations is a third-order monotonic source bifurcation for the period-2 discrete system. (ii1) The period-2 fixed-points are trivial and unstable if ð2Þ

xk þ 2 ¼ xk ¼ ai

ð1Þ

2 04i1 ¼1 fai1 g:

ð3:160Þ

(ii2) The period-2 fixed-points are non-trivial and stable if ð2Þ

ð2Þ

ð2Þ

2 2 0M i1 ¼1 fbi1 ;1 ; bi1 ;2 g:

xk þ 2 ¼ xk ¼ ai

ð3:161Þ

Proof The proof is straightforward through the simple algebraic manipulation. Following the proof of quadratic discrete system, this theorem is proved. ■

3.5.2

Period-Doubling Renormalization

The generalized cases of period-doublization of quartic discrete systems are presented through the following theorem. The analytical period-doubling bifurcation trees can be developed for quartic discrete systems. Theorem 3.3 Consider a 1-dimensional quartic nonlinear discrete system as xk þ 1 ¼ xk þ Ax4k þ Bx3k þ Cx2k þ Dxk þ E

ð3:162Þ

ð1Þ

¼ xk þ a0 *4i¼1 ðxk  ai Þ:

(i) After l-times period-doubling bifurcations, a period-2l ðl ¼ 1; 2; . . .Þ discrete system for the quartic discrete system in Eq. (3.162) is given through the nonlinear renormalization as ð2l1 Þ

42

xk þ 2l ¼ xk þ ½a0

f1 þ

l1

*i1 ¼1 ðxk

2l1

ð2

*i1 ¼1 ½1 þ a0 4

2l1

ð2l1 Þ

¼ xk þ ½a0

*i1 ¼1 ðxk 4

2l1

ð2l1 Þ 4

½ða0

Þ

ð2l1 Þ

¼ xk þ ½a0

Þ

2l

22

l1

l1

l

ð2l Þ

¼ x k þ a0

Þ

42

Þ

l

*i¼1 ðxk

Þ=2

42

Þ=2

l

ð2l Þ

ð2l Þ

ð2l Þ

Þ ð2l Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ

*i¼1 ðxk

 ai Þ

Þg

Þ

ð2l1 Þ

l1

ð2l1 Þ

 ai 2

ðx2k þ Bj2 xk þ Cj2 Þ

 ai1

ð42 42 l1

4

ð2l1 Þ

2l1

Þ

2l1

*i2 ¼1;i2 6¼i1 ðxk

 ai1

*i2 ¼1

ð2l1 Þ 1 þ 42

¼ xk þ ða0

l1

ð4 4

*j1 ¼1

*i1 ¼1; ðxk

ð2l1 Þ 42

½ða0

ð2l1 Þ

 ai1

ð2l Þ

 ai Þ

ð3:163Þ

228

3 Quartic Nonlinear Discrete Systems

with l dxk þ 2l ð2l Þ X42l ð2l Þ 42 ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l d 2 xk þ 2 l ð2l Þ X42l X42l ð2l Þ 42 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . l d r xk þ 2 l X42l ð2l Þ X42l ð2l Þ 42 ¼ a0 i1 ¼1    ir ¼1;i3 6¼i1 ;i2 ir1 *ir þ 1 ¼1;i3 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ r dxk

ð3:164Þ 2l

for r  4 where ð2l Þ

ð2Þ

2l

ð2l Þ 04i¼1 fai g ð2l Þ

bi;1

ð2l Þ

Di

Iqð21

l1

¼

l1

ð2l1 Þ 1 þ 42

a0 ¼ ða0 Þ1 þ 4 ; a0

¼ ða0

Þ

; l ¼ 1; 2; 3;    ;

2l1

ð2l Þ ð2l Þ ð2l Þ 2 sortf04i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð2l Þ

ð2l Þ

,ai  ai þ 1 ; qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ; 2

2

ð2l Þ

ð2l Þ

¼ ðBi Þ2  4Ci Þ

l1

 0 for i 2 0Nq11¼1 Iqð21

Þ

00Nq ¼1 Iqð2 Þ ; 2 2

l

2

¼ flðq1 1Þ 2l1 m1 þ 1 ; lðq1 1Þ 2l1 m1 þ 2 ;    ; lq1 2l1 m1 g f1; 2;    ; M1 g0f∅g;

for q1 2 f1; 2;    ; N1 g; M1 ¼ N1 2l1 m1 ; m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ 2l m1 þ 1 ; lðq2 1Þ 2l m1 þ 2 ;    ; lq2 2l m1 g

ð3:165Þ

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; l

l1

for q2 2 f1; 2;    ; N2 g; M2 ¼ ð42  42 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \ 0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ;    ; lM2 g  fM1 þ 1; M1 þ 2;    ; M2 g0f∅g;

with fixed-points ð2l Þ

l

xk þ 2l ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 42 Þ 2l

2l1

ð2l Þ

ð2l1 Þ

04i¼1 fai g ¼ sortf04i1 ¼1 fai1 l

ð2 Þ

l

ð2 Þ

with ai \ai þ 1 :

ð2l Þ

ð2l Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð3:166Þ

3.5 Forward Quartic Discrete Systems

229 ð2l1 Þ

(ii) For a fixed-point of xk þ 2l1 ¼ xk ¼ ai1

ð2l1 Þ

ði1 2 Iq

Þ, if

l1 dxk þ 2l1 ð2l1 Þ ð2l1 Þ ð2l1 Þ 42 j  ð2l1 Þ ¼ 1 þ a0 *i ¼1;i 6¼i ðai1  ai2 Þ ¼ 1; 2 2 1 dxk xk ¼ai1 d s xk þ 2l1 j  ð2l1 Þ ¼ 0; for s ¼ 2;    ; r  1; xk ¼ai dxsk 1 r d xk þ 2l1 l1 j  ð2l1 Þ 6¼ 0 for 1\ r  42 ; r x ¼a dxk i1 k

ð3:167Þ

with • a rth -order oscillatory sink for d r xk þ 2l1 =dxrk j

ð2l1 Þ

xk ¼ai

• a r th -order oscillatory source for d r xk þ 2l1 =dxrk j

[ 0 and r ¼ 2l1 þ 1;

1 ð2l1 Þ

xk ¼ai

\0 and r ¼ 2l1 þ 1;

1

• a r th -order oscillatory upper-saddle for d r xk þ 2l1 =dxrk j

ð2l1 Þ

xk ¼ai

2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ 2l1 =dxrk j

[ 0 and r ¼

1

ð2l1 Þ

xk ¼ai

\0 and r ¼

1

2l1 ; then there is a period-2l fixed-point discrete system ð2l Þ

x k þ 2 l ¼ x k þ a0

*

ð2l Þ

i1 2Iq

ð2l1 Þ 3

ðxk  ai1

42

Þ

l

*j2 ¼1 ðxk

ð2l Þ

 aj2 Þð1dði1 ;j2 ÞÞ

ð3:168Þ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð3:169Þ

dxk þ 2l d 2 xk þ 2 l j  ð2l1 Þ ¼ 1; j ð2l1 Þ ¼ 0: dxk xk ¼ai1 dx2k xk ¼ai1

ð3:170aÞ

and

ð2l1 Þ

• xk þ 2l at xk ¼ ai1

is a monotonic sink of the third-order if

d xk þ 2l ð2l Þ ð2l1 Þ ð2l1 Þ j ða  ai2 Þ3 * ð2l1 Þ ¼ 6a0 ð2l1 Þ i2 2Iq ;i2 6¼i1 i1 dx3k xk ¼ai1 3

2l

l1

ð2

ði1 2 Iq

Þ

ð2l1 Þ

*4j2 6¼1 ðai1

ð3:170bÞ

; q 2 f1; 2; . . .; N1 gÞ; ð2l1 Þ

and such a bifurcation at xk ¼ ai1 bifurcation. ð2l1 Þ

• xk þ 2l at xk ¼ ai1

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

is a third-order monotonic sink

is a monotonic source of the third-order if

230

3 Quartic Nonlinear Discrete Systems

d 3 xk þ 2l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai2 Þ3 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k 2l

ð2l1 Þ

ði1 2 Iq

ð2l1 Þ

ð3:171Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ [ 0

*4j2 6¼1 ðai1

; q 2 f1; 2; . . .; N1 gÞ ð2l1 Þ

and such a bifurcation at xk ¼ ai1 cation.

is a third-order monotonic source bifur-

(ii1) The period-2l fixed-points are trivial if 2l1

ð2l Þ

xk þ 2l ¼ xk ¼ ai1 2 04i1

ð2l1 Þ

fai1

g;

ð3:172Þ

(ii2) The period-2l fixed-points are non-trivial if ð2l Þ

ð2l Þ

ð2l Þ

2 xk þ 2l ¼ xk ¼ ai1 2 0M j1 ¼1 fbj1 ;1 ; bj1 ;2 g:

ð3:173Þ

Such a period-2l fixed-point is • monotonically unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

• monotonically invariant if dxk þ 2l =dxk j

2 ð1; 1Þ;

1 ð2l Þ

xk ¼ai

¼ 1, which is

1

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx [ 0; k

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \ 0; k

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jxk [ 0;;

– a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \ 0; k

• monotonically stable if dxk þ 2l =dxk j

xk ¼ai

• invariantly zero-stable if dxk þ 2l =dxk j • oscillatorilly stable if dxk þ 2l =dxk j • flipped if dxk þ 2l =dxk j

ð2l1 Þ

xk ¼ai

2 ð0; 1Þ;

ð2l Þ 1

ð2l1 Þ

xk ¼ai

1

ð2l1 Þ

xk ¼ai

¼ 0;

2 ð1; 0Þ;

1

¼ 1, which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx [ 0; k

1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \ 0; k

– an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \ 0; k

– an oscillatory sink the ð2l1 þ 1Þth order with d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, this theorem can be proved.

■

3.5 Forward Quartic Discrete Systems

3.5.3

231

Period-n Appearing and Period-Doublization

The forward period-n discrete system for the quartic nonlinear discrete systems will be discussed, and the period-doublization of period-n discrete systems is discussed through the nonlinear renormalization. Theorem 3.4 Consider a 1-dimensional quartic nonlinear discrete system xk þ 1 ¼ xk þ Ax4k þ Bx3k þ Cx2k þ Dxk þ E

ð3:174Þ

ð1Þ

¼ xk þ a0 *4i¼1 ðxk  ai Þ:

(i) After n-times iterations, a period-n discrete system for the quartic discrete system in Eq. (3.174) is xk þ n ¼ xk þ a0 *4i1 ¼1 ðxk  ai2 Þf1 þ n

ð4 1Þ=3

¼ x k þ a0

ðnÞ

¼ x k þ a0

4n

*i1 ¼1 ðxk 4

*i¼1 ðxk

Xn

j¼1 Qj g ð4n 4Þ=2

 ai1 Þ½*j2 ¼1

ð2l Þ

ð2l Þ

ðx2k þ Bj2 xk þ Cj2 Þ

ðnÞ

 ai Þ ð3:175Þ

with

dxk þ n n ðnÞ X n ¼ 1 þ a0 4i1 ¼1 *4i2 ¼1;i2 6¼i1 ðxk  aðnÞ i2 Þ; dxk d 2 xk þ n n ðnÞ X n X n ¼ a0 4i1 ¼1 4i2 ¼1;i2 6¼i1 *4i3 ¼1;i3 6¼i1 ;i2 ðxk  aðnÞ i3 Þ; dx2k .. . d r xk þ n n X n ðnÞ X n ¼ a0 4i1 ¼1    4ir ¼1;ir 6¼i1 ;i2 ;ir1 *4ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  aðnÞ ir þ 1 Þ r dxk

for r  4n ;

ð3:176Þ

where ðnÞ

a0 ¼ ða0 Þð4 Qn ¼ n

n

1Þ=3

; Q1 ¼ 0; Q2 ¼

*i ¼1 ½1 þ a0 *i ¼1;i 6¼i ðxk 2 1 1 2 4

4

*i ¼1 ½1 þ a0 ð1 þ Qn1 Þ *i ¼1;i 6¼i ðxk n n1 n1 n 4

ðnÞ

4

ð1Þ

ðnÞ

ð1Þ

 ain1 Þ; n ¼ 3; 4;    ;

ðnÞ

04i¼1 fai g ¼ sortf04i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2

2

ð1Þ

 ai1 Þ;

232

3 Quartic Nonlinear Discrete Systems ðnÞ

ðnÞ

ðnÞ

Di2 ¼ ðBi2 Þ2  4Ci2  0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þ n þ 1 ; lðq1Þ n þ 2 ;    ; lq n gf1; 2;    ; Mg0f∅g; for q 2 f1; 2;    ; Ng; M ¼ ð4n  4Þ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flN n þ 1 ; lN n þ 2 ;    ; lM g  f1; 2;    ; Mg0f∅g;

ð3:177Þ

with fixed-points ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; 4n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

04i¼1 fai g ¼ sortf04i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ðnÞ

ð3:178Þ

ðnÞ

with ai \ ai þ 1 : ðnÞ

ðnÞ

(ii) For a fixed-point of xk þ n ¼ xk ¼ ai1 ði1 2 Iq ; q 2 f1; 2; . . .; NgÞ, if dxk þ n n ðnÞ ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *4i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk k i1

ð3:179Þ

d 2 xk þ n n ðnÞ X n ðnÞ ðnÞ j  ðnÞ ¼ a0 4i2 ¼1;i2 6¼i1 *4i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; dx2k xk ¼ai1

ð3:180Þ

with

then there is a new discrete system for onset of the qth - set of period-n fixed-points based on the second-order monotonic saddle-node bifurcation as ðnÞ

xk þ n ¼ x k þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *4j2 ¼1 ðxk  aj2 Þð1dði1 ;j2 ÞÞ n

ð3:181Þ

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð3:182Þ

3.5 Forward Quartic Discrete Systems

233

(ii1) If dxk þ n j  ðnÞ ¼ 1 ði1 2 IqðnÞ Þ; dxk xk ¼ai1 d 2 xk þ n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 2a0 *i 2I ðnÞ ;i 6¼i ðai1  ai1 Þ2 1 q 2 1 dx2k xk ¼ai1 ðnÞ

ð3:183Þ

ðnÞ

*4j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk þ n at xk ¼ ai1 is • a monotonic lower-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ \ 0; k

i1

k

i1

• a monotonic upper-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0. n1

(ii2) The period-n fixed-points ðn ¼ 2 mÞ are trivial if ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f04ii ¼1 fai1 g; 04i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 4n g0f∅g for n 6¼ 2n2 ; ðnÞ

2n1 1 m

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 2 f04i2 ¼1 fai2

)

)

gg

ð3:184Þ

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 4n g0f∅g

for n ¼ 2n2 : (ii3) The period-n fixed-points ðn ¼ 2n1 mÞ are non-trivial if ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 62 f04ii ¼1 fai1 g; 04i2 ¼1 fai2 gg n for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 4 g0f∅g for n 6¼ 2n2 ; ðnÞ

2n1 1 m

ð2n1 1 mÞ

xk ¼ xk þ n ¼ aj1 62 f04i2 ¼1 fai2

)

)

gg

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 4n g0f∅g

for n ¼ 2n2 : Such a forward period-n fixed-point is • monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is k

i1

ð3:185Þ

234

3 Quartic Nonlinear Discrete Systems 1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx [ 0; k

1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \ 0; k

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jxk [ 0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx \0; k

• monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð0; 1Þ; i1

k

• invariantly zero-stable if dxk þ n =dxk jx ¼aðnÞ ¼ 0; k

i1

• oscillatorilly stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 0Þ; k

i1

• flipped if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is k

i1

– an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2lk 1 jxk [ 0; 1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jxk \ 0; – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ. k

ðnÞ

i1

ðnÞ

(iii) For a fixed-point of xk þ n ¼ xk ¼ ai1 ði1 2 Iq ; q 2 f1; 2; . . .; NgÞ, there is a period-doubling of the qth -set of period-n fixed-points if

with

dxk þ n n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 1 þ a0 *4j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; dxk xk ¼ai1 d s xk þ n j  ðnÞ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk ¼ai1 d r xk þ n j  ðnÞ 6¼ 0 for 1\r  4n dxrk xk ¼ai1

ð3:186Þ

• a r th -order oscillatory sink for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ 2l1 þ 1; i1

k

• a r th -order oscillatory source for d r xk þ n =dxrk jx ¼aðnÞ \ 0 and r ¼ 2l1 þ 1; k

i1

• a r th -order oscillatory upper-saddle for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ i1

k

2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ 2l1 .

k

i1

3.5 Forward Quartic Discrete Systems

235

The corresponding period-2 n discrete system of the quartic discrete system in Eq. (3.174) is ð2 nÞ

xk þ 2 n ¼ xk þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

ð2 nÞ ð1dði1 ;j2 ÞÞ

2 n

 ai1 Þ3 *4j2 ¼1 ðxk  aj2

Þ

ð3:187Þ with dxk þ 2 n d 2 xk þ 2 n jx ¼aðnÞ ¼ 1; jx ¼aðnÞ ¼ 0; i1 i1 k k dxk dx2k d 3 xk þ 2 n ð2 nÞ ðnÞ ðnÞ 3 jx ¼aðnÞ ¼ 6a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 1 q 2 1 i1 k dx3k ðnÞ

2 n

ð2 nÞ ð1dði1 ;j2 ÞÞ

*4j2 ¼1 ðai1  aj2 ðnÞ

Þ

ð3:188Þ

:

ðnÞ

Thus, xk þ 2 n at xk ¼ ai1 for i1 2 Iq , q 2 f1; 2;    ; Ng is • a monotonic sink of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ \ 0, k

i1

• a monotonic source of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ [ 0. k

i1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l n discrete system of the quartic discrete system in Eq. (3.174) is ð2l1 nÞ

42

xk þ 2l n ¼ xk þ ½a0

f1 þ

4

ð2

¼ xk þ ½a0

l1

ð2

½ða0

ð2

¼ xk þ ½a0

ð2

2l1 n

nÞ

Þ

4

*i1 ¼1 ð2l1 nÞ

4

nÞ 4

Þ

ð2l1 nÞ 42

¼ xk þ ða0

l

ð2 nÞ

¼ x k þ a0

Þ

4

*i¼1

*i2 ¼1;i2 6¼i1 ðxk 4

ð2

ð4

ð4

4

ð2

4

l

42 n

*i¼1

l1

2l1 n

nÞ

ð2l nÞ

ðx2k þ Bj2

nÞ

Þ

Þ=2

Þg

Þ

ð2l nÞ

ð2l nÞ

xk þ Cj2

Þ

ð2l nÞ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ

ð2l nÞ

l

ð2l1 nÞ

 ai2

Þ=2

ðxk  ai

ð2 nÞ

ðxk  ai

l1

2l1 n

ðxk  ai1 2l n

Þ

2l1 n

ðxk  ai1

*j2 ¼1

l1 n

2l n

Þ

*j1 ¼1

*i1 ¼1 2l1 n

l1

2l n

2l1 n

nÞ

ð2l1 nÞ

ðxk  ai1

½1 þ a0

nÞ 4

l1

l1

ð2

½ða0

2l1 n

*i1 ¼1 l1

l1 n

*i1 ¼1

Þ

Þ ð3:189Þ

with

236

3 Quartic Nonlinear Discrete Systems

dxk þ 2l n ð2l nÞ X42l n 42l n ð2l nÞ ¼ 1 þ a0 Þ; i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 dxk l d 2 xk þ 2l n ð2l nÞ X42l n X42l n ð2l nÞ 42 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 2 dxk .. . l d r xk þ 2l n X42l n ð2l nÞ X42l n ð2l nÞ 42 n ¼ a0 i1 ¼1    ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ r dxk l

for r  42 n ; ð3:190Þ

where ð2 nÞ

a0

ðnÞ

2l n

ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l nÞ

2 n

ð2

l1

g ¼ sortf04i1 ¼1 fai1 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; 2 ð2l nÞ 2

¼ ðBi

for i 2 0Nq11¼1 Iqð21 ð2l1 nÞ

ð2l nÞ

Þ  4Ci

l1

Iq1

ð2l1 nÞ 1 þ 42

¼ ða0

2l1 n

l

ð2 nÞ

04i¼1 fai

Di

ð2l nÞ

¼ ða0 Þ1 þ 4 ; a0

nÞ

Þ

nÞ

l1 n

; l ¼ 1; 2; 3; . . .; ð2l nÞ

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

0 l

00Nq22¼1 Iqð22 nÞ

¼ flðq1 1Þ ð2l1 nÞ þ 1 ; lðq1 1Þ ð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f£g;

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1 ð2l1 nÞ; ð2l nÞ

Iq2

¼ flðq2 1Þ ð2l nÞ þ 1 ; lðq2 1Þ ð2l nÞ þ 2 ; . . .; lq2 ð2l1 nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f£g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ ð42 n  42 n Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi ð2l nÞ ð2l nÞ 2 ð2l nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di i 2 flN ð2l nÞ þ 1 ; lN ð2l nÞ þ 2 ; . . .; lM2 g  f1; 2; . . .; M2 g0f£g with fixed-points

ð3:191Þ

3.5 Forward Quartic Discrete Systems ð2l nÞ

xk þ 2l n ¼ xk ¼ ai 2l n

ð2l nÞ

04i¼1 fai with

237 l

; ði ¼ 1; 2; . . .; 42 n Þ 2l1 n

ð2l1 nÞ

g ¼ sortf04i1 ¼1 fai1

ð2l1 nÞ

2 g; 0M i2 ¼1 fbi2 ;1

ð2l1 nÞ

; bi2 ;2

gg

ð2l nÞ ð2l nÞ \ai þ 1 : ai

ð3:192Þ ð2l1 nÞ

ð2l1 nÞ

(ii) For a fixed-point of xk þ ð2l nÞ ¼ xk ¼ ai1 N1 gÞ, there is a period-2

l1

ði1 2 Iq

; q 2 f1; 2; . . .;

n discrete system if

dxk þ 2l1 n ð2l1 nÞ 42l1 n ð2l1 nÞ ð2l1 nÞ j  ð2l1 nÞ ¼ 1 þ a0  ai2 Þ ¼ 1; *i2 ¼1;i2 6¼i1 ðai1 x ¼a dxk i1 k d s xk þ 2l1 n j  ð2l1 nÞ ¼ 0; for s ¼ 2; . . .; r  1; xk ¼ai dxsk 1 r d xk þ 2l1 n l1 j  ð2l1 nÞ 6¼ 0 for 1\r  42 n xk ¼ai dxrk 1 ð3:193Þ with • a r th -order oscillatory sink for d r xk þ 2l n =dxrk j • a r -order oscillatory source for d th

r

ð2l nÞ

xk ¼ai

[ 0 and r ¼ 2l1 þ 1;

1

xk þ 2l n =dxrk j  ð2l nÞ \0 x ¼a

and r ¼ 2l1 þ 1;

i1

k

• a r th -order oscillatory upper-saddle for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

r ¼ 2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

r ¼ 2l1 .

[ 0 and

1

\ 0 and

1

The corresponding period-2l n discrete system is ð2l nÞ

xk þ 2l n ¼ xk þ a0

*

2l n

ð2l1 nÞ

i1 2Iq

ð2l1 nÞ 3

ðxk  ai1

Þ

ð3:194Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ

*4j2 ¼1 ðxk  aj2

Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2 with

ð2l1 nÞ

¼ ai1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð3:195Þ

238

3 Quartic Nonlinear Discrete Systems

dxk þ 2l n d 2 xk þ 2l n j  ð2l1 nÞ ¼ 1; j  ð2l1 nÞ ¼ 0; xk ¼ai xk ¼ai dxk dx2k 1 1 3 l d xk þ 2l n ð2 nÞ ð2l1 nÞ ð2l1 nÞ 3 j  ð2l1 nÞ ¼ 6a0 ðai1  ai 2 Þ * ð2l1 nÞ 3 xk ¼ai i2 2Iq ;i2 6¼i1 dxk 1 2l n

ði1 2

ð2l1 nÞ Iq ;q

ð2l1 nÞ

*4j2 ¼1 ðai1

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj 2

Þ

6¼ 0

2 f1; 2; . . .; N1 gÞ: ð3:196Þ ð2l1 nÞ

Thus, xk þ 2l n at xk ¼ ai1

is

• a monotonic sink of the third-order if d 3 xk þ 2l n =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order if d 3 xk þ 2l n =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0.

1

(ii1) The period-2l n fixed-points are trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

ð2l1 nÞ

2 f04ii ¼1 fai1 g; 04i2 ¼1 fai2

;

l

for j ¼ 1; 2;    ; 42 n for n 6¼ 2n1 ; ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

9 gg =

2l1 n

ð2l1 nÞ

2 04i2 ¼1 fai2

9 g=

ð3:197Þ

;

l

for j ¼ 1; 2;    ; 42 n for n ¼ 2n1 :

(ii2) The period-2l n fixed-points are non-trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

2l1 n

ð1Þ

for n 6¼ 2n1 ; ð2l nÞ

9 gg = ;

l

for j ¼ 1; 2;    ; 4ð2 nÞ xk þ 2l n ¼ xk ¼ aj

ð2l1 nÞ

62 f04ii ¼1 fai1 g; 04i2 ¼1 fai2

2l1 n

ð2l1 nÞ

62 04i2 ¼1 fai2

9 g=

ð3:198Þ

;

l

for j ¼ 1; 2;    ; 42 n for n ¼ 2n1 : Such a period-2l n fixed-point is • monotonically unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• monotonically invariant if dxk þ 2l n =dxk j

1 ð2l nÞ

xk ¼ai

1

2 ð1; 1Þ; ¼ 1, which is

3.5 Forward Quartic Discrete Systems

239

– a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2lk 1 jxk [ 0 (independent ð2l1 Þ-branch appearance); 1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \ 0 (independent ð2l1 Þ-branch appearance) 1 þ1 jx – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n = dx2l k k [ 0 (dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jx \ 0 k (dependent ð2l1 þ 1Þ-branch appearance from one branch); • monotonically stable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• invariantly zero-stable if dxk þ 2l n =dxk j • oscillatorilly stable if dxk þ 2l n =dxk j • flipped if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð0; 1Þ;

1

¼ 0;

ð2l nÞ

xk ¼ai ð2l nÞ

xk ¼ai

1

2 ð1; 0Þ;

1

¼ 1, which is

1

– an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2lk 1 jxk [ 0; 1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jx \0 k

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jxk \ 0; – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2lk 1 þ 1 jxk [ 0; • oscillatorilly unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, the proof of this theorem is similar to the proof of Theorem 1.11. This theorem can be easily proved. ■

3.6

Backward Quartic Discrete Systems

In this section, the analytical bifurcation scenario for backward quartic discrete systems will be discussed in a similar fashion through nonlinear renormalization techniques, and the backward bifurcation scenario based on the monotonic saddle-node bifurcations will be discussed, which is independent of period-1 fixed-points.

3.6.1

Backward Period-2 Quartic Discrete Systems

After the period-doubling bifurcation of a period-1 fixed-points in the backward quartic nonlinear discrete systems, the backward period-doubled fixed-point systems can be obtained. Consider the period-doubling fixed-points for a backward quartic nonlinear discrete system as follows.

240

3 Quartic Nonlinear Discrete Systems

Theorem 3.5 Consider a 1-dimensional, backward, quartic nonlinear discrete system xk ¼ xk þ 1 þ f ðxk þ 1 ; pÞ ¼ xk þ 1 þ AðpÞx4k þ 1 þ BðpÞx3k þ 1 þ CðpÞx2k þ 1 þ DðpÞxk þ 1 þ EðpÞ ¼ xk þ 1 þ a0 ðpÞ½x2k þ 1 þ B1 ðpÞxk þ 1 þ C1 ðpÞ½x2k þ 1 þ B2 ðpÞxk þ 1 þ C2 ðpÞ ð3:199Þ

where a0 ðpÞ ¼ AðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ;    ; pm ÞT :

ð3:200Þ

Di ¼ B2i  4Ci [ 0; for i ¼ 1; 2 with pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 a1;2 ¼ ðB1 D1 Þ; a3;4 ¼ ðB2 D2 Þ; 2 2

ð3:201Þ

Under

the standard form of such a backward discrete system is xk ¼ xk þ 1 þ a0 ðxk þ 1  a1 Þðxk þ 1  a2 Þðxk þ 1  a3 Þðxk þ 1  a4 Þ:

ð3:202Þ

Thus, a general standard form of such a backward quartic discrete system is xk ¼ xk þ 1 þ Ax4k þ 1 þ Bx3k þ 1 þ Cx2k þ ! þ Dxk þ ! þ E ð1Þ

¼ xk þ 1 þ a0 *4i¼1 ðxk þ 1  ai Þ

ð3:203Þ

where ð1Þ bi;1

¼

1 ð1Þ  ðBi þ 2

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð1Þ ð1Þ ð1Þ ð1Þ Di Þ; bi;2 ¼  ðBi  Di Þ 2

ð1Þ for Di  0; i 2 f1; 2g; ð1Þ ð1Þ ð1Þ ð1Þ 04i¼1 ai ¼ sortf02i¼1 fbi;2 ; bi;2 gg; ai

ðiÞ

 ai þ 1 ; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ 2 2 pffiffiffiffiffiffiffi ð1Þ for Di \0; i 2 f1; 2g; i ¼ 1; ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

a1 ¼ b1;1 ; a2 ¼ b1;2 ; a3 ¼ b2;1 ; a4 ¼ b2;2 :

ð3:204Þ

3.6 Backward Quartic Discrete Systems

241

(i) Consider a backward period-2 discrete system of Eq. (3.199) as ð1Þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2

 ai2 Þg

ð1Þ

ð42 4Þ=2

ð2Þ

xk ¼ xk þ 2 þ ½a0 *4i1 ¼1 ðxk þ 2  ai1 Þf1 þ

4

¼ xk þ 2 þ ½a0 *4i1 ¼1 ðxk þ 2  ai1 Þ½a40 *i2 ¼1 ð1Þ

4

ð2Þ

ð1Þ

ðx2k þ 1 þ Bi2 xk þ 1 þ Ci2 Þ ð2Þ

4 ¼ xk þ 2 þ ½a0 *4j1 ¼1 ðxk þ 2  ai1 Þ½a40 *4j2 ¼1 ðxk þ 2  bj2 Þ 2

ð2Þ

¼ xk þ 2 þ a10 þ 4 *4i¼1 ðxk þ 2  ai Þ 2

ð3:205Þ

where ð2Þ

1 2

ð2Þ

bi;1 ¼  ðBi þ ð2Þ

Di

pffiffiffiffiffiffiffiffi ð2Þ Dð2Þ Þ; bi;2

ð2Þ

qffiffiffiffiffiffiffiffi 1 ð2Þ ð2Þ ¼  ðBi  Di Þ; 2

ð2Þ

¼ ðBi Þ2  4Ci  0; i 2 0Nq11¼1 Iqð21

0

Þ

00Nq ¼1 Iqð2 Þ 2

2 2

2

Iqð21 Þ ¼ flðq1 1Þ 20 m1 þ 1 ; lðq1 1Þ 20 m1 þ 2 ;    ; lq1 20 m1 g 0

 f1; 2;    ; M1 g0f∅g; q1 2 f1; 2;    ; N1 g; M1 ¼ N1 20 m1 ; m1 2 f1; 2g; Iqð22 Þ ¼ flðq2 1Þ 21 m1 þ 1 ; lðq2 1Þ 21 m1 þ 2 ;    ; lq2 21 m1 g 1

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g;

ð3:206Þ

q2 2 f1; 2;    ; N2 g; M2 ¼ ð42  4Þ=2; qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i Dð2Þ Þ; bi;2 ¼  ðBi  i Di Þ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \ 0; i 2 J ð2 Þ ¼ flN2 21 m1 þ 1 ; lN2 21 m1 þ 2 ;    ; lM2 g 1

 fM1 þ 1; M1 þ 2;    ; M2 g

with fixed-points ð2Þ

xk ¼ xk þ 2 ¼ ai ; ði ¼ 1; 2; . . .; 42 Þ ð2Þ

ð1Þ

ð2Þ

ð2Þ

2 04i¼1 fai g ¼ sortf04j1 ¼1 faj1 g; 0M j2 ¼1 fbj2 ;1 ; bj2 ;2 gg 2

ð2Þ

ð3:207Þ

ð2Þ

with ai \ai þ 1 ; M2 ¼ ð42  4Þ=2: ð1Þ

(ii) For a fixed-point of xk ¼ xk þ 1 ¼ ai1 ði1 2 f1; 2; . . .; 4gÞ, if dxk ð1Þ ð1Þ 4 j ð1Þ ¼ 1 þ a0 * i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk þ 1 xk þ 1 ¼ai1 with

ð3:208Þ

242

3 Quartic Nonlinear Discrete Systems

• a r th -order oscillatory lower-saddle-node bifurcation ðd r xk =dxrk þ 1 jx

kþ1

[ 0; r ¼ 2l1 Þ; • a r th -order oscillatory upper-saddle-node bifurcation ðd r xk =dxrk þ 1 jx \ 0; r ¼ 2l1 Þ; • a r th -order oscillatory source bifurcation ðd r xk =dxrk þ 1 jx

kþ1

kþ1

[ 0; r ¼

2l1 þ 1Þ; • a r th -order oscillatory sink bifurcation ðd r xk =dxrk þ 1 jx \ 0; r ¼ 2l1 þ 1Þ; kþ1

then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi1 ; Di1 ¼ ðBi1 Þ2  4Ci1 ¼ 0; 2

ð3:209Þ

and there is a period-2 discrete system of the backward quartic discrete system in Eq. (3.199) as xk ¼ xk þ 2 þ a50

*

ð20 Þ

i1 2Iq1

42

*i ¼1 ðxk þ 2 2

ð1Þ

ðxk þ 2  ai1 Þ3 ð2Þ

 ai2 Þð1dði1 ;i2 ÞÞ

ð3:210Þ

for i1 2 f1; 2; . . .; 4g; i1 6¼ i2 with dxk d 2 xk jx ¼að1Þ ¼ 1; 2 jx ¼að1Þ ¼ 0; dxk þ 2 k þ 2 i1 dxk þ 2 k þ 2 i1

ð3:211Þ

ð1Þ

• xk at xk þ 2 ¼ ai1 is a monotonic sink of the third-order if d 3 xk 5 j ð1Þ ¼ 6a0 dx3k þ 2 xk þ 2 ¼ai1

*

ð1Þ

ð20 Þ

i2 2Iq1 ;i2 6¼i1

ð1Þ 42 *i ¼1 ðai1 3



ð1Þ

ðai1  ai2 Þ3

ð3:212Þ

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ \0;

and the corresponding bifurcations is a third-order monotonic source bifurcation for the period-2 discrete system; ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk 5 j ð1Þ ¼ 6a 0 dx3k þ 2 xk þ 2 ¼ai1

*

ð20 Þ

i2 2Iq1 ;i2 6¼i1

ð1Þ

ð1Þ

ðai1  ai2 Þ3

ð3:213Þ

ð2Þ ð1dði2 ;i3 ÞÞ

*4i3 ¼1 ðað1Þ \0; i1  ai3 Þ 2

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the period-2 discrete system.

3.6 Backward Quartic Discrete Systems

243

(ii1) The backward period-2 fixed-points are trivial and unstable if ðtÞ

xk ¼ xk þ 2 2 U4i1 ¼1 fai g: 1

ð3:214Þ

(ii2) The backward period-2 fixed-points are non-trivial and stable if ð2Þ

ð2Þ

xk ¼ xk þ 2 ¼2 U4i1 ¼1 fbi1 ;1 ; fbi1 ;2 g: 2

ð3:215Þ

Proof The proof is straightforward through the simple algebraic manipulation. Following the proof of quadratic discrete system, this theorem is proved. ■

3.6.2

Backward Period-Doubling Renormalization

The generalized case of period-doublization of a backward quartic discrete system is presented through the following theorem. The backward period-doubling bifurcation trees can be developed for backward quartic discrete systems. Theorem 3.6 Consider a 1-dimensional, backward quartic discrete system as xk ¼ xk þ 1 þ Ax4k þ 1 þ Bx3k þ 1 þ Cx2k þ 1 þ Dxk þ 1 þ E ð1Þ

¼ xk þ a0 *4i¼1 ðxk þ 1  ai Þ:

ð3:216Þ

(i) After l-times period-doubling bifurcations, a period-2l ðl ¼ 1; 2; . . .Þ discrete system for the quartic discrete system in Eq. (3.216) is given through the nonlinear renormalization as ð2l1 Þ

xk ¼ xk þ 2l þ ½a0

f1 þ

ð2l1 Þ

ð2l1 Þ 2 ½ða0 Þ4

l1

ð2l1 Þ

¼ xk þ 2l þ ½a0

ð2l1 Þ 42

½ða0

l1

Þ

2l1

*i1 ¼1 ðxk þ 2l 4

l

ð42 42 *j1 ¼1 22

ð2l Þ

Þ=2

l1

l

ð42 42

l1

*i2 ¼1

ð2l1 Þ 1 þ 42

¼ xk þ 2 l þ a0

l1

*i1 ¼1; ðxk þ 2l

l1

¼ xk þ 2l þ ða0 with

ð2l1 Þ

 ai1

Þ

l1 ð2l1 Þ 42l1 42 *i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk þ 2l

¼ xk þ 2l þ ½a0

42

*i1 ¼1 ðxk þ 2l

Þ

42

l

l1

Þ=2 42

ð2l1 Þ

 ai1

Þg

Þ ð2l Þ

ð2l Þ

ðx2k þ 2l þ Bj2 xk þ 2l þ Cj2 Þ ð2l1 Þ

 ai1

Þ ð2l Þ

ð2l Þ

ðxk þ 2l  bi2 ;1 Þðxk þ 2l  bi2 ;2 Þ

l

*i¼1 ðxk þ 2l

*i¼1 ðxk þ 2l

ð2l1 Þ

 ai 2

ð2l Þ

 ai Þ

ð2l Þ

 ai Þ

ð3:217Þ

244

3 Quartic Nonlinear Discrete Systems

dxk 2l 2l ð2l Þ ð2l Þ ¼ 1 þ a0 R4i1 ¼1 *4i2 ¼1;i2 6¼i1 ðxk þ 2l  ai2 Þ; dxk þ 2l l d 2 xk ð2l Þ X42l X42l ð2l Þ 42 ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l  ai3 Þ; 2 dxk þ 2l

.. .

l l d r xk X42l ð2l Þ X42 ð2l Þ 42 ¼ a0    *ir þ 1 ¼1;i3 6¼i1 ;i2 ...;ir ðxk þ 2l  air þ 1 Þ i ¼1;i ¼ 6 i ;i ...i r i ¼1 r 3 1 2 r1 1 dxk þ 2l l

for r  42 : ð3:218Þ

where ð2l Þ

ð2Þ

ð2l1 Þ 1 þ 42

a0 ¼ ða0 Þ1 þ 4 ; a0 2l

¼ ða0 2l1

l

ð2 Þ

Þ

l1

; l ¼ 2; 3;    ;

l

ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

2 04i¼1 fai g ¼ sortf04i1 ¼1 fai1 g; 0M i2 ¼1 0I ð2l Þ fbi2 ;1 ; bi2 ;2 gg ,ai q qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

2

ð2l Þ Di l1

Iqð21

¼ Þ

ð2l Þ

 ai þ 1 ;

2

ð2l Þ ðBi Þ2



ð2l Þ 4Ci

 0 for i 2 0Nq11¼1 Iqð21

l1

Þ

l

00Nq22¼1 Iqð22 Þ ;

¼ flðq1 1Þ 2l1 m1 þ 1 ; lðq1 1Þ 2l1 m1 þ 2 ;    ; lq1 2l1 m1 g f1; 2;    ; M1 g0f∅g;

for q1 2 f1; 2;    ; N1 g; M1 ¼ N1 2l1 m1 ; m1 2 f1; 2g; l

Iqð22 Þ ¼ flðq2 1Þ 2l m1 þ 1 ; lðq2 1Þ 2l m1 þ 2 ;    ; lq2 2l m1 g fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; l

l1

for q2 2 f1; 2;    ; N2 g; M2 ¼ ð42  42 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ;    ; lM2 g  fM1 þ 1; M1 þ 2;    ; M2 g0f∅g ð3:219Þ

with fixed-points ð2l Þ

l

xk ¼ xk þ 2l ¼ ai ; ði ¼ 1; 2; . . .; 42 Þ 2l

2l1

ð2l Þ

ð2l1 Þ

04i¼1 fai g ¼ sortf04i¼1 fai l

ð2 Þ

l

ð2 Þ

with ai \ai þ 1 :

ð2l Þ

ð2l Þ

2 g; 0M i¼1 fbi;1 ; bi;2 gg

ð3:220Þ

3.6 Backward Quartic Discrete Systems

245 ð2l1 Þ

(ii) For a fixed-point of xk ¼ xk þ 2l1 ¼ ai1 dxk

j

2l1

ð2l1 Þ

ði1 2 Iq

ð2l1 Þ

; q 2 f1; 2; . . .; N1 gÞ, if ð2l1 Þ

 ai 2 dxk þ 2l1 d s xk j ð2l1 Þ ¼ 0; for s ¼ 2; . . .; r  1; dxsk þ 2l1 xk þ 2l1 ¼ai1 d r xk 2l1 j ; ð2l1 Þ 6¼ 0 for 1\r  4 dxrk þ 2l1 xk þ 2l1 ¼ai1 ð2l1 Þ x l1 ¼ai kþ2 1

¼ 1 þ a0 *4i2 ¼1;i2 6¼i1 ðai1

Þ ¼ 1; ð3:221Þ

with • a rth-order oscillatory source for d r xk =dxrk þ 2l1 j 2l1 þ 1; • a rth-order oscillatory sink for d r xk =dxrk þ 2l1 j

ð2l1 Þ

x

k þ 2l1

¼ai

ð2l1 Þ

x

k þ 2l1

2l1 þ 1; • a rth-order oscillatory lower-saddle for d r xk =dxrk þ 2l1 j

¼ai

ð2l1 Þ

k þ 2l1

¼ai

k þ 2l1

[ 0 and

1

ð2l1 Þ

x

r ¼ 2l1 ;

\ 0 and r ¼

1

x

r ¼ 2l1 ; • a rth-order oscillatory upper-saddle for d r xk =dxrk þ 2l1 j

[ 0 and r ¼

1

¼ai

\ 0 and

1

then, there is a backward period-2l fixed-point discrete system ð2l Þ

xk ¼ x k þ 2 l þ a 0

*

ð2l1 Þ

i1 2Iq

2l

ð2l1 Þ 3

ðxk þ 2l  ai1

Þ

ð3:222Þ

ð2l Þ

*4j2 ¼1 ðxk þ 2l  aj2 Þð1dði1 ;j2 ÞÞ where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð3:223Þ

and dxk d 2 xk j j ð2l1 Þ ¼ 1; ð2l1 Þ ¼ 0: x ¼a dxk þ 2l k þ 2l i1 dx2k þ 2l xk þ 2l ¼ai1 ð2l1 Þ

• xk at xk þ 2l ¼ ai1

ð3:224Þ

is a monotonic source of the third-order if

246

3 Quartic Nonlinear Discrete Systems

d 3 xk ð2l Þ ð2l1 Þ ð2l1 Þ j ðai1  ai2 Þ3 * ð2l1 Þ ¼ 6a0 ð2l1 Þ 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk þ 2l k þ 2l i1 2l

ð2l1 Þ

ði1 2 Iq

ð2l1 Þ

ð3:225Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \ 0

*4j2 ¼1 ðai1 ; q 2 f1; 2; . . .; N1 gÞ;

ð2l1 Þ

and such a bifurcation at xk þ 2l ¼ ai1 source bifurcation. ð2l1 Þ

• xk at xk þ 2l ¼ ai1

is a third-order monotonic

is a monotonic sink of the third-order if

d 3 xk ð2l Þ ð2l1 Þ ð2l1 Þ j ðai1  ai2 Þ3 * ð2l1 Þ ¼ 6a0 ð2l1 Þ 3 i2 2Iq ;i2 6¼i1 dxk þ 2l xk þ 2l ¼ai1 2l

ð2l1 Þ

ði1 2 Iq

ð2l1 Þ

ð3:226Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ [ 0

*4j2 ¼1 ðai1 ; q 2 f1; 2; . . .; N1 gÞ

ð2l1 Þ

and such a bifurcation at xk þ 2l ¼ ai1 bifurcation.

is a third-order monotonic sink

(ii1) The period-2l fixed-points are trivial if ð2l Þ

xk ¼ xk þ 2l ¼ ai

2l1

ð2l1 Þ

2 04i1 ¼1 ai1

;

ð3:227Þ

ð2l Þ

ð3:228Þ

(ii2) The period-2l fixed-points are non-trivial if ð2l Þ

xk ¼ xk þ 2l ¼ ai

ð2l Þ

2 2 0M j1 ¼1 fbj1 ;1 ; bj1 ;2 g:

Such a backward period-2l fixed-point is • monotonically stable if dxk =dxk þ 2l j

x

ð2l Þ

k þ 2l

• monotonically invariant if dxk =dxk þ 2l j

¼ai

x

2 ð1; 1Þ;

1

k þ 2l

ð2l Þ

¼ai

¼ 1, which is

1

– a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2lk þ1 2l jx

[ 0;

k þ 2l

th

– a monotonic upper-saddle of the ð2l1 Þ order for

1 d 2l1 xk =dx2l j \ k þ 2l xk þ 2l

– a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

k þ 2l

– a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l 1 jx

0;

[ 0;

k þ 2l

\0;

3.6 Backward Quartic Discrete Systems

247

• monotonically unstable if dxk =dxk þ 2l j • invariantly zero-stable if dxk =dxk þ 2l j • oscillatorilly unstable if dxk =dxk þ 2l j • flipped if dxk =dxk þ 2l j

x

k þ 2l

ð2l Þ

¼ai

2 ð0; 1Þ;

ð2l Þ

x

k þ 2l

¼ai

1

¼ 0;

ð2l Þ

x

k þ 2l

¼ai

1

ð2l Þ

x

k þ 2l

¼ai

2 ð1; 0Þ;

1

¼ 1, which is

1

– an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2lk þ1 2l jx

k þ 2l

th

– an oscillatory upper-saddle of the ð2l1 Þ order for d th

– an oscillatory source of the ð2l1 þ 1Þ order for

2l1

[ 0;

1 xk =dx2l j \ k þ 2l xk þ 2l

d 2l1 þ 1 xk =dxk2lþ1 þ2l1 jx k þ 2l

0;

[ 0;

– an oscillatory sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2lk þ1 þ2l1 jx l \0; kþ2

• oscillatorilly stable if dxk =dxk þ 2l j

ð2l Þ

x

k þ 2l

¼ai

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, this theorem can be proved.

3.6.3

■

Backward Period-n Appearing and Period-Doublization

The period-n discrete system for a backward quartic nonlinear discrete system will be discussed, and the period-doublization of a backward period-n discrete system is discussed through the nonlinear renormalization. Theorem 3.7 Consider a 1-dimensional, backward quartic discrete system as xk ¼ xk þ 1 þ Ax4k þ 1 þ Bx3k þ 1 þ Cx2k þ 1 þ Dxk þ 1 þ E ð1Þ

¼ xk þ 1 þ a0 *4i¼1 ðxk þ 1  ai Þ:

ð3:229Þ

(i) After n-times iterations, a backward period-n discrete system for the backward quartic discrete system in Eq. (3.229) is xk ¼ xk þ n þ a0 *4i1 ¼1 ðxk þ n  ai2 Þ½1 þ n

ð4 1Þ=3

¼ x k þ n þ a0

ð4n 4Þ=2 ½*j2 ¼1 ðx2k þ n ðnÞ

¼ xk þ n þ a0 with

*i1 ¼1 ðxk þ n

4n

4

j¼1 Qj 

 ai1 Þ

ð2n Þ þ Bj2 xk þ n

*i¼1 ðxk þ n

Xn

ðnÞ

 ai Þ

ð2n Þ

þ Cj2 Þ

ð3:230Þ

248

3 Quartic Nonlinear Discrete Systems

dxk n ðnÞ X n ¼ 1 þ a0 4i1 ¼1 *4i2 ¼1;i2 6¼i1 ðxk þ n  aðnÞ i2 Þ; dxk þ n d 2 xk n ðnÞ X n X n ¼ a0 4i1 ¼1 4i2 ¼1;i2 6¼i1 *4i3 ¼1;i3 6¼i1 ;i2 ðxk þ n  aðnÞ i3 Þ; dx2k þ n .. . d r xk n X n ðnÞ X n ¼ a0 4i1 ¼1    4ir ¼1;ir 6¼i1 ;i2 ;ir1 *4ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk þ n  aðnÞ ir þ 1 Þ r dxk þ n

for r  4n ;

ð3:231Þ

where ðnÞ

a0 ¼ ða0 Þð4

n

1Þ=2

ð1Þ

; Q1 ¼ 0; Q2 ¼ *4i2 ¼1 ½1 þ a0 *4i1 ¼1;i1 6¼i2 ðxk þ n  ai1 Þ; ð1Þ

Qn ¼ *4in ¼1 ½1 þ a0 ð1 þ Qn1 Þ *4in1 ¼1;in1 6¼in ðxk þ n  ain1 Þ; n ¼ 3; 4; . . .; n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

04i¼1 fai g ¼ sort½04i1 ¼1 fai1 g; 0Nq¼1 0i 2I ðnÞ fbi2 ;1 ; bi2 ;2 g; 2 q qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2 2 ðnÞ ðnÞ ðnÞ Di2 ¼ ðBi2 Þ2  4Ci2  0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þ n þ 1 ; lðq1Þ n þ 2 ;    ; lq n gf1; 2; . . .; g0f£g; for q 2 f1; 2; . . .; Ng; M ¼ ð4n  4Þ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flN n þ 1 ; lN n þ 2 ; . . .; lM g  f1; 2; . . .; Mg0f£g; ð3:232Þ

with backward fixed-points ðnÞ

xk ¼ xk þ n ¼ ai ; ði ¼ 1; 2; . . .; 4n Þ n

ðnÞ

ð1Þ

ðnÞ

ðnÞ

04i¼1 fai g ¼ sortf04i1 ¼1 fai1 g; 0M i¼1 fbi2 ;1 ; bi2 ;2 gg ðnÞ

ð3:233Þ

ðnÞ

with ai \ ai þ 1 : ðnÞ

ðnÞ

(ii) For a backward fixed-point of xk ¼ xk þ n ¼ ai1 ði1 2 Iq ; q 2 f1; 2; . . .; NgÞ, if dxk ðnÞ 4n ðnÞ ð2l Þ j ðnÞ ¼ 1 þ a 0 *i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk þ n xk þ n ¼ai1 with

ð3:234Þ

3.6 Backward Quartic Discrete Systems

249

d 2 xk n ðnÞ X n ðnÞ ð2l Þ jx ¼aðnÞ ¼ a0 4i2 ¼1;i2 6¼i1 *4i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; 2 i dxk þ n k þ n 1

ð3:235Þ

then there is a new discrete system for onset of the qth -set of period-n fixedpoints based on the second-order monotonic saddle-node bifurcation as ðnÞ

x k ¼ x k þ n þ a0

*i 2I ðnÞ ðxk þ n 1 q

ðnÞ

ðnÞ

 ai1 Þ2 *4j2 ¼1 ðxk þ n  aj2 Þð1dði1 ;j2 ÞÞ ð3:236Þ n

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð3:237Þ

(ii1) If dxk ðnÞ j ðnÞ ¼ 1 ði1 2 Iq Þ; dxk þ n xk þ n ¼ai1 d 2 xk ðnÞ ðnÞ ðnÞ 2 j ðnÞ ¼ 2a 0 *i2 2IqðnÞ ;i2 6¼i1 ðai1  ai2 Þ dx2k þ n xk þ n ¼ai1 ðnÞ

ð3:238Þ

ðnÞ

*4j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 n

ðnÞ

xk at xk þ n ¼ ai1 is • a monotonic upper-saddle of the second-order for d 2 xk =dx2k þ n jx

¼ai

• a monotonic lower-saddle of the second-order for d 2 xk =dx2k þ n jx

¼ai

ðnÞ

kþn

kþn

1

ðnÞ

\ 0; [ 0.

1

(ii2) The backward period-n fixed-points ðn ¼ 2n1 mÞ are trivial if ðnÞ

2n1 1 m

ð1Þ

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 2 f04ii ¼1 fai1 g; 04i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 4n g0f∅g for n 6¼ 2n2 ; ðnÞ

2n1 1 m

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 2 f04i2 ¼1 fai2

gg

)

)

for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 4n g0f∅g

for n ¼ 2n2 : (ii3) The period-n fixed-points ðn ¼ 2n1 mÞ are non-trivial if

ð3:239Þ

250

3 Quartic Nonlinear Discrete Systems ðnÞ

ð1Þ

2n1 1 m

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 62 f04ii ¼1 fai1 g; 04i2 ¼1 fai2

gg

)

for n1 ¼ 1; 2; . . .; m ¼ 2l1 þ 1; j1 2 f1; 2; . . .; 4n g0f∅g

for n 6¼ 2n2 ; ðnÞ

2n1 1 m

ð2n1 1 mÞ

xk þ n ¼ xk ¼ aj1 62 f04i2 ¼1 fai2 gg for n1 ¼ 1; 2; . . .; m ¼ 1; j1 2 f1; 2; . . .; 4n g0f∅g

)

ð3:240Þ

for n ¼ 2n2 : Such a backward period-n fixed-point is • monotonically stable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

• monotonically invariant if dxk =dxk þ n jx

2 ð1; 1Þ;

1 ðnÞ

kþn

¼ai

¼ 1, which is

1

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l k þ n j x

kþn

– a – a – a

[ 0;

th

1 monotonic upper-saddle the ð2l1 Þ order for d 2l1 xk =dx2l k þ n jxk þ n \ 0; monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx [ 0; kþn monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jx kþn

\ 0;

• monotonically unstable if dxk =dxk þ n jx

ðnÞ

kþn

¼ai

2 ð1; 0Þ;

1

• monotonically unstable with infinity eigenvalue if dxk =dxk þ n jx • oscillatorilly source with infinity eigenvalue if dxk =dxk þ n jx • oscillatorilly unstable if dxk =dxk þ n jx • flipped if dxk =dxk þ n jx

kþn

ðnÞ

kþn

ðnÞ

¼ai

¼ai

2 ð1; 0Þ;

kþn

ðnÞ

kþn

¼ai

ðnÞ

¼ai

¼ 0þ ;

1

¼ 0 ;

1

1

¼ 1, which is

1

– an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2lk þ1 n jxk þ n [ 0; 1 – an oscillatory upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ n jx \0; kþn

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2lk þ1 þn 1 jxk þ n [ 0; – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þn 1 jxk þ n \ 0; • oscillatorilly stable if dxk =dxk þ n jx

kþn

ðnÞ

¼ai

2 ð1; 1Þ.

1

ðnÞ

ðnÞ

(iii) For a backward fixed-point of xk ¼ xk þ n ¼ ai1 ði1 2 Iq ;q 2 f1; 2; . . .; NgÞ, there is a period-doubling of the qth -set of period-n fixed-points if

3.6 Backward Quartic Discrete Systems

251

dxk n ðnÞ ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *4j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; i k þ n dxk þ n 1 d s xk j ðnÞ ¼ 0; for s ¼ 2; . . .; r  1; dxsk þ n xk þ n ¼ai1 d r xk n j ðnÞ 6¼ 0 for 1\r  4 dxrk þ n xk þ n ¼ai1

ð3:241Þ

with • a rth-order oscillatory sink for d r xk =dxrk þ n jx • a rth-order oscillatory source for d

r

ðnÞ

\0 and r ¼ 2l1 þ 1;

¼ai 1 xk =dxrk þ n jx ¼aðnÞ i kþn kþn

[ 0 and r ¼

1

2l1 þ 1; • a rth-order oscillatory lower-saddle for d r xk =dxrk þ n jx

ðnÞ

kþn

r ¼ 2l1 ; • a rth-order oscillatory upper-saddle for d r xk =dxrk þ n jx

¼ai

ðnÞ

kþn

r ¼ 2l1 .

[ 0 and

1

¼ai

\0 and

1

The corresponding period-2 n discrete system of the backword quartic discrete system in Eq. (3.229) is ð2 nÞ

xk ¼ xk þ 2 n þ a0

*i 2I ðnÞ ðxk þ 2 n 1 q

ðnÞ

 ai 1 Þ 3

ð3:242Þ

ð2 nÞ ð1dði1 ;j2 ÞÞ

2 n

*4j2 ¼1 ðxk þ 2 n  aj2

Þ

with dxk dxk þ 2 n

jx

ðnÞ

¼ai k þ 2 n

1

¼ 1;

d 2 xk j ðnÞ ¼ 0; dx2k þ 2 n xk þ 2 n ¼ai1

d xk ð2 nÞ ðnÞ ðnÞ 3 j ðnÞ ¼ 6a *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 0 1 q 2 1 dx3k þ 2 n xk þ 2 n ¼ai1 3

2 n

ðnÞ

ð2 nÞ ð1dði1 ;j2 ÞÞ

*4j2 ¼1 ðai1  aj2 ðnÞ

Þ

ð3:243Þ

:

ðnÞ

Thus, xk at xk þ 2 n ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic source of the third-order if d 3 xk =dx3k þ 2 n jx

ðnÞ

k þ 2 n

• a monotonic sink of the third-order if d

3

xk =dx3k þ 2 n jx

k þ 2 n

¼ai

ðnÞ

¼ai

\0,

1

[ 0.

1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l n discrete system of the quartic discrete system in Eq. (3.216) is

252

3 Quartic Nonlinear Discrete Systems ð2l1 nÞ

42

xk ¼ xk þ 2l n þ ½a0

f1 þ

4

2l1 n

*i1 ¼1

ð2

¼ xk þ 2l n þ ½a0 ð2

½ða0

l1

nÞ 4

Þ

ð2l1 nÞ 4

½ða0

Þ

nÞ

4

l1

nÞ

2l1 n

¼

Þ

ð4

*j1 ¼1

ð2l1 nÞ

ðxk þ 2l n  ai1 4

ð2

ðxk þ 2l n  ai1

4

ð2l1 nÞ

4

*i ¼1 1 2l n

ð4

*j2 ¼1

Þ

2l1 n

2l1 n

l

42 n

*i¼1

Þ=2

nÞ

l1

nÞ

ð2l nÞ

xk þ 2l n þ Cj2

Þ

Þ

ð2l nÞ

ð2l nÞ

ðxk þ 2l n  bj2 ;1 Þðxk þ 2l n  bj2 ;2 Þ ð2l nÞ

ðxk þ 2l n  ai

ð2l nÞ 42l n xk þ 2l n þ a0 *i¼1 ðxk þ 2l n

Þg

Þ ð2l nÞ

ð2

4

 ai 2

ðx2k þ 2l n þ Bj2

ðxk þ 2l n  ai1

2l1 n

l1 n

Þ=2

l1

Þ ð2l1 nÞ

*i2 ¼1;i2 6¼i1 ðxk þ 2l n

2l1 n 2l n

ð2l1 nÞ 42

¼ xk þ 2l n þ ða0

l1

*i1 ¼1

ð2l1 nÞ

ð2

¼ xk þ 2l n þ ½a0

ð2

½1 þ a0

l1

l1 n

*i1 ¼1



Þ

ð2l nÞ ai Þ

ð3:244Þ

with dxk

ð2l nÞ X42l n 42l n i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk þ 2l n

ð2l nÞ

¼ 1 þ a0

 ai2 Þ; dxk þ 2l n l d 2 xk ð2l nÞ X42l n X42l n ð2l nÞ 42 n ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk þ 2l n  ai3 2 dxk þ 2l n

.. .

l d r xk X42l n ð2l nÞ X42l n ð2l nÞ 42 n ¼ a0 i1 ¼1    ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk þ 2l n  air þ 1 Þ dxrk þ 2l n l

for r  42 n ;

ð3:245Þ

where ð2l nÞ

a0

2l n

ð2l1 nÞ 1 þ 42

¼ ða0

ð2l nÞ ð2l nÞ

bi;2

ð2l nÞ

l1 n

; l ¼ 1; 2; 3; . . .;

2l1 n

l

ð2 nÞ

04i¼1 fai bi;1

Þ

ð2l1 nÞ

g ¼ sortf04i1 ¼1 fai1 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ¼  ðBi  Di Þ; 2 ð2l nÞ 2

ð2l nÞ

Di

¼ ðBi

for i 2

l1 l 0Nq11¼1 Iqð21 nÞ 00Nq22¼1 Iqð22 nÞ ;

ð2l1 nÞ

Iq1

Þ  4Ci

ð2l nÞ

ð2l nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

0

¼ flðq1 1Þ ð2l1 nÞ þ 1 ; lðq1 1Þ ð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f£g;

3.6 Backward Quartic Discrete Systems

253

for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1 ð2l1 nÞ; ð2l nÞ

Iq2

¼ flðq2 1Þ ð2l nÞ þ 1 ; lðq2 1Þ ð2l nÞ þ 2 ; . . .; lq2 ð2l nÞ g f1; 2; . . .; M1 g0f£g; l

l1

for q2 2 f1; 2; . . .; N2 g; M2 ¼ ð42  42 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi l l l ð2 nÞ ð2 nÞ 2 ð2 nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di l

i 2 J ð2 nÞ ¼ flN2 ð2l nÞ þ 1 ; lN2 ð2l1 nÞ þ 2 ; . . .; lM2 g  f1; 2; . . .; M2 g0f£g

ð3:246Þ

with backward fixed-points ð2l nÞ

xk ¼ xk þ 2l n ¼ ai 2l n

ð2l nÞ

04i¼1 fai l

ð2 nÞ

with ai

l

; ði ¼ 1; 2; . . .; 42 n Þ 2l1 n

g ¼ sortf04i¼1

ð2l1 nÞ

fai

ð2l nÞ

2 g; 0M i¼1 fbi;1

ð2l nÞ

; bi;2

ð3:247Þ

gg

l

ð2 nÞ

\ai þ 1 : ð2l1 nÞ

ð2l1 nÞ

(ii) For a fixed-point of xk ¼ xk þ 2l n ¼ ai1 ði1 2 Iq l there is a backward period-2 n discrete system if

; q 2 f1; 2; . . .; N1 gÞ,

l1 dxk ð2l1 nÞ ð2l1 nÞ 42 n j  ai2 Þ ¼ 1; ð2l1 nÞ ¼ 1 þ a0 *i ¼1;i 6¼i ðai1 2 2 1 dxk þ 2l1 n xk þ 2l1 n ¼ai1 d s xk j ð2l1 nÞ ¼ 0; for s ¼ 2; . . .; r  1; s dxk þ 2l1 n xk þ 2l1 n ¼ai1 d r xk 2l1 n j ð2l1 nÞ 6¼ 0 for 1\r  4 r dxk þ 2l1 n xk þ 2l1 n ¼ai1

ð3:248Þ with • a rth-order oscillatory source for d r xk =dxrk þ 2l1 n j r ¼ 2l1 þ 1; • a rth-order oscillatory sink for d r xk =dxrk þ 2l1 n j

x

ð2l1 nÞ

x

k þ 2l1 n

k þ 2l1 n

[ 0 and

1

ð2l1 nÞ

¼ai

2l1 þ 1; • a rth-order oscillatory lower-saddle for d r xk =dxrk þ 2l1 n j and r ¼ 2l1 ;

¼ai

\0 and r ¼

1

x

k þ 2l1 n

ð2l1 nÞ

¼ai

1

[0

254

3 Quartic Nonlinear Discrete Systems

• a rth-order oscillatory upper-saddle for d r xk =dxrk þ 2l1 n j

ð2l1 nÞ

x

and r ¼ 2l1 .

k þ 2l1 n

¼ai

\0

1

The corresponding period-2l n discrete system is ð2l nÞ

xk ¼ xk þ 2l n þ a0

*

ð2l nÞ

i1 2Iq

2l n

ð2l1 nÞ 3

ðxk þ 2l n  ai1

Þ

ð3:249Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ

*4j2 ¼1 ðxk þ 2l n  aj2

Þ

where ð2l nÞ

ð2l1 nÞ

dði1 ; j2 Þ ¼ 1 if aj2

¼ ai 1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð3:250Þ

with dxk

j

ð2l1 nÞ x l ¼ai k þ 2 n 1

¼ 1;

d 2 xk j ð2l1 nÞ ¼ 0; dx2k þ 2l n xk þ 2l n ¼ai1

dxk þ 2l n d 3 xk ð2l nÞ 42l1 n ð2l1 nÞ ð2l1 nÞ 3 j  ai 2 Þ *i2 ¼1;i2 6¼i1 ðai1 ð2l1 Þ ¼ 6a0 3 x ¼a dxk þ 2l n k þ 2l n i1 2l n

ði1 2

ð2l1 nÞ Iq ;q

ð2l1 nÞ

*4j2 ¼1 ðai1

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

ð3:251Þ

6¼ 0

2 f1; 2; . . .; N1 gÞ: ð2l1 nÞ

Thus, xk at xk þ 2l n ¼ ai1

xk þ 2l n is

• a monotonic source of the third-order if d 3 xk =dx3k þ 2l n j • a monotonic sink of the third-order if d 3 xk =dx3k þ 2l n j

k þ 2l n

ð2l nÞ

ð1Þ

2l1 n

k þ 2l n

ð2l1 nÞ

2 f04ii ¼1 fai1 g; 04i2 ¼1 fai2

¼ai

gg

¼ai

\0;

1

ð2l1 Þ

x

(ii1) The period-2l n fixed-points are trivial if xk ¼ xk þ 2l n ¼ aj

ð2l1 Þ

x

[ 0.

1

)

l

for j ¼ 1; 2; . . .; 4ð2 nÞ for n 6¼ 2n1 ; ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

2l1 n

ð2l1 nÞ

2 04i2 ¼1 fai2

g

)

l

for j ¼ 1; 2; . . .; 42 n for n ¼ 2n1 : (ii2) The backward period-2l n fixed-points are non-trivial if

ð3:252Þ

3.6 Backward Quartic Discrete Systems ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

255 2l1 n

ð1Þ

ð2l1 nÞ

62 f02ii ¼1 fai1 g; 02i2 ¼1 fai2

gg

)

l

for j ¼ 1; 2; . . .; 2ð2 nÞ for n 6¼ 2n1 ; ð2l nÞ

xk ¼ xk þ 2l n ¼ aj

2l1 n

ð2l1 nÞ

62 f02i2 ¼1 fai2

g

ð3:253Þ

)

l

for j ¼ 1; 2; . . .; 22 n for n ¼ 2n1 : Such a backward period-2l n fixed-point for the quartic discrete system is • monotonically unstable if dxk =dxk þ 2l n j

• monotonically invariant if dxk =dxk þ 2l n j

2 ð1; 1Þ;

ð2l nÞ

x

k þ 2l n

¼ai

1

¼ 1, which is

ð2l nÞ

x

k þ 2l n

¼ai

1

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2l j k þ 2l n x

[0

k þ 2l n

(independent ð2l1 Þ-branch appearance); 1 j – a monotonic upper-saddle the ð2l1 Þth order for d 2l1 xk =dx2l k þ 2l n x

\0

(independent ð2l1 Þ-branch appearance); – a monotonic sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l1 n jx

[0

(dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic source the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dx2lk þ1 þ2l1 n jx

\0

k þ 2l n

k þ 2l n

k þ 2l n

(dependent ð2l1 þ 1Þ-branch appearance from one branch); • monotonically unstable if dxk =dxk þ 2l n j

ð2l nÞ

x

k þ 2l n

¼ai

• monotonically infinity-unstable if dxk =dxk þ 2l n j • oscillatorilly infinity-unstable if dxk =dxk þ 2l n j • oscillatorilly unstable if dxk =dxk þ 2l n j • flipped if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

k þ 2l n

ð2l nÞ

x

k þ 2l n

k þ 2l n

¼ai

¼ai

1

ð2l nÞ

x

ð2l nÞ

x

2 ð0; 1Þ;

1

¼ai

¼ 0þ ;

¼ 0 ;

1

2 ð1; 0Þ;

1

¼ 1, which is

1

– an oscillatory lower-saddle of the ð2l1 Þth order for d 2l1 xk =dx2lk þ1 2l n jx

k þ 2l n

th

– an oscillatory upper-saddle of the ð2l1 Þ order for d

2l1

– an oscillatory sink of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk =dxk2lþ1 þ2l1 n jx

k þ 2l n

th

– an oscillatory source the ð2l1 þ 1Þ order for d • oscillatorilly stable if dxk =dxk þ 2l n j

x

k þ 2l n

ð2l nÞ

¼ai

2l1 þ 1

[ 0;

1 xk =dx2l j \ k þ 2l n xk þ 2l n

xk =dxk2lþ1 þ2l1 n jx k þ 2l n

0;

\ 0; [ 0;

2 ð1; 1Þ.

1

Proof Through the nonlinear renormalization, this theorem can be easily proved. ■

256

3 Quartic Nonlinear Discrete Systems

Reference Luo ACJ (2019) The stability and bifurcation of fixed-points in low-degree polynomial systems. J Vib Test Syst Dyn 3(4):403–451

Chapter 4

(2m)th-Degree Polynomial Discrete Systems

In this Chapter, the global stability and bifurcations of period-1 fixed-points in the (2m)thdegree polynomial discrete system are presented. The parallel-appearing, spraying-appearing, sprinkler-spraying-appearing bifurcations for simple and higher-order period-1 fixed-points are presented, and the antenna-switching, straw-bundle-switching bifurcations and flower-bundle-switching bifurcations for simple and higher-order period-1 fixed-points are presented. From the period-doubling bifurcation, the period-2 fixed-point solutions and the corresponding period-doubling renormalization of such a forwarded (2m)th-degree polynomial discrete system are discussed. For multiple iterations, the appearing bifurcations of period-n fixed-points and the corresponding period-doublization of the forward (2m)th-degree polynomial discrete system are presented as well.

4.1

Global Stability and Bifurcations

In a similar fashion in Chaps. 1–3, the global stability and bifurcation of fixed-points in the (2m)th-degree polynomial nonlinear discrete systems are discussed as in Luo (2020a, b). The stability and bifurcation of each individual fixed-point are analyzed from the local analysis. Definition 4.1 Consider a (2m)th-degree polynomial nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ 2m1 ¼ xk þ A0 ðpÞx2m þ    þ A2m2 ðpÞx2k þ A2m1 ðpÞxk þA2m ðpÞ k þ A1 ðpÞxk

¼ xk þ a0 ðpÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ    ½x2k þ Bm ðpÞxk þ Cm ðpÞ ð4:1Þ

© Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3_4

257

4 (2m)th-Degree Polynomial Discrete Systems

258

where A0 ðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm1 ÞT :

ð4:2Þ

(i) If Di ¼ B2i  4Ci \0 for i ¼ 1; 2; . . .; m;

ð4:3Þ

the 1-dimensional nonlinear discrete system with a (2m)th-degree polynomial does not have any period-1 fixed-point, and the corresponding standard form is 1 2

1 4

1 2

1 4

xk þ 1 ¼ xk þ a0 ½ðxk þ B1 Þ2 þ ðD1 Þ    ½ðxk þ Bm Þ2 þ ðDm Þ:

ð4:4Þ

The flow of such a discrete system without fixed-points is called a non-fixedpoint discrete flow. (a) If a0 [ 0, the non-fixed-point discrete flow is called a positive discrete flow. (b) If a0 \0,the non-fixed-point discrete flow is called a negative discrete flow. (ii) If Di ¼ B2i  4Ci [ 0; i ¼ i1 ; i2 ; . . .; il 2 f1; 2; . . .; mg; Dj ¼ B2j  4Cj \0; j ¼ il þ 1 ; il þ 2 ; . . .; im 2 f1; 2; . . .; mg with l 2 f0; 1; . . .; mg;

ð4:5Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has 2l-fixed-points as pffiffiffiffiffi pffiffiffiffiffi ðiÞ ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; xk ¼ b2 ¼ 12ðBi  Di Þ ð4:6Þ i 2 fi1 ; i2 ; . . .; il gf1; 2; . . .; mg: (ii1) If ðiÞ

ðjÞ

br 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; l ð1Þ ð1Þ ðlÞ ðlÞ fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; as \as þ 1 ;

ð4:7Þ

then, the corresponding standard form is 1 2

1 4

2 xk þ 1 ¼ xk þ a0 *li¼1 ðxk  a2i1 Þðxk  a2i Þ *m k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ:

ð4:8Þ (a) If a0 [ 0, the simple fixed-point discrete flow is called a ðmSI-oSO: mSO:. . . : mSI-oSO:mSO) discrete flow.

4.1 Global Stability and Bifurcations

259

(b) If a0 \0, the simple fixed-point discrete flow is called a ðmSO: mSI-oSO:. . . : mSO:mSI-oSO) discrete flow. (ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð4:9Þ

P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2l i¼1 i i¼1 i

with

Pr

s¼1 ls

¼ 2l;

then, the corresponding standard form is 2 xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *m k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ:

1 2

1 4

ð4:10Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ discrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; r, 8 > ð2r  1Þth mSO  ð2rp  1Þth order monotonic source, > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink, > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth mLS  ð2rp Þth order monotonic lower-saddle, > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mUS  ð2rp Þth order monotonic upper-saddle, > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:11Þ where ð2rp  1Þth mSI  mSI-oSO for rp ¼ 1 and ap ¼ (b) For a0 \0 and p ¼ 1; 2; . . .; r,

Xr

s¼p ls :

ð4:12Þ

4 (2m)th-Degree Polynomial Discrete Systems

260

lp th mXX ¼

8 > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source; > > > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; > > ðð2rp  1Þth mUS  2rp Þth order monotonic upper-saddle; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > th th > > > ð2rp  1Þ mLS  ð2rp Þ order monotonic lower-saddle; > > > : for a ¼ 2M ; l ¼ 2r : p

p

p

p

ð4:13Þ (c) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX : . . . : lpb th mXXÞ fixed-points at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1

a Pp1 l i¼1

i þ1

i¼1

6¼    6¼ a Pp1 l i¼1

i þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:14Þ

(iii) If Dj1 ¼ B2j1  4Cj1 ¼ 0; j1 2 fi11 ; i12 ; . . .; i1s1 gfi1 ; i2 ; . . .; il gf1; 2; . . .; mg; Dj2 ¼ B2j2  4Cj2 [ 0; j2 2 fi21 ; i22 ; . . .; i2s2 gfi1 ; i2 ; . . .; il gf1; 2; . . .; mg; Dj3 ¼ B2j3  4Cj3 \0; j3 2 fil þ 1 ; il þ 2 ; . . .; im gf1; 2; . . .; mg;

ð4:15Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has 2l-fixedpoints as 9 1 ðj Þ xk ¼ b1 1 ¼  Bj1 ; > = 2

xk xk

¼

ðj Þ b2 1

1 > ¼  Bj1 ;

¼

ðj Þ b1 2

¼

ðj Þ

xk ¼ b2 2

2 1  ðBj2 2

for j1 2 fi11 ; i12 ; . . .; i1s1 g;

pffiffiffiffiffiffi 9 Dj2 Þ; > = pffiffiffiffiffiffi > for j2 2 fi21 ; i22 ; . . .; i2s2 g: 1 ¼  ðBj2  Dj2 Þ ; 2

þ

ð4:16Þ

4.1 Global Stability and Bifurcations

261

If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð4:17Þ

P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2l i¼1 i i¼1 i

with

Pr

s¼1 ls

¼ 2l;

then, the corresponding standard form is 2 xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *m j¼l þ 1 ½ðxk þ Bij Þ þ ðDij Þ:

1 2

1 4

ð4:18Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ discrete flow. (a) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-point appearance or vanishing is called an lp th mXX appearing bifurcation of fixed-points at a point p ¼ p1 2 @X12 , and the appearing bifurcation condition is 1 2

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ¼  Bip ; i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ; . . .; il gÞ; þ

aPp1 l

i¼1 i

þ

þ1

6¼    6¼ aPp1 l

i¼1 i



þ lp

or aPp1 l

i¼1 i

þ1

ð4:19Þ 

6¼    6¼ aPp1 l

i¼1 i

þ lp

:

(b) The fixed-point of xk ¼ aiq for ðlq [ 1Þ-repeated fixed-points switching is called an lq th XX bifurcation of ðlq1 th mXX : lq2 th mXX :    : lqb th mXXÞ fixed-point switching at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aiq  aPq1 li þ 1 ¼    ¼ aPq1 li þ lq ; i¼1



aPq1 l

i¼1 i

þ1

i¼1



6¼    6¼ aPq1 l

i¼1 i

þ lq

; lq ¼

Xb

i¼1 lqi :

ð4:20Þ

(c) The fixed-point of xk ¼ aip for ðlp1 1Þ-repeated fixed-points appearance/ vanishing and ðlp2 2Þ repeated fixed-points switching of ðlp21 th mXX : lp22 th mXX :    : lp2b th mXXÞ-fixed-point switching is called an lp th mXX bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the flowerswitching bifurcation condition is

4 (2m)th-Degree Polynomial Discrete Systems

262

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp i¼1

i¼1

with Diq ¼ B2iq  4Ciq ¼ 0 ðiq 2 fi1 ; i2 ; . . .; il gÞ þ aP p1 l

i¼1 i

þ j1

þ 6¼    6¼ aP p1 l

i¼1 i

þ jp1

or a Pp1 1 i¼1

li þ j1

6¼    6¼ a Pp1 1 i¼1

li þ jp1

;

for fj1 ; j2 ; . . .; jp1 gf1; 2; . . .; lp g; a Pp1 l

i¼1 i

þ k1

6¼    6¼ a Pp1 l

i¼1 i

ð4:21Þ

þ kp2

for fk1 ; k2 ; . . .; kp2 gf1; 2; . . .; lp g; with lp1 þ lp2 ¼ lp ; lp2 ¼

Pb

i¼1 lp2i

(iv) If Di ¼ B2i  4Ci [ 0 for i ¼ 1; 2; . . .; m

ð4:22Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has (2m) fixed-points as ðiÞ

1 2 1  ðBi 2

pffiffiffiffiffi 9 Di Þ; = for i ¼ 1; 2; . . .; m: pffiffiffiffiffi  Di Þ ;

xk ¼ b1 ¼  ðBi þ ðiÞ

xk ¼ b2 ¼

ð4:23Þ

(iv1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; m ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2 . . .; a2m g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; as \as þ 1 :

ð4:24Þ

The corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þðxk  a3 Þ. . .ðxk  a2m1 Þðxk  a2m Þ: ð4:25Þ Such a discrate flow is formed with all the simple fixed-points. (a) If a0 \0, the simple fixed-point discrete flow is called a ðmSO : mSI-oSO : . . . : mSO : mSI-oSOÞ discrete flow. (b) If a0 [ 0, the simple fixed-point discrete flow is called a ðmSI-oSO : mSO : . . . : mSI-oSO : mSOÞ discrete flow.

4.1 Global Stability and Bifurcations

263

(iv2) If ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2 . . .; a2m g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai1  a1 ¼ . . . ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð4:26Þ

P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2m i¼1 i i¼1 i

with

Xr

s¼1 ls

¼ 2m;

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls :

ð4:27Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ-discrete flow. The fixed-point of xk ¼ aip for lp -repeated fixed-points switching is called an lp th XX bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed-point switching at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1

a Pp1 l

i¼1 i

þ1

i¼1

6¼    6¼ a Pp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:28Þ

Definition 4.2 Consider a 1-dimensional, (2m)th-degree polynomial nonlinear forward discrete system as xk þ 1 ¼ xk þ f ðxk ; pÞ 2m1 ¼ xk þ A0 ðpÞx2m þ    þ A2m2 ðpÞx2k þ A2m1 xk þ A2m ðpÞ k þ A1 ðpÞxk

¼ a0 ðpÞ *ni¼1 ½x2k þ Bi ðpÞxk þ Ci ðpÞqi

ð4:29Þ where A0 ðpÞ 6¼ 0, and p ¼ ðp1 ; p2 ; . . .; pm1 ÞT ; m ¼

Xn

i¼1 qi :

ð4:30Þ

(i) If Di ¼ B2i  4Ci \0 for i ¼ 1; 2; . . .; n;

ð4:31Þ

4 (2m)th-Degree Polynomial Discrete Systems

264

the 1-dimensional nonlinear discrete system with a (2m)th-degree polynomial does not have any fixed-point, and the corresponding standard form is xk þ 1 ¼ xk þ a0 *ni¼1 ½ðxk þ Bi Þ2 þ ðDi Þqi : 1 2

1 4

ð4:32Þ

The discrete flow of such a system without fixed-points is called a non-fixed-point discrete flow. (a) If a0 [ 0, the non-fixed-point discrete flow is called a positive discrete flow. (b) If a0 \0, the non-fixed-point discrete flow is called a negative discrete flow. (ii) If Di ¼ B2i  4Ci [ 0; i 2 fi1 ; i2 ; . . .; il gf1; 2; . . .; ng; Dj ¼ B2j  4Cj \0; j 2 fil þ 1 ; il þ 2 ; . . .; in gf1; 2; . . .; ng;

ð4:33Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has (2l) fixed-points as ðiÞ

1 2 1  ðBi 2

pffiffiffiffiffi 9 Di Þ; = pffiffiffiffiffi  Di Þ ;

xk ¼ b1 ¼  ðBi þ ðiÞ

xk ¼ b2 ¼

ð4:34Þ

i 2 fi1 ; i2 ; . . .; il gf1; 2; . . .; ng: (ii1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; l; ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; as \as þ 1 ;

ð4:35Þ

then, the corresponding standard form is ls n 2 qik xk þ 1 ¼ xk þ a0 *2l s¼1 ðxk  as Þ *k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ

1 2

with ls 2 fqi1 ; qi2 ; . . .; qil g:

1 4

ð4:36Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : l2l th mXXÞ discrete flow.

4.1 Global Stability and Bifurcations

265

(a) For a0 [ 0 and p ¼ 1; 2; . . .; 2l, 8 > ð2r  1Þth mSO  ð2rp  1Þth order monotonic source; > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp  1Þth mLS  ð2rp Þth order monotonic lower-saddle, > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mUS  ð2rp Þth order monotonic upper-saddle, > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:37Þ where ð2rp  1Þth mSI  mSI-oSO for rp ¼ 1 and ap ¼

X2l

s¼p ls :

ð4:38Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; 2l, 8 > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink, > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth mUS  ð2rp Þth order monotonic upper-saddle; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp : ð4:39Þ (ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai 1  a1 ¼    ¼ al 1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2l i¼1 i i¼1 i

with

Pr

s¼1 ls

¼ 2l;

then, the corresponding standard form is

ð4:40Þ

4 (2m)th-Degree Polynomial Discrete Systems

266

xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *nj¼l þ 1 ½ðxk þ Bij Þ2 þ ðDij Þqik : ð4:41Þ 1 2

1 4

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ-discrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; r, 8 > ð2r  1Þth mSO  ð2rp  1Þth order monotonic source; > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp  1Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mUS  ð2rp Þth order monotonic upper-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:42Þ where ap ¼

Xr

s¼p ls :

ð4:43Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; r, 8 > ð2r  1Þth mSI  ð2rp  1Þth order monotonic sink; > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp  1Þth mUS  ð2rp Þth order monotonic upper-saddle; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp : ð4:44Þ (c) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed point switching at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is

4.1 Global Stability and Bifurcations

267

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1



aPp1 l

i¼1 i

þ1

i¼1



6¼    6¼ aPp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:45Þ

(iii) If Di ¼ B2i  4Ci ¼ 0; i 2 fi11 ; i12 ; . . .; i1s gfi1 ; i2 ; . . .; il gf1; 2; . . .; ng; Dk ¼ B2k  4Ck [ 0; k 2 fi21 ; i22 ; . . .; i2r gfi1 ; i2 ; . . .; il gf1; 2; . . .; ng; Dj ¼ B2j  4Cj \0; j 2 fil þ 1 ; il þ 2 ; . . .; in gf1; 2; . . .; ngwith i 6¼ j 6¼ k;

ð4:46Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has (2l) fixedpoints as 9 1 ðj Þ xk ¼ b1 1 ¼  Bj1 ; > = 2

xk ¼

ðj Þ b2 1

1 > ¼  Bj1 ;

xk

ðj Þ b1 2

¼

¼

ðj Þ

xk ¼ b2 2

2 1  ðBj2 2

for j1 2 fi11 ; i12 ; . . .; i1s g;

pffiffiffiffiffiffi 9 Dj2 Þ; > = pffiffiffiffiffiffi >for j2 2 fi21 ; i22 ; . . .; i2r g: 1 ¼  ðBj2  Dj2 Þ ; þ

ð4:47Þ

2

If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2 . . .; a2l g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai 1  a1 ¼    ¼ al 1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð4:48Þ

P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2l i¼1 i i¼1 i

with

Pr

s¼1 ls

¼ 2l;

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *nk¼l þ 1 ½ðxk þ Bik Þ2 þ ðDik Þqik : ð4:49Þ 1 2

1 4

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ-discrete flow.

4 (2m)th-Degree Polynomial Discrete Systems

268

(a) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points appearance or vanishing is called an lp th mXX appearing bifurcation of fixed-points at a point p ¼ p1 2 @X12 , and the appearing bifurcation condition is 1 2

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ¼  Bip i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ; . . .; il gÞ; þ

aPp1 l

i¼1 i

þ

þ1

6¼    6¼ aPp1 l

i¼1 i



þ lp

or aPp1 l

i¼1 i

ð4:50Þ 

þ1

6¼    6¼ aPp1 l

i¼1 i

þ lp

:

(b) The fixed-point of xk ¼ aip for ðlp [ 1Þ- repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ-fixed-points switching at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1

a Pp1 l

i¼1 i

þ1

i¼1

6¼    6¼ a Pp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:51Þ

(iv) If Di ¼ B2i  4Ci [ 0 for i ¼ 1; 2; . . .; n

ð4:52Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has 2n-fixedpoints as ðiÞ

1 2 1  ðBi 2

pffiffiffiffiffi 9 Di Þ; = for i ¼ 1; 2; . . .; n: pffiffiffiffiffi  Di Þ ;

xk ¼ b1 ¼  ðBi þ ðiÞ

xk ¼ b2 ¼

ð4:53Þ

(iv1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; n; ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2 . . .; a2n g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; as \as þ 1 ;

ð4:54Þ

then the corresponding standard form is ls xk þ 1 ¼ xk þ a0 *2n s¼1 ðxk  as Þ with ls 2 fqi1 ; qi2 ; . . .; qin g:

ð4:55Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : l2n th mXXÞ-discrete flow.

4.1 Global Stability and Bifurcations

269

(a) For a0 [ 0 and p ¼ 1; 2; . . .; 2n, 8 > ð2r  1Þth mSO  ð2rp  1Þth order monotonic source, > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink, > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth mLS  ð2rp Þth order lower-saddle, > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp Þth mUS  ð2rp Þth order upper-saddle, > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:56Þ where ð2rp  1Þth mSI  mSI-oSO for rp ¼ 1, and ap ¼

X2n

s¼p ls :

ð4:57Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; 2n, 8 > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink, > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source, > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp  1Þth mUS  ð2rp Þth order montonic upper-saddle, > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp  1Þth mLS  ð2rp Þth order monotonic lower-saddle, > > > : for ap ¼ 2Mp ; lp ¼ 2rp : ð4:58Þ (iv2) If ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2 . . .; a2n g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . P air  aPr1 l þ 1 ¼    ¼ a r1 l þ lr ¼ a2n ; i¼1 i i¼1 i

with

Pr

s¼1 ls

¼ 2n;

then, the corresponding standard form is

ð4:59Þ

4 (2m)th-Degree Polynomial Discrete Systems

270

xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls :

ð4:60Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞ-discrete flow. The fixed-point of xk ¼ aip for lp - repeated fixed-points switching is called a lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed-point at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1

a Pp1 l i¼1

i þ1

i¼1

6¼    6¼ a Pp1 l i¼1

i þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:61Þ

Definition 4.3 Consider a 1-dimensional, (2m)th-degree polynomial nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ 2m1 ¼ xk þ A0 ðpÞx2m þ    þ A2m2 ðpÞx2k þ A2m1 xk þ A2m ðpÞ k þ A1 ðpÞxk

¼ a0 ðpÞ *rs¼1 ðxk  cis ðpÞÞls *ni¼r þ 1 ½x2k þ Bi ðpÞxk þ Ci ðpÞqi ð4:62Þ

where A0 ðpÞ 6¼ 0, and Xr

Xn

¼ ðm  lÞ; p ¼ ðp1 ; p2 ; . . .; pm1 ÞT :

ð4:63Þ

Di ¼ B2i  4Ci \0 for i ¼ r þ 1; r þ 2; . . .; n; fa1 ; a2 ; . . .; ar g ¼ sortfc1 ; c2 ; . . .; cr g with ai \ai þ 1 ;

ð4:64Þ

s¼1 ls

¼ 2l;

i¼r þ 1 qi

(i) If

the 1-dimensional nonlinear discrete system with a (2m)th-degree polynomial have a fixed-point of xk ¼ ais ðpÞ ðs ¼ 1; 2; . . .; r Þ, and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðpÞ *rs¼1 ðxk  ais Þls *ni¼r þ 1 ½ðxk þ Bi Þ2 þ ðDi Þli : 1 2

1 4

ð4:65Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : lr th mXXÞdiscrete flow.

4.1 Global Stability and Bifurcations

271

(a) For a0 [ 0 and s ¼ 1; 2; . . .; r, 8 > ð2r  1Þth mSO  ð2rp  1Þth order monotonic source, > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink, > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth mLS  ð2rp Þth order monotonic lower-saddle, > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp Þth mUS  ð2rp Þth order monotonic upper-saddle, > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:66Þ where ap ¼

Xr

s¼p ls :

ð4:67Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; r, 8 > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink; > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ th th > > ð2rp Þ mUS  ð2rp Þ order monotonic upper-saddle; > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp : ð4:68Þ (ii) If Di ¼ B2i  4Ci [ 0; i ¼ j1 ; j2 ; . . .; js 2 fl þ 1; l þ 2; . . .; ng; Dj ¼ B2j  4Cj \0; j ¼ js þ 1 ; js þ 2 ; . . .; jn 2 fl þ 1; l þ 2; . . .; ng with s 2 f1; . . .; n  lg;

ð4:69Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has 2n2 -fixedpoints as

4 (2m)th-Degree Polynomial Discrete Systems

272 ðiÞ

1 2

xk ¼ b1 ¼  ðBi þ

pffiffiffiffiffi pffiffiffiffiffi 1 ðiÞ Di Þ; xk ¼ b2 ¼  ðBi  Di Þ

ð4:70Þ

2

i 2 fj1 ; j2 ; . . .; jn1 gfl þ 1; l þ 2; . . .; ng: If ðr þ 1Þ

ðr þ 1Þ

ðn Þ

ðn Þ

; b2 ; . . .; b1 1 ; b2 1 g; fa1 ; a2 . . .; a2n2 g ¼ sortfc1 ; c2 . . .; c2l ; b1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qr þ 1 sets

ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

qn1 sets

ð4:71Þ

ain1  aPn1 1 li þ 1 ¼    ¼ aPn1 1 li þ ln ¼ a2n2

with

Pn 1

i¼1

i¼1

1

s¼1 ls ¼ 2n2 ;

then, the corresponding standard form is 1 ðxk  ais Þls *ni¼n2 þ 1 ½ðxk þ Bi Þ2 þ ðDi Þqi : xk þ 1 ¼ xk þ a0 *ns¼1

1 2

1 4

ð4:72Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : ln1 th mXXÞdiscrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; r; r þ 1; . . .; n1 ,

lp th mXX ¼

8 > ð2rp  1Þth mSO  ð2rp  1Þth order monotonc source; > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSI  ð2rp  1Þth order monotonic sink; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; > ð2rp Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > > ð2rp  1Þth mUS  ð2rp Þth order monotonic upper-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp ; ð4:73Þ

where ð2rp  1Þth mSI  mSI-oSO for rp ¼ 1, and ap ¼

Xn1

s¼p ls :

ð4:74Þ

4.1 Global Stability and Bifurcations

273

(b) For a0 \0 and p ¼ 1; 2; . . .; r; r þ 1; . . .; n1 , 8 > ð2r  1Þth mSI  ð2rp  1Þth order monotonic sink; > > p > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > > > ð2rp  1Þth mSO  ð2rp  1Þth order monotonic source; > > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth mUS  ð2rp Þth order monotonic upper-saddle; > > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2rp Þth mLS  ð2rp Þth order monotonic lower-saddle; > > > : for ap ¼ 2Mp ; lp ¼ 2rp : ð4:75Þ   (c) The fixed-point of xk ¼ aip for lp [ 1 -repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed-point at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; a Pp1 l i¼1

i¼1

i¼1 i

þ1

6¼    6¼ a Pp1 l

i¼1 i

þ lp

: ð4:76Þ

(iii) If Di ¼ B2i  4Ci ¼ 0; for i 2 fi11 ; i12 ; . . .; i1s gfil þ 1 ; il þ 2 ; . . .; in2 gfl þ 1; l þ 2; . . .; ng; Dk ¼ B2k  4Ck [ 0; for k 2 fi21 ; i22 ; . . .; i2r gfil þ 1 ; il þ 2 ; . . .; in2 gfl þ 1; l þ 2; . . .; ng; Dj ¼ B2j  4Cj \0; for j 2 fin2 þ 1 ; in2 þ 2 ; . . .; in gfl þ 1; l þ 2; . . .; ng;

ð4:77Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has ð2n2 Þ-fixedpoints as 9 1 ðj Þ xk ¼ b1 1 ¼  Bi ; > = 2 for j1 2 fi11 ; i12 ; . . .; i1s g; 1 > ðj Þ xk ¼ b2 1 ¼  Bi ; 2 ð4:78Þ pffiffiffiffiffiffi 9 1 ðj2 Þ  xk ¼ b1 ¼  ðBj2 þ Dj2 Þ; > = 2 pffiffiffiffiffiffi > for j2 2 fi21 ; i22 ; . . .; i2r g: 1 ðj Þ xk ¼ b2 2 ¼  ðBj2  Dj2 Þ ; 2

4 (2m)th-Degree Polynomial Discrete Systems

274

If ðr þ 1Þ

ðr þ 1Þ

ðn Þ

ðn Þ

; b2 ; . . .; b1 1 ; b2 1 g; fa1 ; a2 . . .; a2n2 g ¼ sortfc1 ; c2 . . .; c2l ; b1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qr þ 1 sets

ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

qn1 sets

ð4:79Þ

ain1  aPn1 1 li þ 1 ¼    ¼ aPn1 1 li þ ln ¼ a2n2

with

Pn 1

i¼1

s¼1 ls

i¼1

1

¼ 2n2 ;

then, the corresponding standard form is 1 xk þ 1 ¼ xk þ a0 *ns¼1 ðxk  ais Þls *ni¼n2 þ 1 ½ðxk þ Bi Þ2 þ ðDi Þqi :

1 2

1 4

ð4:80Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : ln1 th mXXÞdiscrete flow. (a) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points appearance or vanishing is called an lp th mXX appearing bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the appearing bifurcation condition is 1 2

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ¼  Bip i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ; . . .; il gÞ þ aP p1 l

i¼1 i

þ1

þ 6¼    6¼ aP p1 l

i¼1 i

þ lp

or a Pp1 l

i¼1 i

þ1

ð4:81Þ

6¼    6¼ a Pp1 l

i¼1 i

þ lp

:

(b) The fixed-point of xk ¼ aip for ðlp [ 1Þ- repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed-point at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1



aPp1 l

i¼1 i

þ1

i¼1



6¼    6¼ aPp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:82Þ

(c) The fixed-point of xk ¼ aip for ðlp1 1Þ-repeated fixed-points appearance/ vanishing and ðlp2 2Þ repeated fixed-points switching of ðlp21 th mmXX : lp22 th mXX :    : lp2b th mXXÞ is called an lp th mXX switching bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the flower-bundle witching bifurcation condition is

4.1 Global Stability and Bifurcations

275

aip  aPp1 qi þ 1 ¼    ¼ aPp1 qi þ qp i¼1

i¼1

with Diq ¼ B2iq  4Ciq ¼ 0 ðiq 2 fi1 ; i2 ; . . .; il gÞ þ aP p1 l

i¼1 i

þ j1

þ 6¼    6¼ aP p1 l

i¼1 i

þ jp1

or a Pp1 1 l i¼1

i

þ j1

6¼    6¼ a Pp1 1 l i¼1

i

þ jp1

;

for fj1 ; j2 ; . . .; jp1 gf1; 2; . . .; lp g; a Pp1 l

i¼1 i

þ k1

6¼    6¼ a Pp1 l

i¼1 i

þ kp 2

for fk1 ; k2 ; . . .; kp2 gf1; 2; . . .; lp g;

ð4:83Þ

with lp1 þ lp2 ¼ lp :

(iv) If Di ¼ B2i  4Ci [ 0 for i ¼ l þ 1; l þ 2; . . .; n

ð4:84Þ

the 1-dimensional, (2m)th-degree polynomial discrete system has (2m) fixedpoints as ðiÞ

1 2 1  ðBi 2

pffiffiffiffiffi 9 Di Þ; = for i ¼ l þ 1; l þ 2; . . .; n: pffiffiffiffiffi  Di Þ ;

xk ¼ b1 ¼  ðBi þ ðiÞ

xk ¼ b2 ¼

ð4:85Þ

If ðr þ 1Þ

ðr þ 1Þ

ðnÞ

ðnÞ

fa1 ; a2 . . .; a2m g ¼ sortfc1 ; c2 . . .; c2l ; b1 ; b2 ; . . .; b1 ; b2 g; |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

qr þ 1 sets

qn sets

ð4:86Þ

P ain  aPn1 l þ 1 ¼    ¼ a n1 l þ lr ¼ a2m i¼1 i i¼1 i

with

Pn

s¼1 ls

¼ 2m;

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *ns¼1 ðxk  ais Þls :

ð4:87Þ

The fixed-point discrete flow is called an ðl1 th mXX : l2 th mXX :    : ln th mXXÞdiscrete flow. The fixed-point of xk ¼ aip for lp -repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX : lp2 th mXX :    : lpb th mXXÞ fixed-point switching at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is

4 (2m)th-Degree Polynomial Discrete Systems

276

aip  aPp1 li þ 1 ¼    ¼ aPp1 li þ lp ; i¼1



aPp1 l

i¼1 i

4.2

þ1

i¼1



6¼    6¼ aPp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð4:88Þ

Simple Fixed-Point Bifurcations

From the global analysis, the bifurcations of simple fixed-points in the (2m)th-degree polynomial discrete systems are discussed, which include appearing/vanishing bifurcations, switching bifurcations, and switching and appearing bifurcations.

4.2.1

Appearing Bifurcations

Consider a (2m)th-degree polynomial discrete system in a form of xk þ 1 ¼ xk þ a0 Qðxk Þ *ni¼1 ðx2k þ Bi xk þ Ci Þ:

ð4:89Þ

Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are 1 1 pffiffiffiffiffi ðiÞ b1;2 ¼  Bi  Di ; Di ¼ B2i  4Ci 0ði ¼ 1; 2; . . .; nÞ 2 2 ð1Þ ð1Þ ð2Þ ð2Þ ðnÞ ðnÞ fa1 ; a2 ; . . .; a2l g sortfb1 ; b2 ; b1 ; b2 ; . . .; b1 ; b2 g; as  as þ 1  Bi 6¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ at bifurcation: Di ¼ 0ði ¼ 1; 2; . . .; nÞ

ð4:90Þ

The second-order singularity bifurcation is for the birth of a pair of simple fixed points with monotonic sink and monotonic source. There are two appearing bifurcations for i 2 f1; 2; . . .; ng 2

nd

ith quadratic factor

order mUS ! appearing bifurcation

2

nd



mSO; for xk ¼ a2i ; mSI-oSO; for xk ¼ a2i1 :

ith quadratic factor

order mLS ! appearing bifurcation



mSI-oSO; for xk ¼ a2i ; mSO; for xk ¼ a2i1 :

ð4:91Þ ð4:92Þ

4.2 Simple Fixed-Point Bifurcations

277

If Qðxk Þ ¼ 1 and n ¼ m, a set of paralleled different simple monotonic upper-saddle appearing bifurcations in the ð2mÞth degree polynomial nonlinear discrete system is called an m-monotonic-upper-saddle-node (m-mUSN) parallel appearing bifurcation. Such a bifurcation is also called an m-monotonic-uppersaddle-node (m-mUSN) teethcomb appearing bifurcation. At the appearing bifurcation point, Di ¼ 0ði ¼ 1; 2; . . .; mÞ, and the m-mUSN teethcomb appearing bifurcation structure is ( 8 mSO, for xk ¼ a2m ; th > m bifurcation > > mUS ! > > appearing > mSI-oSO, for xk ¼ a2m1 ; > > > > > > > ... > > > > ( > < mSO, for xk ¼ a2i ; ith bifurcation m-mUSN mUS ! appearing > mSI-oSO, for xk ¼ a2i1 ; > > > > > .. > > > . > > > ( > > > mSO, for xk ¼ a2 ; > 1st bifurcation > > mUS ! : appearing mSI-oSO, for xk ¼ a1 :

ð4:93Þ

Similarly, a set of paralleled different simple monotonic lower-saddle appearing bifurcations is called an m-monotonic-lower-saddle-node (m-mLSN) parallel appearing bifurcation for the (2m)th degree polynomial nonlinear discrete system. The monotonic lower-saddle-node bifurcation is called the m-monotonic-lowersaddle-node (m-mLSN) teethcomb appearing bifurcation. At the bifurcation point, Di ¼ 0ði ¼ 1; 2; . . .; mÞ, and the m-mLSN appearing bifurcation structure is ( 8 mSI-oSO; for xk ¼ a2m ; > mth bifurcation > > mLS ! > > appearing > mSO; for xk ¼ a2m1 ; > > > > > . > > > .. > > > ( > < mSI-oSO; for xk ¼ a2i ; ith bifurcation m-mLSN mLS ! appearing > mSO; for xk ¼ a2i1 ; > > > > > .. > > > . > > > ( > > > mSI-oSO; for xk ¼ a2 ; > 1st bifurcation > > mLS ! : appearing mSO; for xk ¼ a1 :

ð4:94Þ

Consider an appearing bifurcation for a cluster of fixed-points with monotonic sink to oscillatory source and monotonic source with the following conditions.

4 (2m)th-Degree Polynomial Discrete Systems

278

Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ Di ¼ 0 ði ¼ 1; 2; . . .; nÞ

) at bifurcation:

ð4:95Þ

Thus, the (2l)th-order appearing bifurcation is for a cluster of fixed-points with simple monotonic sinks to oscillatory sources and monotonic sources. Two (2l)th order appearing bifurcations for l 2 f1; 2; . . .; sg are 8 mSO; for xk ¼ a2sl ; > > > > >  > > > mSI-oSO; for xk ¼ a2sl 1 ; > < cluster of l-quadratics ð2lÞth order mUS ! ... appearing bifurcation > > > > > > mSO; for xk ¼ a2s1 ; > > > : mSI-oSO; for xk ¼ a2s1 1 :

ð4:96Þ

8 mSI-oSO; for xk ¼ a2sl ; > > > > >  > > > mSO; for xk ¼ a2sl 1 ; > < cluster of l-quadratics ð2lÞth order mLS ! ... appearing bifurcation > > > > > > mSI-oSO; for xk ¼ a2s1 ; > > > : mSO; for xk ¼ a2s1 1 :

ð4:97Þ

A set of paralleled, different, higher-order upper-saddle-node bifurcations with multiplicity is a ðð2l1 Þth mUS : ð2l2 Þth mUS :    : ð2ls Þth mUSÞ parallel appearing bifurcation in the (2m)th degree polynomial discrete system. ð2li Þth mUS for ði ¼ 1; 2; . . .; sÞ with monotonic sources and monotonic sinks to oscillatory source is the ð2li Þth -order monotonic upper-saddle with li -pairs of simple monotonic source and monotonic sink to oscillatory source fixed-points. With different orders of li pairs of fixed points with simple monotonic sources and monotonic sinks to oscillatory sources, the ð2li Þth mUSN bifurcation possesses different spraying appearing clusters of fixed points with monotonic sinks to oscillatory sources and monotonic sources. Psi¼1 li ¼ n  m where s; li 2 f0; 1; 2; . . .; mg . If li ¼ 1 for i ¼ 1; 2; . . .; m with n ¼ m, the monotonic upper-saddle-node parallel bifurcation or the monotonic upper-saddle-node teethcomb appearing bifurcation is recovered. Introduce ðð2l1 Þth mUS:(2l2 Þth mUS:    : ð2ls Þth mUSÞ  ð2l1 : 2l2 :    : 2ls Þth mUS: ð4:98Þ At the sprinkler-spraying appearing bifurcation, Di ¼ 0ði ¼ 1; 2; . . .; sÞ and Bi ¼ Bj ð i; j 2 f1; 2; . . .; sg; i 6¼ jÞ: The sprinkler-spraying mUSN appearing bifurcation is

4.2 Simple Fixed-Point Bifurcations

ð2l1 : 2l2 :    : 2ls Þth mUS ¼

279

8 > ð2ls Þth order mUS, > > > > > < .. .

> > ð2l2 Þth order mUS, > > > > : ð2l Þth order mUS: 1

ð4:99Þ

Thus, the ð2l1 : 2l2 :    : 2ls Þth mUS appearing (or vanishing) bifurcation is called a ð2l1 : 2l2 :    : 2ls Þth mUSN sprinkler-spraying appearing (or vanishing) bifurcation. Similarly, a set of paralleled different lower-saddle appearing bifurcations with multiplicity is the ðð2l1 Þth mLS:(2l2 Þth mLS:  :ð2ls Þth mLSÞ appearing bifurcation in the (2m)th degree polynomial system. Thus, the ð2l1 :2l2 :  :2ls Þth mLS appearing (or vanishing) bifurcation is also called a ð2l1 :2l2 :  :2ls Þth mLS sprinkler-spraying appearing (or vanishing) bifurcation. Again, at the mLS sprinkler-spraying bifurcation, Di ¼ 0 ði ¼ 1; 2; . . .; nÞ and Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ: Thus, the sprinkler-spraying mLSN appearing bifurcation is

ð2l1 : 2l2 :    : 2ls Þth mLS ¼

8 > ð2ls Þth order mLS; > > > > > < .. .

> > ð2l2 Þth order mLS; > > > > : ð2l Þth order mLS: 1

ð4:100Þ

Two m-mUSN and m-mLSN teethcomb appearing bifurcations are presented in Fig. 4.1(i) and (ii) for a0 [ 0 and a0 \0, respectively. The set of paralleled ð4th mUS:    : ð2rÞth mUS:    : 4th mUS:6th mUSÞ appearing bifurcations for simple monotonic sinks to oscillatory sources and monotonic sources is presented in Fig. 4.1(iii) for a0 [ 0, where l1 ¼ 2; . . .; li ¼ r; . . .; ls1 ¼ 2; ls ¼ 3 with Psi¼1 li ¼ m: The ð4 :    : 2r :    : 4 : 6Þth -mUSN appearing bifurcation is a mUSN sprinkler-spraying appearing bifurcation. However, for a0 \0, the ð4th mLS:    : ð2rÞth mLS:    : 4th mLS:6th mLS) appearing bifurcations for simple sources and sinks is presented in Fig. 4.1(iv). The ð4 :    : 2r :    : 4 : 6Þth -mLSN appearing bifurcation is a mLSN sprinkler-spraying appearing bifurcation. For a cluster of m-quadratics,Bi ¼ Bj ði; j 2 f1; 2; . . .; mg; i 6¼ jÞ and Di ¼ 0 ði ¼ 1; 2; . . .; mÞ: The (2m)th order monotonic upper-saddle-node appearing bifurcation for m-pairs of fixed points with monotonic sink to oscillatory source and monotonic sources is

4 (2m)th-Degree Polynomial Discrete Systems

280 a0 > 0

b1(i1 )

mSO

a0 < 0

mSI-oSO

P-2

P-2 mUSN

mSI-oSO

LSN

( i1 ) 2

b

mSO

( i2 ) 1

b

mSO

b1(i1 )

mSI-oSO

b2(i1 ) b1(i2 ) P-2

P-2

mUSN

mLSN ( i2 ) 2

b

mSI-oSO •

mSO •

•

•

•

•

mSO

mSI-oSO P-2

P-2

mUSN

mLSN

mSI-oSO • •

b1(im )

mSO mUSN

mSI-oSO

mSI-oSO

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

xk∗

mSO

(i)

(iii) mSO

a0 > 0

b1(i1 )

mSI-oSO

b2(i1 )

6th mLS

( i2 ) 2

b

mSO

P-2

mSI-oSO

• •

b2(i2 )

mSI-oSO

mSO

4th mUS

b1(i1 ) b1(i2 ) P-2

b2(i1 )

6th mUS

mSI-oSO

a0 < 0

b1(i2 )

•

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

b1(im ) P-2

mLSN

P-2

xk∗

mSO

• • •

•

|| p ||

b2(i2 )

4th LS

P-2

mSO

• •

P-2

•

mSO

mSI-oSO P-2

P-2

(2r)th mUS

• • •

4th

mSI-oSO

P-2

mSO

b1(im )

P-2

• • •

mSO

mSI-oSO

mUS

b1(im ) P-2

4th mLS

xk∗

xk∗ mSI-oSO

|| p ||

(2r)th mLS

Δ iq < 0 Δ iq = 0

(ii)

Δ iq > 0

mSO

b2(im ) || p ||

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

(iv)

Fig. 4.1 (i) m-mUSN parallel bifurcation ða0 [ 0Þ, (ii) m-mLSN parallel bifurcation ða0 \0Þ, (iii) ðð2l1 Þth mUS:(2l2 Þth mUS:    : ð2ls Þth mUSÞ parallel bifurcation ða0 [ 0Þ: (iv) ðð2l1 Þth mLS: ð2l2 Þth mLS:    : ð2ls Þth mLSÞ parallel bifurcation ða0 \0Þ in a (2m)th-degree polynomial discrete system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4.2 Simple Fixed-Point Bifurcations

281

8 mSO; for xk ¼ a2m ; > > > > > mSI-oSO; for xk ¼ a2m1 ; > > < cluster of m quadratics ð2mÞth order mUS ! ... appearing bifurcation > > > > > mSO; for xk ¼ a2 ; > > : mSI-oSO; for xk ¼ a1 : 

ð4:101Þ

The (2m)th order lower-saddle-node appearing bifurcation for m-pairs of fixedpoints with monotonic sink to oscillatory source, and monotonic source is 8 mSI-oSO; for xk ¼ a2m ; > > > > > > mSO; for xk ¼ a2m1 ; > > > < cluster of m quadratics ð2mÞth order mLS ! ... appearing bifurcation > > > >  > > > mSI-oSO; for xk ¼ a2 ; > > : mSO; for xk ¼ a1 : 

ð4:102Þ

The (2m)th order monotonic upper-saddle-node appearing bifurcation with mpairs of fixed-points with monotonic sources and monotonic sinks to oscillatory sources is a sprinkler-spraying cluster of the m-pairs of fixed-points with monotonic sources and monotonic sinks to oscillatory sources. The (2m)th order monotonic lower-saddle-node appearing bifurcation with m-pairs of fixed-points is also a sprinkler-spraying cluster of the m-pairs of monotonic sources and monotonic sinks to oscillatory sources. Thus, the (2m)th order mUSN appearing bifurcation ða0 [ 0Þ and (2m)th order mLSN bifurcation ða0 \0Þ are presented in Fig. 4.2(i) and (ii), respectively. The (2m)th order monotonic-upper-saddle-node appearing bifurcation is named a (2m)th order mUSN sprinkler-spaying appearing bifurcation, and the (2m)th order monotonic-lower-saddle-node appearing bifurcation is named a (2m)th order mLSN sprinkler-spraying appearing bifurcation. A series of the monotonic saddle-node bifurcations is aligned up with varying with parameters, which is formed a special pattern. For m-quadratics in the (2m)th order polynomial discrete system, the following conditions should be satisfied. Bi Bj i; j 2 f1; 2; . . .; ng; i 6¼ j; Di [ Di þ 1 ði ¼ 1; 2; . . .; n; n  mÞ; Di ¼ 0 with jjpi jj\jjpi þ 1 jj:

ð4:103Þ

Thus, a series of m-(mUSN-mLSN-mUSN-. . .Þ appearing bifurcations ða0 [ 0Þ and a series of m-(mLSN-mUSN-mLSN-. . .Þ appearing bifurcations ða0 \0Þ are presented in Fig. 4.3(i) and (ii), respectively. The bifurcation scenario is formed by the swapping pattern of mUSN and mLSN appearing bifurcations. Such a bifurcation scenario is like the fish-scale. Thus, such a bifurcation swapping pattern of the mUSN and mLSN is called a fish-scale appearing bifurcation in the (2m)th

4 (2m)th-Degree Polynomial Discrete Systems

282 a0 > 0

a0 < 0

a2m

mSO

mSI-oSO

P-2

P-2 (2m)th mLS

(2m)th mUS P-2

P-2

a2

x∗

Δ iq > 0

Δ iq < 0 Δ iq = 0

a2

x∗

a1

mSI-oSO

|| p ||

a2m

|| p ||

mSO

Δ iq > 0

Δ iq < 0 Δ iq = 0

(i)

a1

(ii)

Fig. 4.2 (i) (2m)th order mUSN bifurcation ða0 [ 0Þ, (ii) (2m)th order mLSN bifurcation ða0 \0Þ in the (2m)th polynomial system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

degree polynomial nonlinear discrete system. There are two swapping bifurcations: (i) the USN-LSN fish-scale appearing bifurcation and (ii) the mLSN-mUSN fish-scale, appearing bifurcation.

4.2.2

Switching Bifurcations

Consider the roots of x2k þ Bi xk þ Ci ¼ 0 as ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ ðiÞ ðiÞ ðiÞ xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ;  Bi 6¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ

ð4:104Þ

The 2nd order singularity bifurcation is for the switching of a pair of fixed point with simple monotonic sink to oscillatory source and monotonic source. There are two switching bifurcations for i 2 f1; 2; . . .; ng ( ith quadratic factor

2 order mUS ! nd

switching bifurcation

ðiÞ

ðiÞ

mSO, for a2i ¼ b2 ! b1 ; ðiÞ

8 ðiÞ < mSI-oSO, for a2i ¼ bðiÞ 2 ! b1 ; 2 order mLS ! ðiÞ ðiÞ switching bifurcation : mSO, for a2i1 ¼ b1 ! b2 : nd

ith quadratic factor

ðiÞ

mSI-oSO, for a2i1 ¼ b1 ! b2 :

ð4:105Þ

ð4:106Þ

4.2 Simple Fixed-Point Bifurcations

283

a0 > 0

P-2

b1( r )

mSO

mUS

mLS

x∗

mUS

• • •

mSI-oSO

|| p ||

mLS

• • •

P-2

Δr < 0

P-2

mUS

P-2

b2( r )

P-2

Δr > 0

Δr = 0

(i) a0 < 0

P-2

P-2

mSI-oSO

mLS

xk∗

|| p ||

mUS

mLS

• • •

mUS

mSO

• • •

mLS

b2( r )

P-2

Δr < 0

Δr = 0

P-2

b1( r )

P-2

Δr > 0

(ii) Fig. 4.3 (i) m-(mUS-mLS-mUS-. . .Þ series bifurcation ða0 [ 0Þ, (ii) mð-(mUS-mLS-mUS-. . .Þ series bifurcation ða0 [ 0Þ in the (2m)th-degree polynomial discrete system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

A set of m-paralleled-pairs of different simple-monotonic-upper-saddle-node switching bifurcations in the (2m)thdegree polynomial nonlinear discrete system is called an m-monotonic-upper-saddle-node (m-mUSN) parallel switching bifurcation. Such a bifurcation is also called an m-monotonic-upper-saddle-node (mðiÞ mUSN) antenna switching bifurcation. For non-switching point, Di [ 0 at b1 6¼ ðiÞ

ðiÞ

ðiÞ

b2 ði ¼ 1; 2; . . .; nÞ: At the bifurcation point, Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ: The m-mUSN parallel switching bifurcation is

284

4 (2m)th-Degree Polynomial Discrete Systems

8 8 < mSO # mSI-oSO; > th > m bifurcation > > mUS ! > > switching bifurcation : > > mSI-oSO " mSO; > > > > > . > .. > > > > 8 > > > < < mSO # mSI-oSO; th i bifurcation m-mUSN mUS ! switching bifurcation : > > mSI-oSO " mSO; > > > > > .. > > > . > > > 8 > > > < mSO # mSI-oSO; > st > 1 bifurcation > > mUS ! > : switching bifurcation : mSI-oSO " mSO;

ðmÞ

¼ a2m # a2m1 ;

ðmÞ

¼ a2m1 " a2m ;

for b2 for b1

ðiÞ

for b2 ¼ a2i # a2i1 ; ðiÞ

for b1 ¼ a2i1 " a2i ;

ð1Þ

for b2 ¼ a2 # a1 ; ð1Þ

for b1 ¼ a1 " a2 : ð4:107Þ

Similarly, a set of paralleled different simple monotonic-lower-saddle bifurcations is called an m-monotonic-lower-saddle-node (m-mLSN) parallel switching bifurcation for the (2m)th degree polynomial nonlinear system. The monotoniclower-saddle-node switching bifurcation is also called an m-monotonic-lowersaddle-node (m-mLSN) antenna switching bifurcation. For non-switching point, ðiÞ ðiÞ ðiÞ ðiÞ Di [ 0 at b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ: At the bifurcation point, Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ: The m-mLSN antenna switching bifurcation is 8 8 < mSI-oSO # mSO, for bðmÞ > th > 2 ¼ a2m # a2m1 ; m bifurcation > > mLS ! > > ðmÞ switching bifurcation : > > mSO " mSI-oSO, for b1 ¼ a2m1 " a2m ; > > > > > .. > > > . > > 8 > > > < < mSI-oSO # mSO, for bðiÞ th 2 ¼ a2i # a2i1 ; i bifurcation m-mLSN mLS ! ðiÞ switching bifurcation : > > mSO " mSI-oSO, for b1 ¼ a2i1 " a2i ; > > > > > . > > > .. > > > 8 > > > < mSI-oSO # mSO, for bð1Þ > st 2 ¼ a2 # a1 ; > 1 bifurcation > > mLS ! > : ð1Þ switching bifurcation : mSO " mSI-oSO, for b1 ¼ a1 " a2 : ð4:108Þ Consider a switching bifurcation for a bundle of fixed-points with monotonic sink to oscillatory source and monotonic source with the following conditions,

4.2 Simple Fixed-Point Bifurcations ðiÞ

285

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ

ðiÞ

ðiÞ

ðiÞ

xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ; ) Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ

ð4:109Þ

Thus, the (2l)th order switching bifurcation can be for a bundle of simple monotonic sinks to oscillatory sources, and monotonic sources. Two (2l)th order monotonic upper- and lower-saddle switching bifurcations for l 2 f1; 2; . . .; sg are 8 mSO; for a2sl > > > > > mSI-oSO, for > > < a bundle of ð2lÞ fixed points ð2lÞth order mUS ! ... switching bifurcation > > > > > mSO, for a2s1 > > : mSI-oSO, for

! b2sl ; a2sl 1 ! b2sl 1 ; ð4:110Þ ! b2s1 ; a2s1 1 ! b2s1 1 :

8 mSI-oSO, for a2sl ! b2sl ; > > > > > > > > mSO, for a2sl 1 ! b2sl 1 ; > < a bundle of ð2lÞ fixed points ð2lÞth order mLS ! ... switching bifurcation > > > > > > mSI-oSO, for a2s1 ! b2s1 ; > > > : mSO, for a2s1 1 ! b2s1 1 :

ð4:111Þ

where Dij ¼ ðai  aj Þ2 ¼ ðbi  bj Þ2 ¼ 0 with Bi ¼ Bj ði; j ¼ 2s1  1; 2s1 ; . . .; 2sl  1; 2sl Þ and fa2s1 1 ; a2s1 ; . . .; a2sl 1 ; a2sl g fb2s1 1 ; b2s1 ; . . .; b2sl 1 ; b2sl g



ð1Þ

before bifurcation



after bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .b1 ; b2 g; ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .b1 ; b2 g:

ð4:112Þ

The ð2l  1Þth order switching bifurcation can be for a bundle of simple fixed-points with monotonic-sinks to oscillatory-sources and monotonic-sources. Two ð2l  1Þth order monotonic sink and monotonic source switching bifurcations for l 2 f1; 2; . . .; sg are 8 mSO, for a2sl 1 ! b2sl 1 ; > > > > > < .. abundle of ð2l1Þ-fixed points ð2l  1Þth order mSO ! . switching bifurcation > > mSI-oSO, for a2s1 ! b2s1 ; > > > : mSO, for a2s1 1 ! b2s1 1 : ð4:113Þ

4 (2m)th-Degree Polynomial Discrete Systems

286

8 mSI-oSO, for a2sl 1 ! b2sl 1 ; > > > > > < .. a bundle of ð2l1Þ-fixed points ð2l  1Þth order mSI ! . switching bifurcation > > mSO, for a2s1 ! b2s1 ; > > > : mSI-oSO, for a2s1 1 ! b2s1 1 : ð4:114Þ where Dij ¼ ðai  aj Þ2 ¼ ðbi  bj Þ2 ¼ 0 with Bi ¼ Bj ði; j ¼ 2s1  1; 2s1 ; . . .; 2sl  1Þ and fa2s1 1 ; a2s1 ; . . .; a2sl 1 g fb2s1 1 ; b2s1 ; . . .; b2sl 1 g



ð1Þ

before bifurcation



after bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .b1 ; b2 g; ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .b1 ; b2 g:

ð4:115Þ

A set of paralleled, different, higher-order upper-saddle-node switching bifurcations with multiplicity is the ðða1 Þth mXX:(a2 Þth mXX:    : ðas Þth mXXÞ parallel switching bifurcation in the (2m)th degree polynomial discrete system. At the straw-bundle switching bifurcation, Di ¼ 0ði ¼ 1; 2; . . .; nÞ and Bi ¼ Bj ð i; j 2 f1; 2; . . .; ng; i 6¼ jÞ: Thus, the parallel straw-bundle switching bifurcation is ðða1 Þth mXX:ða2 Þth mXX:    : ðas Þth mXXÞ-switching 8 ðas Þth order mXX switching, > > > > > > > < ... ¼ > > > ða2 Þth order mXX switching, > > > > : ða1 Þth order mXX switching;

ð4:116Þ

where ai 2 f2li ; 2li  1g withPsi¼1 ai ¼ 2m; and XX 2 fUS; LS; SO; SIg:

ð4:117Þ

The ð2li Þth mUS for ði ¼ 1; 2; . . .; sÞ with monotonic-sinks to oscillatory sources, and monotonic-sources is the ð2li Þth order monotonic-upper-saddle for a switching of li -pairs of simple monotonic-sinks to oscillatory-sources, and monotonic-sources. With different orders of li -pairs of simple monotonic-sinks to oscillatory-sources, and monotonic-sources, the ð2li Þth mUSN switching bifurcation possesses different straw-bundle switching for a bundle of stable and unstable fixed-points. The ð2l1 :

4.2 Simple Fixed-Point Bifurcations

287

2l2 :    : 2ls Þth mUSN bifurcation is called the ð2l1 : 2l2 :    : 2ls Þth mUSN strawbundle switching bifurcation. 8 > ð2ls Þth order mUSN switching, > > > > > < .. . th ð2l1 : 2l2 :    : 2ls Þ mUSN switching ¼ > > ð2l2 Þth order mUSN switching, > > > > : ð2l Þth order mUSN switching: 1

ð4:118Þ If li ¼ 1 for i ¼ 1; 2; . . .; m with n ¼ m, the simple upper-saddle-node parallel switching bifurcation or the upper-saddle-node antenna switching bifurcation is recovered. Similarly, a set of paralleled different monotonic lower-saddle switching bifurcations with multiplicity is a ðð2l1 Þth mLS:(2l2 Þth mLS:    : ð2ls Þth mLSÞ parallel switching bifurcation in the (2m)th degree polynomial discrete system. Thus, the ð2l1 : 2l2 :    : 2ls Þth mLSN switching bifurcation is also called a ð2l1 : 2l2 :    : 2ls Þth mLSN straw-bundle switching bifurcation. Again, at the mLSN straw-bundle switching bifurcation, Di ¼ 0ði ¼ 1; 2; . . .; nÞ and Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ: Thus, the mLSN straw-bundle switching bifurcation is

ð2l1 : 2l2 :    : 2ls Þth mLSN switching ¼

8 th > > > ð2ls Þ order mLSN switching, > > > < .. .

> > ð2l2 Þth order mLSN switching, > > > > : ð2l1 Þth order mLSN switching: ð4:119Þ

The set of m-monotonic-upper-saddle-node (m-mUSN) parallel switching bifurcation is equivalent to the set of ð2 : 2 :    : 2Þnd -mUSN bifurcations. The set of m-lowersaddle-node (m-mLSN) parallel switching bifurcation is equivalent to the set of ð2 : 2 :    : 2Þnd -mLSN bifurcations. Such two sets of parallel switching bifurcations are presented in Fig. 4.4(i) and (ii) for a0 [ 0 and a0 \0, respectively. A set of paralleled ð3rd mSO:2nd mLS:    : 4th mLS:    : 3rd mSIÞ switching bifurcations for mSI-oSO and mSO fixed-points is presented in Fig. 4.4(iii) for a0 [ 0: However, for a0 \0, the set of ð3rd mSI:2nd mUS:    : 4th mUS:    : 3rd mSIÞ switching bifurcations for monotonic-sources and monotonic-sink-to-oscillatorysources is presented in Fig. 4.4(iv).

4 (2m)th-Degree Polynomial Discrete Systems

288 a0 > 0

a2m

a0 < 0

a2 m−1

P-2

P-2

mUS mSI-oSO

P-2

P-2

mLS mSO

a2 m−2

mSO

a2m

mSI-oSO

mSO

a2 m−1 a2 m−2

mSI-oSO

mUS

P-2

mLS mSI-oSO

a2 m−3

mSO

P-2

•

P-2

P-2

a2 m−3

•

• •

• •

mSO

mSI-oSO P-2

mLS

mUS

P-2 mSO

mSI-oSO P-2

•

•

•

P-2

•

•

•

a2

mSO

a2

mSI-oSO

mUS

P-2

mLS mSI-oSO

xk∗

P-2

a1

a1

mSO

xk∗

P-2 P-2

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(iii)

(i) a2m

a0 > 0

a0 < 0

mSO 3rd mSO

mSO

a2 m−2

mSI-oSO

a2 m−3

mSO

P-2

a2 m−1

a2 m−2

mSI-oSO P-2

P-2

mSO

a2 m−3

mUS

P-2

mSI-oSO

• • •

P-2

3rd mSI

P-2

mLS

a2m

mSI-oSO

P-2

a2 m−1

P-2

P-2

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

• •

P-2

mSI-oSO

P-2

mSO

•

P-2 4th mLS

4th

P-2

P-2

mUS

P-2

mSO

mSI-oSO

• • •

P-2 3rd

•

P-2

mSI-oSO

mSO

a2

mSI

xk∗

3rd mSO

a1

mSI-oSO P-2

P-2

• •

P-2

a2

P-2

mSO

x∗

a1

P-2

|| p ||

Δ iq > 0 Δ iq = 0

(ii)

Δ iq > 0

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.4 Stability and bifurcations of fixed-points in a 1-dimensional, (2m)th-degree polynomial discrete system: (i) m-mUSN parallel switching bifurcation ða0 [ 0Þ, (ii) m-mLSN parallel switching bifurcation ða0 \0Þ, (iii) ð3rd mSO:2nd mLS:    : 3rd mSIÞ parallel switching bifurcation ða0 [ 0Þ: (iv) ð3rd mSI: 2nd mUS:    : 3rd mSOÞ parallel switching bifurcation. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4.2 Simple Fixed-Point Bifurcations

4.2.3

289

Switching-Appearing Bifurcations

Consider a (2m)th degree 1-dimensional polynomial discrete system in a form of n2 2 1 xk þ 1 ¼ xk þ a0 Qðxk Þ *2n i¼1 ðxk  ci Þ *j¼1 ðxk þ Bj xk þ Cj Þ:

ð4:120Þ

Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bj xk þ Cj ¼ 0 are ðjÞ

1 2

b1;2 ¼  Bj 

1pffiffiffiffiffi Dj ; Dj 2

¼ B2j  4Cj 0ðj ¼ 1; 2; . . .; n2 Þ;

ð4:121Þ

either     fa 1 ; a2 ; . . .; a2n1 g ¼ sortfc1 ; c2 . . .; c2n1 g; as  as þ 1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

þ g ¼ sortfc1 ; . . .; c2n1 ; b1 ; b2 ; . . .; b1 2 ; b2 2 g; fa1þ ; a2þ ; . . .; a2n 3

asþ

 asþþ 1 ;

ð4:122Þ

n3 ¼ n1 þ n2 after bifurcation;

or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 ; a2 ; . . .; a2n3 g ¼ sortfc1 ; c2 . . .; c2n1 ; b1 ; b2 ; . . .; b1 ; b2 g;

 a s  as þ 1 ; n3 ¼ n1 þ n2 before bifurcation;

þ fa1þ ; a2þ ; . . .; a2n g 1

¼ sortfc1 ; . . .; c2n1 g;

asþ

ð4:123Þ  asþþ 1

after bifurcation;

and 9 Bj1 ¼ Bj2 ¼    ¼ Bjs ðjk1 2 f1; 2; . . .; ng; jk1 6¼ jk2 Þ > > = ðk1 ; k2 2 f1; 2; . . .; sg; k1 6¼ k2 Þ at bifurcation: Dj ¼ 0 ðj 2 U f1; 2; . . .; n2 g > > ; 1 ci 6¼ 2Bj ði ¼ 1; 2; . . .; 2n1 ; j ¼ 1; 2; . . .; n2 Þ

ð4:124Þ

th    th Consider a just before bifurcation of ðða 1 Þ mXX1 : ða2 Þ mXX2 : . . . : th     ða s1 Þ mXXs1 Þ for simple sources and sinks. For ai ¼ 2li  1; mXXi 2    fmSO, mSIg and for ai ¼ 2li ; mXXi 2 fmUS, mLSgði ¼ 1; 2; . . .; s1 Þ. The detailed structures are as follows.

290

4 (2m)th-Degree Polynomial Discrete Systems

9 9 mSI-oSO > mSO > > > > > > > > > > > mSO mSI-oSO > > > > > > = = th th .. .  . ! ð2l ! ð2l  1Þ mSI; and i i  1Þ mSO; . . > > > > > > > > > > > > mSO mSI-oSO > > > > > > ; ; mSI-oSO mSO ð4:125Þ 9 9 mSO mSI-oSO > > > > > > > > > > > mSI-oSO > mSO > > > > > > = = th .. .  th . ! ð2l ! ð2l Þ mUS; and i i Þ mLS: . . > > > > > > > > > > > mSO mSI-oSO > > > > > > > ; ; mSI-oSO mSO th     th  th The bifurcation set of ðða 1 Þ mXX1 ; : ða2 Þ mXX2 : . . . : ðas1 Þ mXXs1 Þ at the same parameter point is called a left-parallel-bundle switching bifurcation Consider a just after bifurcation of ðða1þ Þth mXX1þ :ða2þ Þth mXX2þ :    : ðasþ2 Þth mXXsþ2 Þ for simple fixed-points with monotonic sources and monotonic sinks to oscillatory sources. For aiþ ¼ 2liþ  1; mXXiþ 2 fmSO, mSIg and for aiþ ¼ 2liþ ; mXX i 2 fmUS, mLSg. The four detailed structures are as follows.

8 8 mSI-oSO mSO > > > > > > > > > > > > mSO mSI-oSO > > > > > > < < ; and ð2liþ  1Þth mSO ! ... ; ð2liþ  1Þth mSI ! ... > > > > > > > > > > > > mSO mSI-oSO > > > > > > : : mSI-oSO mSO 8 8 mSO mSI-oSO > > > > > > > > > > > > mSI-oSO mSO > > > > > > < < ð2liþ Þth mUS ! ... ; and ð2liþ Þth mLS ! ... : > > > > > > > > > > > > mSO mSI-oSO > > > > > > : : mSI-oSO mSO ð4:126Þ The bifurcation set of ðða1þ Þth mXX1þ : ða2þ Þth mXX2þ : . . . : ðasþ2 Þth mXXsþ2 Þ at the same parameter point is called a right-parallel-bundle switching bifurcation

4.2 Simple Fixed-Point Bifurcations

291

(i) For the just before and after bifurcation structure, if there exists a relation of th  þ th þ th   þ ða i Þ mXXi ¼ ðaj Þ mXXj ¼ a mXX, for xk ¼ ai ¼ aj

ði 2 f1; 2; . . .; s1 g; j 2 f1; 2; . . .; s2 gÞ; XX 2 fUS, LS, SO, SIg

ð4:127Þ

then the bifurcation is a ath mXX switching bifurcation for simple fixed-points. (ii) Just for the just before bifurcation structure, if there exists a relation of th th    ð2l i Þ mXXi ¼ ð2lÞ mXX, for xk ¼ ai ¼ ai

ði 2 f1; 2; . . .; s1 g; XX 2 fUS, LSg

ð4:128Þ

then, the bifurcation is a ð2lÞth mXX left appearing (or right vanishing) bifurcation for simple fixed-points. (iii) Just for the just after bifurcation structure, if there exists a relation of ð2liþ Þth mXXiþ ¼ ð2lÞth mXX, for xk ¼ aiþ ¼ ai ði 2 f1; 2; . . .; s1 gÞ; XX 2 fUS, LSg

ð4:129Þ

then, the bifurcation is a ð2lÞth mXX right appearing (or left vanishing) bifurcation for simple fixed-points. (iv) For the just before and after bifurcation structure, if there exists a relation of th  þ th þ   þ ða i Þ mXXi 6¼ ðaj Þ mXXj for xk ¼ ai ¼ aj þ XX i ; XXj 2 fUS, LS, SO, SIg

ð4:130Þ

ði 2 f1; 2; . . .; s1 g; j 2 f1; 2; . . .; s2 gÞ; then, there are two flower-bundle switching bifurcations of simple fixed-points: (iv1) for aj ¼ ai þ 2l, the bifurcation is called a ath j mXX right flower-bundle switching bifurcation for ai to aj -simple fixed-points with the appearance (birth) of 2l-simple fixed-points. (iv2) for aj ¼ ai  2l, the bifurcation is called a ath i mXX left flower-bundle switching bifurcation for ai to aj -simple fixed-points with the vanishing (death) of 2l-simple fixed-points. A general parallel switching bifurcation is switching

th   th   th  ðða 1 Þ mXX1 : ða2 Þ mXX2 :    : ðas1 Þ mXXs1 Þ !

ðða1þ Þth mXX1þ : ða2þ Þth mXX2þ :    :

bifurcation þ th þ ðas2 Þ mXXs2 Þ:

ð4:131Þ

292

4 (2m)th-Degree Polynomial Discrete Systems

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th  þ th þ th ða i Þ mXXi ¼ ðai Þ mXXi ¼ a mXX, þ for xk ¼ a i ¼ ai ði ¼ 1; 2; . . .; sg;

ð4:132Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ðða1 Þth mXX:ða2 Þth mXX:    : ðas Þth mXXÞ: If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th   th þ th þ þ th ða i Þ mXXi ¼ ð2li Þ mXX, ðaj Þ mXXj ¼ ð2lj Þ mYY, þ for xk ¼ a i 6¼ ai ði ¼ 1; 2; . . .; sg

ð4:133Þ

mXX 2 fmUS; mLSg; mYY 2 fmUS; mLSg; then the parallel flower-bundle switching bifurcation becomes a combination of two independent left and right parallel appearing bifurcations: th     th  th (i) a ðð2l 1 Þ mXX1 : ð2l2 Þ mXX2 :    : ð2ls1 Þ mXXs1 Þ-left parallel sprinklerspraying appearing (or right vanishing) bifurcation and (ii) a ðð2l1þ Þth mXX1þ : ð2l2þ Þth mXX2þ :    : ð2lsþ2 Þth mXXsþ2 Þ-right parallel sprinkler-spraying appearing (or left vanishing) bifurcation.

The ð6th mUS:4th mLS:    : 4th mUS:mSI-oSOÞ appearing bifurcation for a0 [ 0 is presented in Fig. 4.5(i). Compared to the case of a0 [ 0, the bifurcation and stability conditions of fixed-points for a0 [ 0 will be swapped. The ð6th mLS: 4th mUS:    : 4th mLS:mSOÞ parallel appearing bifurcation is shown in Fig. 4.5(ii). Such a kind of bifurcation is like a waterfall appearing bifurcation. The switching and appearing bifurcations of fixed-points exist at the same parameter. A set of paralleled, different switching and appearing bifurcations of higher-order th th fixed-points is also named an ðlth 1 mXX:l2 mXX:    : ls mXXÞ parallel switching th and appearing bifurcation in the (2m) degree polynomial discrete system. The lth i mXX switching and appearing bifurcation possesses different clusters of stable and unstable fixed-points before and after the bifurcation. The set of ð5th mSI :    : mSO:6th mUSÞ flower-bundle switching bifurcation for mSI-oSO and mSO fixed-points is presented in Fig. 4.5(iii) for a0 [ 0: Such a flower-bundle switching bifurcation is from ðmSI-oSO: mSO : mSI  oSO : mSOÞ to ð5th mSI :    : mSO:6th mUSÞ with a waterfall appearing. The set of ð5th mSO :    : mSI-oSO:6th mLSÞ flower-bundle switching bifurcation for mSI-oSO and mSO

4.2 Simple Fixed-Point Bifurcations a0 > 0

mSO

b1(i1 )

mSI-oSO

mSO

mSI-oSO

4th mLS

a0 < 0

b1(i2 ) b2(i1 )

6th mUS mSO

293

P-2

mSO

4th mUS

P-2 • •

(2r)th mUS

mSI-oSO mSO

P-2

(2r)th LS

P-2 • •

P-2 • •

mSI-oSO

•

mSO

b1(im )

mSI-oSO

P-2

mLS

4th

mSI-oSO

mSO

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

mSO

mSI-oSO

b2(im )

mUS

mSO

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

( i1 ) 1

b

a0 < 0

b1(i2 )

P-2

b2(i1 )

6th mUS mSI-oSO

mSO

b1(im )

(ii) mSO

mSI-oSO

P-2

mSO

xk∗

(i) a0 > 0

mSO

•

mSI-oSO

mSO

|| p ||

mSI-oSO

mSO P-2

xk∗

mSI-oSO P-2

6th LS mSI-oSO

mSI-oSO

b1(i1 ) b1(i2 )

mSO

b2(i2 )

mSO

P-2

b2(i1 )

6th mLS

b2(i2 )

mSO

mSI-oSO

6th mUS P-2

P-2

mSO

• • •

mSI-oSO

mSO

• •

P-2

•

mSO

mSI-oSO mSI-oSO P-2

P-2 (2r)th mUS

(2r)th mLS

P-2 • •

P-2

• •

mSI-oSO

•

mSI-oSO

b1(im )

mSO

mSO

P-2 mSI-oSO

|| p ||

Δ iq < 0 Δ iq = 0

(iii)

mSO

•

5th mSI mSI-oSO

xk∗

P-2

•

P-2 mSI-oSO

P-2

b2(i2 )

mSI-oSO P-2

•

4th

mSO

b2(i1 )

mSO

•

b1(i1 ) b1(i2 )

6th mLS

b2(i2 )

P-2 •

mSI-oSO

Δ iq > 0

b2(im )

b1(im )

5th mSO

xk∗ mSO

|| p ||

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

(iv)

Fig. 4.5 Stability and bifurcation. (i)ð6th mUS:mSO:4th mLS:    : mSI-oSOÞ appearing bifurcation ða0 [ 0Þ: (ii) ð6th mLS:mSI-oSO:4th mUS:    : mSOÞ appearing bifurcation ða0 \0Þ: (iii) ð6th mUS:6th mLS:    : 5th mSIÞ switching-appearing bifurcation ða0 [ 0Þ: (iv) ð6th mLS:mSIoSO : 6th US:    : 5th mSOÞ switching-appearing bifurcation in a (2m)th-degree polynomial discrete system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4 (2m)th-Degree Polynomial Discrete Systems

294

fixed-points is presented in Fig. 4.5(iv) for a0 \0: Such a flower-bundle switching bifurcation is from ðmSO:mSI-oSO:mSO:mSI-oSOÞ to ð5th mSI :    : mSO:6th mUSÞ with a waterfall appearing. After the bifurcation, the waterfall fixed-points birth can be observed. The fixed-points before such a bifurcation are much less than after the bifurcation.

4.3

Higher-Order Fixed-Points Bifurcations

The afore-discussed appearing and switching bifurcations in the (2m)th degree polynomial system are relative to simple monotonic sources and monotonic sinks. As in Luo (2020a), the higher-order singularity bifurcations in the (2m)th degree polynomial discrete system can be for higher-order fixed-points (i.e., monotonic sinks, monotonic sources, monotonic upper-saddles, monotonic lower-saddles).

4.3.1

Appearing Bifurcations

Consider a (2m)th degree polynomial discrete system as xk þ 1 ¼ xk þ a0 Qðxk Þ *si¼1 ðx2k þ Bi xk þ Ci Þai ;

ð4:134Þ

where ai 2 f2l  1; 2lg . Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are ðiÞ

1 2

b1;2 ¼  Bi 

1pffiffiffiffiffi Di ; Di 2

¼ B2i  4Ci 0; ð1Þ

ð1Þ

ðsÞ

ðsÞ

fa1 ; a2 ; . . .; a2s1 ; a2s g ¼ sortfb1 ; b2 ; . . .; b1 ; b2 g; aj  aj þ 1 :

ð4:135Þ

There are four higher-order bifurcations as follows: ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order mUS  ! appearing bifurcation 8 < ð2li  1Þth order mSO, xk ¼ a2i ; : ð2l  1Þth order mSI, x ¼ a ; i 2i1 k

ð4:136Þ

4.3 Higher-Order Fixed-Points Bifurcations

295

ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order mLS  ! appearing bifurcation ( ð2li  1Þth order mSI, xk ¼ a2i ;

ð4:137Þ

ð2li  1Þth order mSO, xk ¼ a2i1 ; ð2li Þth order quadratics

ð2ð2li ÞÞth order mUS ! switching bifurcation 8 < ð2li Þth order mUS, xk ¼ a2i ;

ð4:138Þ

: ð2l Þth order mUS, x ¼ a ; i 2i1 k ð2li Þth order quadratics

ð2ð2li ÞÞth order mLS ! switching bifurcation 8 < ð2li Þth order mLS, xk ¼ a2i ;

ð4:139Þ

: ð2l Þth order mLS, x ¼ a : i 2i1 k (i) For ai ¼ 2li  1; the ð2ð2li  1ÞÞth order monotonic upper-saddle (mUS) appearing bifurcation is for the onset of the ð2li  1Þth order monotonic source (mSO) ðx ¼ a2i Þ and the ð2l  1Þth order monotonic sink (mSI) ðx ¼ a2i1 Þ with a2i [ a2i1 for a0 [ 0: (ii) For ai ¼ 2li  1; the ð2ð2li  1ÞÞth order monotonic lower-saddle (mLS) appearing bifurcation is for the onset of the ð2li  1Þth order monotonic sink (mSI) ðx ¼ a2i Þ and the ð2li  1Þth order monotonic source (mSO) ðx ¼ a2i1 Þ with a2i [ a2i1 for a0 \0: (iii) For ai ¼ 2li ; the ð2ð2li ÞÞth order monotonic upper-saddle (mUS) appearing bifurcation is for the onset of two ð2li Þth order monotonic upper-saddles (mUS) ðx ¼ a2i1 ; a2i Þ with a2i 6¼ a2i1 for a0 [ 0: (iv) For ai ¼ 2li ; the ð2ð2li ÞÞth order monotonic lower-saddle (mLS) appearing bifurcation is for the onset of two ð2li Þth order monotonic lower-saddles (mLS) ðx ¼ a2i1 ; a2i Þ with a2i 6¼ a2i1 for a0 \0: From the higher-order singular bifurcation conditions, in a (2m)th degree polynomial discrete system, the higher-order saddle-node bifurcations for appearing and switching of the higher-order fixed-points are discussed herein. A set of paralleled different higher order monotonic upper-saddle appearing bifurcations in the (2m)th degree polynomial nonlinear discrete system is called a ðð2a1 Þth mUS:(2a2 Þth mUS:    : ð2as Þth mUSÞ parallel appearing bifurcation for a0 [ 0:

4 (2m)th-Degree Polynomial Discrete Systems

296

Define ðð2a1 Þth mUS:(2a2 Þth mUS:    : ð2as Þth mUSÞ ¼ ð2a1 : 2a2 :    : 2as Þth mUS ð4:140Þ where ai 2 f2li  1; 2li g for i ¼ 1; 2; . . .; s: Such an appearing bifurcation is called a ð2a1 : 2a2 :    : 2as Þth mUS teethcomb appearing bifurcation. Similarly, a set of paralleled different higher order lower-saddle appearing bifurcations in the (2m)th degree polynomial nonlinear system is called a ðð2a1 Þth mLS:(2a2 Þth mLS:    : ð2as Þth mLSÞ parallel appearing bifurcation for a0 \0: Define ðð2a1 Þth mLS:(2a2 Þth mLS:    : ð2as Þth mLSÞ ¼ ð2a1 : 2a2 :    : 2as Þth mLS ð4:141Þ where ai 2 f2li  1; 2li g for i ¼ 1; 2; . . .; s: Such an appearing bifurcation is called a ð2a1 : 2a2 :    : 2as Þth mLS teethcomb appearing bifurcation. Consider a 1-dimensional polynomial system as xk þ 1 ¼ xk þ a0 Qðxk Þ *ni¼1 ðx2k þ Bi xk þ Ci Þai

ð4:142Þ

where ai 2 f2ri  1; 2ri g ði ¼ 1; 2; . . .; nÞ: Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are ðiÞ

1 2

xk;1;2 ¼  Bi 

1pffiffiffiffiffi Di ; Di 2

¼ B2i  4Ci 0; ð4:143Þ

Bi ¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðrÞ

ðrÞ

fa1 ; a2 ; . . .; a2l g 2 sortfxk;1 ; xk;2 ; xk;1 ; xk;2 ; . . .; xk;1 ; xk;2 g; ai  ai þ 1 : The higher-order singularity bifurcation can be for a cluster of higher-order monotonic sinks, monotonic sources, monotonic upper-saddles, and monotonic lower-saddles. There are four higher-order bifurcations as follows: For the higher-order upper-saddle appearing bifurcation, the cluster of higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower saddles is given by the following two cases: (i) The ð2ð2l  1ÞÞth order US spraying appearing bifurcation for a cluster of fixed points with higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic-lower-saddles is

4.3 Higher-Order Fixed-Points Bifurcations

297

a cluster of 2nmXX

ð2ð2l  1ÞÞth order mUS ! appearing bifurcation 8 ða Þth order mXX for xk ¼ a2n ; > > > 2n > > > th > < ða2n1 Þ order mXX for xk ¼ a2n1 ;

ð4:144Þ

.. > > > . > > > > : ða1 Þth order mXX for xk ¼ a1 ; where 2ð2l  1Þ ¼ Pni¼1 ai and the minimum and maximum fixed-points satisfy 8 < ð2r2n Þth order mUS, for a2n ¼ 2rn ;

ða2n Þth order mXX =

ða1 Þth order mXX =

: ð2r  1Þth order mSO, for a ¼ 2r  1; 2n 2n n 8 < ð2r1 Þth order mUS, for a1 ¼ 2r1 ;

ð4:145Þ

: ð2r  1Þth order mSI, for a ¼ 2r  1: 1 1 1

(ii) The ð2ð2lÞÞth order mUS spraying-appearing bifurcation for a cluster of fixed points with higher-order monotonic sinks, monotonic sources, monotonic uppersaddles and monotonic lower-saddles is a cluster of 2nmXX

ð2ð2lÞÞth order mUS ! appearing bifurcation 8 ða2n Þth order mXX for xk ¼ a2n ; > > > > > > th > < ða2n1 Þ order mXX for xk ¼ a2n1 ;

ð4:146Þ

.. > > > . > > > > : ða1 Þth order mXX for xk ¼ a1 ; where 2ð2lÞ ¼ Pni¼1 ai and the minimum and maximum fixed-points satisfy ða2n Þth order mXX ¼

ða1 Þth order mXX ¼

8 < ð2r2n Þth order mUS, for a2n ¼ 2rn ;

: ð2r  1Þth order mSO, for a ¼ 2r  1; 2n 2n n 8 th < ð2r1 Þ order mUS, for a1 ¼ 2r1 ; : ð2r  1Þth order mSI, for a ¼ 2r  1: 1 1 1

ð4:147Þ

4 (2m)th-Degree Polynomial Discrete Systems

298

For the higher-order monotonic lower-saddle bifurcation, the cluster of the higher-order fixed-points is given by the following two cases. (iii) The ð2ð2l  1ÞÞth order mLS spraying-appearing bifurcation for a cluster of fixed-points with higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower-saddles is a cluster of 2nmXX

ð2ð2l  1ÞÞth order mLS ! appearing bifurcation 8 ða2n Þth order mXX, for xk ¼ a2n ; > > > > > > th > < ða2n1 Þ order mXX, for xk ¼ a2n1 ;

ð4:148Þ

.. > > > . > > > > : ða1 Þth order mXX, for xk ¼ a1 ; where 2ð2l  1Þ ¼ Pni¼1 ai and the minimum and maximum fixed-points satisfy ða2n Þth order mXX =

ða1 Þth order mXX =

8 < ð2r2n Þth order mLS, for a2n ¼ 2rn ;

: ð2r  1Þth order mSI, for a ¼ 2r  1; 2n 2n n 8 th < ð2r1 Þ order mLS, for a1 ¼ 2r1 ;

ð4:149Þ

: ð2r  1Þth order mSO, for a ¼ 2r  1: 1 1 1

(iv) The ð2ð2lÞÞth order LS spraying-appearing bifurcation for a cluster of fixedpoints with higher-order monotonic sinks, monotonic sources, monotonic uppersaddles and monotonic lower-saddles is a cluster of 2nmXX

ð2ð2lÞÞth order mLS ! appearing bifurcation 8 ða2n Þth order mXX, for xk ¼ a2n ; > > > > > > th > < ða2n1 Þ order mXX, for xk ¼ a2n1 ;

ð4:150Þ

.. > > > . > > > > : ða1 Þth order mXX, for xk ¼ a1 ; where 2ð2lÞ ¼ Pni¼1 ai and the minimum and maximum fixed-points satisfy

4.3 Higher-Order Fixed-Points Bifurcations

ða2n Þth order mXX =

ða1 Þth order mXX =

299

8 < ð2r2n Þth order mLS, for a2n ¼ 2rn ;

: ð2r  1Þth order mSI, for a ¼ 2r  1; 2n 2n n 8 < ð2r1 Þth order mLS, for a1 ¼ 2r1 ;

ð4:151Þ

: ð2r  1Þth order mSO, for a ¼ 2r  1: 1 1 1

A set of paralleled, different, higher-order monotonic upper-saddle-node appearing bifurcations in the (2m)th-degree polynomial system is the ðð2b1 Þth mUS: ð2b2 Þth mUS:    : ð2bs Þth mUSÞ parallel appearing bifurcation for clusters of fixed-points with higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower saddles. For the ð2bi Þth mUS ( th

ð2bi Þ mUS =

ð2ð2li  1ÞÞth order mUS, for bi ¼ 2li  1; ð2ð2li ÞÞth order mUS, for bi ¼ 2li :

ð4:152Þ

Similarly, the following notation is introduced as ðð2b1 Þth mUS:(2b2 Þth mUS:    : ð2bs Þth mUSÞ¼ð2b1 : 2b2 :    : 2bs Þth mUS: ð4:153Þ Thus, the paralleled ð2b1 : 2b2 :    : 2bs Þth mUS spraying appearing bifurcation is called the ð2b1 : 2b2 :    : 2bs Þth mUS sprinkler-spraying appearing bifurcation for the higher-order fixed-points. Similarly, a set of paralleled different lower-saddle appearing bifurcations for higher-order singularity of fixed-points is called the ðð2b1 Þth mLS:(2b2 Þth mLS:    : ð2bs Þth mLSÞ parallel appearing bifurcation in the (2m)th-degree polynomial discrete system. Thus, the paralleled ð2b1 : 2b2 :    : 2bs Þth mLS bifurcation is also called the ð2b1 : 2b2 :    : 2bs Þth mLS sprinklerspraying appearing bifurcation for higher-order fixed-points. The ð2a1 : 2a2 :    : 2an Þth mUS and ð2a1 : 2a2 :    : 2an Þth mLS teethcomb appearing bifurcations for the higher-order singularity of fixed-points are presented in Figs. 4.6(i) and (ii) for a0 [ 0 and a0 \0, respectively. The components of the teethcomb appearing bifurcation are aj1 ¼2rj1

ð2aj1 Þth mUS !

8 < ð2rj1 Þth mUS

ðj ¼ i; n  1;   Þ; : ð2r Þth mUS 1 j1 8 < ð2rj2  1Þth mSO aj2 ¼2rj2 1 ðj2 ¼ 1; n;   Þ; ð2aj2 Þth mUS ! appearing : ð2rj2  1Þth mSI appearing

ð4:154Þ

4 (2m)th-Degree Polynomial Discrete Systems

300 a0 > 0

(2rn − 1) th mSO

b1(i1 )

a0 < 0

(2rn − 1) th mSI

(2(2rn − 1)) th mUS

b2(i1 )

b1(i2 )

(2rn −1 ) th mLS

b1(i2 )

(2rn −1 ) th mLS

b2(i2 )

(2(2rn −1 )) th mLS

2((2rn −1 )) th mUS th

(2rn −1 ) mUS • • •

b2(i2 )

• • •

(2ri ) th mUS

(2ri ) th mLS

(2(2ri )) th mLS

(2(2ri )) th mUS

(2ri ) th mLS

th

(2ri ) mUS •

•

Δ Δ

• •

(2r1 − 1) th mSO

• •

Δ

(2r1 − 1) th mSI

( in ) 1

b

(2(2r1 − 1)) th mUS

x

(2r1 − 1) th mSI

∗ k

x b2(in )

Δ iq > 0

Δ iq < 0 Δ iq = 0

(2r1 − 1) th mSO

b2(in )

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(i)

(ii) (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 1) th mSI

(2rn − 1) th mSO

2((2ls )) th mUS

th

(2(2ls )) mLS

(2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO

(2rn −3 − 1) th mSO

(2rn −3 − 1) th mSI

(2rn − 4 − 1) th mSI

(2rn − 4 − 1) th mSO

(2(2ls −1 )) th mUS

(2(2ls −1 )) th mLS (2rn −3 − 1) th mSO

(2rn −3 − 1) th mSI

(2rn − 4 − 1) th mSI

• • •

(2rn − 4 − 1) th mSO

• • •

th

(2ri ) mUS

(2ri ) th mLS

(2ri −1 ) th mUS (2ri − 2 − 1) th mSO

(2ri −1 ) th mLS (2ri − 2 − 1) th mSI

th

(2ri −3 − 1) mSI 2((2lr )) th mUS

(2ri −3 − 1) th mSO (2(2lr )) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

(2ri −1 ) th mLS

(2ri −1 ) th mUS

th

• •

(2ri −3 − 1) mSI

• •

(2ri ) th mUS

•

(2ri −3 − 1) th mSO (2ri ) th mLS

•

(2r2 ) th mUS

(2r2 ) th mLS

(2r1 − 1) th mSO

(2r1 − 1) th mSI

(2(2l1 − 1)) th mUS

(2(2l1 − 1)) th mLS

xk∗

(2r1 − 1) th mSI

xk∗

(2r1 − 1) th mSO

(2r2 ) th mUS

|| p ||

b1(in )

[2(2r1 − 1)]th mLS

∗ k

|| p ||

b1(i1 )

(2rn − 1) th mSO

(2(2rn − 1)) th mLS

b2(i1 ) (2rn −1 ) th mUS

(2rn − 1) th mSI

Δ iq < 0 Δ iq = 0

(iii)

Δ iq > 0

(2r2 ) th mLS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.6 The teethcomb appearing bifurcations of ð2ð2r1  1Þ :    : 2ð2rn1 Þ : 2ð2rn  1ÞÞth mXX: (i) XX ¼ USða0 [ 0Þ and (ii) XX ¼ LSða0 \0Þ. The sprinkler-spraying appearing bifurcations of ð2ð2l1  1Þ :    : ð2ð2ln1 Þ : 2ð2ln ÞÞth mXX: (iii) XX ¼ USða0 [ 0Þ and (iv) XX ¼ LSða0 \0Þ: mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4.3 Higher-Order Fixed-Points Bifurcations

301

and aj1 ¼2rj1

ð2aj1 Þth mLS !

8 < ð2rj1 Þth mLS

ðj1 ¼ i; n  1;   Þ; ð2rj1 Þth mLS 8 < ð2rj2  1Þth mSI a ¼2r 1 j j 2 2 th ðj2 ¼ 1; n;   Þ: ð2aj2 Þ mLS ! appearing : ð2rj2  1Þth mSO appearing :

ð4:155Þ

The ð2b1 : 2b2 :    : 2bn Þth mUS and ð2b1 : 2b2 :    : 2bs Þth mLS sprinklerspraying appearing bifurcations for the higher-order singularity of fixed-points are presented in Figs. 4.6(iii) and (iv) for a0 [ 0 and a0 \0, respectively. The components of the sprinkler-spraying appearing bifurcation are ð2b1 : 2b2 :    : 2bn Þth mUS ¼ ðð2ð2l1  1Þ :    : 2ð2li Þ :    : 2ð2ln1 Þ : 2ð2ln ÞÞth mUS

ð4:156Þ

and ð2b1 : 2b2 :    : 2bn Þth mLS ¼ ðð2ð2l1  1Þ :    : 2ð2li Þ :    : 2ð2ln1 Þ : 2ð2ln ÞÞth mLS:

ð4:157Þ

For a cluster of m-quadratics, Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ and Di ¼ 0 ði ¼ 1; 2; . . .; nÞ: The (2m)th order monotonic upper-saddle appearing bifurcation for n-pairs of the higher-order singularity of fixed-points is 8 > ða2n Þth order mXX for xk ¼ a2n ; > > > > > < ða2n1 Þth order mXX for xk ¼ a2n1 ; a cluster of 2n-mXX th ð2mÞ order mUS ! . >. appearing bifurcation > . > > > > : ða Þth order mXX for x ¼ a ; 1 1 k ð4:158Þ where 2m ¼ 2ð2lÞ ¼

X2n

i¼1 ai ; 2m

¼ 2ð2l  1Þ ¼

X2n

i¼1 ai :

ð4:159Þ

The (2m)th order monotonic lower-saddle-node appearing bifurcation for higher-order fixed-points is

302

4 (2m)th-Degree Polynomial Discrete Systems

8 > ða2n Þth order mXX for xk ¼ a2n ; > > > > > < ða2n1 Þth order mXX for xk ¼ a2n1 ; a cluster of 2n-mXX th ð2mÞ order mLS ! . appearing bifurcation > >. . > > > > : ða Þth order mXX for x ¼ a : 1 1 k ð4:160Þ The (2m)th order upper-saddle appearing bifurcation with n-pairs of higher-order singularity of fixed-points is a sprinkler-spraying cluster of the n-pairs of higher-order singularity of fixed-points. The (2m)thorder monotonic lower-saddle appearing bifurcation with n-pairs of fixed-points is also a sprinkler-spraying cluster of the n-pairs of higher-order singularity of fixed-points. Thus, the (2m)th order mUS bifurcation ða0 [ 0Þ and (2m)th order mLS bifurcation ða0 \0Þ are presented in Fig. 4.7(i)–(iv), respectively. The (2m)th order monotonic upper-saddle appearing bifurcation for higher-order singularity of fixed-points is called the (2m)th order mUS sprinkler-spaying appearing bifurcation, and the (2m)th order monotonic lower saddle-node appearing bifurcation for higher-order singularity of fixed-points is also called the (2m)th order mLS sprinkler-spraying appearing bifurcation. A series of the monotonic saddle-node bifurcations for higher-order singularity of fixed-points is aligned up with varying with parameters, which is formed a special pattern. For n-quadratics in the (2m)th order polynomial discrete systems, the following conditions should be satisfied. Bi Bj i; j 2 f1; 2; . . .; sg; i 6¼ j; Di [ Di þ 1 ði ¼ 1; 2; . . .; s; s  n\mÞ; Di ¼ 0 with jjpi jj\jjpi þ 1 jj:

ð4:161Þ

The two series of the fish-scale switching bifurcations in Fig. 4.8(i) and (iii) for a0 \0 have the following detailed structures. 8 8 < ð2r1  1Þth mSO, > > th > > ð2ð2r1  1ÞÞ mUS ! > > : ð2r  1Þth mSI; > > 1 > > > 8 > > th > < ð2r2 Þ mLS, > > > < ð2ð2r2 ÞÞth mLS ! : ð2r Þth mLS; ð4:162Þ 2 > > > > .. > > . > > > 8 > > > < ð2rn  1Þth mSO, > > th > > ð2ð2r  1ÞÞ mUS ! n > : : ð2r  1Þth mSI; n

4.3 Higher-Order Fixed-Points Bifurcations

303

(2r1 − 1) th mSO

a0 > 0

(2r1 − 1) th mSI

a0 < 0

(2r2 ) th mLS

(2r2 ) th mUS

(2r3 ) th mLS

(2r3 ) th mUS

(2ri − 1) th mSI

 (2(2l − 1)) th mUS

(2ri − 1) th mSO



th

(2rl − 1) mSO (2(2l − 1)) th mLS





(2ri − 1) th mSI

x∗

(2r2 ) th mLS

(2rl − 1) th mSI

x∗

(2r2 ) th mUS

(2r1 − 1) th mSI

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(2r1 − 1) th mSO

|| p ||

Δ iq > 0

Δ iq < 0 Δ iq = 0

(i)

(ii) (2r1 ) th mUS

a0 > 0

(2r1 ) th mLS

a0 < 0

(2r2 − 1) th mSO

(2r2 − 1) th mSI

(2r3 ) th mLS

(2r3 ) th mUS

(2ri − 1) th mSI



(2ri − 1) th mSO



th

(2rn ) mUS



(2(2l )) th mUS

(2ri − 1) th mSO

(2ri − 1) th mSI

(2r3 ) th mLS

(2r3 ) th mUS

(2r2 − 1) th mSI

(2rn ) th mUS

x∗

(2r2 − 1) th mSO

(2r1 ) th mUS

Δ iq > 0

Δ iq < 0 Δ iq = 0

(iii)

(2rn ) th mLS



(2(2l )) th mLS

(2rn ) th mLS

x∗

|| p ||

(2ri − 1) th mSO

(2r3 ) th mLS

(2r3 ) th mUS (2rl − 1) th mSO

(2rl − 1) th mSI

(2r1 ) th mLS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.7 Spraying appearing bifurcations for higher-order fixed-points in the (2m)th polynomial system: (i) ð2ð2l  1ÞÞth mUS spraying-appearing bifurcation ða0 [ 0Þ, (ii) ð2ð2l  1ÞÞth mLS spraying appearing bifurcation ða0 \0Þ, (iii) ð2ð2lÞÞth mUS spraying appearing bifurcation ða0 [ 0Þ, (iv)ð2ð2lÞÞth mUS spraying appearing bifurcation ða0 \0Þ: mLS: monotonic-lowersaddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

and 8 8 < ð2r1 Þth mUS, > > th > > ð2ð2r1 ÞÞ mUS ! > > : ð2r Þth mUS; > > 1 > > > 8 > > > < ð2r2  1Þth mSO, > > th > < ð2ð2r2  1ÞÞ mUS ! : ð2r  1Þth mSI; 2 > > > > .. > > . > > > 8 > > > < ð2rn  1Þth mSO, > > th > > ð2ð2r  1ÞÞ mUS ! n > : : ð2r  1Þth mSI: n

ð4:163Þ

4 (2m)th-Degree Polynomial Discrete Systems

304 a0 > 0

(2r2 ) th mLS

(2r1 − 1) th mSO

(2(2r1 − 1)) th mUS

(2(2r2 )) th mLS (2(2r3 )) th mLS

(2(2ri − 1)) th mLS

• • •

(2r3 ) th mLS

(2ri − 1) th mSI

(2rn − 1) th mSO

(2(2rn − 1)) th mUS

• • •

x∗ (2r1 − 1) th mSI

|| p ||

(2r2 ) th mLS

Δr < 0

(2r3 ) th mLS

(2ri − 1) th mSO

(2rn − 1) th mSI

Δr > 0

Δr = 0

(i) a0 < 0

[2(2r1 − 1)]th mLS

(2r1 − 1) th mSI

(2(2r2 )) th mUS (2(2r3 )) th mUS

• • •

(2r2 ) th mUS

(2(2ri − 1)) th mUS

(2r3 ) th mUS

(2ri − 1) th mSO

(2rn − 1) th mSI

(2(2rn − 1)) th mLS

• • •

x∗ (2r1 − 1) th mSO

|| p ||

Δr < 0

(2r2 ) th mUS

Δr = 0

(2r3 ) th mUS

(2ri − 1) th mSI

(2rn − 1) th mSO

Δr > 0

(ii) a0 > 0

(2(2r1 )) th mUS

(2r1 ) th mUS

(2(2r2 − 1)) th mUS (2(2r3 )) th mLS

(2(2ri − 1)) th mLS

• • •

(2r3 ) th mLS

(2r2 − 1) th mSO

(2ri − 1) th mSI

(2rn − 1) th mSO

(2(2rn − 1)) th mUS

• • •

x∗ (2r1 ) th mUS

|| p ||

(2r3 ) th mLS

(2r2 − 1) th mSI

Δr < 0

Δr = 0

(2ri − 1) th mSO

(2rn − 1) th mSI

Δr > 0

(iii) a0 < 0

(2(2r1 )) th mLS

th

(2r2 − 1) th mSI

(2r1 ) mLS

(2(2r2 − 1)) th mLS (2(2r3 )) th mUS

• • •

(2(2ri − 1)) th mUS

(2r3 ) th mUS

• • •

(2ri − 1) th mSO

(2rn − 1) th mSI

(2(2rn − 1)) th mLS

x∗ (2r1 ) th mLS

|| p ||

(2r2 − 1) th mSO

Δr < 0

Δr = 0

(2r3 ) th mUS

(2ri − 1) th mSI

(2rn − 1) th mSO

Δr > 0

(iv) Fig. 4.8 The fish-scale appearing bifurcation patterns in a (2m)th-degree polynomial discrete system: (i) ð2ð2r1  1ÞÞth mUS - (2ð2r2 ÞÞth mLS - . . .ða0 [ 0Þ, (ii) ð2ð2r1  1ÞÞth mLS - (2ð2r2 ÞÞth mUS - . . .ða0 \0Þ, (iii) ð2ð2r1 ÞÞth mUS - (2ð2r2 ÞÞth mUS - . . .ða0 [ 0Þ, (iv)ð2ð2r1 ÞÞth mLS ð2ð2r2  1ÞÞth mLS - . . .ða0 [ 0Þ: mLS: monotonic-lower-saddle, mUS: monotonic-uppersaddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4.3 Higher-Order Fixed-Points Bifurcations

305

Two series of fish-scale appearing bifurcations in Fig. 4.8(ii) and (iv) for a0 \0 have the following structures as 8 8 < ð2r1  1Þth mSI, > > th > > ð2ð2r1  1ÞÞ mLS ! > > : ð2r  1Þth mSO; > > 1 > > > 8 > > th > < ð2r2 Þ mUS, > > > < ð2ð2r2 ÞÞth mUS ! : ð2r Þth mUS; ð4:164Þ 2 > > > > . > .. > > > > 8 > > > < ð2rn  1Þth mSI, > > th > > ð2ð2r  1ÞÞ mLS ! n > : : ð2r  1Þth mSO; n and 8 8 < ð2r1 Þth mLS, > > th > > ð2ð2r ÞÞ mLS ! 1 > > : ð2r Þth mLS; > > 1 > > > 8 > > > < ð2r2  1Þth mSI, > > th > < ð2ð2r2  1ÞÞ mLS ! : ð2r  1Þth mSO; 2 > > > > .. > > . > > > 8 > > > < ð2rn  1Þth mSI, > > th > > > : ð2ð2rn  1ÞÞ mLS ! : ð2rn  1Þth mSO:

ð4:165Þ

The four fish-scale appearing bifurcation patterns for higher-order fixed-points are different from the fish-scale appearing bifurcation patterns for simple fixed-points.

4.3.2

Switching Bifurcations

Consider the roots of ðx2k þ Bi xk þ Ci Þai ¼ 0 as

4 (2m)th-Degree Polynomial Discrete Systems

306 ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ

ðiÞ

ðiÞ

ðiÞ

xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ; ) Bi 6¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ

ð4:166Þ

The ath i -order singularity bifurcation is for the switching of a pair of higher order fixed-points (i.e., monotonic sinks, monotonic sources, monotonic-upper-saddles and monotonic-lower-saddles). There are six switching bifurcations for i 2 f1; 2; . . .; ng ðiÞ

ðiÞ

li ¼r1 þ r2 1

ð2li Þth order mUS ! switching bifurcation 8 ðiÞ th < ð2r2  1Þ order mSO # mSI, for bðiÞ 2 ¼ a2i # a2i1 ;

ð4:167Þ

: ð2r ðiÞ  1Þth order mSI " mSO, for bðiÞ ¼ a 2i1 " a2i ; 1 1 ðiÞ

ðiÞ

li ¼r1 þ r2 1

ð2li Þth order mLS ! switching bifurcation 8 ðiÞ < ð2r2  1Þth order mSI # mSO, for bðiÞ 2 ¼ a2i # a2i1 ;

ð4:168Þ

: ð2r ðiÞ  1Þth order mSO " mSI, for bðiÞ ¼ a 2i1 " a2i ; 1 1 ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li Þth order mUS ! switching bifurcation 8 < ð2r2ðiÞ Þth order mUS # mUS, for bðiÞ 2 ¼ a2i # a2i1 ; :

ðiÞ

ð4:169Þ

ðiÞ

ð2r1 Þth order mUS " mUS for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li Þth order mLS ! switching bifurcation 8 ðiÞ th < ð2r2 Þ order mLS # mLS, for bðiÞ 2 ¼ a2i # a2i1 ; :

ðiÞ

ð4:170Þ

ðiÞ

ð2r1 Þth order mLS " mLS for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li  1Þth order mSO ! switching bifurcation 8 ðiÞ th < ð2r2  1Þ order mSO # mSO, for bðiÞ 2 ¼ a2i # a2i1 ; :

ðiÞ

ðiÞ

ð2r1 Þth order mLS " mUS for b1 ¼ a2i1 " a2i ;

ð4:171Þ

4.3 Higher-Order Fixed-Points Bifurcations ðiÞ li ¼r1

307 ðiÞ þ r2

ð2li  1Þth order mSI ! switching bifurcation 8 ðiÞ th < ð2r2  1Þ order mSI # mSI, for bðiÞ 2 ¼ a2i # a2i1 ;

ð4:172Þ

: 2r ðiÞ Þth order mUS " mLS for bðiÞ ¼ a 2i1 " a2i : 1 1 A set of n-paralleled higher-order mXX switching bifurcations is called a ðða1 Þth mXX:ða2 Þth mXX:    : ðan Þth mXXÞ parallel switching bifurcation in the (2m)th degree polynomial nonlinear discrete system. Such a bifurcation is also called a ðða1 Þth mXX:ða2 Þth mXX:    : ðan Þth mXXÞ antenna switching bifurcation. ai 2 f2li ; 2li  1g and XX 2 fSO, SI, US, LSg. For non-switching points, Di [ 0 ðiÞ ðiÞ ðiÞ ðiÞ at b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ: At the bifurcation point, Di ¼ 0 at b1 ¼ b2 th th th ði ¼ 1; 2; . . .; nÞ: The ðða1 Þ mXX:ða2 Þ mXX:    : ðan Þ mXXÞ parallel antenna switching bifurcation is 8 8 ðnÞ ðnÞ < ðr2ðnÞ Þth mXXðnÞ > th > th 2 # mYY1 ; for b2 ¼ a2n # a2n1 ; n bifurcation > > a mXX ! n  > n > ðnÞ ðnÞ ðnÞ switiching : ðnÞ th > > ðr1 Þ mXX1 " mYY2 ; for b1 ¼ a2n1 " a2n ; > > > > > .. > > > . > < 8 ð2Þ ð2Þ < ðr2ð2Þ Þth mXXð2Þ nd 2 # mYY1 ; for b2 ¼ a4 # a3 ; 2 bifurcation > th > a mXX2 ! > > ð2Þ ð2Þ ð2Þ switiching : ð2Þ th > 2 > ðr1 Þ mXX1 " mYY2 ; for b1 ¼ a3 " a4 ; > > > 8 > > ð1Þ ð1Þ > < ðr2ð1Þ Þth mXXð1Þ > > 2 # mYY1 ; for b2 ¼ a2 # a1 ; 1st bifurcation th > > ! > : a1 mXX1  ð1Þ ð1Þ ð1Þ switiching : ð1Þ th ðr1 Þ mXX1 " mYY2 ; for b1 ¼ a1 " a2 : ð4:173Þ Such eight sets of parallel switching bifurcations of ðða1 Þth mXX:ða1 Þth mXX:    : ðan Þth mXXÞ are presented in Fig. 4.9(i, iii, v, vii) and (ii, iv, vi, viii) for a0 [ 0 and a0 \0, respectively. The eight switching bifurcation structures are as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

ðð2l1 Þth mUS:    : ð2ln1  1Þth mSO:(2ln Þth mUSÞ for a0 [ 0, ðð2l1 Þth mLS:    : ð2ln1  1Þth mSI:(2ln Þth mLSÞ for a0 \0, ðð2l1 Þth mLS:    : ð2ln1  1Þth mSI:(2ln  1Þth mSOÞ for a0 [ 0, ðð2l1 Þth mUS:    : ð2ln1  1Þth mSO:(2ln  1Þth mSIÞ for a0 \0, ðð2l1 Þth mLS:    : ð2ln1  1Þth mSI:(2ln  1Þth mSOÞ for a0 [ 0, ðð2l1 Þth mUS:    : ð2ln1  1Þth mSI:(2ln  1Þth mSIÞ for a0 \0, ðð2l1 Þth mUS:    : ð2ln1  1Þth mSO:(2ln Þth mUSÞ for a0 [ 0, ðð2l1 Þth mLS:    : ð2ln1  1Þth mSI:(2ln Þth mLSÞ for a0 \0:

4 (2m)th-Degree Polynomial Discrete Systems

308 a0 > 0

(2r2 n − 1) th mSO

a2n

a0 < 0

(2r2 n − 1) th mSI

th

(2ln ) mUS

(2ln ) mLS (2r2 n −1 − 1) th mSI

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSI

a2 n−1 (2r2 n − 1) th mSI

th

(2r2 n − 2 ) mUS

a2 n−1 (2r2 n − 1) th SO

a2 n−2

(2r2 n −3 − 1) th mSI

a2 n−3

(2ln −1 − 1) th mSI

(2r2 n −3 − 1) th mSO

(2r2 n −3 − 1) th mSI (2r2 n −3 − 1) th mSO

a2 n−3 th

th

(2r2 n − 2 ) mLS

(2r2 n − 2 ) mUS

•

•

•

•

•

•

(2r2i ) th mLS

(2r2i −1 − 1) th mSI

(2r2i ) th mUS

(2r2i −1 − 1) th mSO

(2li − 1) th mSO

(2li − 1) th mSI

(2r2i −1 − 1) th mSI

(2r2i −1 − 1) th mSO

•

th

(2r2i ) mUS

•

th

(2r2i ) mLS

• • th

(2r2 ) mUS

(2r1 ) th mUS

• •

a2 (2r1 ) th mLS

(2l1 ) th mUS

(2r2 ) th mLS

(2r1 ) th mUS

a1

Δ iq > 0

Δ iq > 0 Δ iq = 0

(2r1 ) th mLS

xk∗

|| p ||

Δ iq > 0

Δ iq > 0 Δ iq = 0

(ii)

a0 > 0

(2r2 n ) th mUS

a2n

a0 < 0

(2r2 n ) th mLS

(2ln − 1) th mSO

a2n

(2ln − 1) th mLS (2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSI

(2r2 n −1 − 1) th mSI

a2 n−1 (2r2 n ) th mLS

th

(2r2 n − 2 ) mLS

a2 n−1 (2r2 n ) th mUS

a2 n−2

(2r2 n − 2 ) mUS

a2 n−2

(2r2 n −3 − 1) th mSO

a2 n−3

th

(2ln −1 − 1) th mSI

(2ln −1 − 1) th mSO

(2r2 n −3 − 1) th mSI

(2r2 n −3 − 1) th mSO (2r2 n −3 − 1) th mSI

(2r2 n − 2 ) th mUS

a2 n−3 (2r2 n − 2 ) th mLS

•

•

• •

• •

(2r2i ) th mUS

(2r2i −1 − 1) th mSO

(2r2i ) th mLS

(2r2i −1 − 1) th mSI

(2li − 1) th mSO

(2li − 1) th mSI

(2r2i −1 − 1) th mSO • • •

(2r2i ) th mLS

(2r2i −1 − 1) th mSI

a2 (2r1 ) th mUS

(2l1 ) th mLS

(2r1 ) th mLS

(iii)

(2r2 ) th mUS

a2

(2l1 ) th mUS

(2r2 ) th mLS

Δ iq > 0 Δ iq = 0

• • •

(2r2i ) th mUS

(2r2 ) th mLS

(2r1 ) th mLS

|| p ||

a1

(2r2 ) th mLS

(i)

xk∗

a2

(2l1 ) th mLS

(2r2 ) th mUS

|| p ||

(2r2 n − 2 ) mLS

a2 n−2

th

(2ln −1 − 1) th mSO

xk∗

a2n

th

Δ iq > 0

a1 xk∗

(2r1 ) th mUS

a1

(2r2 ) th mUS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.9 Parallel antenna switching bifurcations for high-order fixed-points in a (2m)th-degree th th polynomial discrete system. ðath 1 mXX:a2 mXX:. . . : an mXXÞ: (i, iii, v, vii) for a0 [ 0: (ii, iv, vi, viii) for a0 \0: mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. Continued

4.3 Higher-Order Fixed-Points Bifurcations a0 > 0

(2r2 n − 1) th mSO

309

a2n

a0 < 0

(2r2 n − 1) th mSI

th

(2ln − 1) mSO

(2ln − 1) mSI (2r2 n −1 ) th mLS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mLS

a2 n−1 (2r2 n − 1) th mSO

th

(2r2 n − 2 ) mLS

a2 n−1 (2r2 n − 1) th mSI

a2 n−2

(2r2 n −3 − 1) th mSI

(2r2 n − 2 ) th mLS

• • •

• •

(2r2i ) th mUS

(2r2i −1 − 1) th mSO

(2r2i ) th mLS

(2r2i −1 − 1) th mSI

(2li − 1) th mSO

(2li − 1) th mSI

(2r2i −1 − 1) th mSO

(2r2i −1 − 1) th mSI

•

•

th

(2r2i ) mUS

• • th

(2r2 ) mLS

(2r1 ) th mLS

• •

a2

(2r1 ) th mLS

a1 x∗

Δ iq > 0

Δ iq > 0 Δ iq = 0

(2r1 ) th mUS

|| p ||

Δ iq > 0

Δ iq > 0 Δ iq = 0

(vi)

a0 > 0

(2r2 n ) th mUS

a2n

a0 < 0

(2r2 n ) th mLS

(2ln ) th mUS

a2n

(2ln ) th mLS (2r2 n −1 ) th mUS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mLS

(2r2 n −1 ) th mLS

a2 n−1 (2r2 n ) th mUS

(2r2 n − 2 ) th mUS

a2 n−1 (2r2 n ) th mLS

a2 n−2

(2r2 n − 2 ) th mLS

a2 n−2

(2r2 n −3 − 1) th mSI

a2 n−3

(2ln −1 − 1) th mSI

(2ln −1 − 1) th mSO (2r2 n −3 − 1) th mSO

(2r2 n −3 − 1) th mSI (2r2 n −3 − 1) th mSO

(2r2 n − 2 ) th mLS

a2 n−3 (2r2 n − 2 ) th mUS

•

•

• •

• •

(2r2i ) th mLS

th

(2r2i −1 − 1) mSI

(2r2i ) th mUS

th

(2r2i −1 − 1) SO

(2li − 1) th mSI

(2li − 1) th SO

(2r2i −1 − 1) th mSI

(2r2i −1 − 1) th mSO

•

(2r2i ) th mUS

(2r2 ) th mUS

(2r1 ) th mLS

a2

(2l1 ) mLS

(2r1 ) th mUS (2r2 ) th mUS

Fig. 4.9 (continued)

(2r2 ) th mLS

th

(2l1 ) mUS

(vii)

• •

a2

th

Δ iq > 0 Δ iq = 0

•

(2r2i ) th mLS

• •

(2r1 ) th mUS

|| p ||

a1

(2r2 ) th mUS

(v)

xk∗

a2

(2l1 ) th mUS

(2r2 ) th mLS

|| p ||

(2r2 ) th mUS

(2r1 ) th mUS

(2l1 ) th mLS

xk∗

a2 n−3

(2r2 n −3 − 1) th mSO

a2 n−3

•

(2r2i ) mLS

(2r2 n −3 − 1) th mSO

(2ln −1 − 1) th mSI

(2r2 n −3 − 1) th mSI

th

(2r2 n − 2 ) mUS

a2 n−2

th

(2ln −1 − 1) th mSI

(2r2 n − 2 ) th mUS

a2n

th

Δ iq > 0

a1 xk∗

(2r1 ) th mLS (2r2 ) th mLS

|| p ||

Δ iq > 0 Δ iq = 0

(viii)

Δ iq > 0

a1

4 (2m)th-Degree Polynomial Discrete Systems

310

The same switching bifurcations with different higher-order fixed-points are illustrated, which is different from the m-mUSN and m-mLSN for simple monotonic sinks and monotonic sources. Consider a switching bifurcation for a cluster of higher-order fixed-points with the following conditions, ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ

ðiÞ

ðiÞ

ðiÞ

xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ; 9 Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ = at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 = b2 ði ¼ 1; 2; . . .; nÞ ;

ð4:174Þ

Thus, the ðai Þth order switching bifurcation can be for a cluster of higher-order fixed-points. The ðai Þth order switching bifurcations for i 2 f1; 2; . . .; sg are ai ¼

Pl i

ðiÞ r j¼1 j

ðai Þth order mXX ! switching bifurcation 8 ðiÞ ðiÞ ðiÞ ðiÞ > ðrsðiÞ Þth order mXXli # mYYli ; for bli # ali ; > > > > > > .. > > > . > > < ðiÞ ðiÞ ðiÞ ðiÞ ðrj Þth order mXXj # mYYj ; for bj # aðiÞ s ; > > > > > .. > > > . > > > > : ðiÞ th ðiÞ ðiÞ ðiÞ ðr1 Þ order mXX1 " mYY1 ; for b1 # aðiÞ s ;

ð4:175Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

fa1 ; a2 ; . . .; ali1 ; ali g ðiÞ

ðiÞ

ðiÞ

ðiÞ

fb1 ; b2 ; . . .; bli1 ; bli g



ð1Þ

before bifurcation



After bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .; b1 ; b2 g; ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ; . . .; b1 ; b2 g:

ð4:176Þ

A set of paralleled, different, higher-order upper-saddle-node switching bifurcations with multiplicity is the ðða1 Þth mXX:(a2 Þth mXX:    : ðas Þth mXXÞ parallel switching bifurcation in the (2m)th degree polynomial discrete system. At the straw-bundle switching bifurcation, Di ¼ 0 ði ¼ 1; 2; . . .; nÞ and Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ: The parallel straw-bundle switching bifurcation for higher order fixed-points is

4.3 Higher-Order Fixed-Points Bifurcations

311

ðða1 Þth mXX:ða2 Þth mXX:    : ðas Þth mXXÞ-switching 8 ðas Þth order mXX switching, > > > > > > > < ... ¼ > > > ða2 Þth order mXX switching, > > > > : ða1 Þth order mXX switching;

ð4:177Þ

ai 2 f2li ; 2li  1g and mXX 2 fmUS; mLS; mSO; mSIg:

ð4:178Þ

where

th Eight parallel straw-bundle switching bifurcations of ðath 1 mXX:a2 mXX:    : are presented in Fig. 4.10 and Fig. 4.11 for a0 [ 0 and a0 \0, respectively.

ath n mXXÞ

4.3.3

Appearing-Switching Bifurcations

Consider a (2m)th degree polynomial discrete system in a form of ai n2 aj 2 1 xk þ 1 ¼ xk þ a0 Qðxk Þ *2n i¼1 ðxk  ci Þ *j¼1 ðxk þ Bj xk þ Cj Þ :

ð4:179Þ

Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bj xk þ Cj ¼ 0 are ðjÞ

1 2

b1;2 ¼  Bj 

1pffiffiffiffiffi Dj ; Dj 2

¼ B2j  4Cj 0

ð j ¼ 1; 2; . . .; n2 Þ;

ð4:180Þ

either     fa 1 ; a2 ; . . .; a2n1 g ¼ sortfc1 ; c2 . . .; c2n1 g; as  as þ 1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

þ g ¼ sortfc1 ; . . .; c2n1 ; b1 ; b2 ; . . .; b1 2 ; b2 2 g; fa1þ ; a2þ ; . . .; a2n 3

asþ

 asþþ 1 ;

ð4:181Þ

n3 ¼ n1 þ n2 after bifurcation;

or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 ; a2 ; . . .; a2n3 g ¼ sortfc1 ; c2 . . .; c2n1 ; b1 ; b2 ; . . .; b1 ; b2 g;

 a s  as þ 1 ; n3 ¼ n1 þ n2 before bifurcation;

þ fa1þ ; a2þ ; . . .; a2n g 1

¼ sortfc1 ; . . .; c2n1 g;

asþ

ð4:182Þ  asþþ 1

after bifurcation;

4 (2m)th-Degree Polynomial Discrete Systems

312 a0 > 0

a2n

(2r2 n − 2 − 1) th mSO

a2 n−1

th

(2ln ) mUS (2r2 n −1 ) th mLS (2r2 n − 1) th mSI

a0 > 0

(2r2 n − 1) th mSO

(2r2 n − 2 − 1) th mSO th

(2ln − 1) mSO

th

(2r2 n −1 ) mLS

a2 n−2

(2r2 n − 2 − 1) th mSI

a2 n−3

(2r2 n −3 ) th mUS

(2r2 n −1 ) th mLS (2r2 n ) th LS

(2ln −1 − 1) mSO

a2 n−3

(2r2 n − 4 − 1) th mSO

(2r2 n −3 ) mLS

• • •

(2r2i −1 − 1) mSI

(2li ) th mLS

• • •

(2r2i ) th mLS (2r1 ) th mLS

(2r2i − 2 ) th mUS

(2r2i − 2 ) th mLS

(2r2i −3 − 1) th mSO

(2r2i −1 − 1) th mSI

a3

a1

(2r2 n − 4 − 1) th mSI

a0 > 0

a2 n−1

(2ln − 1) mSO (2r2 n −1 ) th mUS (2r2 n − 1) th mSO

a2

(2l1 − 1) th SO (2r2 − 1) th mSO

xk∗

a3

a1

(2r3 ) th mUS

(2r2 − 1) th mSO (2r1 ) th mLS

(2r2 ) th mLS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(ii) a2n

th

(2r2i −1 − 1) th mSO

(2r2i − 2 ) th mLS

• • •

(2r2 − 1) th mSI (2r1 ) th mUS

(2r2i ) th mUS

(2r2i −3 − 1) th mSI

(2r1 ) th mUS

Δ iq > 0

Δ iq > 0 Δ iq = 0

(2r2 n − 2 ) th mUS

a0 > 0

(2r2 n − 1) th mSO

(2r2 n − 2 ) th mUS th

(2ln − 1) mSO

th

(2r2 n −1 ) mLS

a2 n−2

(2r2 n − 2 ) th mLS

a2 n−3

(2r2 n −3 ) th mLS

(2r2 n −1 − 1) th mSO (2r2 n ) th mLS

a2n

(2r2 n ) th mUS

a2 n−1

(2r2 n −1 − 1) th mSO

a2 n−2

(2r2 n − 2 ) th mLS

a2 n−3

(2r2 n −3 ) th mLS

th

th

(2ln −1 − 1) mSO

(2ln −1 − 1) mSI (2r2 n − 4 − 1) th mSI

(2r2 n − 4 − 1) th mSI

a2 n−3

(2r2 n − 4 − 1) th mSI

a2 n−3

(2r2 n − 4 − 1) th mSI

th

th

(2r2 n −3 ) mUS

• •

(2r2i −1 − 1) th mSI

(2r1 ) th mUS (2l1 − 1) th mSI

a3 a2 a1

(2r2 ) th mLS

Δ iq > 0 Δ iq = 0

(iii)

Δ iq > 0

(2r2i −1 − 1) th mSO

(2r2i −1 − 1) th mSO

(2r2i − 2 ) th mLS

(2r2i − 2 ) th mLS

(2r2i −3 − 1) th mSI

(2r2i −1 − 1) th mSI

(2li ) th mUS

(2r2i −3 − 1) th mSI

•

(2r1 ) th mUS

(2r2 − 1) th mSO

(2l1 − 1) th mSO

(2r1 ) th mLS

xk∗

(2r2i − 2 ) th mLS

• •

(2r2i ) th US

(2r3 ) th mUS

(2r2 − 1) th mSO

(2r2i ) th mUS

• •

(2r2i −1 − 1) mSO

(2li ) th mLS

(2r2i ) th mUS

•

th

(2r2i −1 − 1) th mSO

• • •

(2r2 n −3 ) mUS

(2r2i ) th mUS

•

|| p ||

(2r2 n −3 ) th mLS

(2li ) th mUS

(i)

xk∗

a2 n−3

• • •

(2r2i ) th mUS

(2r3 ) mLS

(2r2 ) th mUS

(2r2 − 1) th mSO

(2r2 n − 2 − 1) th mSO

(2r2i −1 − 1) th mSO

th

a2

(2l1 − 1) th mSI

(2r2 n −3 ) mUS

th

(2r2i −1 − 1) th mSO

(2r2i − 2 ) th mLS

a2 n−2

a2 n−3

(2r2i ) th mLS

(2r2i −1 − 1) th mSI

|| p ||

(2r2 n −1 ) th mUS

th

th

xk∗

a2 n−1

(2r2 n − 4 − 1) th mSI

(2r2 n − 4 − 1) th mSO

(2r2 − 1) th mSI

(2r2 n ) th mUS

(2ln −1 − 1) th mSI

th

(2r2i − 2 ) th mUS

a2n

a3 a2 a1

(2r3 ) th mUS

(2r2 − 1) th mSO (2r1 ) th mLS

(2r2 ) th mLS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.10 (i)–(iv) Four types of ðr1 th mXX:r2 th mXX:    : rm th mXXÞ parallel switching bifurcation for a0 [ 0 in the (2m)th-degree polynomial system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4.3 Higher-Order Fixed-Points Bifurcations a0 < 0

a2n

(2r2 n − 2 − 1) th mSI

a2 n−1

th

(2ln ) mLS (2r2 n −1 ) th mUS (2r2 n − 1) th mSO

313 a0 < 0

(2r2 n − 1) th mSI

(2r2 n − 2 − 1) th mSI th

(2ln − 1) mSI

th

(2r2 n −1 ) mUS

a2 n−2

(2r2 n − 2 − 1) th mS

a2 n−3

(2r2 n −3 ) th mLS

(2r2 n −1 ) th mUS (2r2 n ) th mUS

(2ln −1 − 1) th mSI

a2 n−3 • •

(2r2 n − 4 − 1) th mS

th

(2r2i −1 − 1) mSO

(2r2i −1 − 1) th mSO

(2li ) mUS

(2r2i − 2 ) th mLS

(2r2i −1 − 1) mSI

th

(2r2i −3 − 1) mSI

•

(2r1 ) mUS

(2r2 n − 4 − 1) th mS

(2r2i ) th mLS

•

(2r2i −1 − 1) th mSI

(2r2i − 2 ) th mUS

th

(2r2i −1 − 1) mSO

(2r2i −3 − 1) th mSO •

a3 a2

th

(2l1 − 1) mSO

a1

•

th

(2r1 ) mLS

(2r1 ) th mLS

(2l1 − 1) mSI (2r2 − 1) th mSI

xk∗

Δ iq > 0

a3 a2

th

(2r2 − 1) th mSO

(2r2 ) mLS

Δ iq > 0 Δ iq = 0

•

(2r2i ) th LS

(2r3 ) th mUS

th

a1

(2r3 ) th mLS

(2r2 − 1) th mSI (2r1 ) th mUS

th

(2r2 ) mUS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(i)

(ii)

a0 < 0 (2r2 n − 2 ) th mLS (2ln − 1) th mSI (2r2 n −1 ) th mLS (2r2 n − 1) th mSI

a2n

(2r2 n − 1) th mSI

a2 n−1

(2r2 n −1 ) th mUS

a0 < 0 (2r2 n − 2 ) th mLS

a2 n−2

(2r2 n − 2 ) th mUS

a2 n−3

(2r2 n −3 ) th mUS

(2ln − 1) th mSI (2r2 n −1 − 1) th mSI (2r2 n ) th mUS

a2n

(2r2 n ) th mLS

a2 n−1

(2r2 n −1 − 1) th mSI

a2 n−2

(2r2 n − 2 ) th mUS

a2 n−3

(2r2 n −3 ) th mUS

th

th

(2ln −1 − 1) mSO

(2ln −1 − 1) mSI

(2r2 n − 4 − 1) th mSO

(2r2 n − 4 − 1) th mSO

a2 n−4

(2r2 n − 4 − 1) th mSO

a2 n−3

(2r2 n − 4 − 1) th mSO

th

th

(2r2 n −3 ) mLS

• •

(2r2 n −3 ) LS

(2r2i ) th mLS

•

th

(2r2i −1 − 1) mSI

(2r2i −1 − 1) th mSI

(2li ) th mLS

th

(2r2i −1 − 1) mSO

• •

(2r2i ) th mLS (2r1 ) th mLS (2l1 − 1) th mSI

a3 a2

(2r2 ) th mUS

(iii)

(2r2i ) th mLS

•

(2r2i −1 − 1) th mSI

(2r2i − 2 ) th mUS

(2r2i − 2 ) th mUS

(2r2i −3 − 1) th mSO

(2r2i −1 − 1) th mSO

(2li ) th mLS

(2r2i − 2 ) th mUS (2r2i −3 − 1) th mSO •

a1

Δ iq > 0 Δ iq = 0

• •

(2r2i −1 − 1) th mSI

•

|| p ||

(2r2 n −3 ) th mUS

(2li ) mLS

(2r2i − 2 ) th mUS

•

th

xk∗

a2 n−3

• •

•

(2r2i ) th mUS

(2r2 − 1) th mSI

(2r2 n − 2 − 1) th mS

th

th

(2r2i − 2 ) th mUS

a2 n−2

(2r2i −1 − 1) th mSI

th

|| p ||

(2r2 n −1 ) th mLS

a2 n−3 (2r2 n −3 ) th mLS

(2r2i ) th mUS

•

xk∗

a2 n−1

(2r2 n − 4 − 1) th mSO

(2r2 n −3 ) th mUS

(2r2 − 1) th mSO

(2r2 n ) th mLS

(2ln −1 − 1) th mSO

(2r2 n − 4 − 1) th mSI

(2r2i − 2 ) th mLS

a2n

Δ iq > 0

• •

(2r2i ) th mLS

(2r3 ) th mLS

(2r1 ) th mLS

(2r2 − 1) th mSI

(2r2 − 1) th mSI

(2l1 − 1) th mSI

th

(2r1 ) mUS

xk∗

a3 a2 a1

(2r3 ) th mLS

(2r2 − 1) th mSI (2r1 ) th mLS

(2r2 ) th mUS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.11 (i)–(iv) Four types of ðr1 th mXX:r2 th mXX:    : rm th mXXÞ parallel switching bifurcation for a0 [ 0 in the (2m)th-degree polynomial discrete system. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4 (2m)th-Degree Polynomial Discrete Systems

314

and 9 Bj1 ¼ Bj2 ¼    ¼ Bjs ðjk1 2 f1; 2; . . .; ng; jk1 6¼ jk2 Þ > > = ðk1 ; k2 2 f1; 2; . . .; sg; k1 6¼ k2 Þ at bifurcation: Dj ¼ 0 ðj 2 U f1; 2; . . .; n2 g > > ; ci 6¼ 12Bj ði ¼ 1; 2; . . .; 2n1 ; j ¼ 1; 2; . . .; n2 Þ

ð4:183Þ

th  th   Consider a just before bifurcation of ððb 1 Þ mXX1 : ðb2 Þ mXX2 :    : th     ðb s1 Þ mXXs1 Þ for higher-order fixed-points. For bi ¼ 2li  1; XXi 2 fSO,SIg    and for ai ¼ 2li ; XXi 2 fUS,LSg ði ¼ 1; 2; . . .; s1 Þ. The detailed structures are as follows.

9 Þth order mXXðiÞ ; xk ¼ aðiÞ ðrsðiÞ s si ; > > i > > > > > .. > > . > > > = bi ¼Psi rðiÞ j¼1 j ðiÞ th ðiÞ  ðiÞ th ðiÞ !ðb ðrj Þ order mXXj ; xk ¼ aj i Þ order mXX switching bifurcation > > > > > .. > > . > > > > > ðiÞ th ðiÞ  ðiÞ ; ðr1 Þ order mXX1 ; xk ¼ a1 ð4:184Þ th  th  th    The bifurcation set of ððb 1 Þ mXX1 : ðb2 Þ mXX2 :    : ðbs1 Þ mXXs1 Þ at the same parameter point is called a left-parallel-straw-bundle switching bifurcation. Consider a just after bifurcation of ððb1þ Þth mXX1þ : ðb2þ Þth mXX2þ :    : ðbsþ2 Þth mXXsþ2 Þ for monotonic sources and monotonic sinks. For biþ ¼ 2liþ  1; XXiþ 2 fSO,SIg and for biþ ¼ 2liþ ; XXiþ 2 fUS,LSg . The detailed structures are as follows.

ðbiþ Þth order mXXðiÞ þ

8 ðiÞ þ th þ þ > ðrsi Þ order mXXðiÞ ; xk ¼ aðiÞ ; si si > > > > > > .. > > . > > > Psi ðiÞ þ < b ¼ r i j¼1 j ðiÞ þ ðiÞ þ ðiÞ þ ! ðrj Þth order mXXj ; xk ¼ aj switching bifurcation > > > > > .. > > . > > > > > : ðiÞ þ th ðiÞ þ ðiÞ þ ðr1 Þ order mXX1 ; xk ¼ a1 : ð4:185Þ

4.3 Higher-Order Fixed-Points Bifurcations

315

The bifurcation set of ððb1þ Þth mXX1þ : ðb2þ Þth mXX2þ :    : ðbsþ2 Þth mXXsþ2 Þ at the same parameter point is called a right-parallel-straw-bundle switching bifurcation (i) For the just before and after bifurcation structure, if there exists a relation of th þ th th  þ   þ ðb i Þ mXXi ¼ ðbj Þ mXXj ¼ bj mXX, for xk ¼ ai ¼ aj

ði; j 2 f1; 2; . . .; kgÞ; XX 2 fUS,LS,SO,SIg

ð4:186Þ

then the bifurcation is a ðbj Þth mXXj switching bifurcation for higher-order fixed-points. (ii) Just for the just before bifurcation structure, if there exists a relation of th th    ð2l i Þ mXXi ¼ ð2li Þ mXX, for xk ¼ ai ¼ ai

ði 2 f1; 2; . . .; s1 g; mXX 2 fmUS,mLSg

ð4:187Þ

then, the bifurcation is a ð2lÞth mXX left appearing (or right vanishing) bifurcation for higher-order fixed-points. (iii) Just for the just after bifurcation structure, if there exists a relation of ð2liþ Þth mXXiþ ¼ ð2li Þth mXX, for xk ¼ aiþ ¼ ai ði 2 f1; 2; . . .; s1 gÞ; XX 2 fUS,LSg

ð4:188Þ

then, the bifurcation is a ð2lÞth mXX right appearing (or left vanishing) bifurcation for higher-order fixed-points. (iv) For the just before and after bifurcation structure, if there exists a relation of th þ th  þ   þ ðb i Þ mXXi 6¼ ðbj Þ mXXj for xk ¼ ai ¼ aj þ XX i ; XXj 2 fUS,LS, SO,SIg

ð4:189Þ

ði 2 f1; 2; . . .; s1 g; j 2 f1; 2; . . .; s2 gÞ; then, two flower-bundle switching bifurcations of higher-order fixed-points are as follows. (iv1) For bj ¼ bi þ 2l, the bifurcation is called a bth j mXX right flower-bundle th switching bifurcation for the bth i mXX to bj mXX switching of higher-

order fixed-points with the appearance (or birth) of ð2lÞth mXX right appearing (or left vanishing) bifurcation. (iv2) For bj ¼ bi  2l, the bifurcation is called a bth i mXX left flower-bundle th th switching bifurcation for the bi mXX to bj mXX switching of higher-order

316

4 (2m)th-Degree Polynomial Discrete Systems

fixed-points with the vanishing (or death) of ð2lÞth mXX left appearing (or right vanishing) bifurcation. A general parallel switching bifurcation is switching

th  th  th    ððb ! 1 Þ mXX1 : ðb2 Þ mXX2 :    : ðbs1 Þ mXXs1 Þ 

ððb1þ Þth mXX1þ : ðb2þ Þth mXX2þ :    :

bifurcation þ th þ ðbs2 Þ mXXs2 Þ:

ð4:190Þ

Such a general, parallel switching bifurcation consists of the left and right parallelbundle switching bifurcations for higher-order fixed-points. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th þ th th  þ ðb i Þ mXXi ¼ ðbi Þ mXXi ¼ b mXX,

þ for xk ¼ a i ¼ ai ði ¼ 1; 2; . . .; sg

ð4:191Þ

then the parallel flower-bundle switching bifurcation becomes a parallel straw-bundle switching bifurcation of ðða1 Þth mXX:ðb2 Þth mXX:    : ðbs Þth mXXÞ: If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th   th þ th þ þ th ða i Þ mXXi ¼ ð2li Þ mXX, ðaj Þ mXXj ¼ ð2lj Þ mYY, þ for xk ¼ a i 6¼ aj ði ¼ 1; 2; . . .; s1 ; j ¼ 1; 2; . . .; s2 Þ;

ð4:192Þ

XX 2 fUS; LSg; YY 2 fUS; LSg: then the parallel flower-bundle switching bifurcation for higher-order fixed-points becomes a combination of two independent left and right parallel appearing bifurcations: th     th  th (i) a ðð2l 1 Þ mXX1 : ð2l2 Þ mXX2 :    : ð2ls1 Þ mXXs1 Þ-left parallel sprinklerspraying appearing (or right vanishing) bifurcation and (ii) a ðð2l1þ Þth mXX1þ : ð2l2þ Þth mXX2þ :    : ð2lsþ2 Þth mXXsþ2 Þ-right parallel sprinkler-spraying appearing (or left vanishing) bifurcation.

The parallel switching and appearing bifurcations for higher-order fixed-points are presented in Fig. 4.12(i)–(iv). The waterfall appearing bifurcations and the flower-bundle switching bifurcations for higher-order fixed-points are presented.

4.3 Higher-Order Fixed-Points Bifurcations

317

(2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 1) th mSI

(2rn − 1) th mSO (2ls ) th mUS

(2ls ) th mLS

(2rn − 2 ) th mLS (2rn − 2 − 1) th mSI

(2ris ) th mUS

(2ris ) th mUS

(2rn − 2 ) th mUS

(2rn − 2 − 1) th mSO

(2ris ) th mLS (2ris ) th mLS

(2rn −3 − 1) th mSO (2rn − 4 − 1) th mSI

(2rn − 4 − 1) th mSO

(2ls −1 ) th mUS

(2ls −1 ) th mLS (2rn −3 − 1) th mSO

• • •

(2rik − 1) th mSO

(2rik − 1) th mSO

(2ri ) th mUS

(2rn −3 − 1) th mSI

• • •

(2rn − 4 − 1) th mSI (2rik − 1) th mSI

(2rn − 4 − 1) th mSO th

(2rik − 1) mSI

(2ri −1 ) th mUS (2ri − 2 − 1) th mSO (2ri −3 − 1) mSI (2lr ) th mUS

(2ri −3 − 1) th mSO (2lr ) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

th

(2ri −1 ) mLS • • •

(2ri −1 ) th mUS

(2ri −3 − 1) th mSI (2ri2 − 1) th mSI

• • •

(2ri ) th mUS

(2ri −3 − 1) th mSO (2ri2 ) th mSO

(2r1 − 1) th mSI

(2r1 − 1) th mSO

th

(2l1 ) mUS

(2l1 ) th mLS th

(2r1 − 1) mSI

(2ri1 ) th mUS th

(2r2 ) th mUS

(2ri1 ) mUS

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(2r1 − 1) th mSO

(2ri1 ) th mUS

xk∗

(2r2 ) th mLS (2ri1 ) th mUS

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(i)

(iii) (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

th

(2rn − 2 ) mUS

(2rn − 1) th mSI

(2rn − 1) th mSO (2ls ) th mUS

(2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2ls ) th mLS

(2rn − 2 − 1) th mSI

(2ris ) th mUS

(2rn − 2 − 1) th mSO

(2ris ) th mLS

th

(2ris ) mUS

(2ris ) th mLS (2rn −3 − 1) th mSI

(2rn −3 − 1) th mSO

(2ls −1 ) th mUS

(2rn − 4 − 1) th mSO

(2ris−2 − 1) th mSO

(2ls −1 ) th mLS

(2ris−2 − 1) th mSI

(2ris−1 − 1) th mSO

(2rn − 4 − 1) th mSI

(2ris−1 − 1) th mSI

(2ris−1 − 1) th mSI (2rn −3 − 1) th mSO

th

(2ris−2 − 1) mSI

• • •

(2rn − 4 − 1) th mSI

(2ris−1 − 1) th mSO (2rn −3 − 1) th mSI

(2ris−2 − 1) th mSO

(2rn − 4 − 1) th mSO

• • •

(2ri ) th mUS

(2ri ) th mLS

(2ri −1 ) th mUS (2ri − 2 − 1) th mSO

(2ri −1 ) th mLS (2ri − 2 − 1) th mSI

th

(2ri −3 − 1) mSI (2lr ) th mUS

(2ri −3 − 1) th mSO (2lr ) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

(2ri −1 ) th mLS • • •

(2ri −1 ) th mUS

(2ri −3 − 1) th mSI

• • •

(2ri ) th mUS

(2r2 ) th mUS

(2ri −3 − 1) th mSO (2ri ) th mLS

(2r2 ) th mLS

(2r1 − 1) th mSO

(2r1 − 1) th mSI

th

(2l1 ) mUS

xk∗

(2l1 ) th mLS

(2ri1 ) th mLS

(2ri1 ) th mLS

(2r1 − 1) th mSI

x∗

(2ri1 ) th mUS

(2ri1 ) th mUS

(2r1 − 1) th mSO

(2r2 ) th mUS

|| p ||

(2ri ) th mLS

(2r2 ) th mLS

(2ri2 − 1) th mSO

(2r2 ) th mUS

(2ri2 − 1) th mSI

(2ri ) th mLS (2ri −1 ) th mLS (2ri − 2 − 1) th mSI

th

xk∗

(2rn −3 − 1) th mSI

Δ iq < 0 Δ iq = 0

(ii)

Δ iq > 0

(2r2 ) th mLS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 4.12 ðr1 th mXX:r2 th mXX:    : rn th mXXÞ parallel switching-appearing bifurcations. ða0 [ 0Þ: (i) without switching, and (ii) with switching. ða0 [ 0Þ: (iii) without switching, and (vi) with switching. mLS: monotonic-lower-saddle, mUS: monotonic-upper-saddle, mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols

4 (2m)th-Degree Polynomial Discrete Systems

318

4.4

Forward Bifurcation Trees

In this section, the analytical bifurcation scenario of a (2m)th-degree polynomial nonlinear discrete system will be discussed. The period-doubling bifurcation scenario will be discussed first through nonlinear renormalization techniques, and the bifurcation scenario based on the monotonic saddle-node bifurcation will be discussed, which is independent of period-1 fixed-points.

4.4.1

Period-Doubled (2m)th-Degree Polynomial Discrete Systems

After the period-doubling bifurcation of a period-1 fixed-point, the period-doubled fixed-points of a (2m)th-degree polynomial nonlinear discrete system can be obtained. Consider the period-doubling solutions of a forward (2m)th-degree polynomial nonlinear discrete system. Theorem 4.1 Consider a (2m)th-degree polynomial nonlinear discrete system 2m1 þ    þ A2m2 ðpÞx2k þ A2m1 ðpÞxk þ A2m ðpÞ xk þ 1 ¼ xk þ A0 ðpÞx2m k þ A1 ðpÞxk

¼ xk þ a0 ðpÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ    ½x2k þ Bm ðpÞxk þ Cm ðpÞ ð4:193Þ where A0 ðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ; . . .; pm1 ÞT :

ð4:194Þ

If Di ¼ B2i  4Ci [ 0; i ¼ i1 ; i2 ; . . .; il 2 f1; 2; . . .; mg0f∅g; Dj ¼ B2j  4Cj \0; j ¼ il þ 1 ; il þ 2 ; . . .; im 2 f1; 2; . . .; mg0f∅g with l 2 f0; 1; . . .; mg;

ð4:195Þ

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

ð4:196Þ

4.4 Forward Bifurcation Trees

319

where ð1Þ

ð1Þ

bi;1 ¼ 12ðBi þ ð1Þ

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ ð1Þ Di Þ; bi;2 ¼ 12ðBi  Di Þ

for Di 0; i 2 f1; 2; . . .; lg0f∅g; ð1Þ ð1Þ ð1Þ ð1Þ ðiÞ l ; b gg; ai  ai þ 1 ; 02l i¼1 fai g ¼ sortf0 i1 ¼1 fb qffiffiffiffiffiffiffiffiffiffi ffi i1 ;2 i1 ;2 qffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ bi;1 ¼ 12ðBi þ i jDi jÞ; bi;2 ¼ 12ðBi  i jDi jÞ p ffiffiffiffiffiffi ffi ð1Þ for Di \0; i 2 fl þ 1; l þ 2; . . .; mg0f∅g; i ¼ 1; ð1Þ ð1Þ ð1Þ m 02m i¼2l þ 1 fai g ¼ f0i1 ¼l þ 1 fbi1 ;2 ; bi1 ;2 gg:

ð4:197Þ

(i) Consider a forward period-2 discrete system of Eq. (4.193) as ð1Þ

xk þ 2 ¼ xk þ ½a0 *2m i1 ¼1 ðxk  ai1 Þf1 þ

*i1 ¼1 ½1 þ a0 *i2 ¼1;i2 6¼i1 ðxk 2m

2m

ð1Þ

ðð2mÞ2 2mÞ=2

ð1Þ

ð2mÞ2 2m

2m ¼ xk þ ½a0 *2m i1 ¼1 ðxk  ai1 Þ½a0 *i2 ¼1

¼ xk þ ½a0 *3j1 ¼1 ðxk  ai1 Þ½a2m 0 *j2 ¼1 ð2mÞ2

ð2Þ

ð1Þ

 ai2 Þg ð2Þ

ðx2k þ Bi2 xk þ Ci2 Þ ð2Þ

ðxk  bj2 Þ

ð2Þ

¼ xk þ a10 þ 2m *i¼1 ðxk  ai Þ ð4:198Þ where

qffiffiffiffiffiffiffiffi 1 ð2Þ pffiffiffiffiffiffiffiffi ð2Þ 1 ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ Dð2Þ Þ; bi;2 ¼  ðBi  Di Þ; 2 2 0 2 ð2Þ ð2Þ ð2Þ Di ¼ ðBi Þ2  4Ci 0; i 2 0Nq11¼1 Iqð21 Þ 00Nq22¼1 Iqð22 Þ Iqð21 Þ ¼ flðq1 1Þ 20 m1 þ 1 ; lðq1 1Þ 20 m1 þ 2 ;    ; lq1 20 m1 g 0

f1; 2;    ; M1 g0f∅g; q1 2 f1; 2;    ; N1 g; M1 ¼ N1 20 m1 ; m1 2 f1; 2;    ; mg; Iqð22 Þ ¼ flðq2 1Þ 21 m1 þ 1 ; lðq2 1Þ 21 m1 þ 2 ;    ; lq2 21 m1 g 1

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; q2 2 f1; 2;    ; N2 g; M2 ¼ ðð2mÞ2  2mÞ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i jDð2Þ jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0; i 2 J ð2 Þ ¼ flN2 21 m1 þ 1 ; lN2 21 m1 þ 2 ;    ; lM2 g fM1 þ 1; M1 þ 2;    ; M2 g 1

ð4:199Þ

4 (2m)th-Degree Polynomial Discrete Systems

320

with fixed-points ð2Þ

xk þ 2 ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; ð2mÞ2 Þ 2

ð2mÞ

ð2Þ

ð1Þ

ð2Þ

ð2Þ

ð4:200Þ

M 0i¼1 fai g ¼ sortf02m j1 ¼1 faj1 g; 0j2 ¼1 fbj2 ;1 ; bj2 ;2 gg

with

ð2Þ ð2Þ ai \ai þ 1 ;

2

M ¼ ðð2mÞ  2mÞ=2: ð1Þ

(ii) For a fixed-point of xk þ 1 ¼ xk ¼ ai1 ði1 2 f1; 2; . . .; 2mgÞ, if dxk þ 1 ð1Þ ð1Þ j  ð1Þ ¼ 1 þ a0 *2m i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1

ð4:201Þ

with • a r th -order oscillatory upper-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ k

i1

k

i1

[ 0; r ¼ 2l1 Þ, • a r th -order oscillatory lower-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ \0; r ¼ 2l1 Þ, • a r th -order oscillatory sink bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ \0; r ¼ k

i1

2l1 þ 1Þ, • a r th -order oscillatory source bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ [ 0; r ¼ k

2l1 þ 1Þ,

i1

then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi1 ; Di1 ¼ ðBi1 Þ2  4Ci1 ¼ 0; 2

ð4:202Þ

and there is a period-2 discrete system of the quartic discrete system in Eq. (4.193) as 1 þ ð2mÞ

xk þ 2 ¼ x k þ a0

*

ð20 Þ i2 2Iq1

ð1Þ

ð2mÞ2

ð2Þ

ðxk  ai2 Þ3 *i3 ¼1 ðxk  ai3 Þð1dði2 ;i3 ÞÞ ð4:203Þ

for i1 2 f1; 2; . . .; 2mg; i1 6¼ i2 with dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i1 dx2k xk ¼ai1

ð4:204Þ

4.4 Forward Bifurcation Trees

321

ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-order if d 3 xk þ 2 ð1Þ ð1Þ j  ð1Þ ¼ 6a10 þ 2m * ð20 Þ ða  ai2 Þ3 i2 2Iq1 ;i2 6¼i1 i1 dx3k xk ¼ai1

ð2mÞ2 ð1Þ *i3 ¼1 ðai1



ð4:205Þ

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ \0;

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the period-2 discrete system; ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk þ 2 ð1Þ ð1Þ j  ð1Þ ¼ 6a10 þ 2m * ð20 Þ ða  ai2 Þ3 i2 2Iq1 ;i2 6¼i1 i1 dx3k xk ¼ai1

ð2mÞ2 ð1Þ *i3 ¼1 ðai1



ð2Þ ai3 Þð1dði2 ;i3 ÞÞ

ð4:206Þ

[ 0;

and the corresponding bifurcations is a third-order monotonic source bifurcation for the period-2 discrete system. (ii1) The period-2 fixed-points are trivial and unstable if ð2Þ

xk þ 2 ¼ xk ¼ ai

ð1Þ

2 02m i1 fai1 g:

ð4:207Þ

(ii2) The period-2 fixed-points are non-trivial and stable if ð2Þ

xk þ 2 ¼ xk ¼ ai

ð2Þ

ð2Þ

2 2 0M i1 ¼1 fbi1 ;1 ; bi1 ;2 g :

ð4:208Þ

Proof The proof is straightforward through the simple algebraic manipulation. Following the proof of quadratic discrete system, this theorem is proved. ■

4.4.2

Renormalization and Period-Doubling

The generalized case of period-doublization of a (2m)th-degree polynomial discrete system is presented through the following theorem. The analytical period-doubling bifurcation trees can be developed for such a (2m)th-degree polynomial discrete systems.

4 (2m)th-Degree Polynomial Discrete Systems

322

Theorem 4.2 Consider a 1-dimensional (2m)th-degree polynomial discrete system as 2m1 þ    þ A2m2 x2k þ A2m1 xk þ A2m xk þ 1 ¼ xk þ A0 x2m k þ A 1 xk

¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

ð4:209Þ

(i) After l-times period-doubling bifurcations, a period-2l discrete system ðl ¼ 1; 2; . . .Þ for the (2m)th-degree polynomial discrete system in Eq. (4.209) is given through the nonlinear renormalization as ð2l1 Þ

ð2mÞ2

xk þ 2l ¼ xk þ ½a0

f1 þ

ð2mÞ

ð2l1 Þ

ð2

ð2mÞ

Þ 4

Þ

ð2l1 Þ

¼ xk þ ½a0

Þ

ð2l Þ

¼ x k þ a0

2l1

ðð2mÞ2 ð2mÞ2

l1

ð2mÞ2

l

l1

ð2mÞ2

*i¼1

l

ð2l1 Þ

 ai 2

Þg

Þ

Þ=2

ð2l1 Þ

ðxk  ai1

*i2 ¼1

Þ

*i¼1

ð2l1 Þ

ðxk  ai1

l

l1

l1

*i ¼1;i 6¼i ðxk 2 2 1

ðð2mÞ ð2mÞ l1

Þ

ð2mÞ2

Þ

*j1 ¼1

ð2l1 Þ 1 þ ð2mÞ2

¼ xk þ ða0

l1

2l

ð2mÞ2 *i1 ¼1;

ð2l1 Þ 42

½ða0

ð2

*i1 ¼1 2l1

ð2l1 Þ

ðxk  ai1

½1 þ a0 2l1

¼ xk þ ½a0

½ða0

2l1

*i1 ¼1

l1

l1

*i1 ¼1

ð2l Þ

ð2l Þ

ðx2k þ Bj2 xk þ Cj2 Þ

ð4:210Þ

Þ

Þ=2

ð2l Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ ð2l Þ

ðxk  ai Þ

ð2l Þ

ðxk  ai Þ

with l l dxk þ 2l ð2l Þ Xð2mÞ2 ð2mÞ2 ð2l Þ ¼ 1 þ a0 * i1 ¼1 i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk l l l d 2 xk þ 2l ð2l Þ Xð2mÞ2 Xð2mÞ2 ð2mÞ2 ð2l Þ ¼ a * 0 i ¼1 i2 ¼1;i2 6¼i1 i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 1 dx2k .. . l l l d r xk þ 2 l Xð2mÞ2 ð2l Þ Xð2mÞ2 ð2mÞ2 ð2l Þ ¼ a0 i1 ¼1    ir ¼1;i3 6¼i1 ;i2 ...ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ...;ir ðxk  air þ 1 Þ r dxk

ð4:211Þ l

for r  ð2mÞ2 where

4.4 Forward Bifurcation Trees

323 ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1 þ 2m ; a0 2l

ð2l Þ

ð2mÞ

ð2l1 Þ 1 þ ð2mÞ2

¼ ða0 ð2mÞ

Þ

2l1

l1

; l ¼ 1; 2; 3;    ;

ð2l Þ

ð2l Þ

ð2l Þ

ð2l Þ

2 0i¼1 fai g ¼ sortf0i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg, ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ bi;1 ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

2

ð2l Þ

Di

Iqð21

l1

2

ð2l Þ

ð2l Þ

¼ ðBi Þ2  4Ci Þ

ð2l Þ

 ai þ 1 ;

l1

0 for i 2 0Nq11¼1 Iqð21

Þ

00Nq ¼1 Iqð2 Þ ; 2 2

l

2

¼ flðq1 1Þ 2l1 m1 þ 1 ; lðq1 1Þ 2l1 m1 þ 2 ;    ; lq1 2l1 m1 g f1; 2;    ; M1 g0f∅g;

for q1 2 f1; 2;    ; N1 g; M1 ¼ N1 2l1 m1 ; m1 2 f1; 2;    ; mg; l

Iqð22 Þ ¼ flðq2 1Þ 2l m1 þ 1 ; lðq2 1Þ 2l m1 þ 2 ;    ; lq2 2l m1 g fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; l

l1

for q2 2 f1; 2;    ; N2 g; M2 ¼ ðð2mÞ2  ð2mÞ2 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN2 2l m1 þ 1 ; lN2 2l m1 þ 2 ;    ; lM2 g

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; ð4:212Þ

with fixed-points ð2l Þ

l

xk þ 2l ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; ð2mÞ2 Þ ð2mÞ2

l

ð2l Þ

ð2mÞ2

0i¼1 fai g ¼ sortf0i1 ¼1 l

ð2 Þ

l

ð2 Þ

l1

ð2l1 Þ

fai1

ð2l Þ

ð2l Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð4:213Þ

with ai \ai þ 1 : ð2l1 Þ

(ii) For a fixed-point of xk þ 2l1 ¼ xk ¼ ai1

ð2l1 Þ

ði1 2 Iq

Þ, if

l1 dxk þ 2l1 ð2l1 Þ ð2mÞ2 ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 1 þ a0  ai2 Þ ¼ 1; *i ¼1;i 6¼i ðai1 2 2 1 dxk xk ¼ai1 d s xk þ 2l1 j ð2l1 Þ ¼ 0; for s ¼ 2;    ; r  1; dxsk xk ¼ai1 l1 d r xk þ 2l1 j  ð2l1 Þ 6¼ 0 for 1\r  ð2mÞ2 ; r x ¼a dxk i1 k

ð4:214Þ

4 (2m)th-Degree Polynomial Discrete Systems

324

with • a rth-order oscillatory sink for d r xk þ 2l1 =dxrk j

ð2l1 Þ

xk ¼ai

• a rth-order oscillatory source for d r xk þ 2l1 =dxrk j • a r

th

r

[ 0 and r ¼ 2l1 þ 1;

1 ð2l1 Þ

xk ¼ai

\0 and r ¼ 2l1 þ 1;

1

-order oscillatory upper-saddle for d xk þ 2l1 =dxrk j

ð2l1 Þ

xk ¼ai

r ¼ 2l1 ; • a rth-order oscillatory lower-saddle for d r xk þ 2l1 =dxrk j r ¼ 2l1 ;

[ 0 and

1

ð2l1 Þ

xk ¼ai

\0 and

1

then there is a period- 2l fixed-point discrete system ð2l Þ

x k þ 2 l ¼ x k þ a0

ð2mÞ2 *j2 ¼1

l

*

ð2l1 Þ

i1 2Iq1

ðxk 

ð2l1 Þ 3

ðxk  ai1

Þ

ð4:215Þ

ð2l Þ aj2 Þð1dði1 ;j2 ÞÞ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð4:216Þ

and dxk þ 2l d 2 xk þ 2 l j  ð2l1 Þ ¼ 1; j ð2l1 Þ ¼ 0: dxk xk ¼ai1 dx2k xk ¼ai1 ð2l1 Þ

• xk þ 2l at xk ¼ ai1

ð4:217Þ

is a monotonic sink of the third-order if

d 3 xk þ 2l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai 2 Þ 3 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k ð2mÞ2

l

ð2l1 Þ

*j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

ð4:218Þ

; q 2 f1; 2; . . .; N1 gÞ; ð2l1 Þ

and such a bifurcation at xk ¼ ai1 bifurcation.

is a third-order monotonic sink

4.4 Forward Bifurcation Trees

325

ð2l1 Þ • xk þ 2l at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk þ 2 l ð2l Þ ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 6a0 * ð2l1 Þ ðai1  ai2 Þ3 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k ð2mÞ2

l

ð2l1 Þ

*j2 ¼1 ðai1 ði1 2 Iqð2

l1

Þ

ð4:219Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ [ 0

; q 2 f1; 2; . . .; N1 gÞ ð2l1 Þ

and such a bifurcation at xk ¼ ai1 bifurcation.

is a third-order monotonic source

(ii1) The period- 2l fixed-points are trivial if ð2l Þ

xk þ 2l ¼ xk ¼ ai

ð2mÞ2

2 0i1 ¼1

l1

ð2l1 Þ

fai1

l1

g for i1 ¼ 1; 2; . . .; ð2mÞ2 :

ð4:220Þ

(ii2) The period- 2l fixed-points are non-trivial if ð2l Þ

xk þ 2l ¼ xk ¼ ai

ð2mÞ2

2 0i1 ¼1

l1

ð2l Þ

ð2l Þ

fbi1 ;1 ; bi1 ;2 g

j1 2 f1; 2; . . .; M2 g0f∅g

:

ð4:221Þ

Such a period-2l fixed-point is • monotonically unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

• monotonically invariant if dxk þ 2l =dxk j

2 ð1; 1Þ;

1 ð2l Þ

xk ¼ai

¼ 1, which is

1

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l = dx2l k jx [ 0; k

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \0; k

1 þ1 – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l = dx2l jx k k [ 0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \0; k

• monotonically stable if dxk þ 2l =dxk j

• invariantly zero-stable if dxk þ 2l =dxk j • oscillatorilly stable if dxk þ 2l =dxk j • flipped if dxk þ 2l =dxk j

ð2l1 Þ

xk ¼ai

2 ð0; 1Þ;

ð2l Þ

xk ¼ai

1 ð2l1 Þ

xk ¼ai ð2l1 Þ

xk ¼ai

1

¼ 0;

2 ð1; 0Þ;

1

¼ 1, which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l = dx2l k jx [ 0; k

1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \0; k

4 (2m)th-Degree Polynomial Discrete Systems

326

1 þ1 – an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk þ 2l = dx2l jx k k \0; th 2l1 þ 1 2l1 þ 1 xk þ 2l = dxk jx – an oscillatory sink the ð2l1 þ 1Þ order with d k [ 0;

• oscillatorilly unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, this theorem can be proved.

4.4.3

■

Period-n Appearing and Period-Doublization

The forward period-n discrete system for the (2m)th-degree polynomial quartic nonlinear discrete systems will be discussed, and the period-doublization of periodn discrete systems is discussed through the nonlinear renormalization. Theorem 4.3 Consider a 1-dimensional (2m)th-degree polynomial discrete system as 2m1 xk þ 1 ¼ xk þ A0 x2m þ    þ A2m2 x2k þ A2m1 xk þ A2m k þ A 1 xk

¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

ð4:222Þ

(i) After n-times iterations, a period-n discrete system for the quartic discrete system in Eq. (4.222) is xk þ n ¼ xk þ a0 *2m i1 ¼1 ðxk  ai2 Þ½1 þ n

ðð2mÞ 1Þ=ð2m1Þ

¼ xk þ a 0

ðnÞ

¼ xk þ a 0

ð2mÞ

*i¼1

n

Xn

*i1 ¼1 ðxk 2m

j¼1

Qj  ðð2mÞn 2mÞ=2

 ai1 Þ½*j2 ¼1

ðnÞ

ðnÞ

ðx2k þ Bj2 xk þ Cj2 Þ

ðnÞ

ðxk  ai Þ

ð4:223Þ with dxk þ n ðnÞ Xð2mÞn ð2mÞn ðnÞ ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk d 2 xk þ n ðnÞ Xð2mÞn Xð2mÞn ð2mÞn ðnÞ ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . d r xk þ n Xð2mÞn ðnÞ Xð2mÞn ð2mÞn ðnÞ ¼ a0 i1 ¼1    ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ r dxk

for r  ð2mÞn ;

ð4:224Þ

4.4 Forward Bifurcation Trees

327

where ðnÞ

a0 ¼ ða0 Þðð2mÞ

n

1Þ=ð2m1Þ

ð1Þ

2m ; Q1 ¼ 0; Q2 ¼ *2m i2 ¼1 ½1 þ a0 *i1 ¼1;i1 6¼i2 ðxk  ai1 Þ; ð1Þ

2m Qn ¼ *2m in ¼1 ½1 þ a0 ð1 þ Qn1 Þ *in1 ¼1;in1 6¼in ðxk  ain1 Þ; n ¼ 3; 4;    ; ð2mÞn

ðnÞ

ð1Þ

ðnÞ

ðnÞ

M 0i¼1 fai g ¼ sortf02m i1 ¼1 fai1 g; 0i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi2 ;1 ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ; 2 2 ðnÞ ðnÞ ðnÞ Di2 ¼ ðBi2 Þ2  4Ci2 0 for i2 2 0Nq¼1 IqðnÞ ;

IqðnÞ ¼ flðq1Þ n þ 1 ; lðq1Þ n þ 2 ;    ; lq n gf1; 2;    ; Mg0f∅g; for q 2 f1; 2;    ; Ng; M ¼ ðð2mÞn  2mÞ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flN n þ 1 ; lN n þ 2 ;    ; lM g f1; 2;    ; Mg0f∅g; ð4:225Þ

with fixed-points ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; ð2mÞn Þ n

ð2mÞ

ðnÞ

ð1Þ

ðnÞ

ðnÞ

M 0i¼1 fai g ¼ sortf02m i1 ¼1 fai1 g; 0i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

with

ð4:226Þ

ðnÞ ðnÞ ai \ai þ 1 :

ðnÞ

ðnÞ

(ii) For a fixed-point of xk þ n ¼ xk ¼ ai1 ði1 2 Iq ; q 2 f1; 2; . . .; NgÞ, if dxk þ n ðnÞ ð2mÞn ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk k i1

ð4:227Þ

d 2 xk þ n ðnÞ Xð2mÞn ð2mÞn ðnÞ ðnÞ jx ¼aðnÞ ¼ a0 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; 2 i1 k dxk

ð4:228Þ

with

then there is a new discrete system for onset of the qth-set of period-n fixed-points based on the second-order monotonic saddle-node bifurcation as ðnÞ

x k þ n ¼ x k þ a0

*i 2I ðnÞ ðxk 1 q

ðnÞ

ð2mÞn

ðnÞ

 ai1 Þ2 *j2 ¼1 ðxk  aj2 Þð1dði1 ;j2 ÞÞ

ð4:229Þ

4 (2m)th-Degree Polynomial Discrete Systems

328

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð4:230Þ

(ii1) If dxk þ n j  ðnÞ ¼ 1 ði1 2 IqðnÞ Þ; dxk xk ¼ai1 d 2 xk þ n ðnÞ ðnÞ ðnÞ j  ðnÞ ¼ 2a0 *i 2I ðnÞ ;i 6¼i ðai1  ai1 Þ2 1 q 2 1 dx2k xk ¼ai1 ð2mÞn

ðnÞ

ð4:231Þ

ðnÞ

*j2 ¼1 ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0 ðnÞ

xk þ n at xk ¼ ai1 is • a monotonic lower-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ \0; i1

k

• a monotonic upper-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0: k

i1

(ii2) The period-n fixed-points ðn ¼ 2n1 sÞ are trivial if xk

¼

xk þ n

¼

ðnÞ aj1

2

ð1Þ ð2mÞ2 f02m ii ¼1 fai1 g; 0i2 ¼1

n1 1 s

ð2n1 1 sÞ fai2 gg

for n1 ¼ 1; 2; . . .; s ¼ 2l1 þ 1; j1 2 f1; 2; . . .; ð2mÞn g0f∅g for n 6¼ 2n2 ; ðnÞ

ð2mÞ2

xk ¼ xk þ n ¼ aj1 2 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

9 = ; ð4:232Þ

9 =

g

for n1 ¼ 1; 2; . . .; s ¼ 1; j1 2 f1; 2; . . .; ð2mÞn g0f∅g

;

for n ¼ 2n2 : (ii3) The period-n fixed-points ðn ¼ 2n1 sÞ are non-trivial if ðnÞ

ð2mÞ2

ð1Þ

xk ¼ xk þ n ¼ aj1 62 f02m ii ¼1 fai1 g; 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

gg

for n1 ¼ 1; 2; . . .; s ¼ 2l1 þ 1; j1 2 f1; 2; . . .; ð2mÞn g0f∅g for n 6¼ 2n2 ; ðnÞ

ð2mÞ2

xk ¼ xk þ n ¼ aj1 62 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

g

for n1 ¼ 1; 2; . . .; s ¼ 1; j1 2 f1; 2; . . .; ð2mÞn g0f∅g for n ¼ 2n2 :

9 = ;

9 = ; ð4:233Þ

4.4 Forward Bifurcation Trees

329

Such a forward period-n fixed-point is • monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is i1

k

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx [ 0; k

1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

1 þ1 – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l jx k k [ 0; – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx \0; k

• monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð0; 1Þ; i1

k

• invariantly zero-stable if dxk þ n =dxk jx ¼aðnÞ ¼ 0; k

i1

• oscillatorilly stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 0Þ; k

i1

• flipped if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is k

i1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx [ 0; k

1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

1 þ1 – an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dx2l jx k k \0; – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ: k

ðnÞ

i1

ðnÞ

(iii) For a fixed-point of xk þ n ¼ xk ¼ ai1 ði1 2 Iq ; q 2 f1; 2; . . .; NgÞ, there is a period-doubling of the qth -set of period-n fixed-points if dxk þ n ðnÞ ð2mÞn ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; dxk k i1 d s xk þ n j  ðnÞ ¼ 0; for s ¼ 2; . . .; r  1; dxsk xk ¼ai1 d r xk þ n j  ðnÞ 6¼ 0 for 1\r  ð2mÞn dxrk xk ¼ai1

ð4:234Þ

with • a r th -order oscillatory sink for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ 2l1 þ 1; k

i1

• a r th -order oscillatory source for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ 2l1 þ 1; k

i1

• a r th -order oscillatory upper-saddle for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ i1

k

2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ k

2l1 .

i1

4 (2m)th-Degree Polynomial Discrete Systems

330

The corresponding period-2 n discrete system of the (2m)th-degree polynomial discrete system in Eq. (4.222) is ð2 nÞ

xk þ 2 n ¼ xk þ a0

ð2mÞ2 n

ðnÞ

ð2 nÞ ð1dði1 ;j2 ÞÞ

 ai1 Þ3 *j2 ¼1 ðxk  aj2

*i 2I ðnÞ ðxk 1 q

Þ

ð4:235Þ with dxk þ 2 n d 2 xk þ 2 n jx ¼aðnÞ ¼ 1; jx ¼aðnÞ ¼ 0; i1 i1 k k dxk dx2k d 3 xk þ 2 n ð2 nÞ ðnÞ ðnÞ 3 jx ¼aðnÞ ¼ 6a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 1 q 2 1 i1 k dx3k ð2mÞ2 n

*j2 ¼1 ðnÞ

ðnÞ

ð2 nÞ ð1dði1 ;j2 ÞÞ

ðai1  aj2

Þ

ð4:236Þ :

ðnÞ

Thus, xk þ 2 n at xk ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic sink of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ \0, i1

k

• a monotonic source of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ [ 0: i1

k

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l n discrete system of the (2m)th-degree polynomial discrete system in Eq. (4.222) is ð2l1 nÞ

ð2mÞ2

xk þ 2l n ¼ xk þ ½a0

f1 þ

2l1 n

ð2mÞ

*i1 ¼1

ð2l1 nÞ

¼ xk þ ½a0 ð2

½ða0

l1

ð2mÞ

2l1 n

ð2mÞ

*i1 ¼1

ð2l nÞ

Þ

2l n

ð2mÞ

ðð2mÞ

ð2mÞ

ð2l1 nÞ

ðxk  ai1

ðð2mÞ2 n ð2mÞ2

l1 n

l

ð2mÞ2 n

*i¼1

l1 n

Þg

Þ

2l1 n

Þ=2

Þ=2

ð2l nÞ

ðx2k þ Bj2

ð2l nÞ

xk þ Cj2

ð2l nÞ

ð2l nÞ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ

ð2l nÞ

ðxk  ai

ð2l nÞ

ðxk  ai

ð2l1 nÞ

 ai2

Þ

*j2 ¼1

2l n

*i¼1

ð2l1 nÞ

l

l1 n

ð2l1 nÞ ð2mÞ2

¼ xk þ ða0

ð2mÞ

*i ¼1;i 6¼i ðxk 2 2 1

*j1 ¼1 2l1 n

Þ

Þ

Þ

2l1 n

ðxk  ai1

ð2l1 nÞ

ð2l1 nÞ ð2mÞ2

½ða0

l1

ð2

*i1 ¼1

Þ

ð2l1 nÞ

ð2l1 nÞ

ðxk  ai1

½1 þ a0

nÞ ð2mÞ

¼ xk þ ½a0

¼ x k þ a0

l1 n

*i1 ¼1

Þ

Þ

ð4:237Þ with

Þ

4.4 Forward Bifurcation Trees

331

l l dxk þ 2l n ð2l nÞ Xð2mÞ2 n ð2mÞ2 n ð2l nÞ ¼ 1 þ a0 Þ; *i ¼1;i 6¼i ðxk  ai2 i1 ¼1 2 2 1 dxk l l l d 2 xk þ 2l n ð2l nÞ Xð2mÞ2 n Xð2mÞ2 n ð2mÞ2 n ð2l nÞ ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 2 dxk .. . l l l d r xk þ 2l n Xð2mÞ2 n ð2l nÞ Xð2mÞ2 n ð2mÞ2 n ð2l nÞ ¼ a0    ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ i1 ¼1 r dxk l

for r  ð2mÞ2 n ;

ð4:238Þ where ð2 nÞ

a0

ðnÞ

l

2 n

ð2mÞ2 n ð2l nÞ 0i¼1 fai g ð2l nÞ bi;1 ð2l nÞ

bi;2

ð2l nÞ

¼ ða0 Þ1 þ ð2mÞ ; a0

ð2l nÞ

¼

¼

ð2l1 nÞ 1 þ ð2mÞ2

¼ ða0 l1 n

ð2mÞ2 sortf0i1 ¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2l nÞ  12 ðBi

þ

Þ

l1 n

; l ¼ 1; 2; 3; . . .;

ð2l1 nÞ ð2l nÞ ð2l nÞ 2 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg;

ð2l nÞ

Di Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffi l l ð2 nÞ ð2 nÞ ¼  12 ðBi  Di Þ; ð2l nÞ

ð2l nÞ

Di ¼ ðBi Þ2  4Ci 0 l ð2l1 nÞ N1 00Nq22¼1 Iqð22 nÞ for i 2 0q1 ¼1 Iq1 ð2l1 nÞ

Iq 1

¼ flðq1 1Þ ð2l1 nÞ þ 1 ; lðq1 1Þ ð2l1 nÞ þ 2 ; . . .; lq1 ð2l1 nÞ g f1; 2; . . .; M1 g0f∅g; for q1 2 f1; 2; . . .; N1 g; M1 ¼ N1 ð2l1 nÞ; ð2l nÞ Iq 2 ¼ flðq2 1Þ ð2l nÞ þ 1 ; lðq2 1Þ ð2l nÞ þ 2 ; . . .; lq2 ð2l1 nÞ g fM1 þ 1; M1 þ 2; . . .; M2 g0f∅g; l l1 ¼ ðð2mÞ2 n  ð2mÞ2 n Þ=2; for q2 2 f1; 2; . . .; N2 g; M q2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l nÞ

bi;1

ð2l nÞ

bi;2

ð2l nÞ

¼  12 ðBi

þi

ð2l nÞ

jDi jÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi l l nÞ ð2

nÞ ð2 ¼  12 ðBi  i jDi jÞ;

pffiffiffiffiffiffiffi ð2l nÞ ð2l nÞ 2 ð2l nÞ Di ¼ ðBi Þ  4Ci \0; i ¼ 1; i 2 flN ð2l nÞ þ 1 ; lN ð2l nÞ þ 2 ; . . .; lM2 g f1; 2; . . .; M2 g0f∅g

ð4:239Þ with fixed-points ð2l nÞ

xk þ 2l n ¼ xk ¼ ai l

ð2mÞ2 n

0i¼1

l

ð2l nÞ

fai

ð2 nÞ

with ai

l

; ði ¼ 1; 2;    ; ð2mÞ2 n Þ ð2mÞ2

g ¼ sortf0i1 ¼1 l

ð2 nÞ

l1 n

ð2l1 nÞ

fai1

ð2 nÞ

ð2 nÞ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

\ai þ 1 : ð4:240Þ ð2l1 nÞ

(v) For a fixed-point of xk þ ð2l nÞ ¼ xk ¼ ai1 N1 gÞ, there is a period- 2

l1

ð2l1 nÞ

ði1 2 Iq

n discrete system if

; q 2 f1; 2; . . .;

4 (2m)th-Degree Polynomial Discrete Systems

332

l1 dxk þ 2l1 n ð2l1 nÞ ð2mÞ2 n ð2l1 nÞ ð2l1 nÞ j  ð2l1 nÞ ¼ 1 þ a0  ai2 Þ ¼ 1; *i ¼1;i 6¼i ðai1 2 2 1 xk ¼ai dxk 1 d s xk þ 2l1 n j  ð2l1 nÞ ¼ 0; for s ¼ 2; . . .; r  1; xk ¼ai dxsk 1 r l1 d xk þ 2l1 n j  ð2l1 nÞ 6¼ 0 for 1\r  ð2mÞ2 n r x ¼a dxk i1 k

ð4:241Þ with • a r th -order oscillatory sink for d r xk þ 2l1 n =dxrk j

ð2l1 nÞ

xk ¼ai

2l1 þ 1; • a r th -order oscillatory source for d r xk þ 2l1 n =dxrk j

[ 0 and r ¼

1

ð2l1 nÞ

xk ¼ai

\0 and r ¼

1

2l1 þ 1; • a r th -order oscillatory upper-saddle for d r xk þ 2l1 n =dxrk j and r ¼ 2l1 ; • a r th -order oscillatory lower-saddle for d r xk þ 2l1 n =dxrk j

ð2l1 nÞ

xk ¼ai

ð2l1 nÞ

xk ¼ai

r ¼ 2l1 .

[0

1

\0 and

1

The corresponding period- 2l1 n discrete system is ð2l nÞ

xk þ 2l n ¼ xk þ a0

*

ð2l1 nÞ i1 2Iq

l

ð2mÞ2 n ðxk *j2 ¼1



ð2l1 nÞ 3

ðxk  ai1

Þ

ð4:242Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ aj 2 Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð4:243Þ with dxk þ 2l n d 2 xk þ 2l n j  ð2l1 nÞ ¼ 1; j  ð2l1 nÞ ¼ 0; xk ¼ai xk ¼ai dxk dx2k 1 1 d 3 xk þ 2l n ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j  ð2l1 Þ ¼ 6a0 ðai1  ai2 Þ * ð2l1 nÞ 3 x ¼a i 2I ;i ¼ 6 i 2 2 1 q dxk i1 k l

ð2mÞð2 nÞ

*j2 ¼1 ði1 2

l1 Iqð2 nÞ ; q

ð2l1 nÞ

ðai1

ð2l nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

6¼ 0

2 f1; 2; . . .; N1 gÞ ð4:244Þ

4.4 Forward Bifurcation Trees

333 ð2l1 nÞ

Thus, xk þ 2l n at xk ¼ ai1

is

• a monotonic sink of the third-order if d 3 xk þ 2l n =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order if d 3 xk þ 2l n =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0:

1

(v1) The period- 2l n fixed-points are trivial if ð2l nÞ

ð2mÞ

xk þ 2l n ¼ xk ¼ aj

ð2mÞ2

ð1Þ

2 f0ii ¼1 fai1 g; 0i2 ¼1

l1 n

ð2l1 nÞ

fai2

l

for j ¼ 1; 2; . . .; ð2mÞð2 nÞ for n 6¼ 2n1 ð2l nÞ

ð2mÞ2

xk þ 2l n ¼ xk ¼ aj

2 f0i2 ¼1

l1 n

ð2l1 nÞ

fai2

l

for j ¼ 1; 2; . . .; ð2mÞ2 n

9 gg = ; ð4:245Þ

9 g= ;

for n ¼ 2n1 : (v2) The period- 2l n fixed-points are non-trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

ð2mÞ2

ð1Þ

62 f02m ii ¼1 fai1 g; 0i2 ¼1

l1 n

l

for j ¼ 1; 2; . . .; ð2mÞ2 n for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

ð2mÞ2

62 f0i2 ¼1

l1 n

ð2l1 nÞ

fai2

l

for j ¼ 1; 2; . . .; ð2mÞ2 n

ð2l1 nÞ

fai2

9 gg = ;

ð4:246Þ

9 g= ;

for n ¼ 2n1 :

Such a period- 2l n fixed-point is • monotonically unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• monotonically invariant if dxk þ 2l n =dxk j

1 ð2l nÞ

xk ¼ai

2 ð1; 1Þ; ¼ 1, which is

1

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk [ 0 (independent ð2l1 Þ-branch appearance); 1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0 (independent ð2l1 Þ-branch appearance) 1 þ1 jx – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l k k [ 0 (dependent ð2l1 þ 1Þ-branch appearance from one branch); 1 þ1 jx \0 – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l k k (dependent ð2l1 þ 1Þ-branch appearance from one branch);

4 (2m)th-Degree Polynomial Discrete Systems

334

• monotonically stable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• invariantly zero-stable if dxk þ 2l n =dxk j • oscillatorilly stable if dxk þ 2l n =dxk j • flipped if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð0; 1Þ;

1

¼ 0;

ð2l nÞ

xk ¼ai ð2l nÞ

xk ¼ai

1

2 ð1; 0Þ;

1

¼ 1, which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk [ 0; 1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jx \0 k

1 – an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0; 1 þ1 jx – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l k k [0

• oscillatorilly unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð1; 1Þ:

1

Proof Through the nonlinear renormalization, the proof of this theorem is similar to the proof of Theorem 5.11. This theorem can be easily proved. ■

References Luo ACJ (2020a) The stability and bifurcations of the (2m)th degree polynomial systems. J Vib Test Syst Dyn 4(1):1–42 Luo ACJ (2020b) Bifurcation and stability in nonlinear dynamical systems. Springer, New York

Chapter 5

(2m + 1)th-Degree Polynomial Discrete Systems

In this chapter, the global stability and bifurcations of period-1 fixed-points in a forward ð2m þ 1Þth -degree polynomial discrete system are presented. The broom-appearing, broom-spraying-appearing and broom-sprinkler-spraying-appearing bifurcations for simple and higher-order period-1 fixed-points are discussed, and the antenna switching, straw-bundle-switching and flower-bundle-switching bifurcations for simple and higher-order period-1 fixed-points are also presented. As in cubic nonlinear discrete systems, the period-2 fixed-point solutions and the corresponding period-doubling renormalization of such a forwarded ð2m þ 1Þth -degree polynomial discrete system are discussed. For multiple iterations, the period-n appearing and perioddoublization of the forward ð2m þ 1Þth -degree polynomial discrete system are discussed.

5.1

Global Stability and Bifurcations

In a similar fashion of low-degree polynomial discrete systems, the global stability and bifurcation of fixed-points in the ð2m þ 1Þth -degree polynomial nonlinear discrete systems are discussed as in Luo (2020a, b). The stability and bifurcation of each individual fixed-point are analyzed from the local analysis. Definition 5.1 Consider a ð2m þ 1Þth -degree polynomial nonlinear forward discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ þ1 2 þ A1 ðpÞx2m ¼ xk þ A0 ðpÞx2m k k þ    þ A2m1 ðpÞxk þ A2m xk þ A2m þ 1 ðpÞ

¼ xk þ a0 ðpÞðxk  aðpÞÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ    ½x2k þ Bm ðpÞxk þ Cm ðpÞ ð5:1Þ © Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3_5

335

5 (2m + 1)th-Degree Polynomial Discrete Systems

336

where A0 ðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ;    ; pm ÞT :

ð5:2Þ

(i) If Di ¼ B2i  4Ci \0 for i ¼ 1; 2; . . .; m;

ð5:3Þ

the ð2m þ 1Þth -degree polynomial discrete system has one fixed-point of xk ¼ a, and the corresponding standard form is 1 2

1 4

1 2

1 4

xk þ 1 ¼ xk þ a0 ðxk  aÞ½ðxk þ B1 Þ2 þ ðD1 Þ    ½ðxk þ Bm Þ2 þ ðDm Þ:

ð5:4Þ

The discrete flow of such a system with one fixed-point is called a singlefixed-point flow. (a) If a0 [ 0, the fixed-point discrete flow with xk ¼ a is called a monotonic source discrete flow for df =dxk jx ¼a 2 ð0; 1Þ. k (b) If a0 \0, the fixed-point discrete flow with x ¼ a is called • • • •

a monotonic sink discrete flow for df =dxk jx ¼a 2 ð1; 0Þ, k an invariant sink discrete flow for df =dxk jx ¼a ¼ 1, k an oscillatory sink discrete flow for df =dxk jx ¼a 2 ð2; 1Þ, k a flipped discrete flow for df =dxk jx ¼a ¼ 2 k

– of the oscillatory upper-saddle (d 2 f =dx2k jx ¼a [ 0), k

– of the oscillatory lower-saddle (d 2 f =dx2k jx ¼a \0). k

• an oscillatory source discrete flow for df =dxk jx ¼a 2 ð1; 2Þ. k

(ii) If Di ¼ B2i  4Ci [ 0; i ¼ i1 ; i2 ;    ; il 2 f1; 2; . . .; mg; Dj ¼ B2j  4Cj \0; j ¼ il þ 1 ; il þ 2 ;    ; im 2 f1; 2; . . .; mg

ð5:5Þ

with l 2 f0; 1; . . .; mg; the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2l þ 1Þfixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; pffiffiffiffiffi ðiÞ ð5:6Þ xk ¼ b2 ¼ 12ðBi  Di Þ for i 2 fi1 ; i2 ;    ; il g  f1; 2;    ; mg:

5.1 Global Stability and Bifurcations

337

(ii1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; l ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2    ; a2l g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g;as \as þ 1 ;

ð5:7Þ

then, the corresponding standard form is 1 2

1 4

2 þ1 m xk þ 1 ¼ xk þ a0 *2l i1 ¼1 ðxk  ai1 Þ *k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ:

ð5:8Þ

(a) If a0 [ 0, the simple fixed-point discrete flow is called a ðmSI-oSO: mSO:    :mSO : mSI-oSOÞ-discrete flow. (b) If a0 \0, the simple fixed-point discrete flow is called a ðmSO:mSIoSO :    :mSI-oSO:mSO)-discrete flow. (ii2) If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2    ; a2l þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . air  aRr1 ¼    ¼ aRr1 ¼ a2l þ 1 i¼1 li þ 1 i¼1 li þ lr

ð5:9Þ

with Rrs¼1 ls ¼ 2l þ 1;

then, the corresponding standard form is 2 xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *m k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ:

1 2

1 4

ð5:10Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    :lr th mXX)-discrete flow. (a) for a0 [ 0 and p ¼ 1; 2; . . .; r, 8 > ð2rp  1Þth order monotonic source, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic sink, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2r Þth order monotonic upper-saddle, > : p for ap ¼ 2Mp ; lp ¼ 2rp ;

ð5:11Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

338

where ap ¼

Xr

s¼p ls :

ð5:12Þ

(b) for a0 \0 and p ¼ 1; 2; . . .; r, 8 > ð2rp  1Þth order monotonic sink, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic source, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic upper-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic lower-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:13Þ

(c) The fixed-point of xk ¼ aip for (lp [ 1)-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX:lp2 th mXX:    : lpb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the corresponding switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l i¼1

i þ1

i¼1

6¼    6¼ a Rp1 l i¼1

i þ lp

; lp ¼

Xb

i¼1 lpi :

ð5:14Þ

(iii) If Di ¼ B2i  4Ci ¼ 0; i 2 fi11 ; i12 ;    ; i1s g fi1 ; i2 ;    ; il g f1; 2;    ; mg; Di ¼ B2i  4Ci [ 0; i 2 fi21 ; i22 ;    ; i2r g fi1 ; i2 ;    ; il g f1; 2;    ; mg; Di ¼ B2i  4Ci \0; i 2 fil þ 1 ; il þ 2 ;    ; im g f1; 2;    ; mg; ð5:15Þ the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2l þ 1Þfixed-points as ðiÞ

xk ¼ b1 ¼ 12Bi ; ðiÞ

xk ¼ b2 ¼ 12Bi

) for i 2 fi11 ; i12 ;    ; i1s g;

pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; for i 2 fi21 ; i22 ;    ; i2r g: pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ

If

ð5:16Þ

5.1 Global Stability and Bifurcations

339 ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2    ; a2l þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð5:17Þ

air  aRr1 ¼    ¼ aRr1 ¼ a2l þ 1 i¼1 li þ 1 i¼1 li þ lr

with Rrs¼1 ls ¼ 2l þ 1;

then the corresponding standard form is 2 xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *m k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ:

1 2

1 4

ð5:18Þ

The fixed-point discrete flow is called an ðl1 th mXX: l2 th mXX:    :lr th mXX)discrete flow. (a) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points appearance or vanishing is called an lp th mXX appearing bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is 1 2

aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ¼  Bip ; i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ;    ; il gÞ; aRþp1 l þ 1 i¼1 i

6¼    6¼

aRþp1 l þ l or p i¼1 i

a Rp1 l þ1 i¼1 i

6¼    6¼

ð5:19Þ a : Rp1 l þ lp i¼1 i

(b) The fixed-point of xk ¼ aiq for ðlq [ 1Þ-repeated fixed-points switching is called an lq th mXX switching bifurcation of ðlq1 th mXX:lq2 th mXX:    : lqb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aiq  aRq1 li þ 1 ¼    ¼ aRq1 li þ lq ; i¼1

a Rq1 l i¼1

i þ1

i¼1

6¼    6¼ a Rq1 l i¼1

i þ lq

; lq ¼

Xb

i¼1 lqi :

ð5:20Þ

(c) The fixed-point of xk ¼ aip for ðln [ 1Þ-repeated fixed-points appearance  or vanishing and ðlp2 2Þ repeated fixed-points switching of lp11 th mXX : lp22 th mXX:    :lp2b th mXX)-fixed-point switching is called an lp th mXX bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the flower-switching bifurcation condition is

5 (2m + 1)th-Degree Polynomial Discrete Systems

340

aip  aRp1 li þ 1 ¼    ¼ aRp1 þ li þ lp i¼1 i¼1  with Dip ¼ B2ip  4Cip ¼ 0 ip 2 fi1 ; i2 ;    ; il g     for j1 ; j2 ;    ; jp1  1; 2; . . .; lp     for k1 ; k2 ;    ; kp2  1; 2; . . .; lp with lp1 þ lp2 ¼ lp ; lp2 ¼

ð5:21Þ

Xb

i¼1 lp2i

(iv) If Di ¼ B2i  4Ci [ 0 for i ¼ 1; 2; . . .; m

ð5:22Þ

the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2m þ 1Þfixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; for i ¼ 1; 2; . . .; m: pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ

ð5:23Þ

(iv1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; m ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2    ; a2m g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 gðas \as þ 1 Þ; ð5:24Þ then, the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  a1 Þðxk  a2 Þ    ðxk  a2m Þðxk  a2m þ 1 Þ: ð5:25Þ This discrete flow is formed with all the simple fixed-points. (a) If a0 [ 0, the fixed-point discrete flow with ð2m þ 1Þ fixed-points is called a ðmSO:mSI-oSO:    :mSI-oSO:mSO)-discrete flow. (b) If a0 \0, the fixed-point discrete flow with ð2m þ 1Þ fixed-points is called a ðmSI-oSO:mSO:    :mSO : mSI-oSO)-discrete flow.

5.1 Global Stability and Bifurcations

341

(iv2) If ð1Þ

ð1Þ

ðmÞ

ðmÞ

fa1 ; a2    ; a2m þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; ai 1  a1 ¼    ¼ al 1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð5:26Þ

air  aRr1 ¼    ¼ aRr1 ¼ a2m þ 1 i¼1 li þ 1 i¼1 li þ lr

with Rrs¼1 ls ¼ 2m þ 1;

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls :

ð5:27Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX :    : lr th mXX)-discrete flow. The fixed-point of xk ¼ aip for lp -repeated fixedpoints switching is called an lp th mXX bifurcation of ðlp1 th mXX: lp2 th mXX:    :lpb th mXX) fixed-point switching at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l i¼1

i þ1

i¼1

6¼    6¼ a Rp1 l i¼1

i þ lp

; lp ¼

Xb

i¼1 lpi :

ð5:28Þ

Definition 5.2 Consider a ð2m þ 1Þth -degree polynomial nonlinear discrete system as xk þ 1 ¼ xk þ f ðxk ; pÞ þ1 2 ¼ xk þ A0 ðpÞx2m þ A1 ðpÞx2m k k þ    þ A2m1 ðpÞxk þ A2m xk þ A2m þ 1 ðpÞ

¼ a0 ðpÞðxk  aðpÞÞ *ni¼1 ½x2k þ Bi ðpÞxk þ Ci ðpÞqi

ð5:29Þ where A0 ðpÞ 6¼ 0; and p ¼ ðp1 ; p2 ;    ; pm ÞT ; m ¼

Xn

i¼1 qi :

ð5:30Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

342

(i) If Di ¼ B2i  4Ci \0 for i ¼ 1; 2; . . .; n

ð5:31Þ

the ð2m þ 1Þth -degree polynomial nonlinear system has one fixed-point of xk ¼ a, and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðxk  aÞ *ni¼1 ½ðxk þ Bi Þ2 þ ðDi Þqi : 1 2

1 4

ð5:32Þ

The discrete flow of such a system with one fixed-point is called a single fixed-point discrete flow. (a) If a0 [ 0, the fixed-point discrete flow of xk ¼ a is called a monotonic source discrete flow for df =dxk jx ¼a 2 ð0; 1Þ: k (b) If a0 \0, the fixed-point discrete flow of xk ¼ a is called • • • •

a monotonic sink discrete flow for df =dxk jx ¼a 2 ð1; 0Þ; k an invariant sink discrete flow for df =dxk jx ¼a ¼ 1; k an oscillatory sink discrete flow for df =dxk jx ¼a 2 ð2; 1Þ; k a flipped discrete flow for df =dxk jx ¼a ¼ 2 with k

– an oscillatory upper-saddle (d f =dx2k jx ¼a [ 0), 2

k

– an oscillatory lower-saddle (d 2 f =dx2k jx ¼a \0), k

• an oscillatory source discrete flow for df =dxk jx ¼a 2 ð1; 2Þ: k

(ii) If Di ¼ B2i  4Ci [ 0; i 2 fi1 ; i2 ;    ; il gf1; 2; . . .; ng; Dj ¼ B2j  4Cj \0; j 2 fil þ 1 ; il þ 2 ;    ; in gf1; 2; . . .; ng

ð5:33Þ

the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2l þ 1Þ fixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; for i 2 fi1 ; i2 ;    ; il gf1; 2; . . .; ng: ð5:34Þ pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ

5.1 Global Stability and Bifurcations

343

(ii1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; i; j ¼ 1; 2; . . .; l ð1Þ

ð1Þ

ðrÞ

ðrÞ

fa1 ; a2    ; a2l þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ð5:35Þ

qr sets

as as þ 1 ; then, the corresponding standard form is ls n 2 qik þ1 xk þ 1 ¼ xk þ a0 *2l s¼1 ðxk  as Þ *k¼l þ 1 ½ðxk þ Bik Þ þ ðDik Þ

1 2

1 4

with ls 2 fqi1 ; qi2 ;    ; qil ; 1g: ð5:36Þ The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    :l2l þ 1 th mXXÞ-discrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; 2l þ 1, 8 > ð2rp  1Þth order montonic source, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > >  1Þth order monotonic sink, ð2r > > < p for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic upper-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp ;

ð5:37Þ

where ap ¼

X2l þ 1

s¼p

ls :

ð5:38Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; 2l þ 1, 8 > ð2r  1Þth order monotonic sink, > > p > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic source, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic upper-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic lower-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:39Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

344

(ii2) If ð1Þ

ð1Þ

ðrÞ

ðrÞ

fa1 ; a2    ; a2l þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

qr sets

ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð5:40Þ

air  aRr1 ¼    ¼ aRr1 ¼ a2l þ 1 i¼1 li þ 1 i¼1 li þ lr with Rrs¼1 ls ¼ 2l þ 1; then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *nk¼l þ 1 ½ðxk þ Bik Þ2 þ ðDik Þqik : 1 2

1 4

ð5:41Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    : lr th mXX)discrete flow. (a) For a0 [ 0 and s ¼ 1; 2; . . .; r, 8 > ð2r  1Þth order monotonic source, > > p > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic sink, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic upper-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp ;

ð5:42Þ

where ap ¼

Xr

s¼p ls :

ð5:43Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; r, 8 > ð2r  1Þth order monotonic sink, > > p > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > th > ð2r > p  1Þ order monotonic source, > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic upper-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic lower-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:44Þ

5.1 Global Stability and Bifurcations

345

(c) The fixed-point of xk ¼ aip for (lp [ 1)-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX:lp2 th XmX:    :lpb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l i¼1

i þ1

i¼1

6¼    6¼ a Rp1 l i¼1

i þ lp

; lp ¼

Xb

i¼1 lpi :

ð5:45Þ

(iii) If Di ¼ B2i  4Ci ¼ 0; i 2 fi11 ; i12 ;    ; i1s gfi1 ; i2 ;    ; il gf1; 2; . . .; ng; Dk ¼ B2k  4Ck [ 0; k 2 fi21 ; i22 ;    ; i2r gfi1 ; i2 ;    ; il gf1; 2; . . .; ng; Dj ¼ B2j  4Cj \0; j 2 fil þ 1 ; il þ 2 ;    ; in gf1; 2; . . .; ng with i 6¼ j 6¼ k; ð5:46Þ the ð2m þ 1Þth -degree polynomial nonlinear system has ð2l þ 1Þ - fixed-points as ) ðiÞ xk ¼ b1 ¼ 12Bi ; for i 2 fi11 ; i12 ;    ; i1s g; ðiÞ xk ¼ b2 ¼ 12Bi ð5:47Þ pffiffiffiffiffiffi ) ðkÞ xk ¼ b1 ¼ 12ðBk þ Dk Þ; for i 2 fi21 ; i22 ;    ; i2r g: pffiffiffiffiffiffi ðkÞ xk ¼ b2 ¼ 12ðBk  Dk Þ If ð1Þ

ð1Þ

ðlÞ

ðlÞ

fa1 ; a2    ; a2l þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . air  aRr1 ¼    ¼ aRr1 ¼ a2l þ 1 i¼1 li þ 1 i¼1 li þ lr with Rrs¼1 ls ¼ 2l þ 1;

ð5:48Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

346

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls *nk¼l þ 1 ½ðxk þ Bik Þ2 þ ðDik Þqik : 1 2

1 4

ð5:49Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    :lr th mXX)discrete flow. (a) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points appearance or vanishing is called an lp th mXX appearing bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is 1 2

aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ¼  Bip i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ;    ; il gÞ; aRþp1 l þ 1 i¼1 i

6¼    6¼

aRþp1 l þ l or p i¼1 i

a l þ1 Rp1 i¼1 i

6¼    6¼

ð5:50Þ a : l þ lp Rp1 i¼1 i

(b) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points switching is called an lp th mXX switching bifurcation lp th mXX of ðlp1 th mXX : lp2 th mXX :    :lpb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the corresponding switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l

i¼1 i

lp ¼

þ1

i¼1

6¼    6¼ a Rp1 l

i¼1 i

þ lp

ð5:51Þ

Xb

i¼1 lpi :

(iv) If Di ¼ B2i  4Ci [ 0 for i ¼ 1; 2; . . .; n

ð5:52Þ

the ð2m þ 1Þth -degree polynomial nonlinear system has ð2n þ 1Þ-fixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; for i ¼ 1; 2; . . .; n: pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ

ð5:53Þ

5.1 Global Stability and Bifurcations

347

(iv1) If ðjÞ bðiÞ r 6¼ bs for r; s 2 f1; 2g; ði; j ¼ 1; 2; . . .; nÞ; ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2    ; a2n þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

ð5:54Þ

qn sets

as as þ 1 ; then, the corresponding standard form is ls þ1 xk þ 1 ¼ xk þ a0 *2n s¼1 ðxk  as Þ with ls 2 fqi1 ; qi2 ;    ; qin ; 1g: ð5:55Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    : l2n þ 1 th mXX)-discrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; 2n þ 1, 8 > ð2rp  1Þth order monotonic source, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic sink, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic upper-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp ;

ð5:56Þ

where ap ¼

X2n þ 1

s¼p

ls :

ð5:57Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; 2n þ 1, 8 > ð2rp  1Þth order montonic sink, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic source, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order montonic upper-saddle, > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > ð2r Þth order monotonic lower-saddle, > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:58Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

348

(iv2) If ð1Þ

ð1Þ

ðnÞ

ðnÞ

fa1 ; a2    ; a2n þ 1 g ¼ sortfa; b1 ; b2 ;    ; b1 ; b2 g; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q1 sets

qn sets

ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. .

ð5:59Þ

air  aRr1 ¼    ¼ aRr1 ¼ a2n þ 1 ; i¼1 li þ 1 i¼1 li þ lr

with Rrs¼1 ls ¼ 2n þ 1;

then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls :

ð5:60Þ

The fixed-point discrete flow is called an ðl1 th mXX: l2 th mXX:    :lr th mXXÞ-discrete flow. The fixed-point of xk ¼ aip for lp -repeated fixed-points switching is called an lp th XX switching bifurcation of ðlp1 th mXX:lp2 th mXX:    :lpb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l

i¼1 i

þ1

i¼1

6¼    6¼ a Rp1 l

i¼1 i

þ lp

; lp ¼

Xb

i¼1 lpi :

ð5:61Þ

Definition 5.3 Consider a 1-dimensional, ð2m þ 1Þth -degree polynomial nonlinear discrete system xk þ 1 ¼ xk þ f ðxk ; pÞ þ1 2 þ A1 ðpÞx2m ¼ xk þ A0 ðpÞx2m k k þ    þ A2m1 ðpÞxk þ A2m xk þ A2m þ 1 ðpÞ

¼ xk þ a0 ðpÞ *rs¼1 ðxk  cis ðpÞÞls *ni¼r þ 1 ½x2k þ Bi ðpÞxk þ Ci ðpÞqi ð5:62Þ where A0 ðpÞ 6¼ 0; and Xr

s¼1 ls

¼ 2l þ 1;

Xn

i¼r þ 1 qi

¼ ðm  lÞ; p ¼ ðp1 ; p2 ;    ; pm ÞT :

ð5:63Þ

5.1 Global Stability and Bifurcations

349

(i) If Di ¼ B2i  4Ci \0 for i ¼ r þ 1; r þ 2; . . .; n; fa1 ; a2 ; . . .; ar g ¼ sortfc1 ; c2 ; . . .; cr g with ai \ai þ 1

ð5:64Þ

the ð2m þ 1Þth -degree polynomial discrete system has fixed-points of xk ¼ ais ðpÞ (s ¼ 1; 2; . . .; r), and the corresponding standard form is xk þ 1 ¼ xk þ a0 ðpÞ *rj¼1 ðxk  aij Þlj *ni¼r þ 1 ½ðxk þ Bi Þ2 þ ðDi Þli : 1 2

1 4

ð5:65Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    : lr th mXX)discrete flow. (a) For a0 [ 0 and s ¼ 1; 2; . . .; r, 8 > ð2rp  1Þth order monotonic source, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2r  1Þth order monotonic sink, > > < p for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > ð2r Þth order monotonic upper-saddle, > : p for ap ¼ 2Mp ; lp ¼ 2rp ;

ð5:66Þ

where ap ¼

Xr

s¼p ls :

ð5:67Þ

(b) For a0 \0 and p ¼ 1; 2; . . .; r, 8 > ð2rp  1Þth order monotonic sink, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic source, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ > ð2rp Þth order monotonic upper-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2r Þth order monotonic lower-saddle, > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:68Þ

(ii) If Di ¼ B2i  4Ci [ 0; i ¼ j1 ; j2 ;    ; js 2 fl þ 1; l þ 2; . . .; ng; Dj ¼ B2j  4Cj \0; j ¼ js þ 1 ; js þ 2 ;    ; jn 2 fl þ 1; l þ 2; . . .; ng with s 2 f1; . . .; n  lg;

ð5:69Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

350

the ð2m þ 1Þth -degree polynomial nonlinear discrete system has 2n2 -fixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ

ð5:70Þ

for i 2 fj1 ; j2 ;    ; jn1 gfl þ 1; l þ 2; . . .; ng: If ðr þ 1Þ

ðr þ 1Þ

ðn Þ

ðn Þ

; b2 ;    ; b1 1 ; b2 1 g; fa1 ; a2    ; a2n2 þ 1 g ¼ sortfc1 ; c2    ; c2l þ 1 ; b1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} qr þ 1 sets

qn1 sets

ai1  a1 ¼    ¼ al1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . ain1  aRn1 1 li þ 1 ¼    ¼ aRn1 1 li þ ln ¼ a2n2 þ 1 i¼1

1 with Rns¼1 ls ¼ 2n2 þ 1;

i¼1

1

ð5:71Þ

then, the corresponding standard form is 1 xk þ 1 ¼ xk þ a0 *ns¼1 ðxk  ais Þls *ni¼n2 þ 1 ½ðxk þ Bi Þ2 þ ðDi Þqi :

1 2

1 4

ð5:72Þ

The fixed-point discrete flow is called an ðl1 th mXX:l2 th mXX:    :ln1 th mXXÞdiscrete flow. (a) For a0 [ 0 and p ¼ 1; 2; . . .; r; r þ 1; . . .; n1 , 8 > ð2r  1Þth order monotonic source, > > p > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic sink, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; ð5:73Þ lp th mXX ¼ > ð2rp Þth order monotonic lower-saddle, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > Þth order monotonic upper-saddle, ð2r > : p for ap ¼ 2Mp ; lp ¼ 2rp ; where ap ¼

Xn1

s¼p ls :

ð5:74Þ

5.1 Global Stability and Bifurcations

351

(b) For a0 \0 and p ¼ 1; 2; . . .; r; r þ 1; . . .; n1 , 8 > ð2rp  1Þth order monotonic sink, > > > > for ap ¼ 2Mp  1; lp ¼ 2rp  1; > > > > ð2rp  1Þth order monotonic source, > > < for ap ¼ 2Mp ; lp ¼ 2rp  1; lp th mXX ¼ th > > > ð2rp Þ order monotonic upper-saddle, > > for ap ¼ 2Mp  1; lp ¼ 2rp ; > > > > > ð2r Þth order monotonic lower-saddle, > : p for ap ¼ 2Mp ; lp ¼ 2rp :

ð5:75Þ

(c) The fixed-point of xk ¼ aip for (lp [ 1)-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX:lp2 th mXX:    : lpb th mXX) fixed-point at p ¼ p1 2 @X12 , and the corresponding switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l þ1 i¼1 i

6¼    6¼

i¼1

a ;l Rp1 l þ lp p i¼1 i

¼

Xb

i¼1 lpi :

ð5:76Þ

(iii) If Di ¼ B2i  4Ci ¼ 0; for i 2 fi11 ; i12 ;    ; i1s gfil þ 1 ; il þ 2 ;    ; in2 gfl þ 1; l þ 2; . . .; ng; Dk ¼ B2k  4Ck [ 0; for k 2 fi21 ; i22 ; . . .; i2r gfil þ 1 ; il þ 2 ;    ; in2 gfl þ 1; l þ 2; . . .; ng; Dj ¼ B2j  4Cj \0; for j 2 fin2 þ 1 ; in2 þ 2 ; . . .; in gfl þ 1; l þ 2; . . .; ng;

ð5:77Þ

the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2n2 þ 1Þfixed-points as ) ðiÞ xk ¼ b1 ¼ 12Bi ; for i 2 fi11 ; i12 ; . . .; i1s g; ðiÞ xk ¼ b2 ¼ 12Bi ð5:78Þ pffiffiffiffiffiffi ) ðkÞ xk ¼ b1 ¼ 12ðBk þ Dk Þ; for i 2 fi21 ; i22 ; . . .; i2r g: pffiffiffiffiffiffi ðkÞ xk ¼ b2 ¼ 12ðBk  Dk Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

352

If

fa1 ; a2    ; a2n2 þ 1 g ¼ sortfa; c1 ; c2    ; c2l ; bð1rÞ ; bð2rÞ ;    ; bð1n1 Þ ; bð2n1 Þ g; |fflfflffl{zfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} qr sets

qn1 sets

ai1  a1 ¼    ¼ al1 ; ai 2  al 1 þ 1 ¼    ¼ al 1 þ l 2 ; .. . ain1  aRn1 1 li þ 1 ¼    ¼ aRn1 1 li þ ln ¼ a2n2 þ 1 i¼1

i¼1

1 with Rns¼1 ls ¼ 2n2 þ 1;

1

ð5:79Þ then, the corresponding standard form is 1 xk þ 1 ¼ a0 *ns¼1 ðxk  ais Þls *ni¼n2 þ 1 ½ðxk þ Bi Þ2 þ ðDi Þqi :

1 2

1 4

ð5:80Þ

The fixed-point discrete flow is called an l1 th mXX:l2 th mXX:    :ln1 th mXXÞdiscrete flow. (a) The fixed-point of x ¼ aip for ðlp [ 1Þ-repeated fixed-points appearance (or vanishing) is called an lp th mXX appearing bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is 1 2

aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ¼  Biq i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ;    ; il gÞ aRþp1 l þ 1 i¼1 i

6¼    6¼

aRþp1 l þ l or p i¼1 i

a Rp1 l þ1 i¼1 i

6¼    6¼

ð5:81Þ a : Rp1 l þ lp i¼1 i

(b) The fixed-point of xk ¼ aip for ðlp [ 1Þ-repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX:lp2 th mXX:    :lpb th mXX) fixed-point at a point p ¼ p1 2 @X12 , and the corresponding switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l

i¼1 i

lp ¼

þ1

i¼1

6¼    6¼ a Rp1 l

i¼1 i

þ lp

;

ð5:82Þ

Xb

i¼1 lpi :

(c) The fixed-point of xk ¼ aip for ðlp1 1Þ-repeated fixed-points appearance (or vanishing) and ðlp2 2Þ repeated fixed-points switching of ðlp21 th mXX: lp22 th mXX :    :lp2b th mXX) is called an lp th mXX switching bifurcation of fixed-point at a point p ¼ p1 2 @X12 , and the corresponding bifurcation condition is

5.1 Global Stability and Bifurcations

353

aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp i¼1

i¼1

with Dip ¼ B2ip  4Cip ¼ 0 ðip 2 fi1 ; i2 ;    ; il gÞ aRþp1 l

i¼1 i

þ j1

6¼    6¼ aRþp1 l

i¼1 i

þ j p1

or a p 1 R1 l i¼1

i

þ j1

6¼    6¼ a p 1 R1 l i¼1

i

þ jp1

; ð5:83Þ

for fj1 ; j2 ;    ; jp1 gf1; 2; . . .; lp g; a Rp1 l þ k1 i¼1 i

a Rp1 l þ k p2 i¼1 i

6¼    6¼

for fk1 ; k2 ; . . .; kp2 gf1; 2; . . .; lp g; with lp1 þ lp2 ¼ lp : (iv) If Di ¼ B2i  4Ci [ 0 for i ¼ l þ 1; l þ 2; . . .; n

ð5:84Þ

the ð2m þ 1Þth -degree polynomial nonlinear discrete system has ð2m þ 1Þ fixed-points as pffiffiffiffiffi ) ðiÞ xk ¼ b1 ¼ 12ðBi þ Di Þ; for i ¼ l þ 1; l þ 2; . . .; n: ð5:85Þ pffiffiffiffiffi ðiÞ xk ¼ b2 ¼ 12ðBi  Di Þ If ðr þ 1Þ

ðr þ 1Þ

ðnÞ

ðnÞ

; b2 ;    ; b1 ; b2 g; fa1 ; a2    ; a2m þ 1 g ¼ sortfc1 ; c2    ; c2l þ 1 ; b1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} qr þ 1 sets

qn sets

ai 1  a1 ¼    ¼ a l 1 ; ai2  al1 þ 1 ¼    ¼ al1 þ l2 ; .. . ain  aRn1 ¼    ¼ aRn1 ¼ a2m þ 1 i¼1 li þ 1 i¼1 li þ lr with Rns¼1 ls ¼ 2m þ 1; ð5:86Þ then, the corresponding standard form is xk þ 1 ¼ xk þ a0 *rs¼1 ðxk  ais Þls :

ð5:87Þ

The fixed-point discrete flow is called an ðl1 th mXX: l2 th mXX:    :lr th mXX)discrete flow. The fixed-point of xk ¼ aip for lp -repeated fixed-points switching is called an lp th mXX switching bifurcation of ðlp1 th mXX:lp2 th mXX :    :

5 (2m + 1)th-Degree Polynomial Discrete Systems

354

lpb th mXX) fixed-point at p ¼ p1 2 @X12 , and the corresponding switching bifurcation condition is aip  aRp1 li þ 1 ¼    ¼ aRp1 li þ lp ; i¼1

a Rp1 l þ1 i¼1 i

5.2

i¼1

a ;l Rp1 l þ lp p i¼1 i

6¼    6¼

¼

Xb

i¼1 lpi :

ð5:88Þ

Simple Fixed-Point Bifurcations

To illustrate the bifurcations in the ð2m þ 1Þth -degree polynomial discrete system, the detailed discussion with graphical illustrations will be presented as follows.

5.2.1

Appearing Bifurcations

Consider a ð2m þ 1Þth -degree polynomial nonlinear discrete system as xk þ 1 ¼ xk þ a0 Qðxk Þðxk  aÞ *ni¼1 ðx2k þ Bi xk þ Ci Þ:

ð5:89Þ

without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are ðiÞ

1 2

b1;2 ¼  Bi 

1pffiffiffiffiffi Di ; Di 2

¼ B2i  4Ci 0ði ¼ 1; 2; . . .; nÞ; ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðnÞ

ðnÞ

fa1 ; a2 ;    ; a2l g  sortfb1 ; b2 ; b1 ; b2 ;    ; b1 ; b2 g; as as þ 1 ;  Bi 6¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ at bifurcation: Di ¼ 0ði ¼ 1; 2; . . .; nÞ

ð5:90Þ

The 2nd order singularity bifurcation is for the birth of a pair of simple monotonic sink to oscillatory source, and simple monotonic source. There are two appearing bifurcations for i 2 f1; 2; . . .; ng ith quadratic factor



2nd order mUS ! appearing bifurcation ith quadratic factor

2nd order mLS ! appearing bifurcation



mSO, for xk ¼ a2i ; mSI-oSO, for xk ¼ a2i1 ;

ð5:91Þ

mSI-oSO, for xk ¼ a2i ; mSO, for xk ¼ a2i1 :

ð5:92Þ

5.2 Simple Fixed-Point Bifurcations

355

If xk ¼ a 6¼  12 Bi ði 2 f1; 2; . . .; mgÞ, the fixed-point of xk ¼ a breaks a cluster of teethcomb appearing bifurcations to two parts. The teethcomb appearing bifurcation generated by the m-pairs of quadratics becomes a broom appearing bifurcation. The two broom appearing bifurcations are l1 þ l2 ¼m

mSO ( xk ¼ aÞ !ðl1 -mLSN:mSO:l2 -mUSNÞ appearing bifurcation 8  > < l2 -mUSN, for x 2 fa2j ; a2j þ 1 ; j ¼ l1 þ 1;    ; l1 þ l2 g; ¼

> :

mSO; for x ¼ a ¼ a2ðl1 þ 1Þ1

ð5:93Þ

l1 -mLSN, for x 2 fa2i1 ; a2i ; i ¼ 1; 2; . . .; l1 g

and l1 þ l2 ¼m

mSI-oSO(xk ¼ aÞ !ðl1 -mUSN:mSI-oSO:l2 -mLSNÞ appearing bifurcation 8  > < l2 -mLSN, for xk 2 fa2j ; a2j þ 1 : j ¼ l1 þ 1;    ; l1 þ l2 g; ¼ mSI-oSO; for xk ¼ a ¼ a2ðl1 þ 1Þ1 > : l1 -mUSN, for xk 2 fa2i1 ; a2i : i ¼ 1; 2; . . .; l1 g ð5:94Þ where the lj -mLSN and lj -mUSN ðj ¼ 1; 2Þ are 8 ( mSO, for xk ¼ a2ðsj þ lj Þ þ d2j ; > ðlj þ sj Þth bifurcation > > mUS ! > > mSI-oSO, for xk ¼ a2ðsj þ lj Þ1 þ d2j ; > appearing > < lj -mUSN .. . ( > > > mSO, for xk ¼ a2sj þ d2j ; > ðsj Þth bifurcation > > mUS ! > : mSI-oSO, for xk ¼ a2sj 1 þ d2j : appearing

ð5:95Þ

8 ( mSI-oSO, for xk ¼ a2ðsj þ lj Þ þ d2j ; > ðlj þ sj Þth bifurcation > > mLS ! > > mSO, for xk ¼ a2ðsj þ lj Þ1 þ d2j ; > appearing > < lj -mLSN .. . ( > > > mSI-oSO, for xk ¼ a2sj þ d2j ; > ðsj Þth bifurcation > > mLS ! > : mSO, for xk ¼ a2sj 1 þ d2j : appearing

ð5:96Þ

for sj 2 f0; 1; 2; . . .; mg and 0 lj m with 0 lj m. Four special broom appearing bifurcations are

5 (2m + 1)th-Degree Polynomial Discrete Systems

356

8 a2m þ 1 ; mSO ! mSO, for xk ¼ a ¼ 8 > 8 > > > mSI-oSO, > > > > > > < > > > > for xk ¼ a2m ; mth bifurcation > > > > mLS ! > > > > mSO, > > appearing > > > > > > : > < for xk ¼ a2m1 ; > < mSO (xk ¼ aÞ ! > m-mLSN ... > 8 > > > > > > mSI-oSO, > > > > > > < > > > > for xk ¼ a2 ; 1st bifurcation > > > > mUS ! > > > > mSO, > > appearing > > > > : : : for xk ¼ a1 ;

ð5:97Þ

8  mSI ! mSI, > 8 for xk ¼ a ¼ a2m8þ 1 ; > > > mSO, > > > > > > < > > th > > for xk ¼ a2m ; m bifurcation > > > > mUS ! > > > > mSI-oSO, > > appearing > > > > > > : > < for xk ¼ a2m1 ; > <  mSI-oSO (xk ¼ aÞ ! . m-mUSN . > > . 8 > > > > > > > > > > mSO, > > < > > > > for xk ¼ a2 ; 1st bifurcation > > > > mUS ! > > > > > > appearing > mSI-oSO, > > > : : : for xk ¼ a1 ; ð5:98Þ and 8 > > > > > > > > > > > > > > >
> > > < > th > for xk ¼ a2m þ 1 ; m bifurcation > > mUS ! > > mSI-oSO, > appearing > > > > : > for xk ¼ a2m ; > < m-mUSN .. . mSO (xk ¼ aÞ ! 8 > > > > mSO, > > > > > > > < > > st > for xk ¼ a3 ; 1 bifurcation > > > > > mUS ! > > > > mSI-oSO, > appearing > > > > > : : > > for xk ¼ a2 ; > :  mSO ! mSO, for xk ¼ a ¼ a1 ;

ð5:99Þ

5.2 Simple Fixed-Point Bifurcations

8 > > > > > > > > > > > > > > >
> > > < > > for xk ¼ a2m þ 1 ; mth bifurcation > > mLS ! > > mSO, > appearing > > > > : > for xk ¼ a2m ; > < m-mLSN .. . mSI-oSO (xk ¼ aÞ ! 8 > > > > > > mSI-oSO, > > > > > < > > st > for xk ¼ a3 ; 1 bifurcation > > > > > mLS ! > > > > mSO, > appearing > > > > > : : > > for xk ¼ a2 ; > :  mSI-oSO ! mSI-oSO, for xk ¼ a ¼ a1 :

357

ð5:100Þ If xk ¼ a ¼ 12Bi (i 2 f1; 2; . . .; mg), the fixed-point of xk ¼ a possess a third-order mSI or mSO switching bifurcation (or pitchfork bifurcation). The teethcomb appearing bifurcation generated by the m-pairs of quadratics becomes a broom appearing bifurcation. The two broom appearing bifurcations are m¼l1 þ l2 þ 1

mSO ( xk ¼ aÞ !ðl1 -mLSN:3rd mSO:l2 -mUSNÞ appearing bifurcation 8 l2 -mUSN, for xk 2 fa2j ; a2j þ 1 ; i ¼ l1 þ 2;    ; l1 þ l2 g; > > 8 > >  > > > < < mSO, for xk ¼ a2ðl1 þ 2Þ1 ¼ 3rd mSO ! mSI-oSO, for xk ¼ a ¼ a2ðl1 þ 1Þ > > > : > mSO, for xk ¼ a2ðl1 þ 1Þ1 > > > : l1 -mLSN, for xk 2 fa2i1 ; a2i ; i ¼ 1; 2; . . .; l1 g

ð5:101Þ

and m¼l1 þ l2 þ 1

mSI-oSO ( xk ¼ aÞ !ðl1 -mUSN:3rd mSI:l2 -mLSNÞ appearing bifurcation 8 l2 -mLSN, for xk 2 fa2j ; a2j þ 1 ; j ¼ l1 þ 2;    ; l1 þ l2 g; > > 8 > >  > > > < < mSI-oSO, for xk ¼ a2ðl1 þ 2Þ1 ¼ 3rd mSI ! mSO, for xk ¼ a ¼ a2ðl1 þ 1Þ > > > : > mSI-oSO, for xk ¼ a2ðl1 þ 1Þ1 > > > : l1 -mUSN, for xk 2 fa2i1 ; a2i ; i ¼ 1; 2; . . .; l1 g:

ð5:102Þ

Consider an appearing bifurcation for a cluster of fixed-points with monotonic sink to oscillatory source, and monotonic source with the following conditions. Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ Dj ¼ 0ði ¼ 1; 2; . . .; nÞ

 at bifurcation:

ð5:103Þ

Thus, the ð2lÞth -order appearing bifurcation is for a cluster of simple monotonic sinks to monotonic-sources and monotonic sources. Two ð2lÞth order appearing bifurcations for l 2 f1; 2; . . .; sg are

5 (2m + 1)th-Degree Polynomial Discrete Systems

358

8 mSO, for xk ¼ a2sl ; > > > > mSI-oSO, for xk ¼ a2sl 1 ; > < cluster of l quadratics ð2lÞth order mUSN ! ... appearing bifurcation > > > > mSO, for xk ¼ a2s1 ; > : mSI-oSO, for xk ¼ a2s1 1 : 8 mSI-oSO, for xk ¼ a2sl ; > > >  > > < mSO, for xk ¼ a2sl 1 ; cluster of l quadratics ð2lÞth order mLSN ! ... appearing bifurcation > > > > mSI-oSO, for xk ¼ a2s1 ; > : mSO, for xk ¼ a2s1 1 :

ð5:104Þ

ð5:105Þ

If xk ¼ a 6¼ 12Bi (i 2 f1; 2; . . .; ng), the fixed-point of xk ¼ a breaks a cluster of sprinkler-spraying appearing bifurcations to two parts. The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broomsprinkler-spraying appearing bifurcation. The two broom-sprinkler-spraying appearing bifurcations are m¼m1 þ m2

mSO ( xk ¼ aÞ !ðr1 -mLSG:mSO:r2 -mUSGÞ appearing bifurcation 8 8 ð2Þ th > > ð2l Þ mUSN (xk ¼ ar2 þ 1 Þ; > > > < r2 > > > > r2 -mUSG ! ... > > > > > > : ð2Þ th > > ð2l1 Þ mUSN (xk ¼ ar1 þ 2 Þ; > < ¼ mSO (a ¼ ar1 þ 1 Þ ! mSO ða ¼ a2ðm1 þ 1Þ1 Þ; > 8 ð1Þ th > > > > ð2lr1 Þ mLSN (xk ¼ ar1 Þ; > > > < > > > > r1 -mLSG ! ... > > > > > > : ð1Þ th : ð2l1 Þ mLSN (xk ¼ a1 Þ;

ð5:106Þ

and m¼m1 þ m2

mSI-oSO ( xk ¼ aÞ !ðr1 -mUSG:mSI-oSO:r2 -mLSGÞ appearing bifurcation 8 8 ð2Þ th > > ð2l Þ mLSN (xk ¼ ar2 þ 1 Þ; > > > < r2 > > . > > > r2 -mLSG ! > .. > > > > : ð2Þ th > > ð2l1 Þ mLSN (xk ¼ ar1 þ 2 Þ; > < ¼ mSI-oSO (a ¼ ar1 þ 1 Þ ! mSI-oSO ða ¼ a2ðm1 þ 1Þ1 Þ; > 8 ð1Þ th > > > > ð2l Þ mUSN (xk ¼ ar1 Þ; > > > < r1 > > > > r1 -mUSG ! ... > > > > > > : : ð1Þ ð2l1 Þth mUSN (xk ¼ a1 Þ;

ð5:107Þ

5.2 Simple Fixed-Point Bifurcations ð1Þ

359

ð2Þ

1 2 for m1 ¼ Pri¼1 li ; m2 ¼ Prj¼1 lj ; and the acronyms USG and LSG are the upper-saddle-node and lower-saddle-node bifurcation groups, respectively. Four special broom-sprinkler-spraying appearing bifurcations are

m¼

Pr

i¼1 li

mSO ( xk ¼ aÞ ! appearing bifurcation 8 mSO ða ¼ a2r þ 1 Þ ! SmO (a ¼ a2m þ 1 Þ; > > > 8 > > < ð2l Þth mLSN (xk ¼ ar Þ; > > < r > r-mLSG ! ... > > > > > > : : ð2l1 Þth mLSN (xk ¼ a1 Þ; m¼

ð5:108Þ

Pr

i¼1 li

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation 8 mSI-oSO ða ¼ a2r þ 1 Þ ! mSI-oSO (a ¼ a2m þ 1 Þ; > > > 8 > > ð5:109Þ < ð2l Þth mUSN (xk ¼ ar Þ; > > < r > r-mUSG ! ... > > > > > > : : ð2l1 Þth mUSN (xk ¼ a1 Þ; and m¼

Pr

i¼1 li

mSO ( xk ¼ aÞ ! appearing bifurcation 8 8 > ð2lr Þth mUSN (xk ¼ ar þ 1 Þ; > > > > < > > < r-mUSG ! .. . > > : > > > ð2l1 Þth mUSN (xk ¼ a2 Þ; > > : mSO ða ¼ a1 Þ ! mSO (a ¼ a1 Þ; m¼

ð5:110Þ

Pr

i¼1 li

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation 8 8 > ð2lr Þth mLSN (xk ¼ ar þ 1 Þ; > > > > < > > < r-mLSG ! .. . > > : > > > ð2l1 Þth mLSN (xk ¼ a2 Þ; > > : mSI-oSO ða ¼ a1 Þ ! mSI-oSO (a ¼ a1 Þ;

ð5:111Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

360

If xk ¼ a ¼ 12Bi (i 2 f1; 2; . . .; lg), the fixed-point of xk ¼ a possesses a ð2l þ 1Þth -order mSI or mSO switching bifurcation (or broom-switching bifurcation). The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying switching bifurcation. The two broom switching bifurcations are m¼m1 þ m2 þ l

rd mSO ( xk ¼ aÞ !ðr1 -mLSG:(2l þ 1Þ mSO:r2 -mUSGÞ switching bifurcation 8 ð2Þ th 8 > ð2l Þ mUSN (xk ¼ ar2 þ r1 þ 1 Þ; > > > < r2 > > > . > > > r2 -mUSG ! > .. > > > > : ð2Þ th > > ð2l1 Þ mUSN (xk ¼ ar1 þ 2 Þ; > < ¼ ð2l þ 1Þth mSO (a ¼ ar1 þ 1 Þ; > 8 ð1Þ th > > > > ð2l Þ mLSN (xk ¼ ar1 Þ; > > > < r1 > > > > r1 -mLSG ! ... > > > > > > : : ð1Þ ð2l1 Þth mLSN (xk ¼ a1 Þ;

ð5:112Þ

and m¼m1 þ m2 þ l

th mSI-oSO ( xk ¼ aÞ !ðr1 -mUSG:(2l þ 1Þ mSI:r2 -mLSGÞ switching bifurcation 8 ð2Þ th 8 > > ð2l Þ mLSN (xk ¼ ar2 þ r1 þ 1 Þ; > > < r2 > > > > r -mLSG ! ... > > > 2 > > > > : ð2Þ th > > ð2l1 Þ mLSN (xk ¼ ar1 þ 2 Þ; > < ð5:113Þ ¼ ð2l þ 1Þth mSI (a ¼ ar1 þ 1 Þ; > 8 ð1Þ th > >  > > ð2l Þ mUSN (xk ¼ ar1 Þ; > > > < r1 > > > > r1 -mUSG ! ... > > > > > > : : ð1Þ th ð2l1 Þ mUSN (xk ¼ a1 Þ;

where 8 mSO, for > > > > > mSI-oSO, > > < cluster of l-quadratics th  ð2l þ 1Þ order mSO (xk ¼ aÞ ! ... appearing bifurcation > > > > > mSI-oSO, > > : mSO, for

xk ¼ a2sl þ 1 ;

for xk ¼ a2sl ; for xk ¼ a2s1 ;

xk ¼ a2s1 1 :

ð5:114Þ

5.2 Simple Fixed-Point Bifurcations

361

and 8 mSI-oSO, > > > > > mSO, for > > < cluster of l quadratics ð2l þ 1Þth order mSI (xk ¼ aÞ ! ... appearing bifurcation > > > > > mSO, for > > : mSI-oSO,

for xk ¼ a2sl þ 1 ;

xk ¼ a2sl ;

xk ¼ a2s1 ;

for xk ¼ a2s1 1 ð5:115Þ

where xk ¼ a 2 fa2s1 1 ;    ; a2sl ; a2sl þ 1 g. In Fig. 5.1i and ii, the simple switching with two teethcomb appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: (i) mSO ! ðl1 -mLSN:mSO:l2 -mUSN), (ii) mSI-oSO ! ðl1 -mUSN:mSI-oSO:l2 -mLSN) with l1 þ l2 ¼ m. In Fig. 5.1iii and iv, the 3rd-order pitchfork switching bifurcation with two teethcomb appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: (iii) mSO ! ðl1 -mLSN:3rd mSO:l2 -mUSN), (iv) mSI-oSO ! ðl1 -mUSN:3rd mSI:l2 -mLSN) with l1 þ l2 ¼ m  1. The period-2 fixed points are sketched as well through red curves. In Fig. 5.2i and ii, the simple switching with two sprinkler-spraying appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: (i) mSO ! ðr1 -mLSG:mSO:r2 -mUSG), (ii) mSI-oSO ! ðr1 -mUSG:mSI-oSO:r2 -mLSG) ð1Þ

ð2Þ

1 2 with r1 þ r2 þ 1 ¼ n and m1 þ m2 ¼ m where m1 ¼ Pri¼1 li ; m2 ¼ Prj¼1 lj . In

Fig. 5.2iii and iv, the ð2l þ 1Þth order broom-switching with two sprinkler-spraying appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: (iii) mSO ! ðr1 -mLSG:(2l þ 1Þth mSO:r2 -mUSG), (iv) mSI-oSO ! ðr1 -mUSG:(2l þ 1Þth mSI:r2 -mLSG) ð1Þ

ð2Þ

1 2 with r1 þ r2 þ 1 ¼ n and m1 þ m2 þ l ¼ m where m1 ¼ Pri¼1 li ; m2 ¼ Prj¼1 lj . The period-2 fixed points are sketched as well through red curves.

5 (2m + 1)th-Degree Polynomial Discrete Systems

362 a0 > 0

b1(i1 )

mSO

a0 < 0

mUSN

mSI-oSO

P-2

b2(i1 )

mLSN

mSO

b1(i2 )

mSO

b2(i1 ) b1(i2 )

mSI-oSO

P-2

P-2

mUSN

b1(i1 )

mSI-oSO

P-2

LSN • • •

mSI-oSO

b2(i2 )

mSO

a

• • •

mSO

b2(i2 )

mSI-oSO

a

mSI-oSO

mSO

mSO

mSI-oSO P-2

mUSN

P-2

mUSN

mSO • • •

•

mSI-oSO

mSI-oSO

• •

b1(im )

b1(im )

mSO

P-2

mUSN

mUSN

xk∗

mSO

|| p ||

mSO

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

x

mSI-oSO

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(ii)

(i) a0 > 0

P-2

∗ k

b1(i1 )

mSO

a0 < 0

b1(i1 )

mSI-oSO

P-2 mUSN

P-2

mSI-oSO

b2(i1 )

mSO

b1(i2 )

mSO

mUSN

mLSN

b2(i1 )

b1(i2 )

mSI-oSO

P-2

P-2

mLSN •

mSI-oSO

b2(i2 )

• •

•

mSO

mSO

mSI-oSO

a

3rd mSI mSI-oSO

P-2 mSO

mSO

• • •

•

P-2 mSI-oSO

P-2 mSI-oSO

a

3rd mSO

P-2

mSI-oSO

• •

b1(im )

b1(im )

mSO

P-2

mUSN

mUSN

xk∗ || p ||

b2(i2 )

mSO

• •

mSO

Δ iq < 0 Δ iq = 0

(iii)

Δ iq > 0

( im ) 2

b

P-2

∗ k

x

|| p ||

mSI-oSO

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

(iv)

Fig. 5.1 Simple broom switching bifurcations: (i) ðmUS:    :mUS : mSO:mLS:    :mLSÞ(a0 [ 0), (ii) ðmLS:    :mLS:mSI-oSO:mUS:    : mUSÞ(a0 \0), (iii) ðmUS:    :mUS : 3rd mSO:mLS :    :mLSÞ (a0 [ 0). (vi) ðLS:    :LS : 3rd SI:US:    :USÞ (a0 \0) in a ð2m þ 1Þth -degree polynomial system. mLS: monotonic lower saddle, mUS: monotonic upper saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: period-2 fixed points

5.2 Simple Fixed-Point Bifurcations mSO

a0 > 0

363 b1(i1 ) b

b2(i1 )

6th mUS mSI-oSO

mSI-oSO

a0 < 0

( i2 ) 1

b2(i1 )

6th mLS

b2(i2 )

mSO

P-2

P-2

4th mLS

4th mUS

mSI-oSO mSO

•

P-2

P-2

mSO

• •

a

mSI-oSO

•

mSI-oSO

mSO

•

mSO

• •

mSI-oSO 4th

P-2

(2r)th mUS

(2r)th mLS

• •

mSI-oSO mSO

b1(im ) 4th

mLS

a

mSO

mSI-oSO

mSO

•

b2(i2 )

mSI-oSO

mSO

• •

b1(i1 ) b1(i2 )

P-2

b1(im )

mUS

xk∗

xk∗ mSO

|| p ||

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

mSI-oSO

|| p ||

(i)

(ii) mSO

a0 > 0

b1(i1 ) b

b2(i1 ) mSI-oSO

mSI-oSO

a0 < 0

( i2 ) 1

6th mUS

b

mSO mSI-oSO

P-2

4th mUS

• • •

mSI-oSO

mSO (2r+1)th mSO • •

4th mLS

• •

P-2

P-2

mSO mSI-oSO

P-2

P-2

a

mSI-oSO

P-2

(2r+1)th mSI

a

P-2

mSO

• • •

•

mSI-oSO

b2(i2 )

P-2

•

mSO

b1(i1 ) b1(i2 ) b2(i1 )

6th mLS

( i2 ) 2

mSO

mSI-oSO mSO

b1(im )

P-2

b1(im )

4th mUS

4th mLS

xk∗

∗ k

x

mSO

|| p ||

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

Δ iq < 0 Δ iq = 0

(iii)

Δ iq > 0

b2(im )

mSI-oSO

|| p ||

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

(iv)

Fig. 5.2 Broom appearing bifurcation: (i) ðr1 -mLSN:mSO:r2 -mUSNÞ(a0 [ 0); (ii)ðr1 -mUSN: mSI-oSO:r2 -mLSNÞ (a0 \0); broom-sprinkler-spraying switching bifurcation: (iii)ðr1 - mUSG : ð2lk þ 1Þth mSO:r2 -mLSGÞ (a0 [ 0). (iv)ðr1 -mLSG:ð2lk þ 1Þth mSI:r2 -mUSGÞ(a0 \0) in a ð2m þ 1Þth degree polynomial system. mLS: monotonic lower-saddle, mUS: monotonic-uppersaddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: period-2 fixed points

5 (2m + 1)th-Degree Polynomial Discrete Systems

364

For a cluster of m-quadratics, Bi ¼ Bj (i; j 2 f1; 2; . . .; mg; i 6¼ j) and Di ¼ 0 (i 2 f1; 2; . . .; mg). The ð2mÞth -order upper-saddle-node appearing bifurcation for m-pairs of fixed-points with monotonic sink to oscillatory source and monotonic source is 8 mSO, for > > > > mSI-oSO, > < cluster of m-quadratics ð2mÞth order mUS ! ... appearing bifurcation > > > > mSO, for > : mSI-oSO,

xk ¼ a2m ; for xk ¼ a2m1 ; ð5:116Þ xk

¼ a2 ; for xk ¼ a1 :

The ð2mÞth -order lower-saddle-node appearing bifurcation for m-pairs of fixed-points with monotonic sink to oscillatory source and monotonic source is 8 mSI-oSO, > > > mSO, for > > < cluster of m-quadratics ð2mÞth order mLS ! ... appearing bifurcation > > > > mSI-oSO, > : mSO, for

for xk ¼ a2m ; xk ¼ a2m1 ; ð5:117Þ xk

for ¼ a2 ; xk ¼ a1 :

There are four simple switching and ð2mÞth -order saddle-node appearing bifurcations: The two switching bifurcations of mSO ! ðð2mÞth mUS:mSOÞ and mSI-oSO ! ðð2mÞth mLS:mSIÞ with two ð2mÞth -order mUSN and mLSN spraying appearing bifurcations are

mSO (xk ¼ aÞ !

8 mSO ! mSO, for xk > > > > > > > < ð2mÞ > > > > > > > :

th

¼8a ¼ a2m þ 1 > mSI-oSO, > > > > < mSO, for order mLSN ! ... > > > > mSI-oSO, > : mSO, for

for xk ¼ a2m ; xk ¼ a2m1 ; for xk ¼ a2 ; xk ¼ a1 ; ð5:118Þ

mSI-oSO (xk ¼ aÞ !

8 mSI-oSO ! mSI-oSO, > > > > > > > < ð2mÞ > > > > > > > :

th

 for 8 xk ¼ a ¼ a2m þ 1 mSO, for xk ¼ a2m ; > > >  > > < mSI-oSO, for xk ¼ a2m1 ; order mUSN ! ... > > > > mSO, for xk ¼ a2 ; > : mSI-oSO, for xk ¼ a1 :

ð5:119Þ

5.2 Simple Fixed-Point Bifurcations

365

and the two switching bifurcations of mSO ! ðmSO : ð2mÞth mUSÞ and mSI-oSO ! ðmSI-oSO : ð2mÞth mLSÞ with two ð2mÞth -order mUSN and mLSN spraying appearing bifurcations are 8 > > > > > > >
> > mSI-oSO, > > < ð2mÞth order mUSN ! ...  mSO (xk ¼ aÞ ! > > > > > > mSO, for > > > : > > mSI-oSO, > : mSO ! mSO, for xk ¼ a ¼ a1

xk ¼ a2m þ 1 ; for xk ¼ a2m ; xk ¼ a3 ; for xk ¼ a2 ; ð5:120Þ

8 > > > > > > >
> >  > > < mSO, for xk ¼ a2m ; th . ð2mÞ order mLSN ! .. mSI-oSO (xk ¼ aÞ ! > > > > > > mSI-oSO, for xk ¼ a3 ; > > > : > > mSO, for xk ¼ a2 ; > : mSI-oSO ! mSI-oSO, for xk ¼ a ¼ a1 : ð5:121Þ The ð2m þ 1Þth order monotonic source broom switching bifurcation is 8 mSO, for > > > > > mSI-oSO, > > < switching mSO(xk ¼ aÞ   !  ð2m þ 1Þth order mSO ... > > > > > mSI-oSO, > > : mSO, for

xk ¼ a2m þ 1 ; for xk ¼ a2m ; for xk ¼ a2 ; xk ¼ a1 : ð5:122Þ

The ð2m þ 1Þth order monotonic sink broom-switching bifurcation is 8 mSI-oSO, > > > > > mSO, for > > < switching th .  mSI-oSO (xk ¼ a1 Þ   !  ð2m þ 1Þ order mSI .. > > > > > mSO, for > > : mSI-oSO,

for xk ¼ a2m þ 1 ;

xk ¼ a2m ; xk ¼ a2 ;

for xk ¼ a1 : ð5:123Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

366

The switching bifurcation consist of a simple switching and the ð2mÞth order monotonic saddle-node appearing bifurcation with m-pairs of monotonic sources and monotonic sink to oscillatory sources. The ð2mÞth order monotonic saddle-node appearing bifurcation is a sprinkler-spraying cluster of the m-pairs of monotonic sources and monotonic sinks to oscillatory sources. Thus, the four switching bifurcations of mSO ! ðð2mÞth mLS:mSOÞ for a0 [ 0; mSI-oSO ! ðð2mÞth mUS:mSI-oSOÞ for a0 \ 0; mSO ! ðmSO : ð2mÞth mUS) for a0 [ 0; mSI-oSO ! ðmSI-oSO:ð2mÞth mLSÞ for a0 \ 0; are presented in Fig. 5.3i–iv, respectively. The ð2m þ 1Þth -order monotonic source switching bifurcation is named the ð2m þ 1Þth mSO broom switching bifurcation and the ð2m þ 1Þth -order monotonic sink switching bifurcation is named the ð2m þ 1Þth mSI broom switching bifurcation. Such a ð2m þ 1Þth mXX broomswitching bifurcation is from simple fixed-point to a ð2m þ 1Þth mXX broomswitching bifurcation. The two broom-switching bifurcations of mSO ! ð2m þ 1Þth mSO for a0 [ 0; mSI-oSO ! ð2m þ 1Þth mSI for a0 \0; are presented in Fig. 5.3v–vi, respectively. The period-2 fixed points are sketched as well through red curves. A series of the third-order monotonic source and monotonic sink bifurcations is aligned up with varying with parameters. Such a special pattern is from m-quadratics in the ð2m þ 1Þth order polynomial systems, the following conditions should be satisfied. 1 2

1 2

aðpi Þ ¼  Bi and aðpj Þ ¼  Bj Bi Bj i; j 2 f1; 2; . . .; ng; i 6¼ j; Di [ Di þ 1 ði ¼ 1; 2; . . .; n; n mÞ;

ð5:124Þ

Di ¼ 0 with jjpi jj\jjpi þ 1 jj: Thus, a series of m-ð3rd mSO-3rd mSI-   Þ switching bifurcations (a0 [ 0) and a series of m-ð3rd mSI-3rd mSO-   Þ switching bifurcations (a0 \0) are presented in Fig. 5.4i and ii. The bifurcation scenario is formed by the swapping pattern of 3rd mSI and 3rd mSO switching bifurcations. Such a bifurcation scenario is like the fish-bone. Thus, such a bifurcation swapping pattern of 3rdmSI and 3rdmSO switching bifurcations is called the fish-bone switching bifurcation in the ð2m þ 1Þth

5.2 Simple Fixed-Point Bifurcations a0 > 0

mSO mSO

mSI-oSO

mSO

367 a a2m

a0 < 0

mSI-oSO mSI-oSO

mSI-oSO P-2

P-2 P-2

(2m)th mLS

(2m)th mUS

P-2

a2

xk∗

mSO

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

a1

P-2

a2

xk∗

mSI-oSO

Δ iq < 0 Δ iq = 0

|| p ||

(i) a0 > 0

Δ iq > 0

a2m

a0 < 0

mSI-oSO

(2m)th mUS

(2m)th mLS

P-2

P-2

P-2

a2 mSI-oSO

a1

mSO

a

mSO

Δ iq < 0 Δ iq = 0

|| p ||

Δ iq > 0

a2

mSI-oSO

xk∗

mSI-oSO

Δ iq < 0 Δ iq = 0

|| p ||

(iii)

mSO

a1

mSI-oSO

a

Δ iq > 0

(iv)

a0 > 0 mSO

a0 < 0

a2m

mSI-oSO

P-2 (2m+1)th mSI

a mSO

xk∗

mSO

Δ iq < 0 Δ iq = 0

(v)

Δ iq > 0

a1

a

mSI-oSO

P-2

a2

a2m P-2

(2m+1)th mSO

|| p ||

a2m P-2

P-2

xk∗

a1

(ii) mSO

mSO

a a2m

mSO

P-2

P-2

a2

xk∗ || p ||

mSI-oSO

Δ iq < 0 Δ iq = 0

a1

Δ iq > 0

(vi)

Fig. 5.3 (i) ðð2mÞth mLS:mSO)-switching bifurcation ða0 [ 0Þ, (ii) ðð2mÞth mUS:mSI-oSOÞswitching bifurcation ða0 \0Þ, (iii) ðmSO:ð2mÞth mUS)-switching bifurcation ða0 [ 0Þ, (iv) ðmSI-oSO:ð2mÞth mLSÞ-switching bifurcation ða0 \0Þ, (v) ð2m þ 1Þth mSO broom appearing bifurcation (ða0 [ 0Þ, (vi) ð2m þ 1Þth mSI-oSO broom appearing bifurcation ða0 \0Þ in the ð2m þ 1Þth degree polynomial system. mLS: monotonic lower saddle, mUS: monotonic upper saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, which are sketched though red curves

5 (2m + 1)th-Degree Polynomial Discrete Systems

368 a0 > 0

P-2

P-2

b1( r )

mSO

3rd mSO mSO

3rd mSI

3rd mSO mSO

P-2

xk∗

3rd mSI

• • •

• • •

a

mSI-oSO

b2( r )

P-2

Δr < 0

P-2

mSO

P-2

mSO

|| p ||

3rd mSO

P-2

Δr > 0

Δr = 0

(i)

a0 < 0

P-2

P-2 mSO

3rd mSI P-2

x∗

3rd mSO

3rd mSI

mSO

• • •3rd

P-2

mSO

3rd mSI

a

P-2

mSO

mSO

mSI-oSO mSO

P-2

|| p ||

• • •

P-2

b1( r )

Δr < 0

Δr = 0

P-2

b2( r )

P-2

Δr > 0

(ii) Fig. 5.4 (i) m-ð3rd mSO-3rd mSI-   Þ series bifurcation ða0 [ 0Þ, (ii) m-ð3rd mSI-3rd mSO-   Þ series switching bifurcation ða0 \0Þ in the ð2m þ 1Þth -degree polynomial system. mSI: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, which are sketched though red curves

degree polynomial nonlinear system. There are two swaps of the 3rdmSI and 3rdmSO bifurcations: (i) the 3rdmSO-3rdmSI fish-bone switching bifurcation and (ii) the 3rdmSI-3rdmSO fish-bone, switching bifurcation. The period-2 fixed-points are presented by P-2, which is sketched by red curves. The period-2 fixed-points are relative to the monotonic sink to the oscillatory source (mSI-oSO), and the monotonic sink to oscillatory source and back to monotonic sink (mSI-oSO-mSI).

5.2 Simple Fixed-Point Bifurcations

5.2.2

369

Switching Bifurcations

In the ð2m þ 1Þth order polynomial discrete system, among the possible ð2m þ 1Þ roots, there are two roots to satisfy x2k þ Bi xk þ Ci ¼ 0 with ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ

ðiÞ

ðiÞ

ðiÞ

xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2;    ; nÞ; ) Bi 6¼ Bj ði; j ¼ 1; 2;    ; n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2;    ; nÞ

ð5:125Þ

The second-order singularity bifurcation is for the switching of a pair of fixedpoints with a simple monotonic sink to oscillatory source, and monotonic source. There are two switching bifurcations for i 2 f1; 2; . . .; ng ( ith quadratic factor

2nd order mUS ! appearing bifurcation

( ith quadratic factor

2 order mLS ! nd

switching bifurcation

ðiÞ

ðiÞ

mSO, for a2i ¼ b2 ! b1 ; ðiÞ ðiÞ mSI-oSO, for a2i1 ¼ b1 ! b2 ; ðiÞ

ðiÞ

mSI-oSO, for a2i ¼ b2 ! b1 ; ðiÞ ðiÞ mSO, for a2i1 ¼ b1 ! b2 : ðiÞ

ð5:126Þ

ð5:127Þ

ðiÞ

For non-switching point, Di [ 0 at b1 6¼ b2 (i ¼ 1; 2; . . .; n). At the bifurcation ðiÞ

ðiÞ

point, Di ¼ 0 at b1 ¼ b2 (i ¼ 1; 2; . . .; n). The l-mUSN antenna switching bifurcation for si 2 f0; 1; . . .; mg (i ¼ 1; 2; . . .; l) is 8 ( > sth mSO # mSI-oSO, > l bifurcation > mUS ! > > > switching mSI-oSO " mSO, > < . l-mUSN . . ( > > > > sth mSO # mSI-oSO, > l bifurcation > > : mUS ! switching mSI-oSO " mSO,

ðs Þ

for b2 l ¼ a2sl # a2sl 1 ; ðs Þ for b1 l ¼ a2sl 1 " a2sl ; ðs Þ

for b2 1 ¼ a2s1 # a2s1 1 ; ðs Þ for b1 1 ¼ a2s1 1 " a2s1 : ð5:128Þ

The l-mLSN antenna switching bifurcation for si 2 f0; 1; . . .; mg (i ¼ 1; 2; . . .; l) is

370

5 (2m + 1)th-Degree Polynomial Discrete Systems

8 ( > sth mSI-oSO # mSO, > l bifurcation > > mLS ! > > switching mSO " mSI-oSO, > < l-mLSN .. . ( > > > > sth mSI-oSO # mSO, > l bifurcation > > : mLS ! switching mSO " mSI-oSO,

ðs Þ

for b2 l ¼ a2sl # a2sl 1 ; ðs Þ for b1 l ¼ a2sl 1 " a2sl ; ðs Þ

for b2 1 ¼ a2s1 # a2s1 1 ; ðs Þ for b1 1 ¼ a2s1 1 " a2s1 : ð5:129Þ

Two antenna switching bifurcation structures exist for the ð2m þ 1Þth -order polynomial discrete system. The ðl1 -mLSN:mSO:l2 -mUSNÞ-switching bifurcation for a0 [ 0 is 8 < l2 -mUSN l1 þ l2 ¼m ðl1 -mLSN:mSO:l2 -mUSNÞ ! mSO ! mSO, : l1 -mLSN;

ð5:130Þ

and the ðl1 -mUSN:mSI-oSO:l2 -mLSNÞ-switching bifurcation for a0 \0 is 8 < l2 -mLSN, l1 þ l2 ¼m ðl1 -mUSN:mSI-oSO:l2 -mLSNÞ ! mSI-oSO ! mSI-oSO, ð5:131Þ : l1 -USN: As in the ð2m þ 1Þth -order polynomial system, consider a switching bifurcation for a bundle of fixed-points with monotonic-sink-to-oscillatory-source and monotonic-source with the following conditions, Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ ðiÞ ðiÞ Di ¼ 0 at b1 = b2 ði ¼ 1; 2; . . .; nÞ

 at bifurcation:

ð5:132Þ

Two ð2lÞth order switching bifurcations for l 2 f1; 2; . . .; sg are 8 mSO, for > > > > mSI-oSO, > < a bundle of ð2lÞ-fixed-points ð2lÞth order mUS ! ... switching bifurcation > > > > mSO, for > : mSI-oSO,

a2sl ! b2sl ; for a2sl 1 ! b2sl 1 ; a2s1 ! b2s1 ; for a2s1 1 ! b2s1 1 : ð5:133Þ

8 mSI-oSO, > > > > > < mSO, for a bundle of ð2lÞ-fixed-points th ð2lÞ order mLS ! ... switching bifurcation > > > > > mSI-oSO, : mSO, for

for a2sl ! b2sl ; a2sl 1 ! b2sl 1 ; ð5:134Þ for a2s1 ! b2s1 ; a2s1 1 ! b2s1 1 :

5.2 Simple Fixed-Point Bifurcations

371

where Dij ¼ ðai  aj Þ2 ¼ ðbi  bj Þ2 ¼ 0 with Bi ¼ Bj ði; j ¼ 2s1  1; 2s1 ;    ; 2sl  1; 2sl Þ and fa2s1 1 ; a2s1 ;    ; a2sl 1 ; a2sl g fb2s1 1 ; b2s1 ;    ; b2sl 1 ; b2sl g

ð1Þ



before bifurcation

ð1Þ



after bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ;    b1 ; b2 ; ag; ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ;    b1 ; b2 ; ag:

ð5:135Þ

Two ð2l þ 1Þth order switching bifurcations for l 2 f1; 2; . . .; sg are 8 mSO, for a2sl þ 1 ! b2sl þ 1 ; > > > > .. < a bundle of ð2l þ 1Þ-fixed-points ð2l þ 1Þth order mSO ! . switching bifurcation > > mSI-oSO, for a2s1 ! b2s1 ; > > : mSO, for a2s1 1 ! b2s1 1 : ð5:136Þ 8 mSI-oSO, for a2sl þ 1 ! b2sl þ 1 ; > > > > .. < a bundle of ð2l þ 1Þ-fixed-points ð2l þ 1Þth order mSI ! . switching bifurcation > > mSO, for a2s1 ! b2s1 ; > > : mSI-oSO, for a2s1 1 ! b2s1 1 : ð5:137Þ where Dij ¼ ðai  aj Þ2 ¼ ðbi  bj Þ2 ¼ 0 with Bi ¼ Bj ði; j ¼ 2s1  1; 2s1 ;    ; 2sl  1Þ and fa2s1 1 ; a2s1 ;    ; a2sl þ 1 g fb2s1 1 ; b2s1 ;    ; b2sl þ 1 g

ð1Þ

ð1Þ

ðnÞ

ðnÞ



sortfb1 ; b2 ;    b1 ; b2 ; ag;



sortfb1 ; b2 ;    b1 ; b2 ; ag:

before bifurcation After bifurcation

ð1Þ

ð1Þ

ðnÞ

ðnÞ

ð5:138Þ

A set of paralleled, different, higher-order, monotonic upper-saddle-node switching bifurcations is the ðða1 Þth mXX:(a2 Þth mXX:    :ðas Þth mXXÞ parallel switching bifurcation in the ð2m þ 1Þth -degree polynomial discrete system. At the strawbundle switching bifurcation, Di ¼ 0ði ¼ 1; 2; . . .; nÞ and Bi ¼ Bj (i; j 2 f1; 2; . . .; ng; i 6¼ j). Thus, the parallel straw-bundle switching bifurcation is ðða1 Þth mXX:ða2 Þth mXX:    :ðas Þth mXXÞ-switching 8 > ðas Þth order mXX switching, > > > > < .. ¼ . > > ða2 Þth order mXX switching, > > > : ða1 Þth order mXX switching;

ð5:139Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

372

where X

ai 2 f2li ; 2li  1gwith si¼1 ai ¼ 2m þ 1; and mXX 2 fmUS; mLS; mSO; mSIg: ðjÞ

ðjÞ

ð5:140Þ

ðjÞ

The ð2l1 : 2l2 :    : 2ls Þth mUSN parallel switching bifurcation is called a

ðjÞ

ðjÞ

ðjÞ

ð2l1 : 2l2 :    : 2ls Þth mUSN parallel straw-bundle switching bifurcation. ðjÞ

ðjÞ

th sj -mUSG ¼ ð2l1 : 2l2 :    : 2lðjÞ sj Þ mUSN switching 8 ðjÞ th > ð2l Þ order mUSN switching, > > s > > < .. ¼ . ðjÞ > > ð2l2 Þth order mUSN switching, > > > : ðjÞ th ð2l1 Þ order mUSN switching: ðjÞ

ðjÞ

ð5:141Þ

ðjÞ

The ð2l1 : 2l2 :    : 2ls Þth mLSN parallel switching bifurcation is called a

ðjÞ

ðjÞ

ðjÞ

ð2l1 : 2l2 :    : 2ls Þth mLSN parallel straw-bundle switching bifurcation. ðjÞ

ðjÞ

th sj -mLSG ¼ ð2l1 : 2l2 :    : 2lðjÞ sj Þ mLSN switching 8 ðjÞ th > ð2l Þ order mLSN switching, > > s > > < .. ¼ . ðjÞ > > ð2l2 Þth order mLSN switching, > > > : ðjÞ th ð2l1 Þ order mLSN switching:

ð5:142Þ

ð2Þ

The ðs1 -mLSG:ð2l1 þ 1Þth mSO:s3 -mUSGÞ-switching bifurcation for a0 [ 0 is 8 ð3Þ ð3Þ th > < ð2l1 ;    ; 2ls2 Þ -mUSN, ð2Þ th ðs1 -mLSG; ð2l1 þ 1Þth mSO; s3 -mUSGÞ ¼ ð2lð2Þ ð5:143Þ 1 þ 1Þ mSO; > : ð1Þ ð1Þ th ð2l1 ;    ; 2ls1 Þ -mLSN; ð2Þ

and the ðs1 -mUSG:ð2l1 þ 1Þth mSI:s3 -mLSGÞ-switching bifurcation for a0 \ 0 is 8 ð3Þ ð3Þ th > < ð2l1 ;    ; 2ls3 Þ -mLSN, ð2Þ th ðs1 -mUSG:ð2l1 þ 1Þth mSI:s3 -mLSGÞ ! ð2lð2Þ 1 þ 1Þ mSI; > : ð1Þ ð1Þ ð2l1 ;    ; 2ls1 Þth -mUSN;

ð5:144Þ

The two ðl1 -mUSN:mSO:l2 -mLSNÞ and ðl1 -mLSN:mSI-oSO:l2 -mUSNÞ parallelswitching bifurcations (l1 þ l2 ¼ m) are presented in Fig. 5.5i and ii for a0 [ 0 and

5.2 Simple Fixed-Point Bifurcations

373 a2m

a0 > 0

a0 < 0

mSO mUSN

P-2 mSI-oSO

P-2

P-2

mLSN

a2 m−1

mSO

P-2

a2 m−2

mSO

a2m

mSI-oSO

a2 m−1 a2 m−2

mSI-oSO

mUSN

P-2

mLSN mSI-oSO

a2 m−3

mSO

P-2

•

P-2

P-2

a2 m−3

• • •

• •

mSO

mSI-oSO

mSO

P-2

mSO

mSI-oSO • • •

• •

P-2

a2

mSI-oSO

•

P-2

mUSN

a1

mSO

xk∗

a2

mSO P-2

mLSN

a1

mSI-oSO

xk∗

P-2 P-2

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(ii)

(i) a2m

a0 > 0

a0 < 0

mSO 3rd mSO

a2 m−1

P-2 mSO

a2 m−2

mSI-oSO

a2 m−3

a2m

mSI-oSO

P-2

P-2

3rd mSI

a2 m−2

mSI-oSO P-2

P-2

mSO

P-2

a2 m−1

a2 m−3

mSO

mSI-oSO P-2 • •

P-2

•

• • •

mSO P-2

mSI-oSO

P-2

4th

P-2

USN

4thmLSN P-2

P-2

mSI-oSO • •

P-2

•

P-2

•

P-2

• •

mSO

P-2

a2

3rd mSO

mSO

xk∗

mSO

P-2

mSI-oSO

a2

3rd mSI

a1

a1

mSI-oSO

xk∗

P-2 P-2

|| p ||

Δ iq > 0 Δ iq = 0

(iii)

Δ iq > 0

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.5 Parallel switching bifurcations: (i) ðl1 -mUSG:mSO:l2 -mUSGÞ ða0 [ 0Þ, (ii) ðl1 -mUSG: mSI  oSO : l2 -mLSGÞ ða0 \0Þ; (iii)ð3rd mSI:    :mUSN:3rd mSOÞ ða0 [ 0Þ, (vi) ð3rd mSO:    : mLSN : 3rd mSIÞ (ða0 \0Þ in the ð2m þ 1Þth -degree polynomial nonlinear system. mLSN: monotonic lower saddle-node, mUSN: monotonic upper saddle-node, mSI-oSO: monotonic sink, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2: period-2 fixed points which is sketched through red curves

5 (2m + 1)th-Degree Polynomial Discrete Systems

374

a0 \ 0, respectively. A set of ð3rd mSO:    :mSI-oSO:3rd mSOÞ parallel, switching bifurcations for mSI-oSO and mSO fixed-points is presented in Fig. 5.5iii for a0 [ 0. However, for a0 \ 0, the set of ð3rd mSI:    :mSO:3rd mSIÞ switching bifurcations for monotonic sources and monotonic sinks is presented in Fig. 5.5iv. The period-2 fixed-points are sketched through red curves, which is relative to the monotonic sink to the oscillatory source.

5.2.3

Switching-Appearing Bifurcations

Consider a ð2m þ 1Þth degree polynomial discrete system in a form of 2n1 þ 1 2 ðxk  ci Þ *nj¼1 ðx2k þ Bj xk þ Cj Þ: xk þ 1 ¼ xk þ a0 Qðxk Þ *i¼1

ð5:145Þ

Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bj xk þ Cj ¼ 0 are ðjÞ

1 2

b1;2 ¼  Bj 

1pffiffiffiffiffi Dj ; Dj 2

¼ B2j  4Cj 0ðj ¼ 1; 2; . . .; n2 Þ;

ð5:146Þ

either   fa 1 ; a2 ;    ; a2n1 þ 1 g ¼ sortfc1 ; c2    ; c2n1 þ 1 g;  a s as þ 1 before bifurcation; ð1Þ

ð1Þ

ðn Þ

ðn Þ

þ fa1þ ; a2þ ;    ; a2n g ¼ sortfc1 ;    ; c2n1 þ 1 ; b1 ; b2 ;    ; b1 2 ; b2 2 g; 3 þ1

ð5:147Þ

asþ asþþ 1 ; n3 ¼ n1 þ n2 after bifurcation; or ð1Þ

ð1Þ

ðn Þ

ðn Þ

2 2   fa 1 ; a2 ;    ; a2n3 þ 1 g ¼ sortfc1 ; c2    ; c2n1 ; b1 ; b2 ;    ; b1 ; b2 ; ag;

 a s as þ 1 ; n3 ¼ n1 þ n2 before bifurcation;

þ fa1þ ; a2þ ;    ; a2n g ¼ sortfc1 ;    ; c2n1 ; ag; 1 þ1

asþ asþþ 1 after bifurcation; ð5:148Þ and 9 Bj1 ¼ Bj2 ¼    ¼Bjs ðjk1 2 f1; 2; . . .; ng; jk1 6¼ jk2 Þ > > = ðk1 ; k2 2 f1; 2; . . .; sg; k1 6¼ k2 Þ at bifurcation: Dj ¼ 0ðj 2 U  f1; 2; . . .; n2 g > > ; ci 6¼ 12Bj ði ¼ 1; 2; . . .; 2n1 ; j ¼ 1; 2; . . .; n2 Þ

ð5:149Þ

th    th  th Consider a just before bifurcation of ðða 1 Þ mXX1 :ða2 Þ mXX2 :   : ðas1 Þ Ps1  mXXs1 Þ with i¼1 ai ¼ 2m1 þ 1 for simple monotonic-sources and monotonic-

5.2 Simple Fixed-Point Bifurcations

375

sink-to-oscillatory-sources in the ð2m þ 1Þth degree polynomial nonlinear discrete      system. For a i ¼ 2li  1; mXXi 2 fmSO,mSIg and for ai ¼ 2li ; mXXi 2 fmUS,mLSg ði ¼ 1; 2; . . .; s1 Þ. The detailed structures are as follows. 9 mSI-oSO > > > > > mSO > = .. th ! ð2l . i  1Þ mSI; and > > > > mSO > > ; mSI-oSO 9 mSO > > > > mSI-oSO > > = .. th ! ð2l and . i Þ mUS; > > > > mSO > > ; mSI-oSO

9 mSO > > > > mSI-oSO > > = .. th ! ð2l . i  1Þ mSO; > > > mSI-oSO > > > ; mSO 9 mSI-oSO > > > > > mSO > = .. th ! ð2l . i Þ mLS: > > > mSI-oSO > > > ; SO ð5:150Þ

th     th  th The bifurcation set of ðða 1 Þ mXX1 :ða2 Þ mXX2 :  :ðas1 Þ mXXs1 Þ at the same parameter point is called a left-parallel-bundle switching bifurcation Consider a just after bifurcation of ðða1þ Þth mXX1þ :ða2þ Þth mXX2þ :   : ðasþ2 Þth 2 mXXsþ2 Þ with Psi¼1 aiþ ¼ 2m2 þ 1 for simple monotonic sources and monotonic sinks

to oscillatory sources in the ð2m þ 1Þth degree polynomial nonlinear discrete system. þ mXXiþ 2 fmSO,mSIg for aiþ ¼ 2liþ  1; and mXX i 2 fmUS,mLSg for ai ¼ þ 2li . The four detailed structures are as follows. 8 mSI-oSO > > > > mSO < . ; ð2liþ  1Þth mSI ! .. > > > > : mSO mSI-oSO 8 mSO > > > mSI-oSO > < . ð2liþ Þth mUS ! .. ; > > > mSO > : mSI-oSO

8 mSO > > > > mSI-oSO < . and ð2liþ  1Þth mSO ! .. ; > > > > : mSI-oSO mSO 8 mSI-oSO > > > mSO > < . þ th and ð2li Þ mLS ! .. : > > > mSI-oSO > : mSO ð5:151Þ

The bifurcation set of ðða1þ Þth mXX1þ :ða2þ Þth mXX2þ :  :ðasþ2 Þth mXXsþ2 Þ at the same parameter point is called a right-parallel-bundle switching bifurcation.

376

5 (2m + 1)th-Degree Polynomial Discrete Systems

(i) For the just before and after bifurcation structure, if there exists a relation of th  þ th þ th   þ ða i Þ mXXi ¼ ðaj Þ mXXj ¼ a mXX, for xk ¼ ai ¼ aj

ði 2 f1; 2;    ; s1 g; j 2 f1; 2;    ; s2 gÞ; mXX 2 fmUS,mLS,mSO,mSIg ð5:152Þ then the bifurcation is a ath mXX switching bifurcation for simple fixed-points. (ii) Just for the just before bifurcation structure, if there exists a relation of th th    ð2l i Þ mXXi ¼ ð2lÞ mXX, for xk ¼ ai ¼ ai ði 2 f1; 2;    ; s1 g; mXX 2 fmUS,mLSg

ð5:153Þ

then, the bifurcation is a ð2lÞth mXX left appearing (or right vanishing) bifurcation for simple fixed-points. (iii) Just for the just after bifurcation structure, if there exists a relation of ð2liþ Þth mXXiþ ¼ ð2lÞth mXX, for xk ¼ aiþ ¼ ai ði 2 f1; 2;    ; s1 gÞ; mXX 2 fmUS,mLSg

ð5:154Þ

then, the bifurcation is a ð2lÞth mXX right appearing (or left vanishing) bifurcation for simple fixed-points. (iv) For the just before and after bifurcation structure, if there exists a relation of th  þ th þ þ ða for xk ¼ a i Þ mXXi 6¼ ðaj Þ mXXj i ¼ aj þ XX i ; XXj 2 fmUS,mLS, mSO,mSIg

ð5:155Þ

ði 2 f1; 2;    ; s1 g; j 2 f1; 2;    ; s2 gÞ; then, there are two flower-bundle switching bifurcations of simple fixedpoints: (iv1) for aj ¼ ai þ 2l, the bifurcation is called a ath j mXX right flower-bundle switching bifurcation for ai to aj -simple fixed-points with the appearance (birth) of 2l-simple fixed-points. (iv2) for aj ¼ ai  2l, the bifurcation is called a ath i mXX left flower-bundle switching bifurcation for ai to aj -simple fixed-points with the vanishing (death) of 2l-simple fixed-points. A general parallel switching bifurcation is

5.2 Simple Fixed-Point Bifurcations

377 switching

th   th   th  ðða  !  1 Þ mXX1 ; ða2 Þ mXX2 ;    ; ðas1 Þ mXXs1 Þ  bifucation

ðða1þ Þth mXX1þ ; ða2þ Þth mXX2þ ;    ; ðasþ2 Þth mXXsþ2 Þ:

ð5:156Þ

Such a general, parallel switching bifurcation consists of the left and right parallel-bundle switching bifurcations. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e., th th  þ th þ ða i Þ mXXi ¼ ðai Þ mXXi ¼ ðai Þ mXXi ;

þ for xk ¼ a i ¼ ai ði ¼ 1; 2;    ; sg

ð5:157Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ðða1 Þth mXX:ða2 Þth mXX:    :ðas Þth mXXÞ. If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th   th þ th þ þ th ða i Þ mXXi ¼ ð2li Þ mXX, ðaj Þ mXXj ¼ ð2lj Þ mYY, þ for xk ¼ a i 6¼ ai ði ¼ 1; 2;    ; sg

ð5:158Þ

mXX 2 f mUS,mLSg ,mYY 2 f mUS,mLSg then the parallel flower-bundle switching bifurcation becomes a combination of two independent left and right parallel appearing bifurcations: th     th  th (i) a ðð2l 1 Þ mXX1 : ð2l2 Þ mXX2 :    : ð2ls1 Þ mXXs1 Þ-left parallel sprinklerspraying appearing (or right vanishing) bifurcation and (ii) a ðð2l1þ Þth mXX1þ : ð2l2þ Þth mXX2þ :    : ð2lsþ2 Þth mXXsþ2 Þ-right parallel sprinklerspraying appearing (or left vanishing) bifurcation.

The ð4th mLS:    :mSO:m6th USÞ parallel appearing bifurcation for a0 [ 0 is presented in Fig. 5.6i. The ð4th mUS:    :mSI-oSO:6th mLSÞ parallel appearing bifurcation for a0 \0 is shown in Fig. 5.6ii. Such a kind of bifurcation is also like a waterfall appearing bifurcation. The ð5th mSO:    :6th mUS:6th mUSÞ parallel, flowerbundle switching bifurcation for mSI and mSO fixed-points is presented in Fig. 5.6iii for a0 [ 0. Such a parallel flower-bundle switching bifurcation is from ðmSO:mSI-oSO:mSOÞ to ð5th mSO:    :6th mUS:6th mUSÞ with a waterfall appearance. The set of ð5th mSI:    :6th mLS:6th mLSÞ flower-bundle switching bifurcation for mSI and mSO fixed-points is presented in Fig. 5.6iv for a0 \0. Such a parallel flower-bundle switching bifurcation is from ðmSI-oSO:mSO:mSI-oSOÞ to ð5th mSI:    :6th mLS:6th mLSÞ with a waterfall appearance. After the bifurcation, the waterfall fixed-points birth can be observed. The fixed-points before such a bifurcation are much less than after the bifurcation.

5 (2m + 1)th-Degree Polynomial Discrete Systems

378 a0 > 0

mSO

mSI-oSO

mSO

mS-oSOI

4th mLSN

b

mSO

SI

SI mSO

P-2

4th mUSN

•

mSO

mSO

(2r)th mLSN

P-2 • •

• •

mSI-oSO

•

mSO

mSO

•

mSI-oSO

mSO

mSI-oSO

mSO mSO

P-2 (2r)th mUSN

P-2

mSI-oSO

• •

mSI-oSO

b2(i2 )

P-2

•

P-2

b1(i1 ) b1(i2 ) b2(i1 )

6th mLSN

( i2 ) 2

P-2 • •

mSI-oSO

a0 < 0

b1(i2 ) b2(i1 )

6th mUSN mSO

b1(i1 )

b1(im )

P-2

mSI-oSO

mSO

b1(im )

4th mUSN

4th mLSN

xk∗

xk∗ mSO

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

b2(im )

mSI-oSO

(i)

(ii) mSO

a0 > 0

( i1 ) 1 ( i2 ) 1

b b

mSI-oSO

b2(i2 )

mSO

P-2

mSI-oSO

a0 < 0

b2(i1 )

6th mUSN

mSO

b2(i1 ) mSO mSI-oSO

P-2

P-2 mSI-oSO

P-2

• • •

P-2 6th mLSN mSO

mSI-oSO

•

mSO

• •

mSO

mSO

P-2 (2r)th mLSN

P-2 (2r)th mUSN

P-2

• •

P-2 • •

mSI-oSO

•

mSI-oSO

b1(im ) 5th

mSO

xk∗

xk∗ mSO

Δ iq < 0 Δ iq = 0

(iii)

mSI-oSO

•

mSI-oSO

|| p ||

b2(i2 )

mSI-oSO

P-2

mSO

b1(i1 ) b1(i2 )

6th mLSN

6th mUSN

5th

b2(im )

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

Δ iq > 0

b2(im )

b1(im )

mSI

mSI-oSO P-2

|| p ||

mSI-oSO

Δ iq < 0 Δ iq = 0

b2(im )

Δ iq > 0

(iv)

Fig. 5.6 Switching and appearing bifurcations. Simple switching: (i)ð4th mLSN:    : mSO : 6th mUSNÞ ða0 [ 0Þ, (ii) ð4th mUSN:    :mSI-oSO:6th mLSNÞ ða0 \0Þ. Higher-order switching: (iii) ð5th mSI:    :6th mUSN : 6th mUSNÞ ða0 [ 0Þ), (vi) ð5th mSO:    :6th mLSN : 6th mLSNÞ ða0 \0Þ in the ð2m þ 1Þth -degree polynomial nonlinear system. mLSN: monotonic lower saddlenode, mUSN: monotonic upper saddle-node, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. The period-2 fixed points are represented by P-2, which are sketched through red curves

5.3 Higher-Order Fixed-Point Bifurcations

5.3

379

Higher-Order Fixed-Point Bifurcations

The afore-discussed appearing and switching bifurcations in the ð2m þ 1Þth degree polynomial discrete system are relative to the simple monotonic sources and monotonic sinks to oscillatory sources. As similar to the ð2mÞth degree polynomial nonlinear discrete system, the higher-order singularity bifurcations in the ð2m þ 1Þth degree polynomial discrete system can be for higher-order monotonic sinks, monotonic sources, monotonic upper-saddles, and monotonic lower-saddles.

5.3.1

Higher-Order Fixed-Point Bifurcations

Consider a ð2m þ 1Þth degree polynomial nonlinear discrete system as xk þ 1 ¼ xk þ a0 Qðxk Þðxk  aÞ *si¼1 ðx2k þ Bi xk þ Ci Þai ;

ð5:159Þ

where ai 2 f2li  1; 2li g. Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are ðiÞ

1 2

b1;2 ¼  Bi 

1pffiffiffiffiffi Di ; Di 2

¼ B2i  4Ci 0; ð1Þ

ð1Þ

ðsÞ

ðsÞ

fa1 ; a2 ;    ; a2s1 ; a2s ; a2s þ 1 g ¼ sortfb1 ; b2 ;    ; b1 ; b2 ; ag;

ð5:160Þ

aj a j þ 1 : For a 6¼ 12Bi ði ¼ 1; 2; . . .; sÞ, there are four higher-order bifurcations as follows: ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order mUS ! appearing bifurcation ( ðiÞ ð2li  1Þth order mSO, xk ¼ b2 ;

ð5:161Þ

ðiÞ

ð2li  1Þth order mSI, xk ¼ b1 ; ð2li 1Þth order quadratics

ð2ð2li  1ÞÞth order mLS ! appearing bifurcation ( ðiÞ ð2li  1Þth order mSI, xk ¼ b2 ; ðiÞ

ð2li  1Þth order mSO, xk ¼ b1 ;

ð5:162Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

380

ð2li Þth -order power of quadratics ð2ð2li ÞÞth order mUS ! appearing bifurcation ( ðiÞ ð2li Þth order mUS, xk ¼ b2 ;

ð5:163Þ

ðiÞ

ð2li Þth order mUS, xk ¼ b1 ; ð2li Þth -order quadratics ð2ð2li ÞÞth order mLS ! appearing bifurcation ( ðiÞ ð2li Þth order mLS, xk ¼ b2 ;

ð5:164Þ

ðiÞ

ð2li Þth order mLS, xk ¼ b1 : (i) For ai ¼ 2li  1; the ð2ð2li  1ÞÞth -order monotonic upper-saddle (mUS) appearing bifurcation is for the onset of the ð2li  1Þth -order monotonic source ðiÞ ðiÞ (mSO) (xk ¼ b2 ) and the ð2li  1Þth -order monotonic sink (mSI) (xk ¼ b1 ) ðiÞ

ðiÞ

ðiÞ

ðiÞ

with b2 [ b1 . (ii) For ai ¼ 2li  1; the ð2ð2li  1ÞÞth -order monotonic lower-saddle (mLS) appearing bifurcation is for the onset of the ð2li  1Þth -order monotonic sink ðiÞ ðiÞ (mSI) (xk ¼ b2 ) and the ð2li  1Þth -order monotonic source (mSO) (xk ¼ b1 ) with b2 [ b1 . (iii) For ai ¼ 2li ; the ð2ð2li ÞÞth -order monotonic upper-saddle (mUS) appearing bifurcation is for the onset of two ð2li Þth -order monotonic upper-saddles (mUS) ðiÞ ðiÞ ðiÞ ðiÞ (xk ¼ b1 ; b2 ) with b2 [ b1 . (iv) For ai ¼ 2li ; the ð2ð2li ÞÞth order monotonic lower-saddle (mLS) appearing bifurcation is for the onset of two ð2li Þth -order monotonic lower-saddles (mLS) ðiÞ ðiÞ ðiÞ ðiÞ (xk ¼ b1 ; b2 ) with b2 [ b1 . The fixed-point of xk ¼ a 6¼ 12Bi ði ¼ 1; 2; . . .; sÞ breaks a cluster of teethcomb appearing bifurcations of higher order fixed-point to two parts. The teethcomb appearing bifurcation generated by the s-pairs of quadratics becomes a broom appearing bifurcation for higher-order fixed-points. The two broom appearing bifurcations for higher-order fixed-points are 8 ð2Þ th ð2Þ th > < ðð2a1 Þ mUS;    ; ð2as2 Þ mUSÞ; j¼1 mSO (xk ¼ aÞ ! mSO; for xk ¼ a ¼ a2ðs1 þ 1Þ1 ; appearing bifurcation > : ð1Þ ð1Þ ðð2a1 Þth mLS;    ; ð2as1 Þth mLS); P2 Ps j

ðjÞ

a ¼m i¼1 i

and

ð5:165Þ

5.3 Higher-Order Fixed-Point Bifurcations

381

8 ð2Þ th ð2Þ th > < ðð2a1 Þ mLS;    ; ð2as2 Þ mLSÞ; j¼1   mSI-oSO ( xk ¼ aÞ ! mSI-oSO; for xk ¼ a ¼ a2ðs1 þ 1Þ1 ; appearing bifurcation > : ð1Þ ð1Þ ðð2a1 Þth mUS;    ; ð2as1 Þth mUS); P2 Ps j

ðjÞ a ¼m i¼1 i

ð5:166Þ where 8 8 th < ðaðjÞ > > sj Þ mXX; th > > ð2aðjÞ Þ mUS ! s > j > : ðaðjÞ Þth mXX; > > sj < ðjÞ th . ðð2a1 Þth mUS;    ; ð2aðjÞ Þ mUSÞ ¼ sj .. > ( ðjÞ th > > > > ða1 Þ mXX; > ðjÞ th > > : ð2a1 Þ mUS ! ðjÞ ða1 Þth mXX; ð5:167Þ 8 8 th < ðaðjÞ > > sj Þ mXX; ðjÞ th > > ð2asj Þ mLS ! > > : ðaðjÞ Þth mXX; > > sj < ðjÞ th . ðð2a1 Þth mLS;    ; ð2aðjÞ Þ mLSÞ ¼ sj .. > ( ðjÞ th > > > > ða1 Þ mXX; > ðjÞ th > > ð2a1 Þ mLS ! : ðjÞ ða1 Þth mXX; ð5:168Þ for j ¼ 1; 2: Four special broom appearing bifurcations for higher-order fixed-points are Ps

i¼1 ai ¼m

mSO ( xk ¼ aÞ !



mSO; for xk ¼ a ¼ a2s þ 1 ; ðð2a1 Þth mLS;    ; ð2as Þth mLS);

appearing bifurcation Ps

i¼1 ai ¼m

mSI-oSO ( xk ¼ aÞ !



appearing bifurcation

ð5:169Þ

mSo-oSO; for xk ¼ a ¼ a2s þ 1 ; ð5:170Þ ðð2a1 Þth mUS;    ; ð2as Þth mUS)

and mSO (

xk

mSI-oSO (

Ps

i¼1 ai ¼m



¼ aÞ ! appearing bifurcation

xk

Ps

i¼1 ai ¼m

ðð2a1 Þth mUS;    ; ð2as Þth mUS), mSO; for xk ¼ a ¼ a1 ;

¼ aÞ ! appearing bifurcation



ð5:171Þ

ðð2a1 Þth mLS;    ; ð2as Þth mLS), ð5:172Þ mSI-oSO; for xk ¼ a ¼ a1 :

5 (2m + 1)th-Degree Polynomial Discrete Systems

382

For a ¼ 12Bi ði 2 f1; 2; . . .; sgÞ, there are four higher-order bifurcations as follows: mSO (xk ¼ aÞ ! ð2ð2li  1Þ þ 1Þth mSO 8 ðiÞ th  > < ð2li  1Þ order mSO, xk ¼ b2 ; ¼ mSI-oSO, xk ¼ a; > : ðiÞ ð2li  1Þth order SO, xk ¼ b1 ;

ð5:173Þ

mSI-oSO (xk ¼ aÞ ! ð2ð2li  1Þ þ 1Þth mSI 8 ðiÞ th  > < ð2li  1Þ order mSI, xk ¼ b2 ; ¼ mSO, x ¼ a; > : ðiÞ ð2li  1Þth order mSI, xk ¼ b1 ;

ð5:174Þ

mSI-oSO (xk ¼ aÞ ! ð2ð2li Þ þ 1Þth mSO 8 ðiÞ th  > < ð2li Þ order mUS, xk ¼ b2 ; ¼ mSO, xk ¼ a; > : ðiÞ ð2li Þth order mLS, xk ¼ b1 ;

ð5:175Þ

mSI-oSO (xk ¼ aÞ ! ð2ð2li Þ þ 1Þth mSI 8 ðiÞ th  > < ð2li Þ order mLS, xk ¼ b2 ; ¼ mSI-oSO, x ¼ a; > : ðiÞ ð2li Þth order mUS, xk ¼ b1 :

ð5:176Þ

(i) For ai ¼ 2li  1, the ð2ð2li  1Þ þ 1Þth order monotonic source (mSO) switchðiÞ ing bifurcation is with the ð2li  1Þth order monotonic source (mSO) (xk ¼ b2 ) ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

and the ð2li  1Þth order monotonic sink (mSI) (xk ¼ b1 ) with b2 [ a [ b1 . (ii) For ai ¼ 2li  1 the ð2ð2li  1Þ þ 1Þth order monotonic sink (mSI) switching ðiÞ bifurcation is with the ð2li  1Þth order monotonic sink (mSI) (xk ¼ b2 ) and

the ð2li  1Þth order monotonic source (mSO) (xk ¼ b1 ) with b2 [ a [ b1 . (iii) For ai ¼ 2li the ð2ð2li Þ þ 1Þth order monotonic source (mSO) switching bifurðiÞ cation is with the ð2li Þth order monotonic upper-saddle (mUS) (xk ¼ b2 ) and ðiÞ

ðiÞ

the ð2li Þth order monotonic upper-saddles (mLS) (xk ¼ b1 ) with b2 [ a ðiÞ

[ b1 .

5.3 Higher-Order Fixed-Point Bifurcations

383

(iv) For ai ¼ 2li the ð2ð2li Þ þ 1Þth order monotonic sink (mSI) switching bifurcation ðiÞ is with the ð2li Þth order monotonic upper-saddle (mLS) (xk ¼ b2 ) and the ðiÞ

ðiÞ

ðiÞ

ð2li Þth order monotonic upper-saddles (mUS) (xk ¼ b1 ) with b2 [ a [ b1 .

If xk ¼ a ¼ 12Bi ði ¼ 1; 2; . . .; mÞ, the fixed-point of xk ¼ a possesses a ð2ð2li  1Þ þ 1Þth and ð2ð2li Þ þ 1Þth -order mSI or mSO switching bifurcations (or pitchfork bifurcations) for higher-order fixed-points. The teethcomb appearing bifurcation generated by the m-pairs of quadratics becomes a broom switching bifurcation. Such a broom switching bifurcation consists of a pitchfork switching bifurcation and two teethcomb appearing bifurcations in the ð2m þ 1Þth -degree polynomial system. Four broom switching bifurcations for higher-order fixed-points are P2 Ps j

ðjÞ a i¼1 i

j¼1

þ 2ls1 þ 1 ¼m

mSO ( xk ¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ > ðð2a1 Þ mUS;    ; ð2as2 Þth mUSÞ; > > 8 > > th > > > < < ð2ls1 þ 1 Þ mUS, th ð2ð2ls1 þ 1 Þ þ 1Þ mSO mSO, x ¼ a; > > > : > > ð2ls1 þ 1 Þth mLS, > > > : ð1Þ ð1Þ ðð2a1 Þth mLS;    ; ð2as1 Þth mLS); mSO (

xk

P2 Ps j j¼1

ðjÞ a i¼1 i

ð5:177Þ

þ 2ls1 þ 1 1¼m

¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ > ðð2a1 Þ mUS;    ; ð2as2 Þth mUSÞ; > > 8 > > th > > > < < ð2ls1 þ 1  1Þ mSO, ð2ð2ls1 þ 1  1Þ þ 1Þth mSO mSI-oSO, xk ¼ a; > > > : > > ð2ls1 þ 1  1Þth mSO, > > > : ð1Þ ð1Þ ðð2a1 Þth mLS;    ; ð2as1 Þth mLS);

ð5:178Þ

and P2 Ps j j¼1

ðjÞ a i¼1 i

þ 2ls1 þ 1 ¼m

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ > ðð2a1 Þ mLS;    ; ð2as2 Þth mLSÞ; > > 8 > > th > > > < < ð2ls1 þ 1 Þ mLS, ð2ð2ls1 þ 1 Þ þ 1Þth mSI mSI-oSO, xk ¼ a; > > > : > > ð2ls1 þ 1 Þth mUS, > > > : ð1Þ ð1Þ ðð2a1 Þth mUS;    ; ð2as1 Þth mUS);

ð5:179Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

384 P2 Psj j¼1

ðjÞ a i¼1 i

þ 2ls1 þ 1 1¼m

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ > ðð2a1 Þ mLS;    ; ð2as2 Þth mLSÞ; > > 8 > > th > > > < < ð2ls1 þ 1  1Þ mSI, ð2ð2ls1 þ 1  1Þ þ 1Þth mSI mSO, xk ¼ a; > > > : > > ð2ls1 þ 1  1Þth mSI; > > > : ð1Þ ð1Þ ðð2a1 Þth mUS;    ; ð2as1 Þth mUS):

ð5:180Þ

Consider a ð2m þ 1Þth degree polynomial nonlinear discrete system as xk þ 1 ¼ xk þ a0 Qðxk Þðxk  aÞ *ni¼1 ðx2k þ Bi xk þ Ci Þai

ð5:181Þ

where ai 2 f2ri  1; 2ri g ði ¼ 1; 2; . . .; nÞ. Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bi xk þ Ci ¼ 0 are ðiÞ

1 2

b1;2 ¼  Bi 

1pffiffiffiffiffi Di ; Di 2

¼ B2i  4Ci 0;

Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðnÞ

ðnÞ

fa1 ; a2 ;    ; a2n þ 1 g  sortfb1 ; b2 ; b1 ; b2 ;    ; b1 ; b2 ; ag; ai ai þ 1 : ð5:182Þ The higher-order singularity bifurcation can be for a cluster of higher-order fixed-points. There are four higher-order bifurcations as follows: (i) The ð2ð2l  1ÞÞth order monotonic upper-saddle (mUS) spraying appearing bifurcation for a cluster of higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower-saddles is ð2bÞth mUS ¼ ð2ð2l  1ÞÞth order mUS 8 > ða2n Þth order mXX for xk ¼ a2n ; > > > > < ða2n1 Þth order mXX for xk ¼ a2n1 ; a cluster of 2n-mXX ! . appearing bifurcation > .. > > > > : ða1 Þth order mXX for xk ¼ a1 ; ð5:183Þ where 2ð2l  1Þ ¼ Pni¼1 ai and

5.3 Higher-Order Fixed-Point Bifurcations

( th

ða2n Þ order mXX ¼ ( ða1 Þth order mXX ¼

385

ð2r2n Þth order mUS, for a2n ¼ 2rn ; ð2r2n  1Þth order mSO, for a2n ¼ 2rn  1;

ð2r1 Þth order mUS, for a1 ¼ 2r1 ; ð2r1  1Þth order mSI, for a1 ¼ 2r1  1: ð5:184Þ

(ii) The ð2ð2lÞÞth order monotonic upper-saddle (mUS) spraying-appearing bifurcation for a cluster of higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower-saddles is ð2bÞth mUS ¼ ð2ð2lÞÞth order mUS 8 > ða Þth order mXX for xk ¼ a2n ; > > 2n > > < ða2n1 Þth order mXX for xk ¼ a2n1 ; a cluster of 2n-mXX ! . appearing bifurcation > > > .. > > : ða1 Þth order mXX for xk ¼ a1 ; ð5:185Þ where 2ð2lÞ ¼ Pni¼1 ai and ( ða2n Þ

th

order mXX ¼ (

ða1 Þth order mXX ¼

ð2r2n Þth order mUS, for a2n ¼ 2rn ; ð2r2n  1Þth order mSO, for a2n ¼ 2rn  1;

ð2r1 Þth order mUS, for a1 ¼ 2r1 ; ð2r1  1Þth order mSI, for a1 ¼ 2r1  1: ð5:186Þ

For the higher-order monotonic lower-saddle bifurcation, the cluster of the higher-order fixed-points is given by the following two cases. (iii) The ð2ð2l  1ÞÞth order monotonic lower-saddle (mLS) spraying-appearing bifurcation for a cluster of higher-order monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower-saddles is ð2bÞth mLS ¼ ð2ð2l  1ÞÞth order mLS 8 > ða2n Þth order mXX, for xk ¼ a2n ; > > > > < ða2n1 Þth order mXX, for xk ¼ a2n1 ; a cluster of 2n-mXX ! . appearing bifurcation > .. > > > > : ða1 Þth order mXX, for xk ¼ a1 ; ð5:187Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

386

where 2ð2l  1Þ ¼ Pni¼1 ai and ( th

ða2n Þ order mXX ¼ ( ða1 Þth order mXX ¼

ð2r2n Þth order mLS, for a2n ¼ 2rn ; ð2r2n  1Þth order mSI, for a2n ¼ 2rn  1;

ð2r1 Þth order mLS, for a1 ¼ 2r1 ; ð2r1  1Þth order mSO, for a1 ¼ 2r1  1: ð5:188Þ

(iv) The ð2ð2lÞÞth -order lower-order spraying-appearing bifurcation for a cluster of higher-order sinks, sources, upper-saddles and lower-saddles is ð2bÞth mLS ¼ ð2ð2lÞÞth order mLS 8 > ða2n Þth order mXX, for xk ¼ a2n ; > > > > < ða2n1 Þth order mXX, for xk ¼ a2n1 ; a cluster of 2n-mXX ! . appearing bifurcation > .. > > > > : ða1 Þth order mXX, for xk ¼ a1 ; ð5:189Þ where 2ð2lÞ ¼ Pni¼1 ai and ( th

ða2n Þ order mXX ¼ ( th

ða1 Þ order mXX ¼

ð2r2n Þth order mLS, for a2n ¼ 2rn ; ð2r2n  1Þth order mSI, for a2n ¼ 2rn  1;

ð2r1 Þth order mLS, for a1 ¼ 2r1 ;

ð5:190Þ

ð2r1  1Þth order mSO, for a1 ¼ 2r1  1:

If xk ¼ a 6¼ 12Bi ði ¼ 1; 2; . . .; nÞ, the fixed-point of xk ¼ a breaks a cluster of sprinkler-spraying appearing bifurcations for higher-order fixed-points to two parts. The sprinkler-spraying appearing bifurcation generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying appearing bifurcation. The two broomsprinkler-spraying appearing bifurcations in the ð2m þ 1Þth -degree polynomial system are m¼m1 þ m2

mSO ( x ¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ th > > < ðð2b1 Þ mUS:    :ð2br2 Þ mUSÞ; mSO (a ¼ ar1 þ 1 Þ ! mSO ða ¼ a2ðm1 þ 1Þ1 Þ; > > ð1Þ : th ðð2b1 Þth mLS:    :ð2bð1Þ r1 Þ mLSÞ;

ð5:191Þ

5.3 Higher-Order Fixed-Point Bifurcations

387

and m¼m1 þ m2

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation 8 ð2Þ th ð2Þ th > > < ðð2b1 Þ mLS:    : ð2br2 Þ mLSÞ; mSI-oSO (a ¼ ar1 þ 1 Þ ! mSO ða ¼ a2ðm1 þ 1Þ1 Þ; > > ð1Þ : th ðð2b1 Þth mUS:    :ð2bð1Þ r1 Þ mUSÞ; ð1Þ

ð5:192Þ

ð2Þ

1 2 for m1 ¼ Pri¼1 bi ; m2 ¼ Prj¼1 bj .

Four special broom-sprinkler-spraying appearing bifurcations the ð2m þ 1Þth degree polynomial nonlinear discrete system are m¼

Pr

i¼1 bi

mSO ( xk ¼ aÞ ! appearing bifurcation ( mSO (a ¼ a2m þ 1 Þ ! mSO ða ¼ a2m þ 1 Þ;

ð5:193Þ

ðð2b1 Þth mLS;    ; ð2br Þth mLSÞ; m¼

Pr

i¼1 bi

mSI-oSO ( xk ¼ aÞ ! appearing bifurcation ( mSI-oSO (a ¼ a2m þ 1 Þ ! mSI-oSO ða ¼ a2m þ 1 Þ;

ð5:194Þ

ðð2b1 Þth mUS;    ; ð2br Þth mUSÞ;

and m¼

Pr

i¼1 bi

mSO ( xk ¼ aÞ ! appearing bifurcation ( ðð2b1 Þth mUS;    ; ð2br Þth mUSÞ;

ð5:195Þ

mSO (a ¼ a1 Þ ! mSO ða ¼ a1 Þ; m¼

Pr

i¼1 bi

mSI-oSO( xk ¼ aÞ ! appearing bifurcation ( ðð2b1 Þth mLS;    ; ð2br Þth mLSÞ; mSI-oSO (a ¼ a1 Þ ! mSI-oSO ða ¼ a1 Þ:

ð5:196Þ

If xk ¼ a ¼ 12Bi (ði ¼ 1; 2; . . .; lÞ, the fixed-point of xk ¼ a possesses a ð2l þ 1Þth -order mSI or mSO switching bifurcation (or broom-switching bifurcation) for higher-order fixed-points. The sprinkler-spraying appearing bifurcation

5 (2m + 1)th-Degree Polynomial Discrete Systems

388

generated by the m-pairs of quadratics becomes a broom-sprinkler-spraying switching bifurcation. The two broom switching bifurcations in the ð2m þ 1Þth degree polynomial system are m¼m1 þ m2 þ b

mSO (xk ¼ aÞ ! switching bifurcation 8 ð2Þ th > ð2b1 ;    ; 2bð2Þ > r2 Þ mUS > > 8 > > ð2Þ th  > > > > ð2br2 Þ mUS (xk ¼ ar1 þ r2 þ 1 Þ; > Pr2 ð2Þ < > m ¼ b > 2 j¼1 j > > ! ... > > > appearing > > > > : ð2Þ > > ð2b1 Þth mUS (xk ¼ ar1 þ 2 Þ; > < ð2b þ 1Þth mSO (a ¼ ar1 þ 1 Þ; > > > ð1Þ th > ð2b1 ;    ; 2bð1Þ > r1 Þ mLS > > > 8 > ð1Þ th  > > > > > ð2br1 Þ mLS (xk ¼ ar1 Þ; > P r1 ð1Þ < > m ¼ b 1 > i¼1 i > > ! ... > > appearing > > > > : : ð1Þ ð2b1 Þth mLS (xk ¼ a1 Þ;

ð5:197Þ

and m¼m1 þ m2 þ b

mSI-oSO (xk ¼ aÞ ! switching bifurcation 8 ð2Þ th > ð2b1 ;    ; 2bð2Þ > r2 Þ mLS > > 8 > > ð2Þ th > > ð2br2 Þ mLS (xk ¼ ar1 þ r2 þ 1 Þ; > > Pr2 ð2Þ > < > m2 ¼ j¼1 bj > > > ! ... > > > appearing > > > > : ð2Þ > > ð2b1 Þth mLS (xk ¼ ar1 þ 2 Þ; > < ð2b þ 1Þth mSI (a ¼ ar1 þ 1 Þ; > > > ð1Þ th > ð2b1 ;    ; 2bð1Þ > r2 Þ mUS > > > 8 > th  > ð2bð1Þ > > r1 Þ mUS (xk ¼ ar1 Þ; > > > P r1 < ð1Þ > m1 ¼ i¼1 bi > > > ! ... > > appearing > > > > : : ð1Þ ð2b1 Þth mUS (xk ¼ a1 Þ; where

ð5:198Þ

5.3 Higher-Order Fixed-Point Bifurcations

389

cluster of l-quadratics ð2b þ 1Þth order mSO(xk ¼ aÞ ! appearing bifurcation 8 > ða2sl þ 1 Þth mXX, for xk ¼ a2sl þ 1 ; > > > > th  > > > ða2sl Þ mXX, for xk ¼ a2sl ; < .. . > > > > > ða2s1 Þth mXX, for xk ¼ a2s1 ; > > > : ða2s1 1 Þth mXX, for xk ¼ a2s1 1 ;

ð5:199Þ

cluster of l-quadratics ð2b þ 1Þth order mSI (xk ¼ aÞ ! appearing bifurcation 8 > ða2sl þ 1 Þth mXX, for xk ¼ a2sl þ 1 ; > > > > th  > > > ða2sl Þ mXX, for xk ¼ a2sl ; < .. . > > > > > ða2s1 Þth mXX, for xk ¼ a2s1 ; > > > : ða2s1 1 Þth mXX, for xk ¼ a2s1 1 :

ð5:200Þ

where xk ¼ a 2 fa2s1 1 ;    ; a2sl ; a2sl þ 1 g and 2b þ 1 ¼ Pli¼1 a2si 1 þ a2si þ a2sl þ 1 : The two appearing bifurcations for the higher-order singularity of fixed-points are (i) mSO ! ðð2a1 Þth mLS:    :ð2ai Þth mLS:mSO:    :ð2an1 Þth mUS:ð2an Þth mUS), and (ii) mSO ! ðð2a1 Þth mUS:    :ð2ai Þth UmS:mSO:    :ð2an1 Þth mLS:ð2an Þth mLS), as presented in Fig. 5.7i and ii for a0 [ 0 and a0 \ 0, respectively. The broom appearing bifurcation for the higher-order fixed-points are illustrated. The components of the broom appearing bifurcation are th

(

aj1 ¼2rj1

ð2aj1 Þ mUS ! appearing

th

aj2 ¼2rj2 1

(

ð2aj2 Þ mUS ! appearing

ð2rj1 Þth mUS ð2rj1 Þth mUS

ðj1 ¼ i; n  1; . . .Þ;

ð2rj2  1Þth mSO ð2rj2  1Þth mSI

ð5:201Þ ðj2 ¼ 1; n; . . .Þ;

and th

aj1 ¼2rj1

(

ð2aj1 Þ mLS ! appearing

th

aj2 ¼2rj2 1

ð2aj2 Þ mLS ! appearing

(

ð2rj1 Þth mLS ð2rj1 Þth mLS

ðj1 ¼ i; n  1;   Þ;

ð2rj2  1Þth mSI ð2rj2  1Þth mSO

ð5:202Þ ðj2 ¼ 1; n; . . .Þ;

The simple fixed-point does not interact with the bifurcation points. The four switching and appearing switching bifurcation

5 (2m + 1)th-Degree Polynomial Discrete Systems

390 a0 > 0

(2rn − 1) th mSO

b1(i1 )

a0 < 0

(2rn − 1) th mSI

[2(2rn − 1)]th mUS

(2rn −1 ) mUS

b2(i1 )

( i2 ) 1

(2rn −1 ) mLS

b1(i2 )

(2rn −1 ) th mLS

b2(i2 )

th

b

(4rn −1 ) th mUS

(4rn ) th mLS th

•

(2rn −1 ) mUS mSO

•

( i2 ) 2

b

• •

a

•

mSI-oSO

•

(2ri ) th mLS

mSO

(2ri ) th mUS

mSI-oSO

(4ri ) th mLS

P-2

(2ri ) th mUS

•

•

• •

•

(2r1 − 1) th mSI

•

b1(in )

th

[2(2r1 − 1)] mLS

(2r1 − 1) th mSO

b1(in )

(2r1 − 1) th mSI

b2(in )

th

[2(2r1 − 1)] mUS

xk∗

(2r1 − 1) th mSO

∗ k

x b2(in )

Δ iq > 0

Δ iq < 0 Δ iq = 0

a0 > 0

|| p ||

(ii) (2rn − 1) th mSO

b1(i1 )

a0 < 0

(2rn − 1) th mSI

[2(2rn − 1)]th mUS

Δ iq > 0

Δ iq < 0 Δ iq = 0

(i)

(2rn − 1) th mSI

b1(i1 )

(2rn − 1) th mSO

[2(2rn − 1)]th mLS

b2(i1 ) (2rn −1 ) th mUS

b2(i1 )

b1(i2 )

(2rn −1 ) th mLS

b1(i2 )

(2rn −1 ) th mLS

b2(i2 )

(2(2rn )) th mLS

(2(2rn −1 )) th mUS th

• • •

(2rn −1 ) mUS

b2(i2 )

• • •

(2ri ) th mUS

(2ri ) th mLS

a

(2(2ri ) + 1) th mSO

a

(2(2rn ) + 1) th mSI

mSO

mSI-oSO mSI-oSO

(2ri ) th mLS • • •

(2r1 − 1) th mSI

x

•

(iii)

(2r1 − 1) th mSO

b1(in )

(2r1 − 1) th mSI

b2(in )

th

[2(2r1 − 1)] mUS

(2r1 − 1) th mSO

Δ iq < 0 Δ iq = 0

(2ri ) th mUS

b1(in )

[2(2r1 − 1)] mLS ∗ k

P-2

• •

P-2

th

|| p ||

a P-2

(4rn ) th mUS

(2ri ) th mLS

|| p ||

b1(i1 )

(2rn − 1) th mSO

[2(2rn − 1)]th mLS

b2(i1 ) th

(2rn − 1) th mSI

Δ iq > 0

∗ k

x b2(in )

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.7 Six bifurcations in a ð2m þ 1Þth -degree polynomial system. (i) and (ii) two broom appearing bifurcations. (iii)–(vi) broom-switching bifurcations. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, which sketched by red curves

5.3 Higher-Order Fixed-Point Bifurcations a0 > 0

(2rn − 1) th mSO

391

b1(i1 )

a0 < 0

(2rn − 1) th mSI

P-2 mSI-oSO

[2(2rn − 1) + 1]th mSO

mSO

(2rn − 1) th mSO

(2rn −1 ) th mUS

a

[2(2rn − 1) + 1]th mSI

a b2(i1 )

mSO

mSI-oSO

b1(i2 )

(2rn − 1) th mSI

b2(i1 )

(2rn −1 ) th mLS

b1(i2 )

(2rn −1 ) th mLS

b2(i2 )

P-2

(4rn ) th mLS

(4rn −1 ) th mUS

•

(2rn −1 ) th mUS

b2(i2 )

•

•

•

•

•

(2ri ) th mUS

(2ri ) th mLS

(4rn ) th mSI

(4ri + 1) th mSO (2ri ) th mLS

(2ri ) th mUS

• • •

(2r1 − 1) th mSI

• • •

b1(in )

(2r1 − 1) th mSO

b1(in )

(2r1 − 1) th mSI

b2(in )

[2(2r1 − 1)]th mUS

[2(2r1 − 1)]th mLS

xk∗

|| p ||

b1(i1 )

(2r1 − 1) th mSO

Δ iq > 0

Δ iq < 0 Δ iq = 0

(v)

b2(in )

xk∗

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(vi)

Fig. 5.7 (continued)

The two broom-sprinkler-spraying switching bifurcations for the higher-order singularity of fixed-points are ðiiiÞ ðð2ð2r1  1Þth mLS:    :ð2ð2rn1 ÞÞth mUS : ð2ð2rn  1ÞÞth mUS) ðivÞ ðð2ð2r1  1Þth mUS:    :ð2ð2rn1 ÞÞth mLS : ð2ð2rn  1ÞÞth mLS) ðvÞ ðð2ð2r1  1Þth mLS:    :ð2ð2rn1 ÞÞth mLS : ð2ð2rn  1Þ þ 1Þth mSO) ðviÞ ð2ð2r1  1Þth mUS:    :ð2ð2rn1 ÞÞth mUS : ð2ð2rn  1Þ þ 1Þth mSI) as presented in Fig. 5.7iii, v and iv, vi for a0 [ 0 and a0 \0, respectively. The ð2ð2ri Þ þ 1Þth mSO and ð2ð2ri Þ þ 1Þth mSI switching bifurcations are 8 th > < ð2ri Þ mUS, aj ¼2rj th ð2ð2ri Þ þ 1Þ mSO ! mSO, appearing > : ð2ri Þth mLS, 8 th > < ð2ri Þ mLS, aj ¼2rj th ð2ð2ri Þ þ 1Þ mSI ! mSI-oSO appearing > : ð2ri Þth mUS,

ð5:203Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

392 (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 1) th mSI

(2rn − 1) th mSO

(2(2ls )) th mUS

(2(2ls )) th mLS

(2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 2 − 1) th mSI

(2rn − 2 − 1) th mSO

(2rn −3 − 1) th mSO

(2rn −3 − 1) th mSI

(2rn − 4 − 1) th mSI

(2rn − 4 − 1) th mSO

(2(2ls −1 )) th mUS

(2(2ls −1 )) th mLS

(2rn −3 − 1) th mSO •

mSO

mSO

(2rn − 4 − 1) th mSO

• •

(2ri ) th mLS (2ri −1 ) th mLS (2ri − 2 − 1) th mSI

mSO

(2rn −3 − 1) th mSI

•

(2rn − 4 − 1) th mSI

• •

P-2

mSI-oSO

(2ri −3 − 1) th mSO (2(2lr )) th mLS

P-2

(2(2lr )) th mUS

(2ri − 2 − 1) th mSO

(2ri −1 ) th mUS

(2ri −1 ) th mLS

th

(2ri −3 − 1) mSO

•

• •

th

(2ri ) mLS

•

(2ri −3 − 1) th mSI

(2ri ) th mUS

•

(2r2 ) th mLS

(2r2 ) th mUS

(2r1 − 1) th mSI

(2r1 − 1) th mSO

th

(2(2l1 − 1)) mLS

th

(2(2l1 − 1)) mUS

xk∗

th

(2r1 − 1) mSO

xk∗

(2r1 − 1) th mSI

(2r2 ) th mLS

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(2r2 ) th mUS

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(i)

(ii) (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO

(2rn − 2 ) mUS

(2rn − 2 ) th mLS

(2rn − 1) th mSO

(2rn − 1) th mSI

th

(2(2ls )) th mUS

(2(2ls )) th mLS

(2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 2 − 1) th mSI

(2rn − 2 − 1) th mSO

(2rn −3 − 1) th mSO

(2rn −3 − 1) th mSI

(2rn − 4 − 1) th mSI

(2rn − 4 − 1) th mSO

(2(2ls −1 )) th LS

(2(2ls −1 )) th mUS (2rn −3 − 1) th mSO •

(2rn −3 − 1) th mSI (2rn − 4 − 1) th mSO

• • •

(2rn − 4 − 1) th mSI

• •

(2ri ) th mUS

(2ri ) th mLS (2ri −1 ) th mLS

(2ri −1 ) th mUS th

mSO

(2ri − 2 − 1) mSO

(2ri − 2 − 1) th mSI

(2ri −3 − 1) th mSI

(2ri −3 − 1) th mSO

mSI-oSO

mSO

mSI-oSO

(2ri − 2 − 1) th mSI

(2(2lr ) + 1) th mSO

(2ri − 2 − 1) th mSO

(2(2lr ) + 1) th mSI

th

(2ri −1 ) mUS

P-2

(2ri −3 − 1) th mSO

• • •

(2ri −1 ) th mLS

•

(2ri −3 − 1) th mSI

• •

(2ri ) th mLS

(2r2 ) th mLS

(2ri ) th mLS (2r2 ) th mLS

(2r1 − 1) th mSI

(2r1 − 1) th mSO

(2(2l1 − 1)) th mUS

(2(2l1 − 1)) th mLS

xk∗

(2r1 − 1) th mSO

xk∗

(2r1 − 1) th mSI

(2r2 ) th mLS

|| p ||

(2ri ) th mUS (2ri −1 ) th mUS (2ri − 2 − 1) th mSO (2ri −3 − 1) th mSI

(2ri − 2 − 1) th mSI •

a

mSI-oSO

Δ iq < 0 Δ iq = 0

(iii)

Δ iq > 0

(2r2 ) th mUS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.8 Six types of bifurcations in a ð2m þ 1Þth -degree polynomial system. (i) and (ii) broom-sprinkler–spraying appearing bifurcations, (iii)–(vi) broom-spraying switching bifurcations with fixed-points clusters. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed-points, sketched by red curves and such period-2 fixed points are relative to the monotonic sinks to oscillatory sources

5.3 Higher-Order Fixed-Point Bifurcations

393

(2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 1) th mSI

(2rn − 1) th mSO

(2(2ls )) th mUS

(2(2ls )) th mLS

(2rn − 2 ) th mLS

(2rn − 2 ) th mUS

(2rn − 2 − 1) th mSI

(2rn − 2 − 1) th mSO

(2rn −3 − 1) th mSO

(2rn −3 − 1) th mSI

(2rn − 4 − 1) th mSI

(2rn − 4 − 1) th mSO (2(2ls −1 )) th mLS

(2(2ls −1 )) th mUS (2rn −3 − 1) th mSO • •

(2rn − 4 − 1) th mSI

•

(2ri ) th mUS

(2rn −3 − 1) th mSI (2rn − 4 − 1) th mSO

• •

(2ri ) th mLS

•

(2ri −1 ) th mUS

(2ri −1 ) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

th

(2ri −3 − 1) mSI (2(2lr ) + 1) th mSO

(2ri −3 − 1) th mSO

(2(2lr )) th mSI

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

(2ri −1 ) th mLS th

•

(2ri −3 − 1) mSI

• •

(2ri ) mUS

(2r2 ) th mUS th

(2r1 − 1) mSO (2(2l1 − 1) + 1) th mSI

(2ri −1 ) th mUS

• • •

th

(2ri −3 − 1) th mSO

(2ri ) th mLS (2r2 ) th mLS (2r1 − 1) th mSI

(2(2l1 − 1) + 1) th mLS

mSI-oSO

mSO mSI-oSO

mSO

xk∗

th

(2r1 − 1) mSO

∗ k

x

P-2

(2r1 − 1) th mSI

(2r2 ) th mLS

|| p ||

Δ iq < 0 Δ iq = 0

(v)

Δ iq > 0

(2r2 ) th mUS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(vi)

Fig. 5.8 (continued)

and the ð2ð2rn  1Þ þ 1Þth mSO and ð2ð2rn  1Þ þ 1Þth mSI switching bifurcations are 8 th > < ð2rn  1Þ mSO, aj ¼2rj th ð2ð2rn  1Þ þ 1Þ mSO ! mSI-oSO, appearing > : ð2rn  1Þth mSO, 8 th > < ð2rn  1Þ mSI, a ¼2r j j ð2ð2rn  1Þ þ 1Þth mSI ! mSO appearing > : ð2rn  1Þth mSI:

ð5:204Þ

In Fig. 5.8i and ii, the simple switching with two sprinkler-spraying appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: ðiÞ mSO ! ðð2b1 Þth mLS:    :ð2bn1 Þth mUS:ð2bn Þth mUS), ðiiÞ mSI ! ðð2b1 Þth mUS:    :ð2bn1 Þth mLS:ð2bn Þth mLS), where m ¼ Pni¼1 bi ;b1 ¼ ð2l1  1Þ;    ; bi ¼ 2li ;    ; bn1 ¼ 2ln1 ;bn ¼ 2ln : In Fig. 5.8iii, v and vi, iv, the ð2l þ 1Þth -order broom-switching with two sprinkler-

5 (2m + 1)th-Degree Polynomial Discrete Systems

394

spraying appearing bifurcations are presented for a0 [ 0 and a0 \0, respectively. The two bifurcation structures are: (iii) (iv) (v) (vi)

mSO ! ðð2b1 Þth mUS:    :ð2bn1 Þth mUS:ð2bn Þth mUS), mSI-oSO ! ðð2b1 Þth mLS:    :ð2bn1 Þth mLS:ð2bn Þth mLS), mSO ! ðð2b1 þ 1Þth mSO:    :ð2bn1 Þth mUS:ð2bn Þth mUS), mSI-oSO ! ðð2b1 þ 1Þth mSI:    :ð2bn1 Þth mLS:ð2bn Þth mLS):

For a cluster of m-quadratics, Bi ¼ Bj ði; j 2 f1; 2; . . .; mg; i 6¼ jÞ and Di ¼ 0 ði 2 f1; 2; . . .; mgÞ. The ð2mÞth -order monotonic upper-saddle-node appearing bifurcation for s-pairs of higher-order fixed-points is 8 ða2s Þth mXX, for xk ¼ a2s ; > > > < ða2s1 Þth mXX, for xk ¼ a2s1 ; cluster of s-quadratics ð2mÞth order mUS ! . appearing bifurcation > > .. > : ða1 Þth mXX, for xk ¼ a1 ;

ð5:205Þ

where 2m ¼ P2s j¼1 aj and 2m ¼ 2ð2l  1Þ; 2ð2lÞ. ( th

ða1 Þ mXX ¼ ða2s Þth mXX ¼

ð2l1 Þth mUS, for a1 ¼ 2l1 ;

ð2l1  1Þth mSI, for a1 ¼ 2l1  1; ( ð2l2s Þth mUS, for a2s ¼ 2l2s ;

ð5:206Þ

ð2l2s  1Þth mSO, for a2s ¼ 2l2s  1:

The ð2mÞth -order monotonic lower-saddle-node appearing bifurcation for mpairs of higher-order fixed points is 8 ða2s Þth mXX, for xk ¼ a2s ; > > > th  < ða cluster of s-quadratics 2s1 Þ mXX, for xk ¼ a2s1 ; ð2mÞth order mLS ! . appearing bifurcation > . > > : . th ða1 Þ mXX, for xk ¼ a1 :

ð5:207Þ

where ( th

ða1 Þ mXX ¼ ( th

ða2s Þ mXX =

ð2l1 Þth mLS, for a1 ¼ 2l1 ; ð2l1  1Þth mSO, for a1 ¼ 2l1  1; ð2l2s Þth mLS, for a2s ¼ 2l2s ; ð2l2s  1Þth mSI, for a2s ¼ 2l2s  1:

ð5:208Þ

5.3 Higher-Order Fixed-Point Bifurcations

395

There are four simple switching and ð2mÞth -order saddle-node appearing bifurcations for higher-order fixed-points: The two switching bifurcations of mSO ! ðð2mÞth mUS:mSO) and mSI-oSO ! ðð2mÞth mLS:mSI-oSO) with two ð2mÞth -order mUSN and mLSN spraying appearing bifurcations in the ð2m þ 1Þth -degree polynomial system are mSO (xk ¼ aÞ ! mSI-oSO (xk ¼ aÞ !





mSO ! mSO, for xk ¼ a ¼ a2m þ 1 ; ð2mÞth order mLS:

mSI-oSO ! mSI-oSO, for xk ¼ a ¼ a2m þ 1 ; ð2mÞth order mUS:

ð5:209Þ ð5:210Þ

and the two switching bifurcations of mSO ! ðmSO:ð2mÞth mUS) and mSI-oSO ! ðmSI-oSO:ð2mÞth mLS) with two ð2mÞth -order mUSN and mLSN spraying appearing bifurcations in the ð2m þ 1Þth -degree polynomial nonlinear discrete system are mSO

mSI-oSO

(xk

(xk

¼ aÞ !

¼ aÞ !

ð2mÞth order mUS, mSO ! mSO, for xk ¼ a ¼ a1 :

ð2mÞth order mLS; mSI-oSO ! mSI-oSO, for xk ¼ a ¼ a1 :

ð5:211Þ ð5:212Þ

The ð2m þ 1Þth order source broom-switching bifurcation for higher-order fixedpoints is 8 > ða2s þ 1 Þth mXX, for xk ¼ a2s þ 1 ; > > > > < ða2s Þth mXX, for xk ¼ a2s ; switching mSO(xk ¼ aÞ ! ð2m þ 1Þth order mSO . > .. > > > > : ða1 Þth mXX, for xk ¼ a1 ;

ð5:213Þ þ1 where m þ 1 ¼ P2s j¼1 aj and m ¼ 2ð2l  1Þ; 2ð2lÞ.

( th

ða1 Þ mXX ¼ th

ða2s Þ mXX ¼

ð2l1 Þth mLS, for a1 ¼ 2l1 ;

ð2l1  1Þth mSI, for a1 ¼ 2l1  1; ( ð2l2s Þth mUS, for a2s ¼ 2l2s ; ð2l2s  1Þth mSO, for a2s ¼ 2l2s  1:

ð5:214Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

396

The ð2m þ 1Þth order monotonic sink broom-switching bifurcation is 8 > ða Þth mXX, for xk ¼ a2s þ 1 ; > > 2s þ 1 > > < ða2s Þth mXX, for xk ¼ a2s ; switching mSI-oSO(xk ¼ aÞ ! ð2m þ 1Þth order mSI . > .. > > > > : ða1 Þth mXX, for xk ¼ a1 ;

ð5:215Þ where ( th

ða1 Þ mXX ¼ th

ða2s Þ mXX ¼

ð2l1 Þth mUS, for a1 ¼ 2l1 ;

ð2l1  1Þth mSO, for a1 ¼ 2l1  1; ( ð2l2s Þth mLS, for a2s ¼ 2l2s ;

ð5:216Þ

ð2l2s  1Þth mSI, for a2s ¼ 2l2s  1:

The switching bifurcation consist of a simple switching and the ð2mÞth order saddle-node appearing bifurcation with s-pairs of fixed-points. The ð2mÞth order saddle-node appearing bifurcation is a sprinkler-spraying cluster of the s-pairs of higher-order fixed-points. Thus, the four switching bifurcations of mSO ! ðð2mÞth mLS:mSO) for higher order fixed-points for a0 [ 0, mSI-oSO ! ðð2mÞth mUS:mSI-oSO) for higher order fixed-points for a0 \0, mSO ! ðmSO:ð2mÞth mUS) for higher order fixed-points for a0 [ 0, mSI-oSO ! ðmSI-oSO:ð2mÞth mLS) for higher order fixed-point for a0 \ 0 are presented in Fig. 5.9i–iv, respectively. The ð2m þ 1Þth -order monotonic source switching bifurcation is named a ð2m þ 1Þth mSO broom-sprinkle-spraying switching bifurcation, and the ð2m þ 1Þth -order monotonic sink switching bifurcation is named a ð2m þ 1Þth mSI broom switching bifurcation. Such a ð2m þ 1Þth mXX broomswitching bifurcation is from simple fixed-point to a ð2m þ 1Þth mXX broomswitching. The two broom-switching bifurcations for higher-order fixed-points of mSO ! ð2m þ 1Þth mSO mSI-oSO ! ð2m þ 1ÞÞth mSI are presented in Fig. 5.9v–vi, respectively.

5.3 Higher-Order Fixed-Point Bifurcations

a0 > 0

397 a0 < 0

mSO (2r1 − 1) th mSI

mSO

(2r2 ) th mUS

a

mSI-oSO mSI-oSO

mSI-oSO

P-2

(2r1 − 1) th mSO (2r2 ) th mLS (2r3 ) th mLS

(2r3 ) th mUS

mSO

(2ri − 1) th mSO

 (2(2l − 1)) th mLS

(2ri − 1) th mSI

P-2



(2rl − 1) th mSI (2(2l − 1)) th mUS



(2rl − 1) th mSO

 (2ri − 1) th mSI

(2ri − 1) th mSO

(2r3 ) mLS

(2r3 ) th mUS

(2rl − 1) th mSI

(2rl − 1) th mSO

th

xk∗

(2r2 ) th mUS

xk∗

(2r2 ) th mLS

(2r1 − 1) th mSO

Δ iq < 0 Δ iq = 0

|| p ||

Δ iq > 0

(2r1 − 1) th mSI

Δ iq < 0 Δ i = 0 q

|| p ||

(i)

Δ iq > 0

(ii) th

(2r1 − 1) mSO

a0 > 0

(2r1 − 1) th mSI

a0 < 0

(2r2 ) th mLS

(2r2 ) th mUS (2r3 ) th mUS

(2r3 ) th mLS

(2ri − 1) th mSO

(2ri − 1) th mSI

 (2(2l − 1)) th mUS



th

(2rl − 1) mSO



(2(2l − 1)) th mLS

(2rl − 1) th mSI



(2ri − 1) th mSO

th

(2ri − 1) mSI

(2r3 ) th mLS

(2r3 ) th mUS

mSO

(2r2 ) th mLS (2r1 − 1) th mSI

xk∗

mSO

P-2

xk∗

(2r2 ) th mUS (2r1 − 1) th mSO

mSI-oSO

mSI-oSO

mSO

Δ iq < 0 Δ i = 0 q

|| p ||

(2rl − 1) th mSI

mSI-oSO

(2rl − 1) th mSO

Δ iq < 0 Δ iq = 0

|| p ||

Δ iq > 0

Δ iq > 0

P-2

(iv)

(iii) (2r1 ) th mUS

a0 > 0

(2r1 ) th mLS

a0 < 0

(2r2 − 1) th mSO

(2r2 − 1) th mSI

(2r3 ) th mLS

(2r3 ) th mUS

(2ri − 1) th mSI



(2m + 1) th mSO

(2ri − 1) th mSO

(2rn − 1) th mSO



(2m + 1) th mSI

mSO

mSI-oSO mSO

a



(2ri − 1) th mSO

xk∗

(2r3 ) mLS

(2r2 − 1) th mSO



mSI-oSO

th

(2rn − 1) th mSI

Δ iq < 0 Δ iq = 0

(v)

Δ iq > 0

(2rn − 1) th mSI (2r3 ) th mUS

P-2

(2rn − 1) th mSO

xk∗

(2r2 − 1) th mSI

(2r1 ) th mLS

|| p ||

(2rl − 1) th mSI

(2r1 ) th mUS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(vi)

Fig. 5.9 Broom-switching bifurcations of fixed-points in ð2m þ 1Þth polynomial discrete system: (i) ðð2mÞth mLS:mSOÞ-appearing bifurcation ða0 [ 0Þ, (ii) ðð2mÞth mUS:mSI-oSOÞ-appearing bifurcation ða0 \0Þ, (iii) ðmSO : ð2mÞth mUSÞ-appearing bifurcation ða0 [ 0Þ, (iv) ðmSI-oSO : ð2mÞth mLSÞ-appearing bifurcation ða0 [ 0Þ. (v) ð2m þ 1Þth mSO switching-appearing bifurcation ða0 [ 0Þ, (vi) ð2m þ 1Þth mSI switching-appearing bifurcation ða0 \0Þ. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed-points, sketched by red curves and such period-2 fixed points are relative to the monotonic sinks to oscillatory sources

5 (2m + 1)th-Degree Polynomial Discrete Systems

398

A series of the ð2ai þ 1Þth -order monotonic source and monotonic sink bifurcations is aligned up with varying with parameters, which is formed a special pattern. Such a special pattern is from n-quadratics in the ð2m þ 1Þth degree polynomial nonlinear discrete system, and the following conditions should be satisfied. 1 2

1 2

aðpi Þ ¼  Bi and aðpj Þ ¼  Bj Bi Bj i; j 2 f1; 2; . . .; sg; i 6¼ j; Di [ Di þ 1 ði ¼ 1; 2; . . .; s; s n\mÞ;

ð5:217Þ

Di ¼ 0 with jjpi jj\jjpi þ 1 jj: Four series of switching bifurcations in the ð2m þ 1Þth degree polynomial nonlinear discrete system are (i) (ii) (iii) (iv)

ð2ð2r1  1Þ þ 1Þth mSO-(2(2r2 Þ þ 1Þth mSI-    ð2ð2rn  1Þ þ 1Þth mSOÞ; ð2ð2r1  1Þ þ 1Þth mSI-(2(2r2 Þ þ 1Þth mSO-    ð2ð2rn  1Þ þ 1Þth mSIÞ; ð2ð2r1 Þ þ 1Þth mSO-(2(2r2  1Þ þ 1Þth mSO-    ð2ð2rn  1ÞÞth mSOÞ; ð2ð2r1 Þ þ 1Þth mSI-(2(2r2  1Þ þ 1Þth mSI-    ð2ð2rn  1Þ þ 1Þth mSIÞ;

as presented in Fig. 5.10i, iii–ii, vi for ða0 [ 0Þ and ða0 \0Þ, respectively. The swapping pattern of higher-order sinks and sources switching bifurcations cannot be observed. Such a bifurcation scenario is like the fish-bone for the higher-order switching bifurcations of higher-order fixed-points.

5.3.2

Switching Bifurcations

Consider the roots of ðx2k þ Bi xk þ Ci Þai ¼ 0 as ðiÞ

ðiÞ

ðiÞ

ðiÞ

Bi ¼ ðb1 þ b2 Þ; Di ¼ ðb1  b2 Þ2 0; ðiÞ

ðiÞ

ðiÞ

ðiÞ

xk;1;2 ¼ b1;2 ; Di [ 0 if b1 6¼ b2 ði ¼ 1; 2; . . .; nÞ; ) Bi 6¼ Bj ði; j ¼ 1; 2; . . .; n; i 6¼ jÞ at bifurcation: ðiÞ ðiÞ Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ

ð5:218Þ

The ðai Þth -order singularity bifurcation is for the switching of a pair of higher order fixed-points (i.e., monotonic sinks, monotonic sources, monotonic upper-saddles and monotonic lower-saddles). There are six switching bifurcations for i 2 f1; 2; . . .; ng

5.3 Higher-Order Fixed-Point Bifurcations a0 > 0

399

(2r1 − 1) th mSO

[2(2r1 − 1) + 1]th mSO

(2(2r2 ) + 1) th mSI

mSO

P-2

(2(2r3 ) + 1) th mSI

(2r2 ) th mLS

(2r3 ) th mLS

[2(2ri − 1) + 1]th mSI

• • •

[2(2rn − 1) + 1]th mSO

mSO

P-2

(2r1 − 1) th mSO

|| p ||

P-2

• • •

P-2

xk∗

(2rn − 1) th mSO

(2ri − 1) th mSI

(2r2 ) th mUS

Δr < 0

a

mSI-oSO

(2r3 ) th mUS

(2ri − 1) th mSI

(2rn − 1) th mSO

Δr > 0

Δr = 0

(i) a0 < 0

(2r2 ) th mUS

(2r1 − 1) th mSI

(2r3 ) th mUS

(2ri − 1) th mSO

(2rn − 1) th mSI

[2(2r1 − 1) + 1]th mSI (2(2r2 ) + 1) th mSO

mSO

(2(2r3 ) + 1) th mUS

[2(2ri − 1) + 1]th mSO

• • •

mSO

• • •

[2(2rn − 1) + 1]th mSI

mSO

P-2

P-2

xk∗

(2r1 − 1) th mSI

|| p ||

(2r3 ) th mLS

(2r2 ) th mLS

Δr < 0

Δr = 0

a

mSO

(2ri − 1) th mSO

(2rn − 1) th mSI

Δr > 0

(ii) a0 > 0

(2r1 ) th mUS

(2r2 − 1) th mSO

(2(2r1 ) + 1) th mSO [2(2r2 − 1) + 1]th mSO (2(2r3 ) + 1) th mSO • • •

mSO

(2r3 ) th mLS

[2(2ri − 1) + 1]th mLS • • •

mSO P-2

xk∗ || p ||

(2rn − 1) th mSO

[2(2rn − 1)]th mUS

P-2

a

mSO

P-2

(2r1 ) th mUS

(2ri − 1) th mSI

mSI-oSO

(2r2 − 1) th mSO

Δr < 0

(2r3 ) th mUS

(2ri − 1) th mSI

(2rn − 1) th mSI

Δr > 0

Δr = 0

(iii) a0 < 0

(2(2r1 ) + 1) th mSI

(2r1 ) th mLS

[2(2r2 − 1) + 1]th mSI

P-2

P-2

xk∗ || p ||

(2(2r3 ) + 1) th mSO

(2r2 − 1) th mSI

• • •

mSO

[2(2ri − 1) + 1]th mSO

(2r3 ) th mUS

• • •

(2ri − 1) th mSO

[2(2rn − 1) + 1]th mSI

mSO P-2

(2r1 ) th mUS

(2rn − 1) th mSI

(2r2 − 1) th mSI

Δr < 0

Δr = 0

(2r3 ) th mLS

mSO

(2ri − 1) th mSO

a

(2rn − 1) th mSI

Δr > 0

(iv) Fig. 5.10 Four series switching bifurcations of fixed-points in a ð2m þ 1Þth polynomial system: (i, iii) for a0 [ 0, (ii, iv) for a0 \0. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI-oSO: monotonic sink to oscillatory source, mSO: monotonic source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed-points, sketched by red curves and such period-2 fixed points are relative to the monotonic sinks to oscillatory sources or the monotonic sinks to oscillatory source back to monotonic sinks

5 (2m + 1)th-Degree Polynomial Discrete Systems

400 ðiÞ

ðiÞ

li ¼r1 þ r2 1

ð2li Þth order mUS ! switching bifurcation ( ðiÞ ðiÞ ð2r2  1Þth order mSO # mSI, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:219Þ

ðiÞ

ð2r1  1Þth order mSI " mSO, for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2 1

ð2li Þth order mLS ! switching bifurcation ( ðiÞ ðiÞ ð2r2  1Þth order mSI # mSO, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:220Þ

ðiÞ

ð2r1  1Þth order mSO " mSI, for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li Þth order mUS ! switching bifurcation ( ðiÞ ðiÞ ð2r2 Þth order mUS # mUS, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:221Þ

ðiÞ

ð2r1 Þth order mUS " mUS for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li Þth order mLS ! switching bifurcation ( ðiÞ ðiÞ ð2r2 Þth order mLS # mLS, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:222Þ

ðiÞ

ð2r1 Þth order mLS " mLS for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li  1Þth order mSO ! switching bifurcation ( ðiÞ ðiÞ th ð2r2  1Þ order mSO # mSO, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:223Þ

ðiÞ

ð2r1 Þth order mLS " mUS for b1 ¼ a2i1 " a2i ; ðiÞ

ðiÞ

li ¼r1 þ r2

ð2li  1Þth order mSI ! switching bifurcation ( ðiÞ ðiÞ ð2r2  1Þth order mSI # mSI, for b2 ¼ a2i # a2i1 ; ðiÞ

ð5:224Þ

ðiÞ

ð2r1 Þth order mUS " mLS for b1 ¼ a2i1 " a2i : A set of n-paralleled higher-order mXX switching bifurcations is called the th th mXX,ath 2 mXX,    ; an mXXÞ parallel switching bifurcation in the ð2m þ 1Þ degree polynomial nonlinear discrete system. Such a bifurcation is also called a th th ðath 1 mXX,a2 mXX,    ; an mXXÞ antenna switching bifurcation. ai 2 f2li ; 2li 1g ðath 1

ðiÞ

and mXX 2 fmSO, mSI, mUS, mLSg. For non-switching points, Di [ 0 at b1 6¼ ðiÞ

ðiÞ

ðiÞ

b2 ði ¼ 1; 2; . . .; nÞ. At the bifurcation point, Di ¼ 0 at b1 ¼ b2 ði ¼ 1; 2; . . .; nÞ.

5.3 Higher-Order Fixed-Point Bifurcations

401

The parallel antenna switching bifurcation for higher-order fixed-points in the ð2m þ 1Þth degree polynomial nonlinear discrete system is 8 ð2Þ th ð2Þ th > < ðða1 Þ mXX:    :ðal2 Þ mXXÞ mSI-oSO ( or mSO), for xk ¼ a ð5:225Þ > : ððað1Þ Þth mXX:    :ðað1Þ Þth mXXÞ 1 l1 where sth i bifurcation

th ðaðiÞ si Þ mXXsi ! switching ( ðsi Þ th ðs Þ ðs Þ ðs Þ ðiÞ ðiÞ ðr2 Þ mXX2 i # mYY1 i ; for b2 i ¼ a2si # a2si 1 ; ðs Þ ðr1 i Þth

ðs Þ mXX1 i

"

ðs Þ mYY2 i ;

for

ðs Þ b1 i

¼

ðiÞ a2si 1

"

ð5:226Þ

ðiÞ a2si ;

ðsi ¼ 1; 2; . . .; li ; i ¼ 1; 2Þ Such eight sets of parallel switching bifurcations for higher-order fixed-points are presented in Fig. 5.11i, iii, v, vii and ii, iv, vi, viii for a0 [ 0 and a0 \0, respectively. The eight switching bifurcation structures are as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

ðð2l1 Þth mUS:    :mSI-oSO:    :ð2ln1  1Þth mSO:(2ln Þth mUSÞ for a0 [ 0, ðð2l1 Þth mLS    :mSO:    ð2ln1  1Þth mSI :(2ln Þth mLSÞ for a0 \0, ðð2l1 Þth mLS    :mSO:    ð2ln1  1Þth mSI :(2ln  1Þth mSOÞ for a0 [ 0, ðð2l1 Þth mUS    :mSI:    ð2ln1  1Þth mSO :(2ln  1Þth mSIÞ for a0 \ 0, ðð2l1 Þth mLS    :mSO:    ð2ln1  1Þth mSI :(2ln  1Þth mSOÞ for a0 [ 0, ðð2l1 Þth mUS    :mSI-oSO:    ð2ln1  1Þth mSI:(2ln  1Þth mSIÞ for a0 \ 0, ðð2l1 Þth mUS    :mSI:    ð2ln1  1Þth mSO:(2ln Þth mUSÞ for a0 [ 0, ðð2l1 Þth mLS    : mSO:    ð2ln1  1Þth mSI:(2ln Þth mLSÞ for a0 \ 0.

The switching bifurcations with different higher-order fixed-points are similar to the ðl1 -mLSN:mSO:l2 -mUSN) and ðl1 -mUSN:mSI-oSO:l2 -mLSN) switching bifurcations for simple sinks and sources. Consider a switching bifurcation for a cluster of higher-order fixed-points with the following conditions, Bi ¼ Bj ði; j 2 f1; 2; . . .; ng; i 6¼ jÞ ðiÞ ðiÞ Di ¼ 0 at b1 = b2 ði ¼ 1; 2; . . .; nÞ

 at bifurcation:

ð5:227Þ

Thus, the ðai Þth order switching bifurcation can be for a cluster of higher-order fixed-points. The ðai Þth order switching bifurcations for i 2 f1; 2; . . .; sg are

5 (2m + 1)th-Degree Polynomial Discrete Systems

402 a0 > 0

a2n

(2r2 n − 1) th mSO

a0 < 0

a2n

(2r2 n − 1) th mSI (2ln ) th mLS

th

(2ln ) mUS (2r2 n −1 − 1) th mSI

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSI

a2 n−1

a2 n−1

(2r2 n − 1) th mSI

(2r2 n − 1) th mSO

a2 n−2

(2r2 n − 2 ) th mUS

a2 n−2

(2r2 n −3 − 1) th mSI

a2 n−3

(2ln −1 − 1) th mSI

(2ln −1 − 1) th mSO (2r2 n −3 − 1) th mSO

(2r2 n −3 − 1) th mSI

a2 n−3

(2r2 n −3 − 1) th mSO (2r2 n − 2 ) th mLS

(2r2 n − 2 ) th mLS

(2r2 n − 2 ) th mUS

•

• •

• •

•

mSO

mSI-oSO

mSO

mSI-oSO

P-2

a

P-2

• • •

• • •

a2

(2r2 ) th mUS

(2r1 ) th mUS

(2r1 ) th mLS

a1

(2r1 ) th mUS

xk∗

(2r2 ) th mUS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

th

(2r2 n ) mUS

a2n

a0 < 0

a2n

(2r2 n ) th mLS

th

th

(2ln − 1) mSO

(2ln − 1) mLS

(2r2 n −1 − 1) th mSO

(2r2 n −1 − 1) th mSO

th

(2r2 n − 2 ) mLS

(2r2 n −1 − 1) th mSI

(2r2 n −1 − 1) th mSI

a2 n−1

(2r2 n ) th mLS

a2 n−2

a2 n−1

(2r2 n ) th mUS

th

(2r2 n − 2 ) mUS

a2 n−2

(2r2 n −3 − 1) th mSO

a2 n−3

th

th

(2ln −1 − 1) mSI

(2ln −1 − 1) mSO

(2r2 n −3 − 1) th mSI th

(2r2 n −3 − 1) mSI

(2r2 n −3 − 1) th mSO

a2 n−3

(2r2 n − 2 ) th mLS

•

• •

• •

•

mSO

mSI-oSO

mSO mSO

mSI-oSO

P-2

a

a

•

• • •

• •

(2r2 ) th mLS

th

(2r1 ) mLS

a2

(2r1 ) th mUS

th

(2l1 ) mLS

(iii)

(2r2 ) th mUS

a2

(2l1 ) mUS

(2r1 ) th mLS

Δ iq > 0 Δ iq = 0

P-2

th

(2r2 ) th mLS

|| p ||

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(ii)

a0 > 0

xk∗

a1

(2r1 ) th mLS (2r2 ) th mLS

(i)

(2r2 n − 2 ) th mUS

a2

(2r2 ) th mLS

(2l1 ) th mLS

(2l1 ) th mUS

xk∗

mSO

a

Δ iq > 0

a1 xk∗

(2r1 ) th mUS

a1

th

(2r2 ) mUS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.11 Antenna parallel switching bifurcation of fixed-points for a ð2m þ 1Þth -degree polynomial nonlinear discrete system. (i, iii, vi, vii) four parallel bifurcations for a0 [ 0. (ii, iv, vi, viii) four parallel bifurcations for a0 \0. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI: monotonic sink, mSO: monotonic source. mSI-oSO: monotonic sink to oscillatory source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, sketched by red curves. The period-2 fixed-points are relative to the monotonic sinks to oscillatory sources

5.3 Higher-Order Fixed-Point Bifurcations a0 > 0

(2r2 n − 1) th mSO

403

a2n

a0 < 0

a2n

(2r2 n − 1) th mSI

(2ln − 1) th mSO

(2ln − 1) th mSI

(2r2 n −1 ) th mLS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mLS

a2 n−1 (2r2 n − 1) th mSO

(2r2 n − 2 ) th mLS

a2 n−1 (2r2 n − 1) th mSI

a2 n−2

(2ln −1 − 1) th mSI

(2ln −1 − 1) th mSI

(2r2 n −3 − 1) th mSI (2r2 n −3 − 1) th mSI (2r2 n − 2 ) th mUS

a2 n−2

(2r2 n − 2 ) th mUS

(2r2 n −3 − 1) th mSO

a2 n−3

a2 n−3

(2r2 n −3 − 1) th mSO (2r2 n − 2 ) th mLS

• • •

• • •

mSO

mSI-oSO

mSO

mSO

a • • •

(2r2 ) th mLS

(2r1 ) th mLS

mSI-oSO

P-2

a

a2

a2

(2l1 ) th mUS

(2r1 ) th mLS

a1 xk∗

(2r2 ) th mLS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(2r2 ) th mUS

(2r1 ) th mUS

(2l1 ) th mLS

xk∗

P-2

• • •

(2r1 ) th mUS

a1

(2r2 ) th mUS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(v)

(vi)

a0 > 0

a2n

(2r2 n ) th mUS

a0 < 0

(2r2 n ) th mLS

a2n

(2ln ) th mLS

(2ln ) th mUS (2r2 n −1 ) th mUS

(2r2 n −1 ) th mUS

(2r2 n −1 ) th mLS

(2r2 n −1 ) th mLS

a2 n−1

a2 n−1 (2r2 n ) th mUS

(2r2 n ) th mLS

a2 n−2

(2r2 n − 2 ) th mUS

(2r2 n − 2 ) th mLS

a2 n−2

(2r2 n −3 − 1) th mSI

a2 n−3

(2ln −1 − 1) th mSI

(2ln −1 − 1) th mSO (2r2 n −3 − 1) th mSO

(2r2 n −3 − 1) th mSI

a2 n−3

(2r2 n −3 − 1) th mSO

th

th

(2r2 n − 2 ) mLS

(2r2 n − 2 ) mUS

•

• •

• •

•

mSO

mSI-oSO

mSO mSO

mSI-oSO

P-2

a

a • • • (2r2 ) th mUS

(2r1 ) th mUS

•

P-2

• •

a2

th

(2r1 ) mLS

(2r1 ) th mUS th

(2r2 ) mUS

|| p ||

Δ iq > 0 Δ iq = 0

(vii) Fig. 5.11 (continued)

a2

(2l1 ) mLS

(2l1 ) mUS

xk∗

(2r2 ) th mLS

th

th

Δ iq > 0

a1

xk∗

(2r1 ) th mLS th

(2r2 ) mLS

|| p ||

Δ iq > 0 Δ iq = 0

(viii)

Δ iq > 0

a1

5 (2m + 1)th-Degree Polynomial Discrete Systems

404 ai ¼

Pl i

ðiÞ r j¼1 j

ðai Þth order mXX ! switching appearing 8 ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ > ðrs Þth order mXXli # mYYli ; for bli # ali ; > > > > > . > > > .. < ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðrj Þth order mXXj # mYYj ; for bj # as ; > > > > .. > > > . > > : ðiÞ th ðiÞ ðiÞ ðiÞ ðiÞ ðr1 Þ order mXX1 " mYY1 ; for b1 # as ;

ð5:228Þ

where ðiÞ

ðiÞ

ðiÞ

ðiÞ

fa1 ; a2 ;    ; ali1 ; ali g ðiÞ

ðiÞ

ðiÞ

ðiÞ

fb1 ; b2 ;    ; bli1 ; bli g



ð1Þ

before bifurcation



After bifurcation

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ;    ; b1 ; b2 g; ð1Þ

ð1Þ

ðnÞ

ðnÞ

sortfb1 ; b2 ;    ; b1 ; b2 g:

ð5:229Þ

A set of paralleled, different, higher-order monotonic upper-saddle-node switching bifurcations with multiplicity is a ðða1 Þth mXX:ða2 Þth mXX:    :ðas Þth mXXÞ parallel switching bifurcation in the ð2m þ 1Þth degree polynomial discrete system. At the straw-bundle switching bifurcation, Di ¼ 0 (i ¼ 1; 2; . . .; n) and Bi ¼ Bj ði; j 2 f1; 2; . . .; ng i 6¼ j). The parallel straw-bundle switching bifurcation for higher order fixed-points is ðða1 Þth mXX:ða2 Þth mXX:    :ðas Þth mXXÞ-switching 8 > ðas Þth order mXX switching, > > > > < .. ¼ . > > ða2 Þth order mXX switching, > > > : ða1 Þth order mXX switching;

ð5:230Þ

ai 2 f2li ; 2li  1g and mXX 2 fmUS, mLS, mSO, mSIg:

ð5:231Þ

where

Thus, 8 ð2Þ th ð2Þ ð2Þ < ðða1 Þ mXX:ða2 Þth mXX:    :ðas2 Þth mXXÞ mSI-oSO (or mSO) : ð1Þ th ð1Þ ð1Þ ðða1 Þ mXX:ða2 Þth mXX:    :ðas1 Þth mXXÞ ath n

ð5:232Þ

th Eight parallel straw-bundle switching bifurcations of ðath 1 mXX : a2 mXX:    : mXXÞ are presented in Figs. 5.12 and 5.13 for a0 [ 0 and a0 \0, respectively.

5.3 Higher-Order Fixed-Point Bifurcations a0 > 0

a2n

(2r2 n − 2 − 1) th mSO

a2 n−1

th

(2ln ) mUS (2r2 n −1 ) th mLS (2r2 n − 1) th mSI

405 a0 > 0

(2r2 n − 1) th mSO

(2r2 n − 2 − 1) th mSO th

(2ln − 1) mSO

th

(2r2 n −1 ) mLS

a2 n−2

(2r2 n − 2 − 1) th mSI

a2 n−3

(2r2 n −3 ) th mUS

(2r2 n −1 ) th mLS (2r2 n ) th mLS

a (2r2 n −3 ) th mLS

• • •

mSO

(2r2 n −3 ) th mUS

• •

(2r2i ) th mLS

a3 a2

(2l1 − 1) th mSI

a1

•

(2r2i − 2 ) th mUS

(2r2i − 2 ) th mLS

(2r2i −3 − 1) th mSO

(2r2i −1 − 1) th mSI

(2r2i −1 − 1) th mSO

(2r2i − 2 ) th mLS

(2r2i −3 − 1) th mSI

Δ iq > 0 Δ iq = 0

• •

(2r2i ) th mUS

(2r3 ) th mLS

•

a0 > 0

th

(2r1 ) mUS

(2r2 − 1) th mSO

xk∗

a1

(2ln − 1) mSO

(2r2 n −1 ) th mUS (2r2 n − 1) th mSO

(2r3 ) th mUS

(2r2 − 1) th mSO (2r1 ) th mLS

(2r2 ) th mLS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(ii) a2 n−1

th

a2

(2l1 − 1) th mSO

(2r2 − 1) th mSI

Δ iq > 0

a2n

(2r2 n − 2 ) th mUS

a3

(2r1 ) th mUS

(2r2 ) th mUS

a0 > 0

(2r2 n − 1) th mSO

(2r2 n − 2 ) th mUS th

(2ln − 1) mSO

th

(2r2 n −1 ) mLS

a2 n−2

(2r2 n − 2 ) th mLS

a2 n−3

(2r2 n −3 ) th mLS

(2r2 n −1 − 1) th mSO (2r2 n ) th mLS

mSI-oSO

a2n

(2r2 n ) th mUS

a2 n−1

(2r2 n −1 − 1) th mSO

a2 n−2

(2r2 n − 2 ) th mLS

a2 n−3

(2r2 n −3 ) th mLS

mSI-oSO mSI-oSO

P-2

a

(2r2 n −3 + 1) th mSI (2r2 n −3 ) th mUS

• • •

mSI-oSO

P-2

(2r2 n −3 ) th mUS

th

(2r2i −3 − 1) mSI

(2r2i − 2 ) th mLS

•

th

(2r1 ) mUS (2l1 − 1) th mSI

a3 a2

(2r2 ) th mLS

(iii)

(2r2i − 2 ) th mLS

(2r2i −3 − 1) th mSI

•

a1

Δ iq > 0 Δ iq = 0

(2li ) th mUS

(2r2i −1 − 1) th mSI

• •

(2r2i ) th mUS

(2r2i −1 − 1) th mSO

(2r2i −1 − 1) mSO

th

(2r2i − 2 ) th mLS

P-2 (2r2i ) th mUS

th

(2r2i −1 − 1) mSO

(2li ) mLS

• • •

(2r2i −1 − 1) th mSO

(2r2i −1 − 1) th mSI

a

(2r2 n −3 + 1) th mSI

P-2 (2r2i ) th mUS

th

|| p ||

P-2 (2r2i ) th mUS

(2li ) th mUS

(i)

xk∗

(2r2 n −3 ) th mLS

(2r2i −1 − 1) mSO

(2r1 ) th mLS

(2r2 − 1) th mSO

a2 n−3

•

•

(2r2i − 2 ) th mLS

(2r2 n − 2 − 1) th mSO

th

(2li ) th mLS

|| p ||

a2 n−2

•

(2r2i −1 − 1) th mSI

(2r2i −1 − 1) th mSO

xk∗

(2r2 n −1 ) th mUS

a

(2r2 n −3 + 1) th mSI

(2r2i −1 − 1) mSI

(2r2 − 1) th mSI

a2 n−1

mSI-oSO

P-2

(2r2i ) th mLS

th

(2r2i − 2 ) th mUS

(2r2 n ) th mUS

mSI-oSO

(2r2 n −3 + 1) th mSO

mSO

a2n

Δ iq > 0

• •

(2r2i ) th mUS

(2r3 ) th mUS

(2r1 ) th mUS (2r2 − 1) th mSO

(2r2 − 1) th mSO

(2l1 − 1) th mSO

(2r1 ) th mLS

xk∗

a3 a2 a1

(2r3 ) th mUS

(2r2 − 1) th mSO (2r1 ) th mLS

(2r2 ) th mLS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.12 (i)–(iv) Four types of ðr1 th mXX:r2 th mXX:    :rm th mXXÞ parallel switching bifurcation for a0 [ 0 in the ð2m þ 1Þth -degree polynomial discrete system. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI: monotonic sink, mSO: monotonic source, mSI-oSO: monotonic sink to oscillatory source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, sketched by red curves, which are relative to the monotonic sinks to oscillatory sources

5 (2m + 1)th-Degree Polynomial Discrete Systems

406 a0 < 0

a2 n−1

th

(2ln ) mLS

(2r2 n −1 ) th mUS

a2 n−2

th

(2r2 n − 1) mSO

a0 < 0

(2r2 n − 1) th mSI

a2n

(2r2 n − 2 − 1) th mSI

a2 n−3

(2r2 n − 2 − 1) th mSI

(2ln − 1) th mSI

th

(2r2 n −1 ) mUS

(2r2 n −1 ) th mUS

th

(2r2 n − 2 − 1) mSO

th

th

(2r2 n ) US

(2r2 n −3 ) mLS

mSI-oSO

a

(2r2 n −3 + 1) th mSI (2r2 n −3 ) th mUS

P-2

•

(2r2i ) mUS

• •

(2r2i −1 − 1) th mSO

(2r2i −1 − 1) th mSO

(2li ) th mUS

(2r2i − 2 ) th mLS

(2r2 n −3 ) th mLS

th

(2r2i −1 − 1) th mSI

x∗

• •

(2r2i − 2 ) th mUS

(2r2i −3 − 1) th mSI

(2r2i −1 − 1) th mSO

a2

•

(2r2i ) th mLS (2r2i −1 − 1) th mSI

(2r1 ) mLS

xk∗

a3 a2

(2l1 − 1) th mSI

(2r2 − 1) th mSI

th

a1

(2r3 ) th mLS

(2r2 − 1) th mSI (2r1 ) th mUS

th

(2r2 ) mUS

Δ iq > 0

Δ iq > 0 Δ iq = 0

|| p ||

(i)

(ii)

(2r2 n − 2 ) th mLS (2ln − 1) th mSI (2r2 n −1 ) th mLS (2r2 n − 1) mSI

(2r2 n − 1) mSI

a2 n−1

(2r2 n −1 ) th mUS

a2 n−3

a0 < 0

th

a2n

a2 n−2

th

(2r2 n − 2 ) th mLS

th

(2r2 n − 2 ) mUS

(2ln − 1) th mSI (2r2 n −1 − 1) th mSI

th

th

(2r2 n −3 ) mUS

(2r2 n ) mUS

th

(2r2 n −3 + 1) mSO

a2n

(2r2 n ) th mLS

a2 n−1

(2r2 n −1 − 1) th mSI

a2 n−2

(2r2 n − 2 ) th mUS

a2 n−3

(2r2 n −3 ) th mUS

a

mSO

th

(2r2 n −3 + 1) mSO

mSO

a mSO

th

th

(2r2 n −3 ) mLS

• •

(2r2 n −3 ) mLS

(2r2i ) th mLS

•

(2li ) th mLS

(2r2i − 2 ) th mUS

(2r2i −1 − 1) mSO

(2r2i −3 − 1) th mSO

•

•

th

(2l1 − 1) mSI

a3 a2

(2r2i − 2 ) th mUS

a1 th

(2r2 ) mUS

Δ iq > 0 Δ iq = 0

(iii)

Δ iq > 0

(2li ) th mLS

(2r2i − 2 ) th mUS

th

(2r2i −1 − 1) mSO

(2r2i −3 − 1) th mSO • •

(2r2i ) th mLS

(2r3 ) th mLS

•

(2r1 ) th mLS

(2r2 − 1) th mSI

(2r2i −1 − 1) th mSI

(2r2i −1 − 1) th mSI

• •

(2r1 ) th mLS

(2r2i ) th mLS

•

(2r2i −1 − 1) mSI

th

(2r2i ) th mLS

•

th

(2r2i −1 − 1) th mSI

|| p ||

mSO

•

(2r2 − 1) th mSO

Δ iq > 0

Δ iq > 0 Δ iq = 0

a0 < 0

xk∗

a

(2r2i −3 − 1) th mSO

(2r2i ) th mLS

(2r2 ) mLS

(2r2 − 1) th mSI

(2r2 n −3 ) th mUS

(2r2i − 2 ) th mUS

(2r1 ) th mLS

a1

(2r2i − 2 ) th mUS

a2 n−3

(2li ) th LS

(2r2i − 2 ) th mLS

th

mSO

(2r2 n − 2 − 1) th mSI

(2r2i −1 − 1) th mSI

(2r3 ) th mUS

a3

(2l1 − 1) th mSO

|| p ||

a2 n−2

•

(2r1 ) th mUS (2r2 − 1) th mSO

(2r2 n −1 ) th mLS

• • •

• (2r2i ) th mUS

(2r2 n ) th mLS

a2 n−1

(2r2 n −3 + 1) th SO

mSO

mSI-oSO

P-2

a2n

(2r2 − 1) th SI

th

(2l1 − 1) mSI

(2r1 ) th mUS

xk∗

a3 a2 a1

(2r3 ) th mLS

(2r2 − 1) th mSI (2r1 ) th mLS

th

(2r2 ) mUS

|| p ||

Δ iq > 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.13 (i)–(iv) Four types of ðr1 th mXX:r2 th mXX:    :rm th mXXÞ parallel switching bifurcation for a0 \0 in the ð2m þ 1Þth -degree polynomial discrete system. mLS: monotonic lowersaddle, mUS: monotonic upper-saddle, mSI: monotonic sink, mSO: monotonic source, mSI-oSO: monotoic sink to oscillatory source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, sketched by red curves. The period-2 fixed-points are relative to the monotonic sinks to oscillatory sources

5.3 Higher-Order Fixed-Point Bifurcations

5.3.3

407

Switching-Appearing Bifurcations

Consider a ð2m þ 1Þth degree 1-dimensional polynomial nonlinear discrete system in a form of 1 2 xk þ 1 ¼ xk þ a0 Qðxk Þ *ni¼1 ðxk  ci Þai *nj¼1 ðx2k þ Bj xk þ Cj Þaj :

ð5:233Þ

1 where Pni¼1 ai ¼ 2s1 þ 1. Without loss of generality, a function of Qðxk Þ [ 0 is either a polynomial function or a non-polynomial function. The roots of x2k þ Bj xk þ Cj ¼ 0 are

ðjÞ

1 2

b1;2 ¼  Bj 

1pffiffiffiffiffi Dj ; Dj 2

¼ B2j  4Cj 0ðj ¼ 1; 2; . . .; n2 Þ;

ð5:234Þ

either     fa 1 ; a2 ;    ; a2n1 þ 1 g ¼ sortfc1 ; c2    ; c2n1 ; ag; as as þ 1 before bifurcation ð1Þ

ð1Þ

ðn Þ

ðn Þ

þ g ¼ sortfc1 ;    ; c2n1 ; a; b1 ; b2 ;    ; b1 2 ; b2 2 g; fa1þ ; a2þ ;    ; a2n 3 þ1

asþ asþþ 1 ; n3 ¼ n1 þ n2 after bifurcation; ð5:235Þ or ð1Þ

ð1Þ

ðn Þ

ðn Þ

  2 2 fa 1 ; a2 ;    ; a2n3 þ 1 g ¼ sortfc1 ; c2    ; c2n1 ; a; b1 ; b2 ;    ; b1 ; b2 g;  a s as þ 1 ; n3 ¼ n1 þ n2 before bifurcation;

þ fa1þ ; a2þ ;    ; a2n g ¼ sortfc1 ;    ; c2n1 ; ag; asþ asþþ 1 after bifurcation; 1 þ1

ð5:236Þ and 9 Bj1 ¼ Bj2 ¼    ¼Bjs ðjk1 2 f1; 2; . . .; ng; jk1 6¼ jk2 Þ > > = ðk1 ; k2 2 f1; 2; . . .; sg; k1 6¼ k2 Þ at bifurcation: Dj ¼ 0ðj 2 f1; 2; . . .; n2 g > > ; 1 ci 6¼ 2Bj ði ¼ 1; 2; . . .; 2n1 ; j ¼ 1; 2; . . .; n2 Þ

ð5:237Þ

th  th  th    A just before bifurcation of ððb 1 Þ mXX1 :ðb2 Þ mXX2 :   : ðb31 Þ mXXsi Þ for    higher-order fixed-points is considered. For bi ¼ 2li  1 mXXi 2 fmSO,mSIg   and for a i ¼ 2li ; mXXi 2 fmUS,mLSg ði ¼ 1; 2; . . .; s1 Þ, the detailed structures are as follows.

408

5 (2m + 1)th-Degree Polynomial Discrete Systems

9 ðiÞ ðiÞ  > ðrsi Þth order mXXðiÞ s ; xk ¼ asi ; > > > > > .. > > > . = bi ¼Psi rðiÞ j¼1 j th ðiÞ th ðiÞ  ðiÞ ðiÞ ! ðb ðrj Þ order mXXj ; xk ¼ aj i Þ order mXX > switching bifurcation > > > .. > > . > > > ðiÞ th ðiÞ  ðiÞ ; ðr1 Þ order mXX1 ; xk ¼ a1

ð5:238Þ th  th  th    The bifurcation set of ððb 1 Þ mXX1 : ðb2 Þ mXX2 :   : ðbs1 Þ mXXs1 Þ at the same parameter point is called a left-parallel-straw-bundle switching bifurcation A just after bifurcation of ððb1þ Þth mXX1þ :ðb2þ Þth mXX2þ :   : ðbsþ2 Þth mXXsþ2 Þ for higher-order singularity fixed-points is considered. For bjþ ¼ 2ljþ  1, mXXiþ 2 fmSO,mSIg and for b þ ¼ 2l þ ; mXXiþ 2 fmUS,mLSg. The detailed structures are as follows. 8 ðiÞ þ ðiÞ þ þ  > ðrsi Þth order mXXðiÞ ; xk ¼ asi ; > si > > > > . > > > .. ðiÞ þ  Ps i < bi ¼ j¼1 rj þ  ðiÞ þ ðbiþ Þth order mXXðiÞ þ ! ðrjðiÞ þ Þth order mXXðiÞ ; x k ¼ aj j switching bifurcation > > > > .. > > > . > > : ðiÞ þ th ðiÞ þ ðiÞ þ ðr1 Þ order mXX1 ; xk ¼ a1 :

ð5:239Þ The bifurcation set of ððb1þ Þth mXX1þ : ðb2þ Þth mXX2þ :    : ðbsþ2 Þth mXXsþ2 Þ at the same parameter point is called a right-parallel-straw-bundle switching bifurcation (i) For the just before and after bifurcation structure, if there exists a relation of th þ th th  þ   þ ðb i Þ mXXi ¼ ðbj Þ mXXj ¼ bj mXX, for x ¼ ai ¼ aj

ði; j 2 f1; 2; . . .; kgÞ; mXX 2 fmUS,mLS,mSO,mSIg

ð5:240Þ

then the bifurcation is a ðbj Þth mXXj switching bifurcation for higher-order fixed-points. (ii) Just for the just before bifurcation structure, if there exists a relation of

5.3 Higher-Order Fixed-Point Bifurcations

409

th th    ð2l i Þ mXXi ¼ ð2li Þ mXX, for x ¼ ai ¼ ai

ði 2 f1; 2; . . .; s1 g; mXX 2 fmUS,mLSg

ð5:241Þ

then, the bifurcation is a ð2lÞth mXX left appearing (or right vanishing) bifurcation for higher-order fixed-points. (iii) Just for the just after bifurcation structure, if there exists a relation of ð2liþ Þth mXXiþ ¼ ð2li Þth mXX, for x ¼ aiþ ¼ ai ði 2 f1; 2; . . .; s1 gÞ; mXX 2 fmUS,mLSg

ð5:242Þ

then, the bifurcation is a ð2lÞth mXX right appearing (or left vanishing) bifurcation for higher-order fixed-points. (iv) For the just before and after bifurcation structure, if there exists a relation of th þ th  þ þ ðb for x ¼ a i Þ mXXi 6¼ ðbj Þ mXXj i ¼ aj þ mXX i ; mXXj 2 fmUS,mLS, mSO,mSIg

ð5:243Þ

ði 2 f1; 2; . . .; s1 g; j 2 f1; 2; . . .; s2 gÞ; then, two flower-bundle switching bifurcations of higher-order fixed-points are as follows. (iv1) For bj ¼ bi þ 2l, the bifurcation is called a ðbj Þth mXX right flowerbundle switching bifurcation for the ðbi Þth mXX to ðbj Þth mXX switching of higher-order fixed-points with the appearance (or birth) of ð2lÞth mXX right appearing (or left vanishing) bifurcation. (iv2) For bj ¼ bi  2l, the bifurcation is called a ðbi Þth mXX left flowerbundle switching bifurcation for the ðbi Þth mXX to ðbj Þth mXX switching of higher-order fixed-points with the vanishing (or death) of ð2lÞth mXX left appearing (or right vanishing) bifurcation. A general parallel switching bifurcation is switching

th  th  th    ððb 1 Þ mXX1 : ðb2 Þ mXX2 :    : ðbs1 Þ mXXs1 Þ !

ððb1þ Þth mXX1þ : ðb2þ Þth mXX2þ :    :

bifurcation þ th þ ðbs2 Þ mXXs2 Þ:

ð5:244Þ

Such a general, parallel switching bifurcation consists of the left and right parallelbundle switching bifurcations for higher-order fixed-points. If the left and right parallel-bundle switching bifurcations are same in a parallel flower-bundle switching bifurcation, i.e.,

5 (2m + 1)th-Degree Polynomial Discrete Systems

410

th th þ th  þ ðb i Þ mXXi ¼ ðbi Þ mXXi ¼ ðbi Þ mXXi

þ for xk ¼ a i ¼ ai ði ¼ 1; 2; . . .; sÞ:

ð5:245Þ

then the parallel flower-bundle switching bifurcation becomes a parallel strawbundle switching bifurcation of ððb1 Þth mXX1 : ðb2 Þth mXX2 :    : ðbs1 Þth mXXs Þ. If the left and right parallel-bundle switching bifurcations are different in a parallel flower-bundle switching bifurcation, i.e., th   th  þ th þ þ th þ ða i Þ mXXi ¼ ð2li Þ mXXi ; ðaj Þ mXXj ¼ ð2lj Þ mXXj þ for xk ¼ a i 6¼ aj ði ¼ 1; 2; . . .; s1 ; j ¼ 1; 2; . . .; s2 Þ;

mXX i

2

fmUS,mLSg; mXXjþ

ð5:246Þ

2 fmUS,mLSg;

then the parallel flower-bundle switching bifurcation for higher-order fixed-points becomes a combination of two independent left and right parallel appearing bifurcations: th     th  th (i) a ð2l 1 Þ mXX1 : ð2l2 Þ mXX2 :    : ð2ls1 Þ mXXs1 -left parallel sprinklerspraying appearing (or right vanishing) bifurcation and (ii) a ð2l1þ Þth mXX1þ : ð2l2þ Þth mXX2þ :    : ð2lsþ2 Þth mXXsþ1 -right parallel sprinklerspraying appearing (or left vanishing) bifurcation.

The parallel switching and appearing bifurcations for higher-order fixed-points are presented in Fig. 5.14i–iv. The waterfall appearing bifurcations and the flower-bundle switching bifurcations for higher-order fixed-points are presented. The period-2 fixed point is presented through red curves.

5.4

Forward Bifurcation Trees

In this section, the analytical bifurcation scenario of a ð2m þ 1Þth -degree polynomial nonlinear discrete system will be discussed. The period-doubling bifurcation scenario will be discussed first through nonlinear renormalization techniques, and the bifurcation scenario based on the saddle-node bifurcation will be discussed, which is independent of period-1 fixed-points.

5.4.1

Period-Doubled ð2m þ 1Þth -Degree Polynomial Systems

After the period-doubling bifurcation of a period-1 fixed-point, the period-doubled fixed-points of a ð2m þ 1Þth -degree polynomial nonlinear discrete system can be obtained. Consider the period-doubling solutions of a forward quartic nonlinear discrete system first.

5.4 Forward Bifurcation Trees

411 (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO (2rn − 2 ) th mLS

(2rn − 2 ) th mUS th

(2rn − 1) mSO (2ls ) th mUS

(2rn − 1) th mSI (2ls ) th mLS

(2rn − 2 ) th LS (2rn − 2 − 1) th mSI

(2ris ) th mUS

(2ris ) th mUS

(2rn − 2 ) th mUS (2rn − 2 − 1) th mSO

(2ris ) th mLS (2ris ) th mLS

(2rn −3 − 1) th mSO (2rn − 4 − 1) th mSI

(2ls −1 ) th mUS

(2ls −1 ) th mLS

(2rn −3 − 1) th mSO

• • •

(2rn −3 − 1) th mSI

•

(2rn − 4 − 1) th mSI

mSO

•

mSI-oSO

•

(2ri ) th mUS

mSO

(2lr ) th mUS

(2ri −1 ) th mLS (2ri − 2 − 1) th mSI

P-2

(2ri −3 − 1) th mSI

(2ri −3 − 1) th mSO (2lr ) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

(2ri −1 ) th mLS (2ri −3 − 1) mSI

•

(2ri ) th mUS

•

(2ri −1 ) th mUS

th

•

(2ri2 − 1) th mSI

• (2ri2 ) th mSO

(2r1 − 1) th mSO

(2r1 − 1) th mSI

(2l1 ) th mLS

(2r1 − 1) th mSI

(2ri1 − 1) th mSO th

(2ri1 − 1) mSO

(2r2 ) th mUS

(2r2 ) th mLS (2ri1 − 1) th mSI

Δ iq > 0

Δ iq < 0 Δ iq = 0

(2r1 − 1) th mSO

(2ri1 − 1) th mSI

xk∗

Δ iq > 0

Δ iq < 0 Δ iq = 0

|| p ||

(i)

(iii) (2rn − 1) th mSO

a0 > 0

(2rn − 1) th mSI

a0 < 0

(2rn −1 − 1) th mSI

(2rn −1 − 1) th mSO

th

(2rn − 2 ) mUS

(2rn − 2 ) th mLS

(2rn − 1) th mSO

(2ls ) th mUS

(2rn − 1) th mSI

(2rn − 2 ) th mLS

(2ls ) mLS

th

(2ris−2 − 1) mSI

oSI-oSO

(2ls −1 ) mLS

(2rn − 4 − 1) th mSO

(2rn − 4 − 1) th mSI

mSI-oSO

mSO

(2rn −3 − 1) th mSO

• •

(2rn −3 − 1) th mSI

(2ris−2 − 1) th mSO

th

P-2 (2ris−2 − 1) th mSI

(2ris ) th mLS

P-2

(2rn −3 − 1) th mSO

mSO

(2rn − 2 − 1) th mSO

(2ris ) th LS

(2ris ) th mUS

(2ls −1 ) th mUS

(2rn − 2 ) th mUS

th

(2rn − 2 − 1) th mSI

(2ris ) th US

(2rn − 4 − 1) th mSI

(2rn −3 − 1) th mSI

(2ris−2 − 1) th mSO

(2rn − 4 − 1) th mSO

• • •

(2ri ) th mUS

•

(2ri −1 ) th mUS (2ri − 2 − 1) th mSO

(2ri ) th mLS (2ri −1 ) th mLS (2ri − 2 − 1) th mSI

th

(2ri −3 − 1) mSI

(2lr ) th mUS

(2ri −3 − 1) th mSO (2lr ) th mLS

(2ri − 2 − 1) th mSO

(2ri − 2 − 1) th mSI

(2ri −1 ) th mLS • •

(2ri −1 ) th mUS

(2ri −3 − 1) th mSI

• • •

(2ri ) th mUS

•

(2r2 ) th mUS

(2ri −3 − 1) th mSO

(2ri ) th mLS (2r2 ) th mLS

(2r1 − 1) th mSO th

(2l1 − 1) mSO

xk∗

(2r1 − 1) th mSI

(2ri1 − 1) th mSI

(2ri1 − 1) th mSO

(2r1 − 1) th mSO

(2l1 − 1) th mSI

xk∗

(2ri1 − 1) th mSO

(2ri1 − 1) th mSI

(2r1 − 1) th mSI

(2r2 ) th mLS

|| p ||

(2ri ) th mLS (2r2 ) th mLS

(2ri2 − 1) th mSO

(2l1 ) th mUS

|| p ||

(2ri −3 − 1) th mSO

• •

(2r2 ) th mUS

(2ri2 − 1) th mSI

(2rn − 4 − 1) th mSO (2ri ) th mLS

mSI-oSO

(2ri −1 ) th mUS (2ri − 2 − 1) th mSO

xk∗

(2rn −3 − 1) th mSI (2rn − 4 − 1) th mSO

Δ iq < 0 Δ iq = 0

(ii)

Δ iq > 0

(2r2 ) th mUS

|| p ||

Δ iq < 0 Δ iq = 0

Δ iq > 0

(iv)

Fig. 5.14 ðr1 th mXX:r2 th mXX:    :rn th mXXÞ parallel bifurcation (a0 [ 0): (i) without switching, and (ii) with switching. The ðr1 th mXX:r2 th mXX:    :rn th mXXÞ parallel bifurcation ða0 \0Þ: (iii) without switching, and (vi) with switching. mLS: monotonic lower-saddle, mUS: monotonic upper-saddle, mSI: monotonic sink, mSO: monotonic source, mSI-oSO: monotonic sink to oscillatory source. Stable and unstable fixed-points are represented by solid and dashed curves, respectively. The bifurcation points are marked by circular symbols. P-2 is for period-2 fixed points, sketched by red curves. The period-2 fixed-points are relative to the monotonic sinks to oscillatory sources

5 (2m + 1)th-Degree Polynomial Discrete Systems

412

Theorem 5.1 Consider a ð2m þ 1Þth -degree polynomial nonlinear discrete system þ1 2 þ A1 ðpÞx2m xk þ 1 ¼ xk þ A0 ðpÞx2m k k þ    þ A2m1 ðpÞxk þ A2m xk þ A2m þ 1 ðpÞ

¼ xk þ a0 ðpÞðxk  aðpÞÞ½x2k þ B1 ðpÞxk þ C1 ðpÞ    ½x2k þ Bm ðpÞxk þ Cm ðpÞ ð5:247Þ where A0 ðpÞ 6¼ 0 and p ¼ ð p1 ; p2 ;    ; pm 1 Þ T :

ð5:248Þ

If Di ¼ B2i  4Ci [ 0; i ¼ i1 ; i2 ; . . .; ii 2 f1; 2; . . .; mg0f£g Dj ¼ B2j  4Cj \0; j ¼ it þ 1 ; il þ 2 ;    ; im 2 f1; 2; . . .; mg0f£g with l 2 f0; 1; . . .; mg

ð5:249Þ

then, the corresponding standard form is ð1Þ

þ1 xk þ 1 ¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

ð5:250Þ

where ð1Þ

1 2

ð1Þ

bi;1 ¼  ðBi þ

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ð1Þ ð1Þ ð1Þ ð1Þ Di Þ; bi;2 ¼  ðBi  Di Þ 2

ð1Þ

for Di 0; i 2 f1; 2; . . .; lg0f£g; ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

l 02l i¼1 fai g ¼ sortf0i1 ¼1 fbi1 ; bi1 ;2 gg; ai ai þ 1 ; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ 2 2 pffiffiffiffiffiffiffi ð1Þ for Di \0; i 2 fl þ 1; l þ 2; . . .; mg0f£g; i ¼ 1; ð1Þ

ð1Þ

ð5:251Þ

ð1Þ

þ1 m 02m i¼2l þ 1 fai g ¼ f0i1 ¼l þ 1 fbi1 ; bi1 ;2 g; ag:

(i) Consider a forward period-2 discrete system of Eq. (5.247) as ðxk  ai1 Þf1 þ

2m þ 1

ðxk  ai1 Þ

*i ¼1 1

¼ xk þ ½a0

*i ¼1 1

þ1

½a2m 0

¼ xk þ ½a0

ðð2m þ 1Þ2 ð2m þ 1ÞÞ=2

2m þ 1

*j ¼1 1

2m þ 1

*i ¼1 1

½1 þ a0

2m þ 1

*i ¼1;i 6¼i ðxk 2 2 1

ð1Þ

 ai2 Þg

ð1Þ

*i ¼1 2

1 þ ð2m þ 1Þ

¼ xk þ a0

ð1Þ

2m þ 1

xk þ 2 ¼ xk þ ½a0

ð1Þ

ð2Þ

þ1 ðxk  ai1 Þ½a2m 0 ð2m þ 1Þ2

*i¼1

ð2Þ

ðx2k þ Bi2 xk þ Ci2 Þ ð2m þ 1Þ2 ð2m þ 1Þ

*j ¼1 2

ð2Þ

ðxk  bj2 Þ

ð2Þ

ðxk  ai Þ ð5:252Þ

5.4 Forward Bifurcation Trees

413

where ð2Þ

ð2Þ

1 2

bi;1 ¼  ðBi þ ð2Þ

Di

qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ð2Þ ð2Þ ð2Þ Dð2Þ Þ; bi;2 ¼  ðBi  Di Þ; 2

ð2Þ

ð2Þ

¼ ðBi Þ2  4Ci 0; i 2 0Nq11¼1 Iqð21 Þ 00Nq22¼1 Iqð22 0

2

Þ

Iqð21 Þ ¼ flðq1 1Þ 20 m1 þ 1 ; lðq1 1Þ 20 m1 þ 2 ;    ; lq1 20 m1 g 0

f1; 2;    ; M1 g0f∅g; q1 2 f1; 2;    ; N1 g; M1 ¼ N1 20 m1 ; m1 2 f1; 2;    ; mg; Iqð22 Þ ¼ flðq2 1Þ 21 m1 þ 1 ; lðq2 1Þ 21 m1 þ 2 ;    ; lq2 21 m1 g 1

ð5:253Þ

fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; q2 2 f1; 2;    ; N2 g; M2 ¼ ðð2mÞ2  2mÞ=2; qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ð2Þ 1 ð2Þ ð2Þ ð2Þ ð2Þ bi;1 ¼  ðBi þ i Dð2Þ Þ; bi;2 ¼  ðBi  i Di Þ; 2 2 pffiffiffiffiffiffiffi ð2Þ ð2Þ ð2Þ i ¼ 1; Di ¼ ðBi Þ2  4Ci \0; i 2 J ð2 Þ ¼ flN2 21 m1 þ 1 ; lN2 21 m1 þ 2 ;    ; lM2 g 1

fM1 þ 1; M1 þ 2;    ; M2 g with fixed-points ð2Þ

xk þ 2 ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; ð2m þ 1Þ2 Þ ð2m þ 1Þ2

0i¼1

ð2Þ

ð2Þ

ð1Þ

ð2Þ

ð2Þ

ð5:254Þ

þ1 M fai g ¼ sortf02m j1 ¼1 faj1 g; 0j2 ¼1 fbj2 ;1 ; bj2 ;2 gg ð2Þ

with ai \ai þ 1 ; M ¼ ðð2m þ 1Þ2  ð2m þ 1ÞÞ=2: ð1Þ

(ii) For a fixed-point of xk þ 1 ¼ xk ¼ ai1 (i1 2 f1; 2; . . .; 2m þ 1g), if dxk þ 1 ð1Þ ð1Þ þ1 j  ð1Þ ¼ 1 þ a0 *2m i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk xk ¼ai1

ð5:255Þ

with • a r th -order oscillatory upper-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ k

i1

k

i1

[ 0, r ¼ 2l1 Þ, • a r th -order oscillatory lower-saddle-node bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ \0, r ¼ 2l1 Þ, • a r th -order oscillatory source bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ \0, r ¼ 2l1 þ 1Þ,

k

i1

5 (2m + 1)th-Degree Polynomial Discrete Systems

414

• a r th -order oscillatory sink bifurcation ðd r xk þ 1 =dxrk jx ¼að1Þ [ 0, r ¼ k

2l1 þ 1Þ,

i1

then the following relations satisfy 1 ð2Þ ð2Þ ð1Þ ð2Þ ð2Þ ai1 ¼  Bi ; Di1 ¼ ðBi Þ2  4Ci1 ¼ 0; 2

ð5:256Þ

and there is a period-2 discrete system of the quartic discrete system in Eq. (5.247) as 1 þ ð2m þ 1Þ

xk þ 2 ¼ x k þ a0

*

ð20 Þ

i2 2Iq1

ð1Þ

ðxk  ai2 Þ3

ð5:257Þ

2

þ 1Þ ð2Þ

*ð2m ðxk  ai3 Þð1dði2 ;i3 ÞÞ i3 ¼1 ð20 Þ

for i1 2 Iq1  f1; 2; . . .; 2m þ 1g; i1 6¼ i2 with dxk þ 2 d 2 xk þ 2 jx ¼að1Þ ¼ 1; j  ð1Þ ¼ 0; dxk k i1 dx2k xk ¼ai1

ð5:258Þ

ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic sink of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a10 þ 2m dx3k xk ¼ai1

*

ð1Þ

ð20 Þ

i2 2Iq1 :i2 6¼i1

ð2m þ 1Þ2 ð1Þ *i ¼1 ðai1 3



ð1Þ

ðai1  ai2 Þ3

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ \

ð5:259Þ 0

and the corresponding bifurcations is a third-order monotonic sink bifurcation for the period-2 discrete system; ð1Þ

• xk þ 2 at xk ¼ ai1 is a monotonic source of the third-order if d 3 xk þ 2 j  ð1Þ ¼ 6a10 þ 2m dx3k xk ¼ai1

*

ð20 Þ

ð1Þ

i2 2Iq1 :i2 6¼i1

ð2m þ 1Þ2 ð1Þ *i ¼1 ðai1 3



ð1Þ

ðai1  ai2 Þ3

ð2Þ ai3 Þð1dði2 ;i3 ÞÞ

ð5:260Þ

[0

and the corresponding bifurcations is a third-order monotonic source bifurcation for the period-2 discrete system. (ii1) The period-2 fixed-points are trivial and unstable if

5.4 Forward Bifurcation Trees

415 ð2Þ

xk þ 2 ¼ xk ¼ ai

ð1Þ

þ1 2 02m i1 ¼1 fai1 g :

ð5:261Þ

(ii2) The period-2 fixed-points are non-trivial and stable if ð2Þ

xk þ 2 ¼ xk ¼ ai

ð2Þ

ð2Þ

2 2 0M i1 ¼1 fbi1 ;1 ; bi1 ;2 g :

ð5:262Þ

Proof The proof is straightforward through the simple algebraic manipulation. Following the proof of quadratic discrete system, this theorem is proved. ■

5.4.2

Renormalization and Period-Doubling

The generalized cases of period-doublization of a ð2m þ 1Þth -degree polynomial discrete system are presented through the following theorem. The analytical period-doubling bifurcation trees can be developed for such a ð2m þ 1Þth -degree polynomial discrete systems. Theorem 5.2 Consider a 1-dimensional ð2m þ 1Þth -degree polynomial discrete system as þ1 2 xk þ 1 ¼ xk þ A0 x2m þ A1 x2m k k þ    þ A2m1 xk þ A2m xk þ A2m þ 1

ð5:263Þ

þ1 ¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

(i) After l-times period-doubling bifurcations, a period- 2l discrete system ðl ¼ 1; 2; . . .Þ for the ð2m þ 1Þth -degree polynomial discrete system in Eq. (5.263) is given through the nonlinear renormalization as ð2l1 Þ

ð2m þ 1Þ2

xk þ 2l ¼ xk þ ½a0

f1 þ

2l1

ð2m þ 1Þ

*i ¼1 1

ð2l1 Þ

ð2

½ða0

Þ 4

Þ

ð2l1 Þ

ð2

½ða0

l1

2l1

ð2l Þ

¼ xk þ a0

ð2m þ 1Þ2

l1

ð2l1 Þ

ðxk  ai1

ð2l1 Þ

ðð2m þ 1Þ ð2m þ 1Þ

ð2m þ 1Þ2

l

l1

ð2m þ 1Þ2

*i¼1

l

Þg

Þ=2

ð2l Þ

ð2l Þ

ðx2k þ Bj2 xk þ Cj2 Þ

Þ

2l1

*i ¼1 2

ð2l1 Þ

ðxk  ai2

Þ

2l1

ðxk  ai1 2l

l1

*i ¼1;i 6¼i 2 2 1

ðð2m þ 1Þ ð2m þ 1Þ

Þ

*i¼1

Þ

*j ¼1 1

ð2l1 Þ 1 þ ð2m þ 1Þ2

¼ xk þ ða0

Þ

2l

*i ¼1; 1

Þ 4

Þ

2l1

ð2m þ 1Þ2

¼ xk þ ½a0

l1

ð2

*i ¼1 1 2l1

ð2l1 Þ

ðxk  ai1

½1 þ a0

ð2m þ 1Þ

¼ xk þ ½a0

l1

l1

*i ¼1 1

Þ=2

ð2l Þ

ð2l Þ

ðxk  bi2 ;1 Þðxk  bi2 ;2 Þ ð2l Þ

ðxk  ai Þ

ð2l Þ

ðxk  ai Þ ð5:264Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

416

with l l dxk þ 2l ð2l Þ Xð2m þ 1Þ2 ð2m þ 1Þ2 ð2l Þ ¼ 1 þ a0 *i ¼1;i 6¼i ðxk  ai2 Þ; i1 ¼1 2 2 1 dxk l l l d 2 xk þ 2 l ð2l Þ Xð2m þ 1Þ2 Xð2m þ 1Þ2 ð2m þ 1Þ2 ð2l Þ ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . l l l d r xk þ 2 l Xð2m þ 1Þ2 ð2l Þ Xð2m þ 1Þ2 ð2m þ 1Þ2 ð2l Þ ¼ a0    ir ¼1;i3 6¼i1 ;i2 ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ i1 ¼1 r dxk l

for r ð2m þ 1Þ2

ð5:265Þ where ð2l Þ

ð2Þ

a0 ¼ ða0 Þ1 þ ð2m þ 1Þ ; a0 2l

ð2l Þ

ð2l Þ Di l1

Iqð21

ð2 Þ

ð2m þ 1Þ

l1

l

; l ¼ 1; 2; 3;    ; ð2l Þ

ð2 Þ

ð2l Þ

ð2l Þ

2 fai g ¼ sortf0i1 ¼1 fai1 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ,ai qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l1 Þ ð2l Þ ¼  ðBi þ Di Þ; bi;2 ¼  ðBi  Di Þ;

0i¼1 bi;1

Þ

2l1

l

ð2m þ 1Þ

ð2l1 Þ 1 þ ð2m þ 1Þ2

¼ ða0

2

¼ Þ

ð2l Þ ðBi Þ2

ð2l Þ

ai þ 1 ;

2



ð2l Þ 4Ci

0 for i 2 0Nq11¼1 Iqð21

l1

Þ

l

00Nq22¼1 Iqð22 Þ ;

¼ flðq1 1Þ 2l1 m1 þ 1 ; lðq1 1Þ 2l1 m1 þ 2 ;    ; lq1 2l1 m1 g f1; 2;    ; M1 g0f∅g;

for q1 2 f1; 2;    ; N1 g; M1 ¼ N1 2l1 m1 ; m1 2 f1; 2;    ; mg; l

Iqð22 Þ ¼ flðq2 1Þ 2l m1 þ 1 ; lðq2 1Þ 2l m1 þ 2 ;    ; lq2 2l m1 g fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; l

l1

for q2 2 f1; 2;    ; N2 g; M2 ¼ ðð2m þ 1Þ2  ð2m þ 1Þ2 Þ=2; qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l Þ 1 ð2l Þ ð2l Þ ð2l Þ ð2l Þ ð2l Þ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ð2l Þ ð2l Þ ð2l Þ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1; l

i 2 J ð2 Þ ¼ flN 2l m1 þ 1 ; lN 2l m1 þ 2 ;    ; lM2 g  fM1 þ 1; M1 þ 2;    ; M2 g0f∅g; ð5:266Þ with fixed-points

5.4 Forward Bifurcation Trees

417

ð2l Þ

l

xk þ 2l ¼ xk ¼ ai ; ði ¼ 1; 2;    ; ð2m þ 1Þ2 Þ ð2m þ 1Þ2

0i¼1

l

ð2 Þ

l

ð2l Þ

ð2m þ 1Þ2

fai g ¼ sortf0i1 ¼1

l1

ð2l1 Þ

fai1

ð2l Þ

ð2l Þ

2 g; 0M i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð5:267Þ

l

ð2 Þ

with ai \ ai þ 1 : ð2l1 Þ

ð2l1 Þ

(ii) For a fixed-point of xk þ 2l1 ¼ xk ¼ ai1

ði1 2 Iq

Þ, if

l1 dxk þ 2l1 ð2l1 Þ ð2m þ 1Þ2 ð2l1 Þ ð2l1 Þ j  ð2l1 Þ ¼ 1 þ a0  ai2 Þ ¼ 1; *i ¼1;i 6¼i ðai1 2 2 1 x ¼a dxk i1 k d s xk þ 2l1 j ð2l1 Þ ¼ 0; for s ¼ 2;    ; r  1; dxsk xk ¼ai1 l1 d r xk þ 2l1 j  ð2l1 Þ 6¼ 0 for 1\r ð2m þ 1Þ2 ; xk ¼ai dxrk 1

ð5:268Þ

with

 • a r th -order oscillatory sink for d r xk þ 2l1 =dxrk 

ð2l1 Þ

xk ¼ai

 • a r th -order oscillatory source for d r xk þ 2l1 =dxrk 

[ 0 and r ¼ 2l1 þ 1;

1

ð2l1 Þ

xk ¼ai

\0 and r ¼

1

2l1 þ 1;  • a r th -order oscillatory upper-saddle for d r xk þ 2l1 =dxrk 

ð2l1 Þ

xk ¼ai

r ¼ 2l1 ;  • a r th -order oscillatory lower-saddle for d r xk þ 2l1 =dxrk 

[ 0 and

1

ð2l1 Þ

xk ¼ai

\0 and

1

r ¼ 2l1 ; then there is a period- 2l fixed-point discrete system ð2l Þ

x k þ 2 l ¼ x k þ a0

*

ð2m þ 1Þ2 *j ¼1 2

l

ð2l1 Þ i1 2Iq1

ðxk 

ð2l1 Þ 3

ðxk  ai1

Þ

ð5:269Þ

ð2l Þ aj2 Þð1dði1 ;j2 ÞÞ

where ð2l Þ

ð2l1 Þ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1

ð2l Þ

ð2l1 Þ

; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1

ð5:270Þ

and dxk þ 2l d 2 xk þ 2l j  ð2l1 Þ ¼ 1; j ð2l1 Þ ¼ 0: dxk xk ¼ai1 dx2k xk ¼ai1

ð5:271Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

418 ð2l1 Þ

• xk þ 2i at xk ¼ ai

is a monotonic sink of the third-order if

d 3 xk þ 2 l ð2l Þ j ð2l1 Þ ¼ 6a0 dx3k xk ¼ai1

ði1 2 Iqð21

l1

Þ

*

ð2l1 Þ

i2 2Iq1

ð2m þ 1Þ2

l

*j ¼1 2

ð2l1 Þ

;i2 6¼i1

ð2l1 Þ

ðai1

ð2l1 Þ

ð2l1 Þ

l1

Þ

Þ

ð5:272Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ \0

is a third-order monotonic sink

is a monotonic source of the third-order if

d 3 xk þ 2l ð2l Þ j  ð2l1 Þ ¼ 6a0 3 x ¼a dxk i1 k ði1 2 Iqð21

ð2l1 Þ 3

 ai 2

; q1 2 f1; 2;    ; N1 gÞ;

and such a bifurcation at xk ¼ ai bifurcation. • xk þ 2l at xk ¼ ai

ðai1

*

ð2l1 Þ i2 2Iq1 ;i2 6¼i1

ð2m þ 1Þ2

*j ¼1 2

l

ð2l1 Þ

ðai1

ð2l1 Þ

ðai1

ð2l1 Þ 3

 ai2

Þ

ð5:273Þ

ð2l Þ

 aj2 Þð1dði2 ;j2 ÞÞ [ 0

; q1 2 f1; 2;    ; N1 gÞ ð2l1 Þ

and such a bifurcation at xk ¼ ai1

is a third-order monotonic source bifurcation.

ðii1 Þ The period- 2l fixed-points are trivial if ð2l1 Þ

xk þ 2l ¼ xk ¼ ai

ð2m þ 1Þ2

2 0i1 ¼1

l1

ð2l1 Þ

fai1

g:

ð5:274Þ

ðii2 Þ The period- 2l fixed-points are non-trivial if ð2l1 Þ

xk þ 2l ¼ xk ¼ ai

ð2l Þ

ð2l Þ

2 2 0M i1 ¼1 fbj1 ;1 ; bj1 ;2 g:

ð5:275Þ

Such a period- 2l fixed-point is • monotonically unstable if dxk þ 2l =dxk j

ð2l Þ

xk ¼ai

• monotonically invariant if dxk þ 2l =dxk j

1 ð2l Þ

xk ¼ai

2 ð1; 1Þ; ¼ 1, which is

1

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx [ 0; k

1 – a monotonic lower-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jx \0; k

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx [ 0; k

– a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \0; k

5.4 Forward Bifurcation Trees

419

• monotonically stable if dxk þ 2l =dxk j

• invariantly zero-stable if dxk þ 2l =dxk j • oscillatorilly stable if dxk þ 2l =dxk j • flipped if dxk þ 2l =dxk j

ð2l1 Þ

xk ¼ai

2 ð0; 1Þ;

ð2l Þ

xk ¼ai

1 ð2l1 Þ

xk ¼ai

1

ð2l1 Þ

xk ¼ai

¼ 0;

2 ð1; 0Þ;

1

¼ 1, which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l k jxk [ 0;  – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l =dx2l1   \0

k

xi

– an oscillatory source of the ð2l1 þ 1Þth order if d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx \0; k

– an oscillatory sink the ð2l1 þ 1Þth order with d 2l1 þ 1 xk þ 2l =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ 2l =dxk j

2 ð1; 1Þ.

ð2l Þ

xk ¼ai

1

Proof Through the nonlinear renormalization, this theorem can be proved.

5.4.3

■

Period-n Appearing and Period-Doublization

The forward period-n discrete system for the quartic nonlinear discrete systems will be discussed, and the period-doublization of period-n discrete systems is discussed through the nonlinear renormalization. Theorem 5.3 Consider a 1-dimensional ð2m þ 1Þth -degree polynomial discrete system as þ1 2 þ A1 x2m xk þ 1 ¼ xk þ A0 x2m k k þ    þ A2m1 xk þ A2m xk þ A2m þ 1 þ1 ¼ xk þ a0 *2m i¼1 ðxk  ai Þ:

ð5:276Þ

(i) After n-times iterations, a period-n discrete system for the quartic discrete system in Eq. (5.276) is xk þ n ¼ x k þ a0

2m þ 1

*i ¼1 1

ð1Þ

ðxk  ai1 Þf1 þ Rnj¼1 Qj g

ðð2m þ 1Þn 1Þ=ð2mÞ

¼ xk þ a0

2m þ 1

*i ¼1 1

n

ð1Þ

ðxk  ai1 Þ

þ 1Þ ð2m þ 1ÞÞ=2 2 ðnÞ ðnÞ

½ *jðð2m ðxk þ Bj2 xk þ Cj2 Þ 2 ¼1 ðnÞ

¼ x k þ a0 with

ð2m þ 1Þn

*i¼1

ðnÞ

ðxk  ai Þ

ð5:277Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

420

dxk þ n ðnÞ Xð2m þ 1Þn ð2m þ 1Þn ðnÞ ¼ 1 þ a0 i1 ¼1 *i2 ¼1;i2 6¼i1 ðxk  ai2 Þ; dxk d 2 xk þ n ðnÞ Xð2m þ 1Þn Xð2m þ 1Þn ð2m þ 1Þn ðnÞ ¼ a0 i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 Þ; 2 dxk .. . d r xk þ n Xð2m þ 1Þn ðnÞ Xð2m þ 1Þn ð2m þ 1Þn ðnÞ ¼ a0 i1 ¼1    ir ¼1;ir 6¼i1 ;i2 ;ir1 *ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ r dxk

for r ð2m þ 1Þn ;

ð5:278Þ where n

ðnÞ

a0 ¼ ða0 Þðð2m þ 1Þ 1Þ=ð2mÞ ; Q1 ¼ 0; Q2 ¼ Qn ¼

2m þ 1

*i ¼1 n

ð2m þ 1Þn

ðnÞ

ðnÞ

2m þ 1 ðxk n1 ¼1;in1 6¼in

*i

ð1Þ

½1 þ a0

2m þ 1

*i ¼1;i 6¼i ðxk 1 1 2

ð1Þ

 ai1 Þ;

ð1Þ

 ain1 Þ; n ¼ 3; 4;    ;

ðnÞ

ðnÞ

þ1 M fai g ¼ sortf02m i1 ¼1 fai1 g; 0i2 ¼1 fbi2 ;1 ; bi2 ;2 gg ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ¼  ðBi2 þ Di2 Þ; bi2 ;2 ¼  ðBi2  Di2 Þ;

0i¼1 bi2 ;1

½1 þ a0 ð1 þ Qn1 Þ

2m þ 1

*i ¼1 2

ðnÞ

2

2

ðnÞ

ðnÞ

Di2 ¼ ðBi2 Þ2  4Ci2 0 for i2 2 0Nq¼1 IqðnÞ ; IqðnÞ ¼ flðq1Þ n þ 1 ; lðq1Þ n þ 2 ;    ; lq n gf1; 2;    ; Mg0f∅g; for q 2 f1; 2;    ; Ng; M ¼ ðð2m þ 1Þn  ð2m þ 1ÞÞ=2; qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 1 ðnÞ 1 ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ bi;1 ¼  ðBi þ i jDi jÞ; bi;2 ¼  ðBi  i jDi jÞ; 2 2 pffiffiffiffiffiffiffi ðnÞ ðnÞ ðnÞ Di ¼ ðBi Þ2  4Ci \0; i ¼ 1 i 2 flN n þ 1 ; lN n þ 2 ;    ; lM g  f1; 2;    ; Mg0f∅g; ð5:279Þ

with fixed-points ðnÞ

xk þ n ¼ xk ¼ ai ; ði ¼ 1; 2; . . .; ð2m þ 1Þn Þ ð2m þ 1Þn

0i¼1

ðnÞ

ðnÞ

ð1Þ

ðnÞ

ðnÞ

þ1 M fai g ¼ sortf02m i1 ¼1 fai1 g; 0i2 ¼1 fbi2 ;1 ; bi2 ;2 gg

ð5:280Þ

ðnÞ

with ai \ai þ 1 : ðnÞ

ðnÞ

(ii) For a fixed-point of xk þ n ¼ xk ¼ ai1 (i1 2 Iq , q 2 f1; 2; . . .; Ng), if dxk þ n ðnÞ ð2m þ 1Þn ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *i2 ¼1;i2 6¼i1 ðai1  ai2 Þ ¼ 1; dxk k i1 with

ð5:281Þ

5.4 Forward Bifurcation Trees

421

d 2 xk þ n ðnÞ Xð2m þ 1Þn ð2m þ 1Þn ðnÞ ðnÞ jx ¼aðnÞ ¼ a0 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðai1  ai3 Þ 6¼ 0; 2 i k dxk 1

ð5:282Þ

then there is a new discrete system for onset of the qth-set of period-n fixed-points based on the second-order monotonic saddle-node bifurcation as ðnÞ

x k þ n ¼ x k þ a0

*i 2I ðnÞ ðxk 1 q

ð2m þ 1Þn

ðnÞ

 ai1 Þ2 *j2 ¼1

ðnÞ

ðxk  aj2 Þð1dði1 ;j2 ÞÞ

ð5:283Þ

where ðnÞ

ðnÞ

ðnÞ

ðnÞ

dði1 ; j2 Þ ¼ 1 if aj2 ¼ ai1 ; dði1 ; j2 Þ ¼ 0 if aj2 6¼ ai1 :

ð5:284Þ

(ii1) If dxk þ n j  ðnÞ ¼ 1 ði1 2 IqðnÞ Þ; dxk xk ¼ai1 d 2 xk þ n ðnÞ ðnÞ 2 j  ðnÞ ¼ 2a0 *i 2I ðnÞ ;i 6¼i ðaðnÞ i1  ai2 Þ 2 q 2 1 dx2k xk ¼ai1

ð2m þ 1Þn

*j ¼1 2

ðnÞ

ð5:285Þ

ðnÞ

ðai1  aj2 Þð1dði2 ;j2 ÞÞ 6¼ 0

ðnÞ

xk þ n at xk ¼ ai1 is – a monotonic lower-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ \0; k

i1

– a monotonic upper-saddle of the second-order for d 2 xk þ n =dx2k jx ¼aðnÞ [ 0. k

i1

n1

(ii2) The period-n fixed-points ðn ¼ 2 sÞ are trivial n o n1 1 s ðnÞ ð2m þ 1Þ2 ð2n1 1 sÞ þ 1 ð1Þ xk ¼ xk þ n ¼ aj1 2 02m fa g; 0 fa g i1 i2 i2 ¼1 ii ¼1 for n1 ¼ 1; 2; . . .; s ¼ 2l1 þ 1; j1 2 f1; 2; . . .; ð2m þ 1Þn g0f£g for n 6¼ 2n2 ; ðnÞ

ð2m þ 1Þ2

xk ¼ xk þ n ¼ aj1 2 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

g

9 = ;

)

for n1 ¼ 1; 2; . . .; s ¼ 1; j1 2 f1; 2; . . .; ð2m þ 1Þn g0f£g for n ¼ 2n2 : ð5:286Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

422

ðii3 Þ The period-n fixed-points ðn ¼ 2n1 sÞ are non-trivial if ðnÞ

ð2m þ 1Þ2

ð1Þ

þ1 xk ¼ xk þ n ¼ aj1 62 f02m ii ¼1 fai1 g; 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

gg

)

for n1 ¼ 1; 2; . . .; s ¼ 2l1 þ 1; j1 2 f1; 2; . . .; ð2mÞn g0f£g for n 6¼ 2n2 ; ð2m þ 1Þ2

ðnÞ

xk ¼ xk þ n ¼ aj1 62 0i2 ¼1

n1 1 s

ð2n1 1 sÞ

fai2

g

ð5:287Þ

)

for n1 ¼ 1; 2; . . .; s ¼ 1; j1 2 f1; 2; . . .; ð2m þ 1Þn g0f£g

for n ¼ 2n2 : Such a forward period- n fixed-point is • monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ; i1

k

• monotonically invariant if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is i1

k

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx [ 0; k

1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \ 0; k

– a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx [ 0; k

– a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx \ 0; k

• monotonically unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð0; 1Þ; i1

k

• invariantly zero-stable if dxk þ n =dxk jx ¼aðnÞ ¼ 0; k

i1

• oscillatorilly stable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 0Þ; k

i1

• flipped if dxk þ n =dxk jx ¼aðnÞ ¼ 1, which is k

i1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx [ 0; k

1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ n =dx2l k jx \0; k

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx \ 0; k

– an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ n =dxk2l1 þ 1 jx [ 0; k

• oscillatorilly unstable if dxk þ n =dxk jx ¼aðnÞ 2 ð1; 1Þ. k

ðnÞ

i1

ðnÞ

For a fixed-point of xk þ n ¼ xk ¼ ai1 ði1 2 Iq , q 2 f1; 2; . . .; NgÞ, there is a period-doubling of the qth -set of period-n fixed-points if dxk þ n ðnÞ ð2mÞn ðnÞ ðnÞ jx ¼aðnÞ ¼ 1 þ a0 *j2 ¼1;j2 6¼i1 ðai1  aj2 Þ ¼ 1; dxk k i1 d s xk þ n j  ðnÞ ¼ 0; for s ¼ 2;    ; r  1; dxsk xk ¼ai1 d r xk þ n j  ðnÞ 6¼ 0 for 1\r ð2mÞn dxrk xk ¼ai1

ð5:288Þ

5.4 Forward Bifurcation Trees

423

with • a r th -order oscillatory sink for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ 2l1 þ 1; i1

k

• a r th -order oscillatory source for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ 2l1 þ 1; i1

k

• a r th -order oscillatory upper-saddle for d r xk þ n =dxrk jx ¼aðnÞ [ 0 and r ¼ 2l1 ; i1

k

• a r th -order oscillatory lower-saddle for d r xk þ n =dxrk jx ¼aðnÞ \0 and r ¼ 2l1 . i1

k

The corresponding period- 2 n discrete system of the ð2m þ 1Þth -degree polynomial discrete system in Eq. (5.276) is ð2 nÞ

xk þ 2 n ¼ xk þ a0

*i 2I ðnÞ ðxk 1 q

ð2m þ 1Þ2 n

ðnÞ

 ai1 Þ3 *j2 ¼1

ð2 nÞ ð1dði1 ;j2 ÞÞ

ðxk  aj2

Þ

ð5:289Þ with dxk þ 2 n d 2 xk þ 2 n jx ¼aðnÞ ¼ 1; jx ¼aðnÞ ¼ 0; i1 i1 k k dxk dx2k d 3 xk þ 2 n ð2 nÞ ðnÞ ðnÞ 3 jx ¼aðnÞ ¼ 6a0 *i 2I ðnÞ ;i 6¼i ðai1  ai2 Þ 1 q 2 1 i1 k dx3k ð2m þ 1Þ2 n

*j2 ¼1 ðnÞ

ðnÞ

ð5:290Þ

ð2 nÞ ð1dði1 ;j2 ÞÞ

ðai1  aj2

Þ

:

ðnÞ

Thus, xk þ 2 n at xk ¼ ai1 for i1 2 Iq , q 2 f1; 2; . . .; Ng is • a monotonic sink of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ \0, k

i1

• a monotonic source of the third-order if d 3 xk þ 2 n =dx3k jx ¼aðnÞ [ 0. k

i1

(iv) After l-times period-doubling bifurcations of period-n fixed points, a period2l n discrete system of the ð2m þ 1Þth degree polynomial discrete system in Eq. (5.276) is ð2l1 nÞ

xk þ 2l n ¼ xk þ ½a0

f1 þ

ð2m þ 1Þ

ð2l1 nÞ

ð2

2l1 n

*i1 ¼1

¼ xk þ ½a0

½ða0

ð2m þ 1Þ2

l1

ð2l1 nÞ

ðxk  ai1 ð2l1 Þ

½1 þ a0

ð2m þ 1Þ

2l1 n

*i1 ¼1

nÞ ð2m þ 1Þ

Þ

l1 n

*i1 ¼1

ð2l1 nÞ

ð2m þ 1Þ

*i ¼1;i 6¼i 2 2 1 ð2l1 nÞ

ðxk  ai1 ðð2m þ 1Þ

*j1 ¼1

Þ

2l1 n

2l n

ð2l1 nÞ

ðxk  ai2

Þg

Þ

ð2m þ 1Þ2

l1 n

Þ=2

ð2l nÞ

ðx2k þ Bj2

ð2l nÞ

xk þ Cj2

Þ

5 (2m + 1)th-Degree Polynomial Discrete Systems

424 ð2l1 nÞ

¼ xk þ ½a0 ð2

½ða0

l1

ð2m þ 1Þ2

nÞ ð2m þ 1Þ

Þ

ð2l nÞ

ð2m þ 1Þ

ð2m þ 1Þ2 n

*i¼1

ð2l nÞ

ðxk  ai

Þ

ð2m þ 1Þ2

l

l1 n

2l n

*i¼1

ðð2m þ 1Þ

*j2 ¼1

Þ

¼ xk þ a 0

ð2l1 nÞ

ðxk  ai1

2l n

2l1 n

ð2l1 nÞ ð2m þ 1Þ2

¼ xk þ ða0

l1 n

*i1 ¼1

l1 n

Þ=2

ð2l nÞ

ðxk  ai

ð2l nÞ

ð2l nÞ

ðxk  bj2 ;1 Þðxk  bj2 ;2 Þ

Þ

Þ ð5:291Þ

with l l dxk þ 2l n ð2l nÞ Xð2m þ 1Þ2 n ð2m þ 1Þ2 n ð2l nÞ ¼ 1 þ a0 Þ; *i ¼1;i 6¼i ðxk  ai2 i1 ¼1 2 2 1 dxk l l l d 2 xk þ 2l n ð2l nÞ Xð2m þ 1Þ2 n Xð2m þ 1Þ2 n ð2m þ 1Þ2 n ð2l nÞ ¼ a0 Þ; i1 ¼1 i2 ¼1;i2 6¼i1 *i3 ¼1;i3 6¼i1 ;i2 ðxk  ai3 2 dxk .. . l l l d r xk þ 2l n Xð2m þ 1Þ2 n ð2l nÞ Xð2m þ 1Þ2 n ð2m þ 1Þ2 n ð2l nÞ ¼ a    * i1 ¼1 0 ir ¼1;ir 6¼i1 ;i2 ;ir1 ir þ 1 ¼1;ir þ 1 6¼i1 ;i2 ;ir ðxk  air þ 1 Þ r dxk l

for r ð2m þ 1Þ2 n ; ð5:292Þ where ð2 nÞ

ðnÞ

2 n

ð2l nÞ

2l1 n

ð2l1 nÞ

¼ ða0 Þ1 þ ð2m þ 1Þ ; a0 ¼ ða0 Þ1 þ ð2m þ 1Þ ; l ¼ 1; 2; 3; . . .; 2l n 2l1 n     l l1 ð2m þ 1Þ ð2 nÞ ð2m þ 1Þ ð2 nÞ ð2l nÞ ð2l nÞ  2 fai g ¼ sort 0i1 ¼1 0i¼1 ai1 ; 0M g; i2 ¼1 bi2 ;1 ; bi2 ;2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ Di Þ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  Di Þ; 2

a0

ð2l nÞ

Di

l1

Iqð21

ð2l nÞ 2

¼ ðBi

nÞ

ð2l nÞ

Þ  4Ci

l

0 ; for i 2 0Nq22¼1 Iqð22 nÞ

¼ flðq1 1Þ ð2l1 nÞ þ 1 ; lðq1 1Þ ð2l1 nÞ þ 2 ;    ; lq1 ð2l1 nÞ g f1; 2;    ; M1 g0f£g;

5.4 Forward Bifurcation Trees

425

for q1 2 f1; 2;    ; N1 g; M1 ¼ N1 ð2l1 nÞ; l

Iqð22 nÞ ¼ flðq2 1Þ ð2l nÞ þ 1 ; lðq2 1Þ ð2l nÞ þ 2 ;    ; lq2 ð2l1 nÞ g fM1 þ 1; M1 þ 2;    ; M2 g0f£g; l

l1

for q2 2 f1; 2;    ; N2 g; M2 ¼ ðð2m þ 1Þ2 n  ð2m þ 1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ ¼  ðBi þ i jDi jÞ; bi;1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2l nÞ ð2l nÞ ð2l nÞ bi;2 ¼  ðBi  i jDi jÞ; 2 pffiffiffiffiffiffiffi ð2l nÞ ð2l nÞ 2 ð2l nÞ ¼ ðBi Þ  4Ci \0; i ¼ 1; Di

n

Þ=2;

i 2 flN2 ð2l nÞ þ 1 ; lN2 ð2l nÞ þ 2 ;    ; lM2 g  f1; 2;    ; M2 g0f£g

ð5:293Þ

with fixed-points ð2l nÞ

l

xk þ 2l n ¼ xk ¼ ai ; tði ¼ 1; 2; . . .; ð2m þ 1Þ2 n Þ l l1  ð2l nÞ ð2l nÞ  ð2m þ 1Þ2 n  ð2l nÞ  ð2m þ 1Þ2 n ð211 nÞ  2 fai1 0i¼1 ai ¼ sortf0i1 ¼1 ; 0M g i2 ¼1 bi2 ;1 ; bi2 ;2 ð2i nÞ

with ai

ð2l nÞ \ai þ 1 ð5:294Þ ð2l1 nÞ

(v) For a fixed-point of xk þ ð2l nÞ ¼ xk ¼ ai1

ð2l1 nÞ

ði1 2 Iq

; q 2 f1; 2; . . .; N1 gÞ,

there is a period- 2l1 n discrete system if l1 dxk þ 2l1 n ð2l1 nÞ ð2m þ 1Þ2 n ð2l1 nÞ ð2l1 nÞ j  ð2l1 nÞ ¼ 1 þ a0 ðai1  ai2 Þ ¼ 1; *i ¼1;i 6¼i 2 2 1 xk ¼ai dxk 1 d s xk þ 2l1 n j  ð2l1 nÞ ¼ 0; for s ¼ 2;    ; r  1; xk ¼ai dxsk 1 r l1 d xk þ 2l1 n j  ð2l1 nÞ 6¼ 0 for 1\r ð2m þ 1Þ2 n r x ¼a dxk i1 k

ð5:295Þ with • a r th -order oscillatory sink for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

[ 0 and r ¼ 2l1 þ 1;

1

• a r th -order oscillatory source for d r xk þ 2l n =dxrk j

ð2l nÞ

xk ¼ai

2l1 þ 1; • a r th -order oscillatory upper-saddle for d r xk þ 2l n =dxrk j r ¼ 2l1 ;

\0 and r ¼

1

ð2l nÞ

xk ¼ai

1

[ 0 and

5 (2m + 1)th-Degree Polynomial Discrete Systems

426

• a r th -order oscillatory lower-saddle for d r xk þ 2l n =dxrk j r ¼ 2l1 .

ð2l nÞ

xk ¼ai

\0 and

1

The corresponding period- 2l n discrete system is ð2l nÞ

xk þ 2l n ¼ xk þ a0

*

ð2l1 nÞ i1 2Iq

l

ð2m þ 1Þ2 n ðxk *j2 ¼1



ð2l1 nÞ 3

ðxk  ai1

Þ

ð5:296Þ

ð2l nÞ ð1dði1 ;j2 ÞÞ aj2 Þ

where ð2l nÞ

dði1 ; j2 Þ ¼ 1 if aj2

ð2l1 nÞ

¼ ai1

ð2l nÞ

; dði1 ; j2 Þ ¼ 0 if aj2

ð2l1 nÞ

6¼ ai1

ð5:297Þ

with dxk þ 2l n d 2 xk þ 2l n j  ð2l1 nÞ ¼ 1; j  ð2l1 nÞ ¼ 0; x ¼a xk ¼ai dxk dx2k i1 k 1 d 3 xk þ 2l n ð2l nÞ ð2l1 nÞ ð2l1 nÞ 3 j  ð2l1 Þ ¼ 6a0 ðai1  ai 2 Þ * ð2l1 nÞ 3 xk ¼ai i2 2Iq ;i2 6¼i1 dxk 1 ð2l nÞ

ð2m þ 1Þ

*j2 ¼1 ði1 2 Iqð2

l1

nÞ

l1

ð2

ðai1

nÞ

l

ð2 nÞ ð1dði2 ;j2 ÞÞ

 aj2

Þ

ð5:298Þ

6¼ 0

; q 2 f1; 2; . . .; N1 gÞ ð2l1 nÞ

Thus, xk þ 2l n at xk ¼ ai1

is

• a monotonic sink of the third-order if d 3 xk þ 2l n =dx3k j

ð2l1 Þ

xk ¼ai

• a monotonic source of the third-order if d 3 xk þ 2l n =dx3k j

\0;

1 ð2l1 Þ

xk ¼ai

[ 0.

1

(v1) The period- 2l n fixed-points are trivial if ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

ð1Þ

ð2m þ 1Þ2

þ1 2 f02m ii ¼1 fai1 g; 0i2 ¼1

l1 n

l

for j ¼ 1; 2;    ; ð2m þ 1Þð2 nÞ for n 6¼ 2n1 ð2l nÞ

xk þ 2l n ¼ xk ¼ aj

ð2m þ 1Þ2

2 f0i2 ¼1 l

for j ¼ 1; 2;    ; ð2m þ 1Þ2 n

l1 n

ð2l1 nÞ

fai2

g

ð2l1 nÞ

fai2

gg

9 = ;

9 = ;

for n ¼ 2n1 : ð5:299Þ

5.4 Forward Bifurcation Trees

427

(v2) The period- 2l n fixed-points are non-trivial if xk þ 2l n

¼

xk

¼

ð2l nÞ aj

62

ð1Þ ð2mþ 1Þ2 þ1 f02m ii ¼1 fai1 g; 0i2 ¼1

l1 n

9

= ð2l1 nÞ fai2 gg ;

l

for j ¼ 1; 2;    ; ð2m þ 1Þ2 n for n 6¼ 2n1 xk þ 2l n

¼

xk

¼

ð2l nÞ aj

62

ð2mþ 1Þ2 f0i2 ¼1

l1 n

9

ð2l1 nÞ = fai2 g

;

l

for j ¼ 1; 2;    ; ð2m þ 1Þ2 n for n ¼ 2n1 :

ð5:300Þ Such a period- 2l n fixed-point is • monotonically unstable if dxk þ 2l n =dxk j

2 ð1; 1Þ;

ð2l nÞ

xk ¼ai

• monotonically invariant if dxk þ 2l n =dxk j

1 ð2l nÞ

xk ¼ai

¼ 1, which is

1

1 – a monotonic upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk [ 0 (independent ð2l1 Þ-branch appearance); 1 – a monotonic lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jxk \0 (independent ð2l1 Þ-branch appearance) – a monotonic source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jx [ 0 k (dependent ð2l1 þ 1Þ-branch appearance from one branch); – a monotonic sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jx \0 k (dependent ð2l1 þ 1Þ-branch appearance from one branch);

• monotonically stable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

• invariantly zero-stable if dxk þ 2l n =dxk j • oscillatorilly stable if dxk þ 2l n =dxk j • flipped if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

2 ð0; 1Þ;

1

¼ 0;

ð2l nÞ

xk ¼ai ð2l nÞ

xk ¼ai

1

2 ð1; 0Þ;

1

¼ 1, which is

1

1 – an oscillatory upper-saddle of the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jx [ 0; k

1 – an oscillatory lower-saddle the ð2l1 Þth order for d 2l1 xk þ 2l n =dx2l k jx \0; k

– an oscillatory source of the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dxk2l1 þ 1 jx \0;  k – an oscillatory sink the ð2l1 þ 1Þth order for d 2l1 þ 1 xk þ 2l n =dx2l1 þ 1   [ 0; k

• oscillatorilly unstable if dxk þ 2l n =dxk j

ð2l nÞ

xk ¼ai

1

2 ð1; 1Þ.

xi

428

5 (2m + 1)th-Degree Polynomial Discrete Systems

Proof Through the nonlinear renormalization, the proof of this theorem is similar to the proof of Theorem 1.11. This theorem can be easily proved. ■

References Luo ACJ (2020a) The stability and bifurcation of the (2m + 1)th-degree polynomial systems. J Vibr Test Syst Dynam 4(2):93–144 Luo ACJ (2020b) Bifurcation and stability in nonlinear dynamical system. Springer, New York

Index

A Antenna switching bifurcation, 284, 307, 369, 401 Appearing bifurcation, 9, 11, 276, 294, 346, 352, 354 B Backward bifurcation tree, 75 Backward cubic nonlinear discrete system, 150 Backward period-1 appearing bifurcation, 28 Backward period-1 switching bifurcation, 39 Backward period-2 quadratic discrete system, 75 Backward period-2 quartic discrete system, 240 Backward period-3 cubic discrete system, 148 Backward period-doubling renormalization, 79, 153, 243 Backward period-n appearing, 82, 157, 247 Backward period-n bifurcation tree, 92 Backward quadratic discrete system, 28 Backward quartic discrete system, 239 Broom appearing bifurcation, 355, 357, 380, 381 Broom-sprinkle-spraying appearing bifurca-tion, 358–360, 386, 388, 391 C Constant adding discrete system, 2, 5 Cubic nonlinear discrete system, 93 D (2m)th -degree polynomial discrete system, 258

F Flower-bundle switching bifurcation, 290–292, 316, 376, 377, 409 Forward bifurcation tree, 44, 318 Forward cubic discrete system, 121 Forward quadratic discrere system, 7 Forward quartic discrete system, 223 I Instant fixed-point, 2 Invariant sink, 2 L Linear backward discrete system, 5 Linear discrete system, 1 lpth mXX appearing bifurcation, 261, 268, 274, 339 lpth mXX switching bifurcation, 260, 261, 263, 266, 268, 270, 273–275, 338, 339, 341, 345, 348, 351–354 M Monotonically stable node, 2, 6 Monotonically unstable node, 3, 6 Monotonic backward saddle discrete flow, 29 Monotonic lower-saddle, 9, 18, 25, 30, 40, 100, 101, 103 Monotonic lower-saddle discrete flow, 119, 169, 170, 192, 212, 217, 219, 222 Monotonic lower-saddle-node appearing bifurcation, 9, 20, 103–105, 169, 171, 173, 194, 196, 206

© Higher Education Press 2020 A. C. J Luo, Bifurcation Dynamics in Polynomial Discrete Systems, Nonlinear Physical Science, https://doi.org/10.1007/978-981-15-5208-3

429

430 Monotonic lower-saddle-node bundle-switching bifurcation, 205, 214, 219, 221 Monotonic lower-saddle-node flower-bundle-switching bifurcation, 208 Monotonic lower-saddle-node switching bifurcation, 7, 18, 26, 27, 43, 100, 101, 170, 173, 195, 212, 217 Monotonic saddle, 22, 40 Monotonic saddle discrete flow, 9, 95–97 Monotonic saddle-node appearing bifurcation, 9, 18 Monotonic saddle-node switching bifurcation, 22, 25, 43 Monotonic saddle switching, 3, 6 Monotonic sink, 2, 6, 10 Monotonic sink bundle-switching bifurcation of the third-order, 213 Monotonic sink discrete flow, 96, 97, 171, 213, 218 Monotonic sink switching bifurcation of the third-order, 171, 193, 194, 207, 218 Monotonic source, 3, 6, 10 Monotonic source bundle-switching bifurcation of the third-order, 213 Monotonic source discrete flow, 96, 97, 171, 213, 218 Monotonic source switching bifurcation of the third-order, 171, 193, 207, 215 Monotonic upper-saddle, 9, 18, 24, 29, 100–102, 106 Monotonic upper-saddle discrete flow, 119, 169, 170, 173, 192, 212, 217, 219 Monotonic upper-saddle-node appearing bifurcation, 9, 20, 30, 38, 102, 106, 107, 169, 171, 173, 194, 196, 206 Monotonic upper-saddle-node bundle-switching bifurcation, 205, 214, 215 Monotonic upper-saddle-node flower-bundle-switching bifurcation, 194, 208 Monotonic upper-saddle-node switching bifurcation, 18, 26, 27, 101, 170, 173, 194 N Negative backward discrete flow, 28 Negative discrete flow, 8, 9, 18, 168, 195, 258

Index O Oscillatorilly stable node, 2, 6 Oscillatorilly unstable node, 3, 6 Oscillatory lower-saddle, 10, 19, 22 Oscillatory saddle switching, 3, 6 Oscillatory sink, 2, 6 Oscillatory source, 3, 6 Oscillatory upper-saddle, 10, 19 P Period-1 appearing bifurcation, 7, 167 Period-1 cubic discrete system, 93 Period-1 quartic discrete system, 224 Period-1 switching bifurcation, 21 Period-2 appearing bifurcation, 44 Period-doubled cubic discrete system, 121 Period-doubling renormalization, 53, 128, 227 Period-doublization, 62, 82, 138, 157, 231, 326 Period-n appearing, 62, 138, 231, 326, 419 Period-n appearing bifurcation, 147 Period-n bifurcation tree, 71 Permanent invariant discrete system, 2, 5 Positive backward discrete flow, 28 Positive discrete flow, 8, 9, 18, 168, 195, 258 Q Quadratic nonlinear discrete system, 1 Quartic nonlinear discrete system, 167 S Sink, 2, 6 Source, 2 Spraying appearing bifurcation, 278, 279, 281, 296, 299, 301 Sprinkler-spraying appearing bifurcation, 278, 279, 281, 299, 301 Stable node, 2, 6 Straw-bundle switching bifurcation, 286, 310, 315, 371, 404, 408 Switching-appearing bifurcation, 289, 311, 374, 407 Switching bifurcation, 21, 24, 39, 95, 114, 282, 305, 369, 398 T Teethcomb appearing bifurcation, 277, 296 U Unstable node, 2