2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach (World Scientific Series on Nonlinear Science: Series A 73)

2-D Quadratic Maps and 3-D ODE Systems A Rigorous Approach

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WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

MONOGRAPHS AND TREATISES*

Volume 57:

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science L. O. Chua

Volume 58:

New Methods for Chaotic Dynamics N. A. Magnitskii & S. V. Sidorov

Volume 59:

Equations of Phase-Locked Loops J. Kudrewicz & S. Wasowicz

Volume 60:

Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods J. Awrejcewicz & M. M. Holicke

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A Gallery of Chua Attractors (with CD-ROM) E. Bilotta & P. Pantano

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Numerical Simulation of Waves and Fronts in Inhomogeneous Solids A. Berezovski, J. Engelbrecht & G. A. Maugin

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Advanced Topics on Cellular Self-Organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems R. Caponetto, L. Fortuna & M. Frasca

Volume 64:

Control of Chaos in Nonlinear Circuits and Systems B. W.-K. Ling, H. H.-C. Lu & H. K. Lam

Volume 65:

Chua’s Circuit Implementations: Yesterday, Today and Tomorrow L. Fortuna, M. Frasca & M. G. Xibilia

Volume 66:

Differential Geometry Applied to Dynamical Systems J.-M. Ginoux

Volume 67:

Determining Thresholds of Complete Synchronization, and Application A. Stefanski

Volume 68:

A Nonlinear Dynamics Perspective of Wolfram’ New Kind of Science (Volume III) L. O. Chua

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Modeling by Nonlinear Differential Equations P. E. Phillipson & P. Schuster

Volume 70:

Bifurcations in Piecewise-Smooth Continuous Systems D. J. Warwick Simpson

Volume 71:

A Practical Guide for Studying Chua’s Circuits R. Kiliç

Volume 72:

Fractional Order Systems: Modeling and Control Applications R. Caponetto, G. Dongola, L. Fortuna & I. Petráš

Volume 73:

2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach E. Zeraoulia & J. C. Sprott

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NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON

Series A

Vol. 73

Series Editor: Leon O. Chua

2-D Quadratic Maps and 3-D ODE Systems A Rigorous Approach Elhadj Zeraoulia University of Tébessa, Algeria

Julien Clinton Sprott University of Wisconsin-Madison, USA

World Scientific NEW JERSEY

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

World Scientific Series on Nonlinear Science, Series A — Vol. 73 2-D QUADRATIC MAPS AND 3-D ODE SYSTEMS A Rigorous Approach Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4307-74-1 ISBN-10 981-4307-74-2

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To my family: My father Tayeb and my brothers Zohir, Samra, Mourad, Saida, and Bachlila. To my sweet wife Houda and to all who helped me write this book. Elhadj

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Preface

This book was written on the basis of research carried out in the past three decades on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case of the H´enon map, and on 3-D piecewise linear ODEs, especially Chua’s circuit. In addition, we have recently published work in the field of general 2-D quadratic maps, especially their classification into equivalence classes and finding regions of chaos, hyperchaos, and nonchaos in the space of bifurcation parameters. The first chapter is the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems. Chapter 2 is focused on the study of the invertible case of the 2-D quadratic map, where the most significant works in the literature are oriented toward the H´enon mapping. Chapter 3 classifies the 2-D quadratic case into 30 maps with well known formulas. Two proofs for the regions of chaos, hyperchaos, and nonchaos in the space of bifurcation parameters are provided using a technique based on the second-derivative test and bounds for Lyapunov exponents. Chapter 4 is concerned with the rigorous proof of chaos in the piecewise linear Chua’s system using two methods, the first of which is based on construction of the Poincar´e map, and the second is based on a computer-assisted proof. Finally, Chapter 5 discusses the rigorous analysis of bifurcation phenomena in the piecewise linear Chua’s system using both an analytical 2-D and a 1-D approximate Poincar´e mapping in addition to other analytical methods. Zeraoulia Elhadj Julien Clinton Sprott March 2010

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Contents

Preface

vii

Acknowledgements

xiii

1. Tools for the rigorous proof of chaos and bifurcations 1.1 1.2 1.3

1.4

1.5

1.6

1.7 1.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A chain of rigorous proof of chaos . . . . . . . . . . . . . . Poincar´e map technique . . . . . . . . . . . . . . . . . . . 1.3.1 Characteristic multiplier . . . . . . . . . . . . . . 1.3.2 The generalized Poincar´e map . . . . . . . . . . . 1.3.3 Interval methods . . . . . . . . . . . . . . . . . . . 1.3.4 Mean value form . . . . . . . . . . . . . . . . . . The method of fixed point index . . . . . . . . . . . . . . 1.4.1 Periodic points of the T S-map . . . . . . . . . . . 1.4.2 Existence of semiconjugacy . . . . . . . . . . . . . Smale’s horseshoe map . . . . . . . . . . . . . . . . . . . . 1.5.1 Some basic properties of Smale’s horseshoe map . 1.5.2 Dynamics of the horseshoe map . . . . . . . . . . 1.5.3 Symbolic dynamics . . . . . . . . . . . . . . . . . The Sil’nikov criterion for the existence of chaos . . . . . 1.6.1 Sil’nikov criterion for smooth systems . . . . . . . 1.6.2 Sil’nikov criterion for continuous piecewise linear systems . . . . . . . . . . . . . . . . . . . . . . . . The Marotto theorem . . . . . . . . . . . . . . . . . . . . The verified optimization technique . . . . . . . . . . . . . 1.8.1 The checking routine algorithm . . . . . . . . . . 1.8.2 Efficacy of the checking routine algorithm . . . . . ix

1 1 3 7 7 8 10 13 14 16 17 19 20 22 23 26 26 27 28 30 30 31

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1.9

1.10 1.11

1.12 1.13 1.14

Shadowing lemma . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Shadowing lemmas for ODE systems and discrete mappings . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Homoclinic orbit shadowing . . . . . . . . . . . . Method based on the second-derivative test and bounds for Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . The Wiener and Hammerstein cascade models . . . . . . . 1.11.1 Algorithm based on the Wiener model . . . . . . 1.11.2 Algorithm based on the Hammerstein model . . . Methods based on time series analysis . . . . . . . . . . . A new chaos detector . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. 2-D quadratic maps: The invertible case 2.1 2.2 2.3 2.4 2.5

2.6

2.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Equivalences in the general 2-D quadratic maps . . . . . Invertibility of the map . . . . . . . . . . . . . . . . . . The H´enon map . . . . . . . . . . . . . . . . . . . . . . Methods for locating chaotic regions in the H´enon map 2.5.1 Finding Smale’s horseshoe maps . . . . . . . . . 2.5.2 Topological entropy . . . . . . . . . . . . . . . . 2.5.3 The verified optimization technique . . . . . . . 2.5.4 The Wiener and Hammerstein cascade models . 2.5.5 Methods based on time series analysis . . . . . . 2.5.6 The validated shadowing . . . . . . . . . . . . . 2.5.7 The method of fixed point index . . . . . . . . . 2.5.8 A new chaos detector . . . . . . . . . . . . . . . Bifurcation analysis . . . . . . . . . . . . . . . . . . . . 2.6.1 Existence and bifurcations of periodic orbits . . 2.6.2 Recent bifurcation phenomena . . . . . . . . . . 2.6.3 Existence of transversal homoclinic points . . . 2.6.4 Classification of homoclinic bifurcations . . . . . 2.6.5 Basins of attraction . . . . . . . . . . . . . . . . 2.6.6 Structure of the parameter space . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Classification of chaotic orbits of the general 2-D quadratic map

33 35 36 38 39 39 42 43 46 47 49

. 49 . 50 . 59 . 63 . 64 . 64 . 65 . 68 . 69 . 70 . 71 . 72 . 72 . 73 . 73 . 74 . 76 . 94 . 99 . 100 . 103

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Contents

3.1

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Analytical prediction of system orbits . . . . . . 3.1.1 Existence of unbounded orbits . . . . . . 3.1.2 Existence of bounded orbits . . . . . . . A zone of possible chaotic orbits . . . . . . . . . 3.2.1 Zones of stable fixed points . . . . . . . . Boundary between different attractors . . . . . . Finding chaotic and nonchaotic attractors . . . . Finding hyperchaotic attractors . . . . . . . . . . Some criteria for finding chaotic orbits . . . . . . 2-D quadratic maps with one nonlinearity . . . . 2-D quadratic maps with two nonlinearities . . . 2-D quadratic maps with three nonlinearities . . 2-D quadratic maps with four nonlinearities . . . 2-D quadratic maps with five nonlinearities . . . 2-D quadratic maps with six nonlinearities . . . . Numerical analysis . . . . . . . . . . . . . . . . . 3.13.1 Some observed catastrophic solutions in namics of the map . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the dy. . . . .

4. Rigorous proof of chaos in the double-scroll system 4.1 4.2

4.3

4.4

4.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . Piecewise linear geometry and its real Jordan form . 4.2.1 Geometry of a piecewise linear vector field in 4.2.2 Straight line tangency property . . . . . . . 4.2.3 The real Jordan form . . . . . . . . . . . . . 4.2.4 Canonical piecewise linear normal form . . . 4.2.5 Poincar´e and half-return maps . . . . . . . . The dynamics of an orbit in the double-scroll . . . . 4.3.1 The half-return map π 0 . . . . . . . . . . . . 4.3.2 Half-return map π 1 . . . . . . . . . . . . . . 4.3.3 Connection map Φ . . . . . . . . . . . . . . . Poincar´e map π . . . . . . . . . . . . . . . . . . . . . 4.4.1 V1 portrait of V0 . . . . . . . . . . . . . . . . 4.4.2 Spiral image property . . . . . . . . . . . . . Method 1: Sil’nikov criteria . . . . . . . . . . . . . . 4.5.1 Homoclinic orbits . . . . . . . . . . . . . . . 4.5.2 Examination of the loci of points . . . . . . 4.5.3 Heteroclinic orbits . . . . . . . . . . . . . . . 4.5.4 Geometrical explanation . . . . . . . . . . .

105 105 107 109 111 112 123 131 139 140 148 149 151 153 153 154 155 159

. . . . R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

159 164 164 166 168 171 175 176 177 185 192 194 195 196 197 197 202 210 214

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4.6 4.7 4.8

4.9

4.5.5 Dynamics near homoclinic and heteroclinic orbits Subfamilies of the double-scroll family . . . . . . . . . . . The geometric model . . . . . . . . . . . . . . . . . . . . . Method 2: The computer-assisted proof . . . . . . . . . . 4.8.1 Estimating topological entropy . . . . . . . . . . . 4.8.2 Formula for the topological entropy in terms of the Poincar´e map . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Rigorous analysis of bifurcation phenomena 5.1 5.2 5.3 5.4

5.5

5.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic stability of equilibria . . . . . . . . . . . . . . Types of chaotic attractors in the double-scroll . . . . . . Method 1: Rigorous mathematical analysis . . . . . . . . 5.4.1 The pull-up map . . . . . . . . . . . . . . . . . . . 5.4.2 Construction of the trapping region for the doublescroll . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Finding trapping regions using confinors theory . 5.4.4 Construction of the trapping region for the R¨ ossler-type attractor . . . . . . . . . . . . . . . . 5.4.5 Macroscopic structure of an attractor for the double-scroll system . . . . . . . . . . . . . . . . . 5.4.6 Collision process . . . . . . . . . . . . . . . . . . . 5.4.7 Bifurcation diagram . . . . . . . . . . . . . . . . . Method 2: One-dimensional Poincar´e map . . . . . . . . . 5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.5.2 Construction of the 1-D Poincar´e map . . . . . . 5.5.3 Properties of the 1-D Poincar´e map π ∗ . . . . . . 5.5.4 Numerical examples for the 1-D Poincar´e map π ∗ 5.5.5 Periodic points of the 1-D Poincar´e map π ∗ . . . . 5.5.6 Bifurcation diagrams using confinors theory . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 219 220 229 230 236 238 239 239 240 244 245 246 247 252 257 265 268 279 281 281 281 289 291 292 307 312

Bibliography

315

Index

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Acknowledgements

We thank L.O.Chua, John Milnor, Frederick Marotto, Michal Misiurewicz, Leonardo Mora, Carsten Knudsen, Huseyin Kocak, Peter Grassberger, Ernest Fontich, Zbigniew Nitecki, Zbigniew Galias, and Piotr Zgliczynski for providing some of their old papers concerning the topics in this book.

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Chapter 1

Tools for the rigorous proof of chaos and bifurcations

1.1

Introduction

In this section, we discuss the notion of rigorous proof . This notion is very problematic because there is no unified definition of a proof. However, there are several opinions as to what constitutes a proof, what is a rigorous proof, and more generally, what are the elements that constitute a proof. The so-called four-color theorem is an example of this problem since its “proof ” is done by exhaustive computer testing of many individual cases and cannot easily be verified by hand. Such proofs are the subject of much controversy in the mathematical world. Some mathematicians think that the so-called computer-assisted proofs are valid and that the loss of the human verifiability of such proofs can be replaced by proving that the proof program itself is valid. Others disagree because of the non-verifiability of the different steps in the proof. Hardy says in [Hardy (1999), pp. 15-16] that “all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply.” Hardy’s assertion given above was echoed by Feynman [Derbyshire (2004), pp. 291] who is reported to have commented, “A great deal more is known than has been proved.” However, we accept the following definitions: Definition 1.1. (a) A proof is a rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. 1

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(b) A proof is called rigorous if the validity of each step and the connections between the steps is explicitly made clear in such a way that the result follows with certainty. In this book, the rigorous proofs that concern us are those dealing with the problem of locating orbits in a dynamical system, especially for the 2-D quadratic maps and the so-called Chua’s circuit [Chua et al. (1986)]. These proofs can be classified into two types: The first is purely mathematical and use only a chain of logical steps that can be verified by hand, and the second is a mixture of mathematical tools and computer calculations. This latter type of rigorous proof is the one most used in the world of nonlinear sciences, especially for proving the existence and properties of different kinds of orbits, such as chaotic orbits, in a dynamical system because of their relevance to real applications. Note that the materials and tools presented here are independent of notation, applying equally to maps and to continuous-time systems. Strange attractors can be classified into three principal classes [Anishchenko and Strelkova (1997), Chua et al. (1986), Plykin (2002)]: hyperbolic, Lorenz-type, and quasi-attractors. The hyperbolic attractors are the limit sets for which Smale’s “axiom A” is satisfied and are structurally stable. Periodic orbits and homoclinic orbits are dense and are of the same saddle type, which is to say that they have the same index1 . However, the Lorenz-type attractors are not structurally stable, although their homoclinic and heteroclinic orbits are structurally stable (hyperbolic), and no stable periodic orbits appear under small parameter variations, as for example in the Lorenz system [Lorenz (1963)]. The quasi-attractors are the limit sets enclosing periodic orbits of different topological types (for example, stable and saddle periodic orbits and structurally unstable orbits). For example, the attractors generated by Chua’s circuit [Chua et al. (1986)] associated with saddle-focus homoclinic loops are quasi-attractors. Note that this type is more complex than the former two attractors and thus are not suitable for potential applications of chaos such as secure communications and signal masking. For further information about these types of chaotic attractors, see [Anishchenko and Strelkova (1997)]. In strange attractors of the hyperbolic type, all orbits in phase space are of the saddle type, and the invariant sets of trajectories approach the original one in forward or backward time, i.e., the stable and unstable manifolds intersect transversally. Generally, most known physical systems do not be1 The

same dimensions for their stable and unstable manifolds.

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long to the class of systems with hyperbolic attractors [Anishchenko and Strelkova (1997)]. The type of chaos in them is characterized by chaotic trajectories and a set of stable orbits of large periods, not observable in computations because of their extremely small basins of attraction. Hyperbolic strange attractors are robust (structurally stable) [Mira (1997)]. Thus, both from the point of view of fundamental studies and of applications, it would be interesting to find physical examples of hyperbolic chaos. For example, the Smale–Williams attractor [Kuznetsov and Seleznev (2006)] is constructed for a three-dimensional map, and the composed equations given by  ′ 1 3   x = −2πu′ + (h1 + A1 cos 2πτ /N ) x − 3 x  u = 2π (x + ε2 y cos 2πτ ) (1.1) ′  y = −4πv + (h2 − A2 cos 2πτ
/N ) y − 31 y 3   v ′ = 4π y + ε1 x2

are obtained by applying the so-called equations of Kirchhoff [Archibald (1988)], where the variables x and u are normalized voltages and currents in the LC circuit of the first self-oscillator (U1 and I1 , respectively), and y and v are normalized voltages and currents in the second oscillator (U2 and I2 ). Time is normalized to the period of oscillations of the first LC oscillator, and the parameters A1 and A2 determine the amplitude of the slow modulation of the parameter responsible for the Andronov–Hopf bifurcation in both self-oscillators. The parameters h1 and h2 determine a map of the mean value of this parameter from the bifurcation threshold, and ε1 and ε2 are coupling parameters. The system (1.1) has been constructed as a laboratory device [Kuznetsov and Seleznev (2006)], and an experimental and numerical solution were found. This example of a physical system with a hyperbolic chaotic attractor is of considerable significance since it opens the possibility for real applications. For further details, see [Kuznetsov and Seleznev (2006)]. The rigorous proof of chaos has a very long history because of the rich variety of dynamical systems, namely discrete and continuous-time, with and without known equations (dynamical system, time series, results of experiments, ...).

1.2

A chain of rigorous proof of chaos

In this section, we give as a main introduction for this book, many examples of dynamical systems with rigorous proofs of the chaos in their parameter

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space. We concentrate on the examples of 2-D and 3-D real dynamical systems, i.e., systems with well known equations and well known domains of potential application. As a Lorenz-type system, consider the original Lorenz system [Lorenz (1963)] given by  ′  x = σ (y − x) y ′ = rx − y − xz (1.2)  ′ z = −bz + xy These equations have proved to be very resistant to rigorous analysis and also present obstacles to numerical study. A very successful approach was taken in [Guckenheimer and Williams (1979), Afraimovich et al. (1982)] where they constructed so-called geometric models (these models are flows in three dimensions) for the behavior observed by Lorenz for which one can rigorously prove the existence of a robust attractor. Another approach through rigorous numerics [Hasting and Troy (1992), Hassard et al. (1994), Mischaikow and Mrozek (1995-1998)] showed that the equations exhibit a suspended Smale horseshoe. In particular, they have infinitely many closed solutions. A computer assisted proof of chaos for the Lorenz equations is given in [Tucker (1999), Stewart (2000), Franceschini et al. (1993), Galias and Zgliczynski (1998), Mischaikow and Mrozek (1995), Sparrow (1982)]. In [Tucker (1999)] a rigorous proof was provided that the geometric model does indeed give an accurate description of the dynamics of the Lorenz equations, i.e., it supports a strange attractor as conjectured by Lorenz in 1963. This conjecture was listed by Steven Smale as one of several challenging mathematical problems for the 21st century [Smale (19982000)]. Also a proof was given that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. This proof is based on a combination of normal form theory and rigorous numerical computations. The robust chaotic Lorenz attractor is shown in Fig. 1.1. As a general result, it was proved in [Araujo et al. (2005)] that the so-called singular-hyperbolic attractor (or Lorenz-like attractor) of a three-dimensional flow is chaotic in two different strong senses: Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparameterization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. In particular, these results show that both the flow defined by the Lorenz equations and the geometric Lorenz flows are expansive.

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Fig. 1.1 The Lorenz chaotic attractor obtained from (1.2) for σ = 10, r = 28, b = [Lorenz (1963)].

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Another proof of the robustness of the Lorenz attractor is given in [Franceschini et al. (1993)] where the chaotic attractors of the Lorenz system for r = 28 and r = 60 were characterized in terms of their unstable periodic orbits and eigenvalues. While the Hausdorff dimension is approximated to very good accuracy in both cases, the topological entropy was computed in an exact sense only for r = 28. A general method for proving the robustness of chaos in a set of systems called C 1 -robust transitive sets with singularities for flows on closed 3-manifolds is given in [Morales et al. (2004)]. The elements of the set C 1 are partially hyperbolic with a volume-expanding central direction and are either attractors or repellors. In particular, any C 1 -robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has a positive Lyapunov exponent for every orbit, showing that any C 1 -robust attractor resembles a geometric Lorenz attractor. A new topological invariant (Lorenz-manuscript) leading to the existence of an uncountable set of topologically different attractors is proposed in [Klinshpont et al. (2005)] where a new definition of the hyperbolic properties of the Lorenz system close to singular hyperbolicity is introduced, as well as a proof that small nonautonomous perturbations do not lead to the appearance of stable solutions. Other than the Lorenz attractor, there are some works that focus on the proof of chaos and its robustness in 3-D continuous-time systems. For example the set C 1 introduced in [Morales et al. (2004)], and a characterization of maximal transitive sets with singularities for generic C 1 -vector

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Fig. 1.2 The classic double-scroll attractor obtained from (1.3) for α = 9.35, β = 14.79, m0 = − 17 , m1 = 27 [Chua et al. (1986)].

fields on closed 3-manifolds in terms of homoclinic classes associated with a unique singularity is given and applied to some special cases. However, no robust chaos occurs in quasi-attractor-type systems because the complexity of quasi-attractors is essentially due to the existence of structurally unstable homoclinic orbits in the system itself and in any system close to it. This results in a sensitivity of the attractor structure to small variations of the parameters of the generating dynamical equation, i.e., quasi-attractors are structurally unstable. Therefore, this type of system cannot generate robust chaotic attractors in the sense of this section [Mira (1997)]. Attractors generated by Chua’s circuits [Chua et al. (1986)], given by  ′  x = α (y − h (x)) (1.3) y′ = x − y + z  z ′ = −βy

where

1 h(x) = m1 x + (m0 − m1 ) (|x + 1| − |x − 1|) 2

(1.4)

are associated with saddle-focus homoclinic loops and are quasi-attractors. The corresponding, nonrobust, double-scroll attractor is shown in Fig. 1.2. Note that the chaos in Eq. (1.3) has been analytically proved by two independent methods [Chua et al. (1986), Matsumoto et al. (1988)] that are described in detail in Chapter 4 and constitute a major theme of this book.

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1.3

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7

Poincar´ e map technique

Let f : R × Ω −→ Ω be a continuous flow, where Ω ⊂ Rn is open. Let us consider the following n-dimensional continuous-time system given by x′ = f (t, x) (1.5) Let ϕt denote the corresponding flow of the system (1.5). Definition 1.2. A map P is called a C r -diffeomorphism if both P and P −1 its inverse are bijective and are r times continuously differentiable. Let γ be a periodic orbit through a point p, and let U be an open and + connected neighborhood of p. For any x ∈ Ω, let I(x) = ]t− x , tx [ be an open interval in the real numbers. Then one has the following definitions: Definition 1.3. (a) A positive semi-orbit through x is the set γ + x = − {f (t, x) , t ∈ ]0, t+ [} , and a negative semi-orbit through x is the set γ x x = − {f (t, x) , t ∈ ]tx , 0[} . (b) A Poincar´e section through a point p is a local differentiable and transversal section S of f through the point p. Hence the Poincar´e map is defined by the following: Definition 1.4. A function P : U −→ S is called a Poincar´e map for the orbit γ on the Poincar´e section S through point p if: (1) P (p) = p. (2) P (U ) is a neighborhood of p and P : U → P (U ) is a diffeomorphism. (3) For every point x in U , the positive semi-orbit of x intersects S for the first time at P (x). The relation between limit sets of the Poincar´e map P and limit sets of the flow for the considered system (1.5) can be summarized as follows: A limit cycle of ϕt is a fixed point of P , and a period-m closed orbit of P is a subharmonic solution (relative to the considered section S) of ϕt . A chaotic orbit of P corresponds to a chaotic solution of ϕt . Also, a stable periodic point of P corresponds to a stable periodic orbit of ϕt , and an unstable periodic point of P corresponds to a saddle-type periodic orbit of ϕt . 1.3.1

Characteristic multiplier

Assume that the system (1.5) has a limit cycle Γ and a Poincar´e map P associated with this cycle at some arbitrary fixed point x∗ located on this

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cycle. Then the local behavior of the map P near x∗ is determined by linearizing the map at x∗ . Especially, the system δxk+1 = DP (x∗ ) δxk

(1.6)

where DP (x∗ ) is the Jacobian matrix of P evaluated at x = x∗ , governs the evolution of the perturbation δx0 in a neighborhood of the fixed point x∗ . Definition 1.5. The eigenvalues of DP (x∗ ) are called characteristic multipliers of the periodic solution Γ. The notion of characteristic multipliers was used to predict some routes to chaos in Chua’s circuit as shown in Sec. 5.6.5.3. 1.3.2

The generalized Poincar´ e map

In this section, assume that the function f given in (1.5) is a continuous piecewise linear vector field. Let us denote by Σ1 , ..., Σp the hyperplanes separating the linear regions Ui of f , so their union is the set Ω. Let us denote by ϕ(t, x) the trajectory of the system starting at the point x. Definition 1.6. The generalized Poincar´e map H : Ω → Ω is defined by H(x) = ϕ(τ (x), x), where τ (x) is the time needed for the trajectory ϕ (t, x) to reach Ω. Note that the generalized Poincar´e map H has the same properties given above for the map P. For evaluation of H in regions where H is continuous, one can use the analytical formulas for solutions of linear systems. In order to evaluate the generalized Poincar´e map H on a box X ⊂ Ω, we first find the return time for all points in X, i.e., the interval {τ (x), x ∈ X} ⊂ τ (X) and then use analytic solutions to compute ϕ(τ (X), X). H(X) is enclosed in the intersection of ϕ(τ (X), X) with Ω. The Jacobian of H at X can be expressed in terms of the return time τ (x), the start box X, and the image H(X). Some important properties of the Poincar´e maps and their generalizations are as follows: (1) The Poincar´e map P is sometimes called the first recurrence map because the method of analysis is to consider a periodic orbit with initial conditions on the Poincar´e section S and observe the point at which this orbit first returns to the section S.

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(2) The Poincar´e map is the intersection of a periodic orbit of the considered continuous-time system with the transversal Poincar´e section2 S in one dimension smaller than that of the original continuous dynamical system. (3) The map P is used to analyze the original system because it preserves many properties of the periodic and quasiperiodic orbits in the original continuous-time system. (4) There is no general method for constructing a Poincar´e map. The majority of methods used here are numeric. In [H´enon (1982)], a method for accurately finding the intersections of a numerically integrated trajectory of a system of ordinary differential equations with a surface of section is given. A generalization of the stopping procedure described by H´enon is given in [Tucker (2002b)]. In [Tsuji and Ido (2002)], a computational method based on parallel computation of data tables and interpolation is given for calculating the Poincar´e map. This method was successfully applied to the so-called chaotic torus magnetic field line caused by the perturbation coil . In [Fujisaka and Sato (1997)], a numerical method based on the Poincar´e map, the second map constructed from the Poincar´e map, and the topological degree is presented to compute the number and stability of fixed points of Poincar´e maps of ordinary differential equations. The computation of the topological degree of the second map is equivalent to the calculation of the number of fixed points of the Poincar´e map in a given domain of a Poincar´e section. In some special cases, the Poincar´e map was constructed rigorously by [Chua et al. (1986), Chua and Tichonicky (1991), Kuznetsov and Satayev (1994)] and references therein. (5) The Poincar´e map P is not defined for points, trajectories of which never come back to the section S defining the map. Hence the image of a given set under P can be computed rigorously only if it is continuous on this set. Conversely, if the map P is well defined, this does not imply its continuity. One example of that is a point belonging to a stable manifold of an equilibrium of (1.5) where its trajectory converges to the fixed point and hence never comes back to the Poincar´e plane S. A second example is where the map P is well defined but not continuous at points for which the flow is parallel to section S at this point or at the image P (S). (6) If the map P is not continuous at a point, then in a close neighborhood 2 This

means that any periodic orbits starting on the subspace flow through S are not parallel to it.

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of such a point rigorous evaluation of P becomes very difficult, the sets which have to be studied become smaller, and the computation time becomes longer. All these problems are circumvented if one knows the regions where the map P is not continuous.

1.3.3

Interval methods

In the interval Newton’s method [Alefeld and Herzberger (1983), Neumaier (1990)], the existence of zeros of a function f in an n-dimensional interval X is established by evaluating the so-called interval Newton operator N (X) given by N (X) = x0 − (Df (X))−1 f (x0 ) (1.7) where Df (X) is the interval matrix containing all Jacobian matrices of f for x ∈ X and x0 is an arbitrary point belonging to X. Hence the following theorem [Alefeld and Herzberger (1983), Neumaier (1990)] was obtained: Theorem 1.1. If N (X) ⊂ X, then there exists exactly one zero of f in X. If N (X) ∩ X = ∅, then there are no zeros of f in X. Hence the interval Newton’s method can be used to prove the existence and uniqueness of zeros. For computation of the expression (Df (X))−1 f (x0 ), one can use for example the Gaussian algorithm3 . Periodic orbits of a piecewise linear system can be rigorously studied by means of the interval Newton’s method [Galias (2002a-2002b)]. The methods are based on the concept of a Poincar´e map and the interval Newton’s method to find regions where the generalized Poincar´e map H is well defined and continuous and for locating all low-period cycles in this region. An example can be found in Sec. 4.8.1. 1.3.3.1

Existence of periodic orbits

The existence of periodic solutions for continuous-time piecewise linear systems can be carried out using the generalized Poincar´e map H associated with the continuous-time flow f given in (1.5). To prove the existence of a period-m orbit of H, one applies the interval Newton’s method to the map m m G : (Rn ) −→ (Rn ) defined by [G (z)]k = x(k+1) mod m − H (xk ) , 0 < k < m

(1.8)

3 The Gaussian elimination method (some elementary row operations) can be used to determine the solutions of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix [Lipschutz and Lipson (2001)].

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where z = (x0 , ...xm−1 ) . We remark that G (z) = 0 if and only if x0 is a fixed point of H m . Note that this technique is limited to the subsets of Ω where the Poincar´e map H can be rigorously and effectively evaluated, and the problem of existence of periodic orbits for the system (1.5) is translated into the problem of existence of zeros of the higher-dimensional function G. 1.3.3.2

Interval arithmetic

Consider the following elements of the theory of interval arithmetic:  X = [a, b] = {x ∈ R : a ≤ x ≤ b}  (1.9) V = (X1 , X2 , ..., Xn )  X1 ♦X2 = {x = x1 ♦x2 ∈ R, : x1 ∈ X1 , x2 ∈ X2 }

where: X : is an interval, i.e., a closed bounded set of real numbers. V : is called an n-dimensional interval vector , and it consists of n closed intervals Xi , i = 1, ..., n. ♦ : is any of the following usual numeric operators: +, −, ×, and /, where all operations but division are defined for arbitrary intervals. For division, we assume that the interval X2 does not contain the number 0 because a real number a can be treated as a degenerate interval a = [a, a]. Generally, all the steps of this method are summarized as follows [Galias (2001)]: (1) Reduce the continuous system (1.5) to the discrete one using the Poincar´e map P . (2) Apply the interval Newton’s operator G defined in (1.8). (3) Evaluate P (X) from Eq. (1.5) using direct integration by the Lohner method, which helps to reduce the wrapping effect 4 [Moore (1966), Lohner (1992)]. (4) Find the image of P (X), i.e., the intersection of Ω and the trajectory computed by the rigorous integration procedure in step 3. (5) Find P´ (X), the Jacobian matrix of P (X) by solving the variational equation ∂f dD = (x (t)) D dt ∂x 4 The

(1.10)

wrapping effect occurs when a solution set of a differential equation that is not a box (a parallelepiped with edges that are parallel to the axes of an orthogonal coordinate system) is enclosed or wrapped by a box on each integration step. A result of this wrapping phenomenon is that the computed bounds become unacceptably large.

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t where D (t, x0 ) = ∂ϕ ∂x0 (t, x0 ) with the initial condition D (0, x0 ) = I, the unit matrix. (6) Calculate the enclosure for the Jacobian matrix of P at x ∈ X using the formula:   f (y) hT D (1.11) P ′ (X) = I − T h f (y) where: D is the enclosure for the solution of the variational equation

{D (t, x) , x ∈ X, t = τ (x)}

(1.12)

h is a vector orthogonal to Ω and y is the enclosure for the set {P (x) , x ∈ X}. ˜ for the Poincar´e map P . Note that (7) Construct the trapping region5 Γ this region can be found by choosing a polygon enclosing trajectories of the Poincar´e map P generated by the computer. ˜ by boxes of a specified size. (8) Cover the trapping region Γ (9) Apply a combination of the interval Newton’s method and the generalized bisection technique to find all periodic orbits. This can be done in eight steps: (9-1) Find the graph representation of the dynamics of the system6 . (9-2) Compute the image of each box. (9-3) Find the set of admissible transitions (the so-called transitions represent edges of the directed graph) between boxes. (9-4) Reduce the graph by removing vertices corresponding to boxes having no intersection with the invariant part of the trapping region. In this case, a box is removed if its image has no intersection with other boxes or if it has no intersection with images of all boxes. (9-5) Repeat this procedure until no more boxes can be removed. (9-6) Find all period-m cycles in the resulting graph. (9-7) Evaluate the interval operator for each period-m cycle on the corresponding interval vector Z, and check what is the position of N (Z) with respect to Z, i.e., If N (Z) ⊂ Z then there exists exactly one period-m cycle of f in Z. If N (Z) ∩ Z = ∅, then there are no period-m cycles of f in Z. 5 A set Γ ˜ is said to be a trapping region, if it is positively invariant under the action of ˜ for all x ∈ Γ. ˜ the map f , i.e., f (x) ∈ Γ 6 For finding the graph representation of the dynamics of the system, the trapping region ˜ must be covered by ǫ-boxes of the form v = [k1 ǫ1 , (k1 + 1)ǫ1 ] × [k2 ǫ2 , (k2 + 1)ǫ2 ] where Γ ki are integer numbers, ǫi are fixed positive real numbers, and ǫ = (ǫ1 , ǫ2 ). ǫ-boxes define vertices, and admissible connections between boxes define edges of the graph.

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(9-8) If the step (9-7) does not hold, then one option is to divide the interval vector Z into smaller parts and to evaluate the interval operator on each of them. 1.3.4

Mean value form

For evaluation of the Poincar´e map H of a piecewise linear system, the socalled mean value form [Galias (2002b)] was used to eliminate some types of over-estimations of the resulting solutions set called the wrapping effect. Assume that in the linear regions Σk the system (1.5) has the form x′ = Ak (x − pk )

(1.13)

ϕ (t, x0 ) = exp (Ak t) (x0 − pk ) + pk

(1.14)

Then its solution is given by

The following theorem was used to describe the method based on mean value form techniques: Theorem 1.2. Assume that H : Rn −→ Rm is a C 1 map. Then for all x, y ∈ Rn , and for all i = 1, 2, ..., m, there exists a point zi in the interval xy such that Hi (x) − Hi (y) =

j=m X j=1

∂Hi (zi ) (xj − yj ) ∂xj

(1.15)

Then, the mean value form techniques can be done in the following steps: (1) Take an interval vector X, fix a point y ∈ X as the center of X, and choose another point x ∈ X. (2) Apply Theorem 1.2, and deduce that there exists a point zi ∈ X such that  j=m   H (x) ∈ H (y) + X ∂Hi (z ) (x − y ) i j j i i ∂xj (1.16) j=1   H (x) ∈ H (y) + DH (X) (x − y)

(3) Deduce that

{H (x) ∈ X} ⊂ H (y) + DH (X) (x − y)

(1.17)

Clearly, more stringent computational results are obtained when the Poincar´e map H is computed using Eq. (1.16). An example of the application of this method for Chua’s circuit (1.3) is given in Sec. 4.8.1.

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The method of fixed point index

The existence of the chaotic H´enon map [H´enon (1976)] was proved in [Zgliczynski (1997a)] using the method based on fixed point index. This method is described in Sec. 2.5.7, and it is done by introducing horseshoetype mappings which are geometrically similar to Smale’s horseshoes. The existence of chaotic dynamics and the semi-conjugacy introduced in Sec. 1.4.2 to the shift map on a finite number of symbols7 are obtained for such mappings using the so-called fixed point index. This method is a purely topological one and does not require any assumptions concerning derivatives, and its assumptions can be rigorously verified by computer-assisted computations. For a good presentation of this method, we need the following definitions: Let (X, ρ) be a metric space. Let Z ⊂ X and x ∈ X. Let int(Z), cl(Z), bd(Z) denote the interior, the closure, and the boundary of the set Z, respectively. Let f : X → X be any continuous map and N ⊂ X. Let G denote the class of pairs (f, Z), such that Z is an open and bounded set and cl(Z) ⊂ X. Let f |N denote the map obtained by restricting the domain of f to the set N . F ixf denotes the set of fixed points of f . Assume that f has no fixed points on bd(X). The fixed point index theory introduced here using algebraic topology was developed in [Dold (1980)]. In particular, we use the property of the fixed point index which establishes the existence of a fixed point if the fixed point index is not zero. Here we give only the axiomatic definition: Definition 1.7. The fixed point index is an integer valued function I : G → Z satisfying the following axioms: (1) If W is an open set such that F ixf ∩ Z ⊂ W , then I (f, Z) = I (f, W ) . (2) If f is constant, then I (f, Z) = 1 if f (Z) ∈ Z, and I (f, Z) = 0 if f (Z) ∈ / Z. (3) If Z is a sum of a finite number of open sets Zi , i = 1, ..., m, such Pm that Zi ∩ Zj ∩ F ixf = ∅ for i 6= j, then I (f, Z) = i=1 I (f, Zi ). (4) If f : X → X, f ′ : X ′ → X ′ are continuous maps and (f, Z) , (f ′ , Z ′ ) belong to the class G, then I ((f, f ′ ) , Z × Z ′ ) = I (f, Z) I (f ′ , Z ′ ). (5) if Ft : X × [0, 1] → Rn is a homotopy, Z ⊂ X, and F ixf ∩ bd(Z) = ∅ 7 The

class of TS-maps (the topological shifts), which includes as particular cases Smale’s horseshoes [Smale (1967)], is used in this case.

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for every t ∈ [0, 1], then I (F0 , Z) = I (F1 , Z). Also, we need the definition of the maximal invariant part: Definition 1.8. The maximal invariant part of N with respect to f is defined by −i Inv (N, f ) = ∩i∈Z f N (Z) . (1.18)

For any set, P = ∪k Pk = ∪k [ak , bk ] × [ck , dk ] ⊂ R2 , where Pk are disjoint rectangles. Let us define the sets L (P ), R (P ), V (P ), H (P ) that are equal to the union of left vertical, right vertical, vertical, and horizontal edges in P , respectively:  L (P ) = ∪k {ak } × [ck , dk ]    R (P ) = ∪k {bk } × [ck , dk ] (1.19)  V (P ) = L (P ) ∪ R (P )   H (P ) = ∪k ([ak , bk ] × {ck } ∪ [ak , bk ] × {dk }) Let us fix u, d ∈ R, u > d and a sequence a−1 = −∞ < a0 < a1 < ... < a2K−2 < a2K−1 < a2K = ∞, where ai ∈ R, for i = 0, 1, ..., 2K − 1. Let:  Ni = [a2i , a2i+1 ] × [d, u] , for i = 0, ..., K − 1    Ei = [a2i−1 , a2i ] × [d, u] , for i = 0, ..., K (1.20) N = N0 ∪ N1 ∪ ... ∪ Nk−1    E = E0 ∪ E1 ∪ ... ∪ Ek−1 ∪ Ek

Then one has the following result [Zgliczynski (1997)]:

Lemma 1.1. The sets Ei , Ni are contained in the horizontal strip (−∞, ∞) × [d, u] in the following order (if one compares x-coordinates): E0 < N0 < E1 < N1 < ... < Ek−1 < Nk−1 < Ek For i = 0, ..., K, let us define the sets Ei′ as follows:  ′  Ei = Ei ∩ (−∞, ∞) × [d, u] ′ cl (Ei ) ∩ (N −
V (N )) = ∅  cl (Ei′ ) ∩ cl Ej′ = ∅ for i 6= j

(1.21)

(1.22)

and suppose that there exist continuous homotopies hi : [0, 1] × Ei′ → Ei′ such that  hi (0, p) = p for p ∈ Ei′  (1.23) hi (1, p) ∈ Ei′ for p ∈ Ei′  ′ hi (t, p) = p for p ∈ Ei and t ∈ [0, 1]

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E ′0 E0

Fig. 1.3

E ′3

E ′1 N0

E1

N1 E ′2 = E2 N2

E3

An example of sets Ni , Ei , and Ei′ for K = 3.

Then one has the following result [Zgliczynski (1997b)]: Lemma 1.2. (a) The set Ei′ can be continuously deformed to the set of Ei without any intersection with the set 8 N . (b) If ′ E ′ = E0′ ∪ E1′ ∪ ... ∪ EK

(1.24)

Ei′ ∩ Nj = ∅ for i, j = 0, 1, ..., K − 1

(1.25)

then

Fig. 1.3 presents a schematic drawing of the sets Ni , Ei , and Ei′ for K = 3. Definition 1.9. Let the sets Ni , Ei , and Ei′ be as above. Let D be an open set such that N ⊂ D and the map f : D → R2 be continuous. We say that f is a TS-map (topological shift) (relative to the sets N, E, E ′ ) if there exist functions l, r : {0, 1, ..., K − 1} → {0, 1, ..., K} such that the following conditions hold:  ′   f (L (Ni )) ⊂ El(i) ′ (1.26) f (R (Ni )) ⊂ Er(i)   f (N ) ⊂ E ′ ∪ N. 1.4.1

Periodic points of the T S-map

In this section, the periodic points of the T S-map f are characterized by periodic infinite sequences c = (ci )i∈N of symbols 0, 1, ..., K − 1 with the 8 In this case, the set E is called the deformation retract of E ′ , where a deformation i i retraction is a map that captures the idea of continuously shrinking a space into a subspace.

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property f i (x) ∈ Nci , for i ∈ N. Indeed, let ( Z ΣK = {0, 1, ..., K − 1} N + ΣK = {0, 1, ..., K − 1}

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(1.27)

Then (see also Sec. 2.5.7) [Zgliczynski (1997b)]: Lemma 1.3. (a) The sets ΣK , Σ+ K are topological spaces with a Tichonov topology. (b) On ΣK , ΣK + , the shift map σ is given by σ ((c))i = ci+1 .

(1.28)

Let A = (αij ) be a K × K matrix, αij ∈ R+ ∪ {0} , i, j = 0, 1, ..., K − 1. + We define ΣA ⊂ ΣK and Σ+ A ⊂ ΣK by   ΣA = c = (ci )i∈Z : αci ci+1 > 0 (1.29) Σ+ A = c = (ci )i∈N : αci ci+1 > 0

Then one has the following result [Zgliczynski (1997b)]:

Lemma 1.4. The sets ΣA and Σ+ A are invariant under σ. Let f be a T S-map. To relate the dynamics of f on Inv (N, f ) with shift dynamics on Σ+ K , we introduce the transition matrix of f denoted by A (f ). We define A (f )ij , where i, j = 0, 1, ..., K − 1:  1, if El(i) < Nj < Er(i) or El(i) > Nj > Er(i) A (f )ij = (1.30) 0, otherwise. Then one has the following lemma [Zgliczynski (1997b)]: Lemma 1.5. If Nj lies between the images9 of vertical edges of Ni , then A (f )ij 6= 0, for all i, j = 0, 1, ..., K − 1. 1.4.2

Existence of semiconjugacy

For i ∈ N, we define the map π i : Inv (N, f ) → {0, 1, ..., K} given by π i (x) = j if and only if f i (x) ∈ Nj .

(1.31)

Now we define the map π : Inv (N, f ) → Σ+ K by π (x) = (π i (x))i∈N . 9 In

this case, one can deform the image by the homotopies h if necessary.

(1.32)

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The map π assigns to the point x the indices of the rectangles Ni that its trajectory goes through. Then one has [Zgliczynski (1997b)]: Lemma 1.6. (a) We have π◦f =σ◦π

(1.33)

(b) If f is also a homeomorphism, then the definition of π i can be extended to all integers, and the domain of π is ΣK . Lemma 1.6(a) indicates the existence of a semi-conjugacy between f and σ, and this semi-conjugacy is not a sign of complicated dynamics because the set Inv (N, f ) is finite or empty. However, the dynamics is complicated if the set π (Inv (N, f )) is infinite as confirmed by the following theorem that gives the characterization of this set for TS-maps [Zgliczynski (1997)]: Theorem 1.3. Let f be a TS-map. Then Σ+ A(f ) ⊂ π (Inv (N, f )). The + preimage of any periodic sequence from ΣA(f ) contains periodic points of f . If we additionally suppose that f is a homeomorphism, then ΣA(f ) ⊂ π (Inv (N, f )). If we have the following definition: Definition 1.10. Let N ⊂ Rd be a compact set and f : N → Rd be a continuous map. The set N is called an isolating neighborhood if and only if Inv (N, f ) ⊂ Int (N ) .

(1.34)

Then the following theorem was proved in [Zgliczynski (1996)]: d Theorem 1.4. Let N = ∪K−1 i=0 Ni , where Ni ⊂ R are compact and disn joint. Let F : [0, 1] × N → R be a continuous map such that N is an isolating neighborhood for Fλ for λ ∈ [0, 1]. Then for every finite sequence n+1 (σ 0 , σ 1 , ..., σ n ) ∈ {0, 1, ..., K} , the fixed point index
I Fλn+1 , Nσ0 ∩ Fλ−1 (Nσ1 ) ∩ ... ∩ Fλ−n (Nσn ) (1.35)

is defined and does not depend on λ.

An application of this method to the H´enon map [H´enon (1976)] is given in Sec. 2.5.7.

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Smale’s horseshoe map

For defining the so-called Smale’s Horseshoe map, we must define the unit square: Definition 1.11. A unit square D with side lengths 1 is the one with coordinates (0, 0), (1, 0), (1, 1), (0, 1) in the real plane, or 0, 1, 1 + i, i in the complex plane. Thus Smale’s horseshoe map f [Smale (1967), Cvitanovi´c et al. (1988)] consists of the following sequence of operations as shown in Fig. 1.4 on the unit square D in blue and the following operations in red: (a) Stretch in the y direction by more than a factor of two. (b) Compress in the x direction by more than a factor of two. (c) Fold the resulting rectangle and fit it back onto the square, overlapping at the top and bottom, and not quite reaching the ends to the left and right and with a gap in the middle. Hence the action of f is defined through the composition of the three geometrical transformations defined above. (d) Endlessly repeating this procedure generates the horseshoe attractor with a Cantor set structure. From Fig. 1.4, one see that horseshoe maps cross the original square in a linear fashion, but in most applications, horseshoe maps are rarely so regular, although the behavior can be very similar. Mathematically, the above actions can be translated as follows [Smale (1967)]: (a) Contract the square D by a factor of λ in the vertical direction, where 0 < λ < 21 , such that D is mapped into the set [0, 1] × [0, λ]. (b) Expand the rectangle obtained by a factor of µ in the horizontal direction, where 2 + ǫ < µ, and the set [0, 1] × [0, λ] is mapped into the set [0, µ] × [0, λ] (the need for this ǫ factor is explained in step 3). (c) Steps 1 and 2 produce a rectangle f (D) of dimensions µ × λ. This rectangle crosses the original square D in two sections after being bent as shown in Fig. 1.4. The ǫ in step 2 indicates the extra length needed to create this bend as well as any extra on the other side of the square. (d) This process is then repeated, only using f (D) rather than the unit square. The nth iteration of this process will be called f k (D), k ∈ N.

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Fig. 1.4

Smale’s horseshoe map f after a single iteration.

A picture showing some higher iterations of this process in Smale’s horseshoe map f can be found in [Casselman (2005)]. 1.5.1

Some basic properties of Smale’s horseshoe map

Before stating some important properties of Smale’s horseshoe map f , we need the following definitions [Abraham and Marsden (1978)]: Let f : Ω ⊂ Rn −→ Rn be a real function that defines a discrete map also called f . Definition 1.12. (a) A point x is a nonwandering point for the map f if for every neighborhood U of x there is a k ≥ 1 such that f k (U ) ∩ U is nonempty. (b)The set of all nonwandering points is called the nonwandering set of f. (c) An f -invariant subset Λ of Rn satisfies f (Λ) ⊂ Λ. (d) If f is a diffeomorphism defined on some compact smooth manifold Ω ⊂ Rn , an f -invariant subset Λ of Rn is said to be hyperbolic if there exists a 0 < λ1 < 1 and a c > 0 such that (d-1) TΛ Ω = E s (+) E u , where (+) means the direct algebraic sum. (d-2) Df (x) Exs = Efs (x) , and Df (x) Exu = Efu(x) for each x ∈ Λ.

k

Df v ≤ cλk1 kvk , for each v ∈ E s and k > 0. (d-3)

(d-4) Df −k v ≤ cλk1 kvk , for each v ∈ E u and k > 0.

where E s , E u are the stable and unstable submanifolds of the map f , i.e., the two Df -invariant submanifolds, and Exs , Exu are the two Df (x)invariant submanifolds. When Λ = Ω, then the diffeomorphism f is called an Anosov diffeomorphism.

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Definition 1.13. The map f is an “Axiom A” diffeomorphism if (a) the nonwandering set Ω(f ) has a hyperbolic structure, and (b) the set of periodic points of f is dense in Ω(f ), i.e., P er (f ) = Ω(f ), the closure is the nonwandering set itself. Note that the “Axiom A” diffeomorphism serves as a model for the general behavior at a transverse homoclinic point where the stable and unstable manifolds of a periodic point intersect, and it plays a crucial role in the study of homoclinic bifurcations. Then Smale’s horseshoe map f has the following properties given in the form of a lemma [Smale (1967)]: Lemma 1.7. (a) The horseshoe map f is a diffeomorphism defined from the unit square D of the plane into itself. (b) The horseshoe map is one-to-one. (c) The domain of f −1 is f (D). (d) The horseshoe map f is an Axiom A diffeomorphism. Proof. The proof of this lemma is based on the geometrical formations of horseshoes as shown in Fig. 1.4 and its repetitions.  The presence of a horseshoe in a dynamical system implies the following important properties: (1) There is an infinite number of periodic orbits, especially those with arbitrarily long periods. (2) The number of periodic orbits grows exponentially with the period. (3) In any small neighborhood of any point of the fractal invariant set Λ, there is a point of a periodic orbit. The proof of the above statements can be carried out using symbolic dynamics, and similar proofs have been given for Cantor’s set. If one considers Smale’s horseshoe map as a set of topological operations, then the method for construction of an attractor of a dynamical system is Smale’s horseshoe map itself. This method consists of two operations. The first is the stretching which gives sensitivity to initial conditions, and the second is the folding which gives the attraction. An example can be found in [Galias (1997b)]: ˜0 = [−1, 1] × [−1, −0.5] , N ˜1 = [−1, 1] × [0.5, 1] , P1 = Example 1.1. Let N ˜0 and N ˜ 1 , M− = [−1, 1] × R, be the smallest vertical stripe containing N [−1, 1] × (−∞, −1) , M0 = [−1, 1] × (−0.5, 0.5) , M+ = [−1, 1] × (1, ∞) M− ,

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˜0 and M0 , and M+ are subsets of P1 lying below, between, and above N ˜ N1 , respectively. Let N0D , N0U be the lower and upper horizontal edges and N0L , N0R be the left and right vertical edges of N0 , and similarly N1D , ˜1 , respectively. N1U , N1L , N1R be the lower, upper, left, and right edges of N ˜ ˜ Then Smale’s horseshoe map is a map linear on N0 and N1 defined by

 1 1 3 ˜ 4 x − 2 , ±5 y + 4

, if (x, y) ∈ N0 (1.36) f (x, y) = 1 3 1 ˜1 , if (x, y) ∈ N 4 x + 2 , ±5 y − 4 Note that the Jacobian of f n is given by:   n (0.25) 0 n Df (x, y) = n 0 (±5)

(1.37)

The study of this example is done in Exercise 1.8 below. 1.5.2

Dynamics of the horseshoe map

The role of the horseshoe map is the reproduction of the chaotic dynamics of a flow in a small disk ∆ perpendicular to a period-T orbit Γ. The important features in the dynamics of the map can be summarized as follows: (1) When the system evolves, some orbits diverge, and the points in this disk remain close to the given periodic orbit Γ, tracing out orbits that eventually intersect the disk once again. (2) The intersection of the disk ∆ and points in its neighborhood with the given period-T orbit Γ comes back to itself every period T . When this neighborhood returns, its shape is transformed. (3) The points inside the disk ∆ consist of some that will leave the disk neighborhood and others that will continue to return. (4) The set of points that never leaves the neighborhood of the given periodT orbit Γ forms a fractal. The above features imply that Smale’s horseshoe map consists of the iterated application of both f (D) and f −1 (D). Here some important results that can be found in [Smale (1967)]: i Theorem 1.5. (a) The set Π+ = ∩i=∞ i=0 f (D) is an interval cross a Cantor-like set. −i (b) The set Π− = ∩i=∞ (D) is a Cantor-like set cross an interval. i=0 f (c) The set Π = Π+ ∩ D ∩ Π− is a Cantor-like set cross a Cantor-like set.

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(d) If we consider the map f (D), where f is the horseshoe map that contracts by λ and expands by µ,where 0 <  λ < 21 and µ > 2 + ǫ, the Hausdorff dimension of Π is log 2 log1 µ − log1 λ .

(e) The limit set Π forms an uncountable, nowhere dense set (the interior of its closure is empty) in R2 . (f ) f (D) is equivalent to the shift map in the symbolic space.

For more detail, see [Banks and Dragan (1994)]. These principal features give the idea that symbolic names can be given to all the orbits that remain in the neighborhood. If the initial disk ∆ can be divided into a small number of regions Di , i = 1, .., m and if one knows the sequence of points {x0 , x1 , ...} in which the orbit visits these regions Di , then the visitation sequence sˆ = {ˆ s0 ; sˆ1 ; ...; sˆj ; ...} composed of symbols sˆj = i if xj = f j (x0 ) ∈ Di , i = 1, .., m of the orbits provide a symbolic representation of the dynamics, known as symbolic dynamics introduced in Sec. 1.5.3. However, we have the following definition: Definition 1.14. A dynamical system is chaotic in the sense of Smale if it has horseshoes of the Smale type. Definition 1.14 with the so-called shadowing lemma [Stoffer and Palmer (1999)] was used to prove that the H´enon [H´enon (1976)] map is chaotic in the sense of Smale as shown in Sec. 2.5.6. Generally, there are no rigorous methods for finding Smale horseshoes in a dynamical system with relatively small dimension [Smale (1967), Banks and Dragan (1994)], but a few works are concerned with this topic. An example can be found in [Zgliczynski (1997b)] and is presented in Sec. 2.5.1. 1.5.3

Symbolic dynamics

Let Sm = {0, 1, ..., m − 1} be the set of non-negative successive integers from 0 to m − 1. Let Σm be the collection of all bi-infinite sequences of Sm with their elements, i.e., every element s of Σm is of the form s = (..., s−n , ..., s−1 , s0 , s1 , ..., sn , ...) , si ∈ Σm . Let s ∈ Σm be another sequence given by s = (..., s−n , ..., s−1 , s0 , s1 , ..., sn , ...) , si ∈ Σm . Then: Definition 1.15. The distance between s and s is defined as ∞ X |si − si | 1 . d (s, s) = |i| 1 + |s − s | 2 i i −∞

(1.38)

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Hence the set Σm with the distance defined as (1.38) is a metric space, and one has the following results proved in [Robinson (1995)]: Theorem 1.6. The space Σm is compact, totally disconnected, and perfect. A set having the three properties given in Theorem 1.6 is a Cantor set, which frequently appears in the characterization of the complex structure of an invariant set in a chaotic dynamical system. However, let an m-shift map σ : Σm → Σm be defined by: σ (s)i = si+1

(1.39)

Then the map σ has the following properties proved in [Robinson (1995)]: Theorem 1.7. (a) σ(Σm ) = Σm , and σ is continuous. (b) The shift map σ as a dynamical system defined on Σm has the following properties: (b-1) σ has a countable infinity of periodic orbits consisting of orbits of all periods; (b-2) σ has an uncountable infinity of nonperiodic orbits; and (b-3) σ has a dense orbit. The major result of statements (b) of Theorem 1.7 is that the dynamics generated by the shift map σ are sensitive to initial conditions, and therefore it is chaotic in the commonly accepted sense. Now, let X be a separable metric space and f be a continuous map ˜ → X, where Q ˜ ⊂ X is locally connected and compact. f :Q Assumption A [Kennedy and York (2001)]. Suppose the following assumptions hold: ˜ denoted by Q ˜ 1 and Q ˜ 2 , respectively. The A1. There exist two subsets of Q, ˜ ˜ sets Q1 and Q2 are disjoint and compact, ˜ intersects both Q ˜ 1 and Q ˜ 2, A2. Each connected component of Q ˜ with respect to f is not less than 2, A3 The cross number m of Q where the cross number m is defined as follows: ˜ for Q ˜ 1 and Q ˜ 2 is a compact subset Definition 1.16. (a) A connection Γ ˜ that intersects both Q ˜ 1 and Q ˜ 2. of Q ˜ for which (b) A preconnection γ˜ is a compact connected subset of Q f (˜ γ ) is a connection. (c) The cross number m is the largest number such that every connection contains at least m mutually disjoint preconnections.

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Using these preliminaries, the following fundamental result was proved in [Kennedy and York (2001)]: Theorem 1.8. Let f be the map satisfying Assumption A. Then there ex I ˜ ˜ ists a compact invariant set Q ⊂ Q for which f Q˜ 1 is semiconjugate to an m-shift map10 .

A more applicable version of the above result was given in [Yang and Tang (2004)] for piecewise continuous maps as follows: ˜ be a compact subset of X, and f : Q ˜ → X be a map Theorem 1.9. Let Q satisfying the following conditions: ˜ and (a) There exist m mutually disjoint subsets D1 , ..., and Dm of Q, the restriction of f to each Di , i.e., f |Di is continuous. (b)

∪m (1.40) i=1 Di ⊂ f (Dj ) , j = 1, 2, .., m ˜ Then there exists a compact invariant set K ⊂ Q such that f |K is semiconjugate to m-shift dynamics. Definition 1.16 can be generalized to an m domain D1 , ..., Dm−1 and Dm as follows: ˜ such that for each 1 ≤ i ≤ Definition 1.17. Let γ˜ be a compact subset of Q m; γ˜i = γ˜∩, Di is nonempty and compact. Then γ˜ is called a connection with respect to D1 , ..., Dm−1 and Dm . Let F˜ be a family of connections γ˜s with respect to D1 , ..., Dm−1 and Dm satisfying the following property: γ˜ ∈ F˜ =⇒ f (˜ γ ) ⊂ F˜ (1.41) i

Then F˜ is said to be an f -connected family with respect to D1 , ..., Dm−1 and Dm .

Hence the following result was obtained and proved in [Yang and Tang (2004)]: Theorem 1.10. Suppose that there exists an f -connected family F˜ with respect to D1 , ..., Dm−1 and Dm . Then there exists a compact invariant set ˜ such that f |K is semi-conjugate to m-shift dynamics. K ⊂Q A version of Theorem 1.8 was used in Sec. 4.8.2 to estimate the topological entropy of Chua’s circuit in term of half-Poincar´e maps. 10 This

σ ◦ h.

˜ I → Σm such that h ◦ f = holds if there exists a continuous and onto map h : Q

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The Sil’nikov criterion for the existence of chaos

Homoclinic and heteroclinic orbits arise in the study of bifurcations and chaos, as well as in their applications to mechanics, biomathematics, and chemistry [Aulbach and Flockerzi (1989), Balmforth (1995), Feng (1998)]. In some cases, it is necessary to determine the nature or the type of chaotic behavior resulting from a dynamical system. One of the commonly used analytic criteria for proving chaos in autonomous systems is based on the work of Sil’nikov [Sil’nikov (1965-1970), Silva (2003)], whose role is similar to that of the Li–York lemma [Li and York (1975), Kennedy et al. (2001)] in the discrete case. The resulting chaos is called horseshoe type or Sil’nikov chaos. These horseshoes gives extremely complicated behavior typically observed in chaotic systems [Guckenheimer and Holmes (1983)] as shown in the previous section. A more detailed discussion of homoclinic bifurcations is given in Sec. 2.6.3 in connection with the H´enon map [H´enon (1976)]. 1.6.1

Sil’nikov criterion for smooth systems

Consider the third-order autonomous system x′ = f (x)

(1.42)

where the vector field f : R3 −→ R3 belongs to class C r (r ≥ 1), x ∈ R3 is the state variable of the system, and t ∈ R is the time. Suppose that f has at least one equilibrium point P . Definition 1.18. (a) The point P is called a hyperbolic saddle focus for system (1.42) if the eigenvalues of the Jacobian A = Df (P ) are γ, ρ + iω, where ρ, γ < 0, and ω 6= 0. (b) A homoclinic orbit γ (t) refers to a bounded trajectory of system (1.42) that is doubly asymptotic to an equilibrium point P of the system, i.e., limt−→+∞ γ (t) =limt−→−∞ γ (t) = P . (c) A heteroclinic orbit δ (t) is similarly defined except that there are two distinct saddle foci P1 and P2 being connected by the orbit, one corresponding to the forward asymptotic time limit, and the other to the reverse asymptotic time limit, i.e., limt−→+∞ δ (t) = P1 , and limt−→−∞ γ (t) = P2 . The homoclinic and heteroclinic Sil’nikov method, namely, the Sil’nikov criterion for the existence of chaos, is summarized in the following theorem [Sil’nikov (1965-1970)]: Theorem 1.11. Assume the following:

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(i) The equilibrium point P is a saddle focus, and |γ| > |ρ| . (ii) There exists a homoclinic orbit based at P . Then (1) The Sil’nikov map, defined in a neighborhood of the homoclinic orbit of the system, possesses a countable number of Smale horseshoes in its discrete dynamics. (2) For any sufficiently small C 1 -perturbation g of f , the perturbed system x′ = g (x)

(1.43)

has at least a finite number of Smale horseshoes in the discrete dynamics of the Sil’nikov map defined near the homoclinic orbit. (3) Both the original system (1.42) and the perturbed system (1.43) exhibit horseshoe chaos. Similarly, there is also a heteroclinic Sil’nikov theorem [Sil’nikov (19651970) ] given by: Theorem 1.12. Suppose that two distinct equilibrium points, denoted by P1 and P2 , respectively, of system x ` = f (x) are saddle foci whose characteristic values γ k , ρk + iωk , (k = 1, 2) satisfy the following Sil’nikov inequalities: ρ1 ρ2 > 0, or γ 1 γ 2 > 0. Suppose also that there exists a heteroclinic orbit joining P1 and P2 . Then the system x′ = f (x) has both Smale horseshoes and the horseshoe type of chaos. 1.6.2

Sil’nikov criterion for continuous piecewise linear systems

Because we are interested in proving the existence of chaos in a continuous piecewise linear vector field, namely the so-called double-scroll system [Chua et al. (1986)], we must state a piecewise linear version [Silva (2003)] of Theorem 1.11 as follows: Theorem 1.13. Let ξ be a continuous piecewise linear vector field associated with a third-order autonomous system x′ = ξ (x), where x ∈ R3 . Assume the origin is an equilibrium point with a pair of complex eigenvalues (ρ + iω, ρ < 0, ω 6= 0) and a real eigenvalue γ > 0 satisfying |ρ| < γ. If in addition, ξ has a homoclinic orbit through the origin, then ξ can be infinitesimally perturbed into a nearby vector field ξ ′ with a countable set of horseshoes. Using this version, [Chua et al. (1986)] proved that the double-scroll family studied in Chapter 4 is chaotic by showing that the conditions of

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Sil’nikov’s Theorem 1.13 is satisfied. In particular, for proving the existence of a homoclinic orbit, they proved that there exist parameters such that the trajectory along the unstable real eigenvector associated with the origin enters the complex stable eigenspace, and hence returns to the origin as shown in Sec. 4.5. The piecewise linear version of the heteroclinic Theorem 1.12 is posed as a challenge for the reader in Exercise 1.14. 1.7

The Marotto theorem

The Marotto theorem [Marotto (1978)], is best suited for predicting and analyzing discrete chaos in higher-dimensional difference equations because in practice, the homoclinic orbit or other techniques used for predicting chaos in dynamical systems are extremely difficult to compute, whereas the snap-back repellors are relatively easy, often needing only a small number of iterations. First, consider the following one-dimensional discrete dynamical system: xk+1 = f (xk ), xk ∈ R, k = 0, 1, 2, ...

(1.44)

where the map f : R −→ R is continuous. Then we have the following definition [Li and Yorke (1975)]: Definition 1.19. A map f is chaotic (in the sense of Li and Yorke) if f has arbitrarily large periods and there exists an uncountable set S, called a scrambled set for f , such that for every x, y ∈ S and every periodic point z we have:

(1) limk−→+∞ sup

f k (x) − f k (y)

> 0 (2) limk−→+∞ inf

f k (x) − f k (y)

= 0 (3) limk−→+∞ sup f k (x) − f k (z) > 0.

If the map is n-dimensional, then we introduce the so-called snap-back repellor [Marotto (1978)] which is a generalization of the scrambled set given the Li-York definition of chaos in a one-dimensional discrete dynamical system. Indeed, consider the following nonlinear n-dimensional dynamical system: Xk+1 = f (Xk ), Xk ∈ Rn , k = 0, 1, 2, ...

(1.45)

where the map f : Rn −→ Rn is continuous. Denote by Br (P ) the closed ball in Rn of radius r centered at a point P ∈ Rn . Let f be differentiable in Br (P ). The point P ∈ Rn is an expanding fixed point of f in Br (P ) if

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f (P ) = P and all eigenvalues of Df (X) exceed 1 in absolute value for all X ∈ Br (P ). Then one has the following definition [Marotto (1978)]: Definition 1.20. (Snap-back repellor): Assume that P is an expanding fixed point of f in Br (P ) for some r > 0. Then P is said to be a snap-back repellor of f if there exists a point P0 ∈ Br (p) with P0 6= P such that f m (P0 ) = P and the determinant |Df m (P0 )| = 6 0 for an integer m > 0. Thus the Marotto theorem [Marotto (1978)] is given by: Theorem 1.14. If f is differentiable and has a snap-back repellor, the map (1.45) is chaotic in the sense of Li-Yorke, and (a) there is a positive integer N such that for each integer p ≥ N , f has a point of period-p, and (b) there is a “scrambled set” of f , i.e., an uncountable set S containing no periodic points of f such that: (b-1) f (S) ⊂ S, (b-2) for every XS ; YS ∈ S with XS 6= YS , limk−→+∞ sup f k (XS ) − f k (YS ) > 0, (b-3) for every XS ∈ S and any periodic point Yper of f ,

limk−→+∞ sup f k (XS ) − f k (Yper ) > 0; (c) there is an uncountable subset S0 of S such that for every XS0 ; YS0 ∈ S0 :

limk−→+∞ sup f k (XS0 ) − f k (YS0 ) = 0. Now consider the map



xk+1 = f (xk , byk ) yk+1 = xk

(1.46)

where f : R2 → R is differentiable. The map (1.46) can be reduced to the one-dimensional equation xk+1 = f (xk , 0)

(1.47)

when b is close to 0. Hence the following theorem was proved in [Marotto (1979a)]: Theorem 1.15. Suppose (1.47) has a snap-back repellor. Then (1.46) has a transversal homoclinic orbit for all |b| < ǫ for some ǫ > 0. Theorem 1.15 was used in [Marotto (1979b)] to prove that the H´enon map is chaotic as shown in Sec. 2.6.3.

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The verified optimization technique

The verified optimization technique was introduced in [Tibor et al. (2006)], where a new version of this method was employed with some sufficient conditions to find chaotic regions for the H´enon map [H´enon (1976)]. The method is as follows: (1) Check the set theoretical conditions of a respective theorem in a reliable way by computer programs. (2) Introduce optimization problems that provide a model to locate chaotic regions. (3) Prove the correctness of the underlying checking algorithms and the optimization model. The method given in [Zgliczynski (1997a)] works only with human interaction because of difficulties related to some mathematical relations. The actual method [Tibor et al. (2006)] finds chaotic regions automatically without any human assistance, where the techniques used are a combinations of interval arithmetic [Alefeld and Herzberger (1983), Neumaier (1990)] and adaptive branch-and-bound subdivision of the region of interest [Dellnitz and Junge (2002)]. 1.8.1

The checking routine algorithm

For state this algorithm, we need the following inputs: Υ: the respective mapping, ǫ: the user set limit size of subintervals, Q′ : the argument set to be proved, O′ : the aimed set for which Υ(Q′ ) ⊂ O′ is to be checked, and the following definition: Definition 1.21. A mapping F : I n → I m is an inclusion mapping of the mapping f : Rn → Rm if for ∀Y ∈ I n and ∀y ∈ Y : f (y) ∈ F (Y ), where I stands for the set of all closed real intervals. Hence the algorithm for an interval arithmetic based inclusion function [Ratschek and Rokne (1988)] is given by: Algorithm 1.1. 1. Calculate the initial interval I that contains the regions of interest. 2. Push the initial interval onto the stack.

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3. while the stack is nonempty, 4. Pop an interval v off the stack. 5. Calculate the width of v. 6. Determine the widest coordinate direction. 7. Calculate the transformed interval w = Υ(v). 8. if v ∩ Q′ 6= ∅ and the condition w ⊂ O′ does not hold, then 9. if the width of interval v is less than ǫ then 10. print that the condition is hurt by v and stop, 11. else bisect v along the widest side: v = v1 ∪ v2 and 12. push the subintervals onto the stack 13. endif 14. endif 15. end while 16. print that the condition is proven and stop. 1.8.2

Efficacy of the checking routine algorithm

Note that each step in the checking routine algorithm needs some additional explanations in view of optimization theory [Dellnitz and Junge (2002)]. Here are some theorems that guarantee the efficacy of the checking routine algorithm [Tibor et al. (2006)]: Theorem 1.16. Assume that the underlying mapping Υ is given by an inclusion mapping T and that the algorithm returns that the checked condition Υ(Q′ ) ⊂ O′ is fulfilled. Then the checking routine algorithm generates a subdivision of the initial interval I around the search region in such a way that for all subintervals either, (i) the subinterval does not contain a point of the argument region Q′ , or (ii) the transformed subinterval is a subset of the respective set of the condition O′ . Theorem 1.17. Assume that the underlying mapping Υ is given by an inclusion function T that has the zero convergence property, ǫ = 0, and that T (Q′ ) ⊂ O′ holds. Then the checking routine algorithm concludes after a finite number of iteration steps that the condition of chaotic behavior is fulfilled. Theorem 1.18. Assume that the underlying mapping Υ is given by an inclusion function T , ǫ = 0, and there exist a point x ∈ Q′ such that

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T (x) ∈ / O′ . Then the checking routine algorithm cannot conclude after a finite number of iteration steps whether the condition of chaotic behavior is fulfilled. An example of the application of the checking routine algorithm can be found in Sec. 2.5.3. The most crucial role of Smale’s horseshoe is its relationship to the homoclinic tangency11 discovered by Poincar´e while investigating the threebody problem of celestial mechanics [Tufillaro et al. (1987)]. Definition 1.22. (a) The invariant manifold of a map f is a set of points X such that f (X) ⊂ X 12 . (b) Given a fixed point Y0 in the invariant manifold, the manifold Y is called stable if ∀y ∈ Y , limn→∞ f n (y) → Y0 . Similarly, a manifold is called unstable if ∀y ∈ Y , limn→∞ f −n (y) → Y0 . (c) A fixed point is hyperbolic if it is the intersection of one or more stable manifolds and one or more unstable manifolds. (c) A homoclinic point is a point x, different from a fixed point, that lies on both a stable manifold and an unstable manifold of the same fixed point Q. Theorem 1.19. (Poincar´e, 1890). If there exists a single homoclinic point on a stable and an unstable invariant manifold corresponding to a particular hyperbolic fixed point, then there exist an infinite number of homoclinic points on the same invariant manifolds. Proof. The proof is done by induction on the number of homoclinic points. Assume that there exist n homoclinic points for these invariant manifolds. Let W s (Q) be the stable manifold and W u (Q) be the unstable manifold, and let x be the homoclinic point farthest from the fixed point Q along the unstable manifold W u (Q). Since W s (Q) and W u (Q) are invariant manifolds, f (x) ∈ W s (Q) ∩ W u (Q). Thus f (x) is either a homoclinic point or the fixed point Q. Since f takes a point on the unstable manifold W u (Q) away from the fixed point Q, then f (x) cannot be the fixed point. Thus it is a homoclinic point. For the same reason, it is not one of the n considered homoclinic points. Thus there exist n + 1 homoclinic points. 11 The homoclinic tangency is the tangled intersection of such invariant manifolds with a homoclinic point as shown in Fig. 1.5. 12 Of course every fixed point, with the exception of centers, will be an element of some invariant manifold.

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x f (x)

Q W s (Q) W u (Q) Fig. 1.5 Part of a homoclinic tangency: Q is the hyperbolic fixed point, W s (Q) , W u (Q) are the stable and unstable manifolds, respectively, x is a homoclinic point, and thus f (x) is also a homoclinic point.

Therefore, the number of homoclinic points on the corresponding invariant manifolds is infinite.  Hence the basic theorem about the rigorous proof of chaos for a dynamical system f : Rn → Rn was proved by [Smale (1967)]: Theorem 1.20. If f is a diffeomorphism and has a transversal homoclinic point, then there exists a Cantor set Λ ⊂ Rn in which f m is topologically equivalent to the shift automorphism for some m. The existence of the shift automorphism implies the existence of a dense set of periodic orbits and an uncountably infinite collection of asymptotically aperiodic points of the map f within the set Λ. It is not difficult to show that the homoclinic tangency has the same topological structure as the horseshoe map [Weisstein (2002)] using some geometrical illustrations. 1.9

Shadowing lemma

It is well known that there are rounding errors at every step when calculating a trajectory of a dynamical system. If the dynamical system has a chaotic attractor, these errors will grow exponentially, and the resulting orbit will differ wildly from the exact one. The so-called, shadowing lemma introduced in [Palmer (1988)] eliminates this problem and ensures

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that, under quite general conditions, there is a trajectory of the true system starting from a slightly perturbed initial state that remains in the vicinity of the computed one for all time. The general Shadowing lemma can be applied on an arbitrary Riemannian manifold as follows: Theorem 1.21. Let Ω be a Riemannian manifold, f : Ω −→ Ω, a diffeo˜ ⊂ Ω be a compact hyperbolic set for f . Then there is morphism, and let Λ ˜ such that for every δ > 0 there is an ǫ > 0 for which a neighborhood U of Λ every ǫ-orbit in U is δ-shadowed by an orbit of f . Moreover, there is a δ 0 > 0 such that, if δ < δ 0 and if the pseudo-orbit is ˜ has a local product bi-infinite, then the shadowing orbit is unique, and if Λ ˜ structure, then the shadowing orbit is in Λ. Generally, the standard use of the shadowing lemma in dynamical systems theory is to prove density of periodic points, and because our interest is the study of the shadowing lemma in real systems, namely 2-D quadratic maps and the Chua systems as an example of ODE dynamics, then Ω = Rn , n = 2, 3, and M is a Riemannian manifold. However, the following definitions help in understanding Theorem 1.21: Definition 1.23. (a) An ǫ-pseudo-orbit for f is a sequence {xn , n ∈ Z}, such that |xn − f (xn )| < ǫ for all n ∈ Z. (b) A sequence {xk , k ∈ Z} is said to be an ǫ-pseudo-periodic-orbit with period-N of f if |xk+1 − f (xk )| < ǫ and xk+N = xk for k ∈ Z13 . (c) Let {xk , k ∈ Z} and {yk , k ∈ Z} be two ǫ-pseudo-periodic-orbits. A sequence {zk , k ∈ Z} is said to be an ǫ-pseudo-connecting-orbit connecting {xk , k ∈ Z} to {yk , k ∈ Z} if (i) |zk+1 − f (zk )| < ǫ, for k ∈ Z. (ii) zk = xk for k ≤ p, and zk = yk for k ≥ q for some integers p < q. (d) In the case {xk , k ∈ Z} = {yk+τ , k ∈ Z} for some τ , the ǫ-pseudoconnecting-orbit {zk , k ∈ Z} is called an ǫ-pseudo-homoclinic-orbit. ˜ is a compact invariant set of a diffeomorphism (e) A hyperbolic set Λ ˜ admits an invariant splitf such that the tangent space at every x ∈ Λ ting that satisfies the contraction and expansion conditions given in Definition 1.12(d). 13 More

precisely, a periodic orbit {xn , n ∈ Z} is a finite set of points.

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1.9.1

Shadowing lemmas for ODE systems and discrete mappings

The definition of a shadow of an ODE system is given by: Definition 1.24. An approximate trajectory y = {yn }n∈Z with time steps {hn }n∈Z is ǫ-shadowed by a true solution if there exists a sequence of points x = {xn }n∈Z with time steps {τ n }n∈Z such that xn+1 = ϕτ n (xn ) where ϕ∆t is the ∆t-flow of the system, and |yn − xn | < ǫ and |τ n − hn | < ǫ. Now define the set l∞ (Z, Rn ) as the space of Rn -valued bounded sequences x = {xn }n∈Z with norm kxk = supn∈Z |xn |2 , and the set C 1,Lip (Ω, Ω) as the ensemble of C 1 -valued Lipschizian functions on Ω ⊂ Rn . A discrete version of Theorem 1.21 was given with its detailed proof in [Stoffer and Palmer (1999)] by: Theorem 1.22. Let Ω ⊂ Rn be open, f ∈ C 1,Lip (Ω, Ω) be injective, y = {yk }k∈Z ∈ ΩZ be a given sequence, let {Ak }k∈Z be a bounded sequence of k × k matrices, and let δ, δ 1 , m be positive constants. Assume that the operator L : l∞ (Z, Rn ) −→ l∞ (Z, Rn ) , defined by is invertible and that

Then the numbers

(Lz)k = zk+1 − Ak zk ,

−1

L ≤

r0 =

r1 =

1 kL−1 k

− δ1 +

1 kL−1 k

− δ1 +

(1.48)

1 √ . δ 1 + 2mδ

r

r

(1.49)

2δ 2

1 kL−1 k

− δ1

1 kL−1 k

− δ1

2

(1.50) − 2mδ − 2mδ

(1.51) m satisfy 0 < r0 ≤ r1 . Let ρ ∈ [r0 , r1 ]. Moreover, assume that the set ∪n∈Z Bρ (yn ) (the closure) is in Ω and that for every n ∈ Z |yk+1 − f (yk )| < δ

(1.52)

|Ak − Df (yk )| < δ 1

(1.53)

|Df (u) − Df (v)| < m |u − v| , u, v ∈ Bρ (yk )

(1.54)

|˜ x − y| < ρ.

(1.55)

Then there is a unique r0 -shadowing-orbit x = {xk }n∈Z of y. Moreover, there is no orbit x ˜ other than x with

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Here are three important things to note about the shadow of an orbit: (1) The numerical solution is shadowed if it closely follows the path of a true solution. (2) The linear growth of errors for the considered system is due to a lack of hyperbolicity in the direction of the flow in phase space [Eric and Vleck (1995)]. (3) Since the shadowing lemmas used for proving the existence of a shadow require the computation of the Jacobian of the map or solving the variational equations of the ODE’s and then estimating hyperbolicity, this method is direct [Palmer (1988), Chow and Palmer (1992)]. This method has been successfully applied to the H´enon map. See Sec. 2.5.6. 1.9.2

Homoclinic orbit shadowing

In [Coomes et al. (2005)], a new computer-assisted technique with two main components for the rigorous proof of the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rn is presented. The computation of a suitable pseudo (approximate) homoclinic orbit to a pseudo-periodic orbit was done using the global Newton’s method introduced in Sec. 1.3.3.1. Then a homoclinic shadowing theorem (Theorem 1.23, 1.24 given below) is applied to prove the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo-orbits of the H´enon map [H´enon (1976)]14 . See Sec. 2.5.6. Theorem 1.23. (Connecting Orbit Shadowing Theorem) Suppose f : Rn → Rn is a C 2 diffeomorphism, and {xk , k ∈ Z} to {yk , k ∈ Z} are two ǫ-pseudo-periodic-orbits with periods N and N ′ , respectively, of f . Let z = {zk , k ∈ Z} be an ǫ-pseudo-connecting-orbit of f connecting {xk , k ∈ Z} to {yk , k ∈ Z}. Suppose that the operator Lz defined in (1.48) is invertible and set

ǫz = 2 L−1 z ǫ (1.56)

and

 Mz = sup D2 f (u) : u ∈ Rn , ku − zk k ≤ ǫz for some k ∈ Z

14 Also,

(1.57)

note that it was shown in [Coomes et al. (2005)] that all the quantities in the hypotheses of Theorem 1.23, 1.24 are computable.

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Then if

2 2Mz L−1 z ǫ < 1,

(1.58)

(i) The pseudo-periodic orbits {xk , k ∈ Z} to {yk , k ∈ Z} are ǫz -shadowed by the unique true hyperbolic periodic orbits {vk , k ∈ Z} of period-N and {wk , k ∈ Z} of period-N ′ . (ii) The pseudo-periodic orbit {zk , k ∈ Z} is ǫz -shadowed by a unique true transversal connecting orbit {ek , k ∈ Z} connecting the true periodic orbit {vk , k ∈ Z} to the periodic orbit {wk , k ∈ Z}. In fact, limk→−∞ kek − uk k = 0, and limk→∞ kek − vk k = 0. Theorem 1.24. (Homoclinic Orbit Shadowing Theorem) Suppose f : Rn → Rn is a C 2 diffeomorphism and {xk , k ∈ Z} is an ǫ-pseudo-periodicorbit of f with period-N . Let {zk , k ∈ Z} be an ǫ-pseudo-homoclinic-orbit of f connecting {xk , k ∈ Z} to {yk+τ , k ∈ Z}, where 0 ≤ τ < N . Suppose that the operator Lz is invertible, and the set

ǫz = 2 L−1 z ǫ (1.59)

and

 Mz = sup D2 f (u) : u ∈ Rn , ku − zk k ≤ ǫz for some k ∈ Z

Then if

(

2 2Mz L−1 z ǫ < 1 kxk − xj k > 2ǫz , for j 6= k, k < N,

(1.60)

(1.61)

(i) The pseudo-periodic orbit {xk , k ∈ Z} is ǫz -shadowed by a unique true hyperbolic periodic orbit {vk , k ∈ Z} of minimal period-N . (ii) The pseudo-homoclinic orbit {zk , k ∈ Z} is ǫz -shadowed by a unique true orbit {ek , k ∈ Z}. When τ > 0, the point e0 is a transversal homoclinic point to the periodic orbit {vk , k ∈ Z} with phase shift τ . When τ = 0, the point e0 is a transversal homoclinic point to the periodic orbit {vk , k ∈ Z} with phase shift 0 provided that kzk − xk k > 2ǫz for some k with p < k < q. For the proof of Theorems 1.23, 1.24, two lemmas are required, where the first lemma establishes infinite-time shadowing of a pseudo-orbit by a true hyperbolic orbit without any uniform hyperbolicity requirement for the considered diffeomorphism. The second lemma gives a hyperbolicity-type condition under which two orbits shadowing the same pseudo-orbit must be asymptotic to each other [Coomes et al. (2005)]. An example showing how to how apply Theorem 1.23, 1.24 to the H´enon map is given in Sec. 2.6.3.

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Method based on the second-derivative test and bounds for Lyapunov exponents

This method based on the second-derivative test [Thomas and Finney (1992)] and bounds for Lyapunov exponents [Li and Chen (2004)] was introduced first in [Zeraoulia and Sprott (2008a)] for the rigorous proof of the nonexistence and existence of chaos and hyperchaos in the general 2-D quadratic map shown in Chapter 3. We give in this section the definition of the so-called second-derivative test and its properties, and we state the theorems determining the upper and lower bounds for the Lyapunov exponents of a discrete mapping. Let h : R2 −→ R, be a real function, and assume that h has continuous partial derivatives at least in the region of interest. The problem of determining the extreme values of a function, i.e., maxima or minima, is encountered in several fields such as geometry, mechanics, physics, as well as the present situation. Definition 1.25. The critical points of the function h (x, y) are solutions of the equations ∂h (x, y) ∂h (x, y) = 0, =0 (1.62) ∂x ∂y Note that Eqs. (1.62) must be solved simultaneously. However, not all critical points are maxima or minima. For example, the function h(x) = x3 has critical point at x = 0, but it is neither a maximum nor a minimum, but rather it is an inflection point. Let (xc , yc ) be a critical point, and define 

2 ∂ 2 h (x, y) (xc , yc ) . ∂x∂y (1.63) Then we have the following theorem [Thomas and Finney (1992)]: dh (xc , yc ) =

∂ 2 h (x, y) ∂ 2 h (x, y) (xc , yc ) (xc , yc ) − 2 ∂x ∂y 2

2

Theorem 1.25. (1) If dh (xc , yc ) > 0 and ∂ h(x,y) (xc , yc ) < 0, then h (x, y) ∂x2 has a relative maximum at (xc , yc ). 2 (xc , yc ) > 0, then h (x, y) has a relative (2) If dh (xc , yc ) > 0 and ∂ h(x,y) ∂x2 minimum at (xc , yc ). (3) If dh (xc , yc ) < 0, then h (x, y) has a saddle point at (xc , yc ). (4) If dh (xc , yc ) = 0, then the second-derivative test is inconclusive, and higher order tests must be used.

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The second theorem determining the upper and the lower bounds for all the Lyapunov exponents of a given n-dimensional discrete map was given in [Li and Chen (2004)] by: Theorem 1.26. If a system xk+1 = f (xk ) , xk ∈ Ω ⊂ Rn , and q kDf (x)k = kJk = λmax (J T J) ≤ N < +∞,

with a smallest eigenvalue of J T J that satisfies
λmin T J ≥ θ > 0,

(1.64)

(1.65)

where N 2 ≥ θ, then for any x0 ∈ Ω, all the Lyapunov exponents at x0 are located inside ln2θ , ln N . That is, ln θ ≤ li (x0 ) ≤ ln N, i = 1, 2, ..., n, 2

(1.66)

where li (x0 ) are the Lyapunov exponents for the map f . 1.11

The Wiener and Hammerstein cascade models

A method based on the Wiener and Hammerstein cascade models [Hunter (1986), Greblicki (1994)] is capable of determining the existence of chaos in dynamical systems. The method uses the following notions of neural networks: The three-layer feed-forward artificial neural network, the nonlinear static subsystem, the linear plant, and training of the neural network. Computer simulation given in [Xu et al. (2001)] confirms the effectiveness of this method when applied to the H´enon map [H´enon (1976)]. The basic idea of the method is to determine the parameters of the ANN by a model-reference adaptive control scheme using the available outputs of the underlying chaotic system whose formulation may be unknown. 1.11.1

Algorithm based on the Wiener model

The architecture of the Wiener model is shown in Fig. 1.6(a), and the structure of the ANN within the Wiener model that is used for the simulation is shown in Fig. 1.8. If: HN is the number of hidden units, zi,j is the weight of the connection from the ith input unit to the j th hidden node,

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ωj is the weight on the connection from the j th hidden node to the output, h is the output of the linear plant (with time delay), x (k) and y (k) are the outputs of the plant and the chaotic system, respectively, θ is either ω i or zi,j , ρ is the learning rate parameter, η is a dynamical factor, then the algorithm based on the Wiener model is as follows: Algorithm 1.2. (1) Choose a simple and completely controllable linear plant15 . (2) Choose a static nonlinear subsystem16 . (3) Calculate the following quantities:       

XHN ωj rj x=  Xj=1 n+1 zi,j vi rj = f i=0      h (k − i) , i = 0, .., n   vi = −1.0, i = n + 1

(1.67)

(4) Train the ANN by a model-reference adaptive control method using the framework shown in Fig. 1.9(a). (5) Update the weights of the ANN by minimizing the following leastsquares matching measure:

J=

1 2 (y (k) − x (k)) 2

(1.68)

15 This plant can be in the first canonical form, with the same dimension as the unknown chaotic system to be identified. If the dimension of the chaotic system is unknown, then it is possible to increase the dimension during the training process until satisfactory results are obtained. 16 Generally, one can use a simple three-layer feed-forward ANN with the sigmoidal function f (x) = 1+e1−x and a linear activation function for the hidden and output layer neurons, respectively. Fig. 1.8 shows that the inputs of the ANN are the signals from the plant outputs with a time delay, and the outputs of the ANN are trained to synchronize to the given chaotic system.

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Fig. 1.6

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(a) The Wiener model. (b) The Hammerstein model.

Fig. 1.7

The learning mechanism of the Wiener model.

according to the following equations:               

θ = θ + ρ (k) [(1 − η) D (k) + ηD (k − 1)] ρ (k) = 2λ ρ (k − 1) λ = sgn [D (k) D (k − 1)] D (k) = −∆ (k) ∂x(k) ∂θ ∆ (k) = y (k) − x (k)     rj , if θ is ωj ∂x(k)   , j = 1, HN  ∂θ =  ω r (1 − rj ) vi , if θ is zi,j  j j      h (k − i) , i = 0, .., n   vi = −1.0, i = n + 1

(1.69)

(6) If an error at the output layer is obtained, then it can be propagated backwards to update the weight of the hidden layers. An example of how to apply an algorithm based on the Wiener model to the H´enon map is discussed in Sec. 2.5.4.

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Fig. 1.8

1.11.2

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The three-layer feed-forward ANN.

Algorithm based on the Hammerstein model

The architecture of the Hammerstein model is shown in Fig. 1.9(b). Similar to the Wiener model, a simple linear plant is chosen to be completely controllable and to have the same dimension as the given chaotic system, and a simple three-layer feed-forward ANN is used to approximate the static nonlinearity, which is the same as that shown in Fig. 1.8. Again the inputs of this ANN are the feedback signals from the chaotic system outputs with time delay. Suppose the linear subsystem of the Hammerstein model can be represented as: x (k) =

n X i=1

ai x (k − i) + u (k − 1) .

(1.70)

The algorithm based on the Hammerstein model is as follows: Algorithm 1.3. (1) Choose a simple and completely controllable linear plant. (2) Choose a static nonlinear subsystem. (3) Calculate the following quantities: Xn   ai x (k − i) + u (k − 1)  x (k) =  i=1 X  HN    ωj rj x=    Xj=1 n+1 (1.71) z v r = f  i,j i j  i=0      h (k − i) , i = 0, .., n    vi = −1.0, i = n + 1

(4) Train the ANN by a model-reference adaptive control method through the framework shown in Fig. 1.8.

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(5) Update the weights of the ANN by minimizing the following leastsquares matching measure: 1 J = (y (k) − x (k))2 (1.72) 2 according to the following equations:   θ = θ − ρ∆ (k) η (k)     ∆ (k) = x (k) − y (k)   Xn    ai η (k − i) + ∂u(k−1) η (k) =  ∂θ i=1  (1.73) r , if θ is ω j j ∂u  = , j = 1, HN  ∂θ  ωj rj (1 − rj ) vi , if θ is zi,j       h (k − i) , i = 0, .., n   vi =  −1.0, i = n + 1

(6) If an error at the output layer is obtained, then it can be propagated backwards to update the weight of the hidden layers. An example of how apply an algorithm based on the Hammerstein model to the H´enon map is discussed in Sec. 2.5.4. 1.12

Methods based on time series analysis

An example of these methods is given in [Bhattacharya and Kanjilal (1999)] where the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with those derived from its surrogate counterpart to assess its chaoticity, which manifests itself in the decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process. This method is applied to the H´enon map in Sec. 2.5.5 where some criteria are used to distinguish chaos from other dynamical behaviors based on the fact that the chaotic process has a finite number of modes compared to the infinitely large number of modes in its stochastic counterpart. Let {x (k)} = {x (1) , x (2) , ...} be the series in question. The series {x (k)} is configured into matrices with different row lengths n for analysis using singular value decomposition (SVD)17 . In each case, successive n-long segments of the series are arranged into successive rows of the m × n matrix A, and A is singular-value decomposed. The quantity n is varied over a 17 Singular

value decomposition (SVD) is a factorization in the form of the product of a rectangular real or complex matrix.

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wide range to capture the embedded dynamics in {x (k)}. The matrix A with any dimension m × n is given by:   x (1) x (2) x (3) ... x (n)  x (n + 1) x (n + 2) x (n + 3) ... x (2n)   (1.74) A=  ... ... ... ... ...  x (1 + n (m − 1)) x (2 + n (m − 1)) ... ... x (nm)

Before the algorithm for this method is given and discussed, we need the following results [Golub and VanLoan (1989), Kanjilal (1995)]:

Definition 1.26. Singular value decomposition of an m × n matrix A is given by A = U ΣV T where U = [u1 , ..., um ] ∈ Rm×m and V = [v1 , ..., vn ] ∈ Rn×n are orthogonal matrices such that U T AV = Σ = [diag [σ 1 , ..., σ p ] : 0] ∈ Rm×n ,

(1.75)

where p = min (m; n), and σ 1 ≥ σ 2 ≥ .... ≥ σ p ≥ 0 are the singular values of A which are non-negative. U and V are the left and the right singular vector matrices, respectively. The decomposed conjugate vector pairs u1 and v1 have related physical meaning [Good (1969), Golub and VanLoan (1989), Kanjilal (1995)]. Theorem 1.27. (a) The left and right singular vectors form a basis for the row-space and the column-space of A, respectively. (b) The singular values of A are the positive square roots of the eigenvalues of AT A or AAT . (c) The number of nonzero singular values gives the rank of the matrix A. The matrix configuration A given by Eq. (1.74) offers some inherent advantages in characterization of the time-series {x (k)}: Indeed, Lemma 1.8. (a) If the series {x (k)} is nearly periodic with a repeating periodic pattern having periodicity n, then the m × n matrix A has nearly rank one because σ 1 ≫ σ 2 . (b) If the series {x (k)} is truly random, then A has the full rank with the singular values having comparable magnitudes. In the case where {x (k)} is nearly periodic, the vector v1 gives the pattern over periodic segments, and the elements of u1 give the scaling factors for the successive periodic segments [Kanjilal (1995)].

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Hence the distinguishing features between deterministic and nondeterministic processes are expected to be reflected in the distribution of the nonprime singular values. The proposed algorithm [Bhattacharya and Kanjilal (1999)] is as follows: Algorithm 1.4. (1) Generate a surrogate series {xsurr (k)} from the given series {x (k)} such that {xsurr (k)} is the nondeterministic counterpart of {x (k)}18 . (2) Find the different configurations of m × n matrices A and Asurr from {x (k)} and {xsurr (k)}. (3) Generate and analyze the scaled distributions of the respective singular values as follows: (3-1) Normalize the total energy in A = (aij ) given by X XX (1.76) σ 2i a2ij = QA = i

j

i

for each configuration A, preserving the Frobenius norm (where kAkF = QA ) of A. (3-2) Choose a value19 R close (which is not a limitation) to the minimum value of the maximum rank20 p. (3-3) Calculate the mean singular values21 σ m (i) , i = 1, R. (3-4) Plot the scaled distribution i2 σ m (i) versus i for i = 1, R for both {x (k)} and {xsurr (k)} allowing for the detection of determinism in {x (k)}.

First, for a purely stochastic series {x (k)}, all the singular values are isotropically distributed, and the scaled distribution i2 σ m (i) will gradually increase, tending to saturate at a high value since the singular values are arranged in a nonincreasing order. Second, for a chaotic series with increasing i, the singular values σ i will have significantly decreasing magnitudes, tending to become vanishingly small, and hence the scaled distribution i2 σ m (i) will eventually decrease, tending to saturate asymptotically at a low value. The distribution of i2 σ m (i) versus i = 1, R for {x (k)} and {xsurr (k)} are compared using the Mann-Whitney (M-W) rank-sum statistic (Z)2 [Zar (1984)], which is a nonparametric test for assessing whether two samples of observations come from the same distribution. The null hypothesis is that the two samples are drawn from a single population and therefore that their 18 The unwindowed amplitude adjusted Fourier transform (AAFT) surrogate generator [Theiler et al. (1992), Kennel and Isabelle (1992)] can be used to generate {xsurr (k)} . 19 For each configuration of A, the total energy Q is linearly mapped to R normalized A singular values; thus the normalized energy is conserved. 20 Since the maximum rank p is different for different configurations of A. 21 M different values of row length n, and M sets of R singular values are obtained.

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Fig. 1.9 (a) The learning mechanism of the Wiener model. (b) The learning mechanism of the Hammerstein model.

probability distributions are equal. Consider two distributions, namely B and D with length N1 and N2 , respectively; compute U as W =

N2 N1 X X i=1 j=1

χ (Bi − Dj )

(1.77)

where χ is a Heaviside function (i.e., χ (x) = 1 for x > 0, and χ (x) = 0 for x ≤ 0). For large N2 (which in practical terms means a few tens), the Mann-Whitney (M-W) rank-sum statistic is given by

Z=q

W − 12 N1 N2

1 12 N1 N2

.

(1.78)

(N1 + N2 + 1)

Hence one has the following result: Theorem 1.28. The Mann–Whitney rank-sum statistic Z is normally distributed with zero mean and unit variance under the null hypothesis that the two observed samples came from the same distribution. Then if we observe a |Z| value greater than 1.96 [Zar (1984)], we can reject the null hypothesis at the 95% confidence level. An example to how to apply the algorithm based on time series analysis to the H´enon map is given in Sec. 2.5.5. 1.13

A new chaos detector

The term ‘chaos detector’ was introduced in [Mcdonough et al. (1995)], and it denotes a new criterion for detecting chaos in dynamical systems just like

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the largest Lyapunov exponent and the different types of dimensions and entropies defined for dynamical systems. An example of its application to the H´enon map is given in Sec. 2.5.8. The main ideas of this method include the search for short periodic sequences, the mean squared errors, and a two-point reference set {0, 1}. Chaotic behavior is indicated by the presence of sharp isolated peaks in the resulting histogram. Note that this method has the following advantages: (1) It is simpler, numerically less intensive, and more robust than previous methods. (2) It works well with short data sets and with severe data quantization even in the presence of noise. The algorithm for this method is called MESAH because it uses the so-called MEan SquAred error histogram, and is given by the following: (1) (2) (3) (4)

Begin with a data set22 x (i) , i = 1, 2, ..., N. Let xmax = max {x (i) , i = 1, 2, ..., N }. Let xmin = min {x (i) , i = 1, 2, ..., N }. For i = 1 to N − 1 i 1h 2 2 (x (i) − xmax ) + (x (i + 1) − xmin ) . value (i) = 2

(1.79)

(5) Plot a histogram of the value array23. 1.14

Exercises

(1) (a) Find the equilibrium points of the system (1.1) modelling a threedimensional Smale-Williams attractor. (b) Determine the corresponding stable and unstable manifolds for each equilibrium point. (c) Choose an appropriate Poincar´e section S, and draw the corresponding bifurcation diagram for a chosen set of parameters showing a route to chaos. (d) Search for a hyperchaotic attractor. (e) Compare the behavior of Eq. (1.1) with the original Smale-Williams attractor. 22 x(i)

here is a component of a map. is the MESAH array of the scaled squared distances in R2 , i.e., x (i + 1) versus x (i) which gives an attractor in the sense of dynamical systems. 23 This

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(2) (3) (4) (5)

(6) (7) (8)

(9) (10) (11) (12)

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Derive the relation (1.14). Derive the Mean value form of Theorem 1.2. Prove Lemma 1.1 and Lemma 1.2. Show that the image of the vertical edges defined in (1.19) does not intersect the set N given in (1.20) and that the image of N is contained in the set which can be continuously deformed to the horizontal strip without any intersection with the horizontal edges of N . Prove Lemmas 1.3, 1.4 and 1.6(a). Prove Lemma 1.7 using the geometrical formation of Smale’s horseshoe map f . (1) Draw all the sets included in the definition of the map (1.36). ˜0 and N ˜1 . (2) Find the images of the sets N (3) Find the fixed points of the map (1.36). (4) Using the linearity of the horseshoe map, show that for every n ∈ N there is still a period-n point (u, v) of the map (1.36). (5) Show that the Jacobian of f n is given by (1.37). (6) Show that there exists a neighborhood U of the point (u, v) which does not contain other fixed points of f n . (7) Calculate the fixed point index of the pair (f n , U ), and deduce that there are 2n fixed points of the map f n . Prove Theorem 1.5 and 1.7. Find a piecewise linear version of the heteroclinic Theorem 1.12. Find bounds for Lyapunov exponents of the H´enon map using Theorem 1.26 for a = 1.4 and b = 0.3. Show the algebraic Theorem 1.27.

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Chapter 2

2-D quadratic maps: The invertible case

2.1

Introduction

The best known chaotic two-dimensional map is the H´enon map [H´enon (1976)] proposed by the French astronomer Michel H´enon in 1976. This map is a special case of the general 2-D quadratic map, and it has a very long history and is a popular example for studying dynamical systems. The most general 2-D quadratic map is given by   a0 + a1 x + a2 y + a3 x2 + a4 y 2 + a5 xy = f (x, y)  , (2.1) f1 (x, y) =  b0 + b1 x + b2 y + b3 x2 + b4 y 2 + b5 xy = g (x, y)

where (ai , bi )0≤i≤5 ∈ R12 are the bifurcation parameters. Some special cases of the map (2.1) have been used in several different ways for applications and studies [Grassi and Mascolo (1999), Newcomb and Sathyan (1983), Miller and Grassi (2001)]. Some important results about the dynamical properties, bifurcations, and stability of some special cases of the 2-D map (2.1) are given in [H´enon (1969-1976), Aronson et al. (1982), Newhouse et al. (1983), Friedland and Milnor (1989), Benedicks and Carleson (1991), Yoccoz (1990-1991), Kathryn et al. (1998), Gomez and Meiss (20032004)]. However, there are a few papers that focus on the general case of this map. For example, in [Sprott (1994)], some solutions of low-dimensional, low-order polynomial maps were classified numerically as either fixed point, limit cycle, chaotic, or unstable using Lyapunov exponent calculations, with the result that a few percent are chaotic. For the 2-D quadratic maps, this percentage is about 11.10 ± 0.36%. Furthermore, in [Sprott (1993c)], the correlation dimension was calculated for the strange attractors obtained numerically for some cases of the map (2.1), and it was found that the average correlation dimension scales approximately as the square root of 49

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the dimension of the system with a small variation. In [Sprott (1993a1993b)], a systematic search for chaotic orbits of the general 2-D quadratic map (2.1) with randomly chosen coefficients was performed using a simple computer program that produces an enormous collection of strange attractors. Some simple special cases of the general 2-D quadratic map (2.1) were studied in detail in [H´enon (1969-1976), Aronson et al. (1982), Benedicks and Carleson (1991), Sprott (2003b), Zeraoulia and Sprott (2008a)], with analytical results in [H´enon (1969-1976), Aronson et al. (1982), Benedicks and Carleson (1991)] and references therein. Some chaotic attractors for this map are illustrated in Figs. 2.1, 2.2, and 2.3, 2.4, and 2.5, where the letters A to Y denote the values of the parameters (ai , bi )0≤i≤5 ∈ R12 from −1.2 to 1.2 in steps of 0.1 [Sprott (1993a-1993b)]. These chaotic attractors represent a small selection of the approximately 1.6% of all the possible 2512 = 6 × 1016 such maps that are chaotic [Sprott (1993a,1993b)]. 2.2

Equivalences in the general 2-D quadratic maps

In this section, we discuss in some detail, with examples, the equivalence relations between the elements of the map (2.1). Let m be the number of nonzero quadratic terms in map (2.1). The analysis shows that it is possible for a map with 1 ≤ m ≤ 6 quadratic nonlinearities to be equivalent to a map with all 1 ≤ m ≤ 6 quadratic nonlinearities. First, interchanging x and y produces two kinds of equivalent maps. Thus one has the following theorem: Theorem 2.1. The two maps   xk+1 = f (xk , yk ) xk+1 = g (yk , xk ) , yk+1 = g (xk , yk ) yk+1 = f (yk , xk )

(2.2)

are topologically equivalent. This theorem facilitates study of the map (2.1) since we want to classify the attractors of the map (2.1) according to the number of nonlinearities. Since there are six nonlinearities in the map (2.1), the number of possible different forms of maps with 1 ≤ m ≤ 6 nonlinearities is 2Nm , but due to this equivalence criterion, this number will be reduced to Nm . It is clear that Nm does not indicate the number of maps because for each m there are infinity many maps since the parameters can be varied continuously.

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2-D quadratic maps: The invertible case

Fig. 2.1

Strange attractors obtained from the map (2.1) [Sprott (1993b)].

On the other hand, if we consider another different quadratic map given by

g1 (x, y) =



c0 + c1 x + c2 y + c3 x2 + c4 y 2 + c5 xy d0 + d1 x + d2 y + d3 x2 + d4 y 2 + d5 xy



,

(2.3)

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Fig. 2.2

Strange attractors obtained from the map (2.1) [Sprott (1993b)].

then we say that f1 and g1 are affinely conjugate if there exists an affine transformation h such that g1 ◦ h (x, y) = h ◦ f1 (x, y) , for all (x, y) ∈ R2 .

(2.4)

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Fig. 2.3

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Strange attractors obtained from the map (2.1) [Sprott (1993b)].

The transformation h is defined by      x e1 e2 s1 h (x, y) = + l1 l2 y s2 with the condition d = e1 l2 − e2 l1 6= 0.

(2.5)

(2.6)

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Fig. 2.4

Strange attractors obtained from the map (2.1) [Sprott (1993b)].

Note that if such a transformation h exists, then there is an equivalence relation and the set of all maps is divided into classes of topologically conjugate maps. This implies that f1 and g1 have identical topological properties, in particular, they have the same number of fixed and periodic points of the same stability types. If f1 and g1 are invertible, then the order of the

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Fig. 2.5

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Strange attractors obtained from the map (2.1) [Sprott (1993b)].

points is preserved, and if the maps are noninvertible, then the order of points is also preserved but the map h maps forward orbits of f1 onto the corresponding forward orbits of g1 . After some tedious calculations, the expressions for the coefficients (ci , di )0≤i≤5 of the map (2.3) in terms of the coefficients (ai , bi )0≤i≤5 of

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the map (2.1) are given by:          

c0 = s1 + e1 a0 + e2 b0 + αd0 + 5 c1 = αd3 + α4d+α 2 α7 +α8 α6 c2 = d + d 2

α1 +α2 d2

−l l (e a +e b )+l2 (e a +e b )+l2 (e a +e b )

c3 = 1 2 1 5 2 5 2 1d23 2 3 1 1 4 2 4     e2 (e a +e b )+e2 (e a +e b )−e e (e a +e b )  c4 = 2 1 3 2 3 1 1 4d2 2 4 1 2 1 5 2 5     c = −2e1 l1 (e1 a4 +e2 b4 )−2e2 l2 (e1 a3 +e2 b3 )+(e1 a5 +e2 b5 )(e1 l2 +e2 l1 ) 5 d2

(2.7)

and                     

2 d0 = s2 + a0 l1 + b0 l2 + βd0 + β 1d+β 2 β 4 +β 5 β3 d1 = d + d2 8 d2 = βd6 + β 7d+β 2 2 3 2 3 (a5 l1 l2 −b3 l2 −a3 l1 l2 −a4 l1 −b4 l21 l2 +b5 l1 l22 ) d3 = − d2

e21 a4 l1 −e1 e2 b5 l2 −e1 e2 a5 l1 +e22 a3 l1 +e21 b4 l2 +e22 b3 l2 d2 2(e2 a3 l1 l2 +e1 b4 l1 l2 +e1 a4 l21 +e2 b3 l22 )−e1 a5 l1 l2 −e2 b5 l1 l2 −e2 a5 l21 −e1 b5 l22 − d2

d4 =

d5 =

(2.8)

where  α0 = −e1 (−s2 e2 (b2 − a1 + s1 (a2 l1 − a1 l2 )))
+ γ 0      γ 0 = −s1 e2 (b2 l1 − b1 l2 ) − s2 e22 b1 − e21 a2     α1 = (e1 a5 + e2 b5 ) (l2 s1 − e2 s2 ) (e1 s2 − l1 s1 )    2  α2 = (e1 a4 + e2 b4 ) (e1 s2 − l1 s1 ) + γ 1      γ 1 = (e1 a3 + e2 b3 ) (e2 s2 − l2 s1 )2     α3 = −l2 (e1 a1 + e2 b1 ) + l1 (e1 a2 +
e2 b2 )  α4 = (e1 a4 + e2 b4 ) 2e1 l1 s2 − 2l12 s1 +
γ 2    γ 2 = (e1 a3 + e2 b3 ) 2e2 l2 s2 − 2l22 s1     α5 = (e1 a5 + e2 b5 ) (2l1 l2 s1 − e2 l1 s2 − e1 l2 s2 )      α6 = −e1 (e1 a2 + e2 b2 ) + e2 (e1 a1 +
e2 b1 )    2  α 7 = (e1 a4 + e2 b4 ) 2e1 l1 s1 − 2e1 s2 +  
γ3  2  γ = (e a + e b ) 2e l s − 2e s  1 3 2 3 2 2 1 3 2 2   α8 = (e1 a5 + e2 b5 ) (2e1 e2 s2 − e1 l2 s1 − e2 l1 s1 )

(2.9)

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and                    

β 0 = s2 e1 (a2 l1 + b2 l2 ) − s2 e2 (a1 l1 + b1 l2 )
+ γ 4 γ 4 = −s1 l2 l1 (b2 − a1 ) + s1 b1 l22 − a2 l12 β 1 = (a5 l1 + b5 l2 ) (l2 s1 − e2 s2 ) (e1 s2 − l1 s1 ) + γ 5 2 γ 5 = (a4 l1 + b4 l2 ) (e1 s2 − l1 s1 )
2 2 β 2 = (a3 l1 + b3 l2 ) −2e2 l2 s1 s2 + e2 s2 +
l22 s21 β 3 = − a1 l1 l2 − b2
l1 l2 − a2 l12 + b1 l22
   β 4 = (a4 l1 + b4 l2 ) 2e1 l1 s2 − 2l12 s1 + (a3 l1 + b3 l2 ) 2e2 l2 s2 − 2l22 s1     β 5 = (a5 l1 + b5 l2 ) (−e1 l2 s2 − e2 l1 s2 + 2l1 l2 s1 )     β 6 = e2 a1 l1 − e1 a2 l
1 − e1 b2 l2 + e2 b1 l2  
  2 2  β = (a l + b l 4 1 4 2 ) 2e1 l1 s1 − 2e1 s2 + (a3 l1 + b3 l2 ) 2e2 l2 s1 − 2e2 s2  7   β 8 = (a5 l1 + b5 l2 ) (2e1 e2 s2 − e1 l2 s1 − e2 l1 s1 ) (2.10) According to the values of e1 , e2 , l1 , l2 , s1 , and s2 and the condition (2.6), there exists an infinity of affine transformations that convert the general quadratic map of the plane (2.1) to another one of the form (2.3). Hence we have proved the following theorem: Theorem 2.2. If (2.6) holds and the vector (ci , di )0≤i≤5 ∈ R12 has the values given in (2.7) and (2.8), then the map f1 (x, y) given by (2.1) is conjugate to the map g1 (x, y) given by (2.3). Note that if e1 = 0, e2 = 1, l1 = 1, l2 = 0, and si = 0, i = 1, 2, we get Theorem 2.1. If e1 = 0, e2 = 1, l1 = −1, l2 = 0, and si = 0, i = 1, 2, then the resulting attractors are rotated 90 degrees counterclockwise from the original given by the map (2.1). If e1 = −1, e2 = 0, l1 = 0, l2 = −1, and si = 0, i = 1, 2, then the resulting attractors are rotated through 180 degrees, and through 270 degrees if e1 = 0, e2 = −1, l1 = −1, l2 = 0, and si = 0, i = 1, 2, . More generally, if h(x, y) is a rotation by angle θ, then the attractors are also rotated by the same angle θ. As a test for Theorem 2.2, consider the H´enon map [H´enon (1976)] given by  xk+1 = 1 + a2 yk + a3 x2k (2.11) yk+1 = xk as shown in Fig. 2.6(a). Then all maps of the form  (−e1 a2 l1 +e2 l2 )xk +(e21 a2 −e22 )yk e l2 a x2 +e e2 a y 2 −2e e l a x y  + 1 2 3 k 1 2 3d2k 1 2 2 3 k k xk+1 = e1 + d (−a2 l21 +l22 )xk +(e1 a2 l1 −e2 l2 )yk l l2 a x2 +e2 l a y 2 −2e a l l x y  y =l + + 12 3 k 21 3 k 2 312 k k k+1

1

d

d2

(2.12)

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(a)

(b) Fig. 2.6 (a) Chaotic attractor of the H´enon map with a3 = −1.4 and a2 = 0.3. (b) Chaotic attractor of the map (2.15) with a4 = 0.59948 and a1 = 1. [Zeraoulia and Sprott (2008a)].

are affinely conjugate to the H´enon map (2.11). As a verification of the formulas (2.12), we choose e1 = 1, e2 = 0, l1 = 0, l2 = 1, and si = 0, i = 1, 2, i.e., the identity transformation, for which rigorous substitutions in the map (2.12) gives the H´enon map (2.11). Note that any quadratic planar map with a constant determinant of the Jacobian matrix can be put into the form of the H´enon map [H´enon (1969-1976)] by an affine coordinate transformation. For example, the parameters (ai , bi )0≤i≤5 ∈ R12 lead to

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the following conditions: a3 b 5 = b 3 a5 , a5 b 4 = a4 b 5 a3 b4 = a4 b3 , a1 b2 − a2 b1 6= 0 2a3 b2 − 2a2 b3 + a1 b5 − b1 a5 = 0 2a1 b4 − 2b1 a4 − a2 b5 + b2 a5 = 0.

(2.13)

Indeed, the determinant of the Jacobian matrix of the map (2.12) has a constant value (−a2 ) . On the one hand, Theorem 2.2 indicates that it is possible that a quadratic map with m nonlinearities is topologically equivalent to another map with j nonlinearities, where 1 ≤ m, j ≤ 6. For example, the H´enon map (2.11) is topologically equivalent to a map with one nonlinearity x2 if e1 l2 6= 0, l1 = 0, e2 = 0, and si = 0, i = 1, 2, and to a map with six nonlinearities if e1 6= 0, e2 6= 0, l1 6= 0, l2 6= 0, e1 l2 − e2 l1 6= 0, and si = 0, i = 1, 2, and so on. On the other hand, all maps of the form  (e1 a1 l2 +e2 l2 )xk −(e1 e2 a1 +e22 )yk e a l2 x2 +e3 a y 2 −2e2 a l x y  + 1 4 1 k 1 4d2k 1 4 1 k k xk+1 = e1 + d (a1 l1 l2 +l22 )xk −(e2 a1 l1 +e2 l2 )yk a l3 x2 +e2 a l y 2 −2e a l2 x y  y + 41 k 1 41 k 1 41 k k =l + k+1

1

d

d2

(2.14) are affinely conjugate to the following map given in [Zeraoulia and Sprott (2008a)]:  xk+1 = 1 + a1 xk + a4 yk2 . (2.15) yk+1 = xk

As a verification of the formulas (2.14), we choose e1 = 1, e2 = 0, l1 = 0, l2 = 1, and si = 0, i = 1, 2, i.e., the identity transformation, which gives by rigorous substitutions in the map (2.14) the formula for the map (2.15). A chaotic attractor of the map (2.15) is shown in Fig. 2.6(b).

2.3

Invertibility of the map

In this section, we discuss the invertibility of the map (2.1), and state the major known theorems discussing conditions about the existence and forms of the reversors. For this purpose we use the so-called Jacobian conjecture, which states that the 2-D quadratic map of the form (2.1) is an automorphism if and only if the determinant of the Jacobian matrix (2.18) given below is a nonzero element of R, namely, we have the following definition: Definition 2.1. A map f : Rn −→ Rn is said to be a Cremona map if it is a polynomial map with constant Jacobian.

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and the following theorem: Theorem 2.3. (Real Jacobian conjecture) Let f : Rn −→ Rn be a Cremona map. Then f is bijective and has a polynomial inverse. If the real Jacobian conjecture is true, then the degree of the inverse of a Cremona map has an upper bound of 2n−1 according to the following theorem proved in [Bass et al. (1982)]: Theorem 2.4. Let f : Rn −→ Rn be a Cremona map, then the upper bound of the degree of its inverse is 2n−1 . In our case of interest with n = 2, the inverse of a quadratic areapreserving mapping is quadratic as was discussed by [H´enon (1976)]. This result was generalized by [Moser (1994)] who showed that quadratic symplectic mappings in any dimension have quadratic inverses, especially, any quadratic symplectic mappings of the plane have quadratic inverses. Also, note that the condition of the real Jacobian conjecture is easily shown to be necessary, but proving sufficiency has been an open problem since [Keller (1939)]. The Jacobian conjecture is one of Smale’s problems [Smale (19982000)]. There have been some published incorrect proofs for this conjecture [Bass (1989), Becker et al. (1993), Hochster (2004)]. For example, the map (x′ , y ′ ) = (tan(x), y) has a nonzero Jacobian, sec2 x, and yet it is not invertible. The actual statement of the map (2.1) requires consideration of polynomial maps on R2 since there the Jacobian is not zero, but it must be constant. Another example is the map x′ = x(1 + x2 ), which has a nonzero Jacobian on R and is invertible, but its Jacobian is not constant. According to the Friedland–Milnor classification theorem [Friedland and Milnor (1989)], each polynomial automorphism of R2 is either conjugate to a polynomial automorphism of R2 with dynamical degree 1 and trivial dynamics, or to a finite composition of generalized H´enon mappings g = g1 ◦ ... ◦ gm with dynamical degree greater than 1 and nontrivial dynamics. Indeed, let G be the set of polynomial automorphisms of R2 , A be the set of affine automorphisms of R2 , and S be the set of automorphisms having the form:   αx + p (y) (x, y) → , (2.16) βy + γ where α, β ∈ R∗ and p is a real polynomial function. To state the FriedlandMilnor theorem, we need the following definition:

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Definition 2.2. A generalized H´enon mapping Hg is a polynomial diffeomorphism of the form   p (x) + y Hg (x, y) = , (2.17) bx where b ∈ R∗ and p is a real polynomial function with degree greater than or equal 2. Hence one has the following result [Friedland and Milnor (1989)]: Theorem 2.5. Each polynomial automorphism of R2 is either conjugate to a polynomial automorphism of A∪S, or to a finite composition of generalized H´enon mappings Hg = H1 ◦ ... ◦ Hm (m ≥ 1) of the form (2.17). Another important result that deals with the form of the inverse is given in [Gomez and Meiss (2004)] where it was shown that reversible automorphisms also have at least two basic reversors that are also either elementary or affine, namely the following result: Theorem 2.6. Suppose f is a nontrivial reversible automorphism. Then f possesses a reversor of order 2n in G and is conjugate to one of the following classes: −1 (1) τ −1 ω ◦ Hg ◦ τ ω ◦ Hg −1 −1 (2) τ ω ◦ Hg ◦ e2 ◦ Hg −1 (3) e−1 1 ◦ t ◦ Hg ◦ e 2 ◦ H g ◦ t where the map Hg is a composition of generalized H´enon transformations (2.17), τ ω is the affine reversor, τ ω (x, y) = (ωy, x), such that ω is a primitive nth root of unity, and the maps e1 , e2 are elementary reversors, ei (x, y) = (pi (y) − δ i x, ǫi x) , i = 1, 2, where pi (ǫi ) = δ i pi (y) , and ǫ2i , δ 2i are primitive nth roots of unity, i.e., they are elements of the subset Un = {z ∈ C, z n = 1}, and t (x, y) = (y, x) is the simple affine permutation. A detailed proof of Theorem 2.6 with many examples can be found in [Gomez and Meiss (2004)]. For the map (2.1), the Jacobian matrix is given by:   a1 + 2a3 x + a5 y a2 + 2a4 y + a5 x J (x, y) = . (2.18) b1 + 2b3 x + b5 y b2 + 2b4 y + b5 x Then one has det J (x, y) = δ = ξ 1 x2 + ξ 2 y 2 + ξ 3 xy + ξ 4 x + ξ 5 y + ξ 6 ,

(2.19)

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where

        

ξ 1 = 2a3 b5 − 2b3 a5 ξ 2 = 2a5 b4 − 2a4 b5 ξ 3 = 4a3 b4 − 4a4 b3  ξ 4 = 2a3 b2 − 2a2 b3 + a1 b5 − b1 a5     ξ  5 = 2a1 b4 − 2b1 a4 − a2 b5 + b2 a5   ξ 6 = a1 b 2 − a2 b 1 .

(2.20)

Thus, the map (2.1) is invertible if and only if ξ 1 x2 + ξ 2 y 2 + ξ 3 xy + ξ 4 x + ξ 5 y + ξ 6 6= 0, for all (x, y) ∈ R2 , and this is possible if and only if ξ 1 x2 + ξ 2 y 2 + ξ 3 xy + ξ 4 x + ξ 5 y + ξ 6 > 0 or ξ 1 x2 + ξ 2 y 2 + ξ 3 xy + ξ 4 x + ξ 5 y + ξ 6 < 0 for all (x, y) ∈ R2 . Thus one has the following theorem: Theorem 2.7. The map (2.1) is invertible if and only if one of the following conditions holds:  ξ 1 > 0, ξ 23 − 4ξ 1 ξ 2 < 0 (2.21) 4ξ 1 ξ 2 ξ 6 + ξ 3 ξ 4 ξ 5 − ξ 1 ξ 25 − ξ 2 ξ 24 − ξ 23 ξ 6 > 0 or 

ξ 1 < 0, ξ 23 − 4ξ 1 ξ 2 < 0 4ξ 1 ξ 2 ξ 6 + ξ 3 ξ 4 ξ 5 − ξ 1 ξ 25 − ξ 2 ξ 24 − ξ 23 ξ 6 > 0.

(2.22)

An important result of Theorem 2.7 is that if conditions (2.21) or (2.22) holds, then the H´enon family [H´enon (1969-1976)] is invertible. Two important questions arise in this situation. The first is about the cases where the inverse map is also quadratic, and the second is about where the inverse map is conjugate to the original map. It is well known that the H´enon map satisfies these criteria [H´enon (1969-1976)]. The investigation of these two questions is not an easy problem because it depends on the solvability of the two equations f (x, y) = z and g (x, y) = u, where (x, y) are the coordinates of the map (2.1) and (z, u) are the coordinates of its inverse map in cases where it exists. For the investigation of some special cases, see [Friedland and Milnor (1989), Yoccoz (1990-1991), Gomez and Meiss (2003), Gomez and Meiss (2004)] discussed above. In [Gomez and Meiss (2003)], a normal form composed of a sequence of generalized H´enon maps together with two simple involutions is obtained for automorphisms of the plane that are reversible by an involution that is also in the group of polynomial automorphisms. Also the coefficients in the normal form are unique up to finitely many choices.

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The H´ enon map

In this section, we discuss the dynamics of the H´enon map [H´enon (1976)] as a unique representation of the subset of Cremona maps in R2 up to affine transformations. However, the famous H´enon map given by   1 − ax2 + y h (x, y) = (2.23) bx is the subject of a many works that deal with its important and unusual properties. The map (2.23) is a simplified model of the Poincar´e map for the Lorenz model [H´enon (1976)]. In this section, several forms of the H´enon map are used to state the corresponding results, and the these results for the map (2.23) can be deduced using affine or linear transformations. Lemma 2.1. If a < −(1 − b)2 /4, then all orbits tend to infinity, and therefore, there are no interesting dynamics in bounded domains. Proof.

Use Theorem 2.7.



We summarize in the following some of these properties, and the reader can reconsider them as advanced exercises. First, note that the H´enon map has two fixed points, it is invertible and conjugate to its inverse, and its inverse is also a quadratic map. The parameter b is a measure of the rate of area contraction (dissipation), and the H´enon map is the most general 2-D quadratic map with the property that the contraction is independent of the variables x and y. For b = 0, the H´enon map reduces to the quadratic map, which is conjugate to the logistic map. Bounded solutions exist for the H´enon map over a range of a and b values, and a portion of this range (about 6%) yields chaotic solutions. For a = 1.4 and b = 0.3, the H´enon map has the chaotic attractor shown in Fig. 2.6(a) with a correlation dimension 1.25 ± 0.02 [Grassberger and Procaccia (1983)] and capacity dimension 1.261 ± 0.003 [Russell et al. (1980)]. A numerical algorithm for estimating generalized dimensions for large negative q for the H´enon map was given in [Pastor-Satorras and Riedi (1996)] because the standard fixed-size box-counting algorithms are inefficient for computing generalized fractal dimensions in the range of q < 0. Some improved results for the calculation of the correlation dimension, the largest Lyapunov exponent, and Kaplan–Yorke dimension can be found in the web pages: http://sprott.physics.wisc.edu/chaos/henongp.htm and http://sprott.physics.wisc.edu/chaos/henondky.htm, respectively.

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Methods for locating chaotic regions in the H´ enon map

In addition to the classical numerical methods used to predict chaotic behavior in dynamical systems, especially well known physical criteria (which are omitted here) used for proving chaos in dynamical systems, we present in this section some new and recent methods concerning the prediction of chaos in the H´enon map. An introduction to these methods was given in Chapter 1. 2.5.1

Finding Smale’s horseshoe maps

In this section, we state the result given in [Zgliczynski (1997b)] where a computer-assisted proof of horseshoe dynamics1 for the 7th iterate of the classical H´enon map (2.23) is presented by calculation of the Lipschitz constant. The determination of an upper bound for the rounding error reduces the task to investigating a finite number of points on a dense enough grid which is defined by a shift map and some inequalities. The introduction of this method is done in Sec. 1.5, with the remark that the proof is a combination of topological results and fixed point index introduced in Sec. 1.4 with computer-assisted computations introduced in Sec. 1.3.3. Z Let us reconsider the notations of Sec. 1.5. Let Σ2 = {0, 1} . On Σ2 , we consider the shift map σ given by σ (α)i = αi+1 . Let S0 , S1 ⊂ R2 be two parallelograms with vertices  p1 = (0.46, 0.01) , p2 = (0.595, 0.28) (2.24) p3 = (0.691, 0.28), p4 = (0.556, 0.01) and  q1 = (0.588, 0.01) , q2 = (0.723, 0.28) (2.25) q3 = (0.755, 0.28) , q4 = (0.62, 0.01) , respectively. Let us put S = S0 ∪ S1 .

(2.26)

Then the following theorem was proved in detail in [Zgliczynski (1997b)]: Theorem 2.8. For all parameter values in a sufficiently small neighborhood of (a, b) = (1.4, 0.3), the mapping
π : Inv S, h7 → Σ2 given by (π (x))i = j if h7i ∈ Sj , for i ∈ Z, j = 0, 1

(2.27)

1 This

orbits.

implies the existence of symbolic dynamics and an infinite number of periodic

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is onto. Moreover the preimage of any periodic sequence contains periodic points of h7 . Throughout the proof of Theorem 2.8, two parallelograms intersecting the numerically observed strange attractor were found. This indicates the existence of horseshoe dynamics for the 7th iterate of h. 2.5.2

Topological entropy

The existence of an infinite number of homoclinic and heteroclinic trajectories for the H´enon map (2.23) for a = 1.4, b = 0.3 was first given in [G´omez and Sim´o (1983)] using standard numerical calculations, and in [Galias and Zgliczynski (2001)] using a linking of topological tools with a local hyperbolic behavior. Also a method for computation of the lower bound of the topological entropy based on the covering relations involving different iterations of the map gives that the topological entropy of the H´enon map (2.23) is larger than 0.3381. Now, if f : X → X is a map, then one has the following definition: Definition 2.3. A set E ⊂ X is called (n, ǫ)-separated if for every two different points x, y ∈ E, there exists a 0 ≤ j < n such that the distance between f j (x) and f j (y) is greater than ǫ. Let us define the number sn (ǫ) as the cardinality of a maximum (n, ǫ)-separated set: sn (ǫ) = max {card E : E is (n, ǫ) − separated} (2.28) The number 1 (2.29) H (f ) = lim lim sup log sn (ǫ) ǫ→0 n→∞ n 2 is called the topological entropy of f . In a topological sense, a dynamical system is called chaotic if its topological entropy is positive. For the H´enon map (2.23), the upper and lower bounds for the topological entropy were estimated based on work given in [Misiurewicz and Szewc (1980), Zgliczynski (1997a), Galias (1998b), Stoffer and Palmer (1999)] using the interval technique introduced in Sec. 1.3.3 with the following result: Theorem 2.9. The topological entropy H (h) of the H´enon map (2.23) is located in the interval 0.3381 < H (h) ≤ log 2 < 0.6932. (2.30) 2 If

f is an Axiom A diffeomorphism, then H (f ) = limn→∞ sup is the number of fixed points of f n [Bowen (1971)].

log C(f n ) , n

where C (f n )

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More generally, let f be a continuous two-dimensional map. Let N1 , N2 , ..., Np , p represent pairwise disjoint quadrangles. For each Ni , we choose two opposite horizontal edges, and the two others are called vertical [Galias (2001)]: Definition 2.4. We say that Ni f -covers Nj , and we use the notation f

Ni =⇒ Nj if (i) the image of Ni under f has an empty intersection with the horizontal edges of Nj , and (ii) the images of the vertical edges of Ni have empty intersections with Nj and they are located geometrically on the opposite sides of Nj . Thus, the following theorem was proved in [Galias (2001)]: Theorem 2.10. Let N1 , N2 , ..., Np be pairwise disjoint quadrangles. Let A = (aij )pi,j=1 be a square matrix, where

aij =

(

f

1, if Ni =⇒ Nj 0, otherwise.

(2.31)

Then f is semiconjugate with the subshift on p symbols, with the transition matrix A. Theorem 2.11. The topological entropy of the map f is not smaller than the logarithm of the dominant eigenvalue of the matrix A defined by Eq. (2.31): H (f ) ≤ log λ1

(2.32)

In [Galias (2001)], the same method of interval arithmetic was used for determining all the periodic orbits of period n ≤ 30 of the H´enon map (2.23). Indeed, if Qn is the number of periodic orbits with period-n, Pn is the number of fixed points of hn , Q≤n is the number of periodic orbits with period smaller or equal to n, P≤n is the number of fixed points of hi for i ≤ n, and Hn = n−1 log(Pn ) is an estimate of topological entropy based

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on Pn , then the following results are obtained: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Qn 1 1 0 1 0 2 4 7 6 10 14 19 32 44

Pn 1 3 1 7 1 15 29 63 55 103 155 247 417 647

Q≤n 1 2 2 3 3 5 9 16 22 32 46 65 97 141

P≤n 1 3 3 7 7 19 47 103 147 257 411 639 1055 1671

Hn 0.00000 0.54931 0.00000 0.48648 0.00000 0.45134 0.48104 0.51789 0.44526 0.46347 0.45849 0.45912 0.46408 0.46231

(2.33)

and n 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Qn 72 102 166 233 364 535 834 1225 1930 2902 4498 6806 10518 16031 24740 37936

Pn 1081 1695 2823 4263 6917 10807 17543 27107 44391 69951 112451 177375 284041 449516 717461 1139275

Q≤n 213 315 481 714 1078 1613 2447 3672 5602 8504 13002 19808 30326 46357 71097 109033

P≤n 2751 4383 7205 11399 18315 29015 46529 73479 117869 187517 299967 476923 760909 1209777 1927237 3065317

Hn 0.46571 0.46471 0.46739 0.46432 0.46535 0.46440 0.46535 0.46398 0.46525 0.46481 0.46521 0.46485 0.46507 0.46485 0.46495 0.46486

(2.34)

Also, the diameter of intervals for which the existence and uniqueness of periodic orbits was proved and given in Table 3 of [Galias (2001)]. Also, the following results were proved:

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Fig. 2.7 Estimation of the topological entropy of the H´ enon map for a = 1.4, b = 0.3 based on the number of low-period cycles given in (2.33) and (2.34). Adapted from [Galias (2001)].

Lemma 2.2. (a) There are no period-three and period-five orbits for the H´enon map within the trapping region for the values a = 1.4, b = 0.3. (b) There are exactly 109033 periodic orbits with period n ≤ 30 and there are 3065317 points belonging to these orbits for the H´enon map with the values a = 1.4, b = 0.3. From (2.33) and (2.34) one can see that H n (h) is almost constant for n ≥ 10 as shown in Fig. 2.7. Thus: Lemma 2.3. The topological entropy of the H´enon map is close to 0.465.

2.5.3

The verified optimization technique

This technique is due to Tibor et al. [Tibor et al. (2006)], and it was introduced in Sec. 1.8 where a new version of this method was employed with some sufficient conditions to find chaotic regions for the H´enon map. The method is as follows: First, check the set theoretical conditions of a respective theorem in a reliable way by computer programs, and second, introduce optimization problems that provide a model to locate chaotic regions, and third, prove the correctness of the underlying checking algorithms and the optimization model. This method was applied successfully to the H´enon map (2.23) where regions of chaotic behavior were checked and located for the usual parameters. As a result of the application of the checking routine algorithm, the following theorem was proved [Tibor et al. (2006)]:

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Theorem 2.12. The mapping h7 is [1.377599, 1.401300]× [0.277700, 0.310301].

2.5.4

chaotic

when

(a, b)

∈

The Wiener and Hammerstein cascade models

This method was introduced in Sec. 1.11, and it is based on the Wiener and Hammerstein cascade models and is capable of determining chaos in dynamical systems. The method uses the following notions of neural networks: The three-layer feed-forward artificial neural network, the nonlinear static subsystem, the linear plant, and training of the neural network. Computer simulation given in [Xu et al. (2001)] confirms the effectiveness of this method when it is applied to the H´enon map. Indeed, the system is described by the equation: y (k + 1) = −ay 2 (k) + by (k − 1) + 1, k = 1, 2, ...

(2.35)

For a = 1.4, b = 0.3, the map (2.35) has a strange attractor as shown in Fig. 2.8(a). The simple linear plant was taken here as x (k + 1) = px (k) + qx (k − 1) , k = 1, 2, ...,

(2.36)

where the constants p and q should satisfy some stability condition and may also be adjusted to obtain a desired dynamic. The p and q were chosen such that the plant is stable and has one equilibrium point (0, 0) . If the Wiener model was used to identify the H´enon system with p = 0.3, q = 0.01, then a 4 × N × 1 neural network with inputs h (k) , h (k − 1) , h (k − 2) and −13 was chosen, and N was determined by trial and error. If the Hammerstein model was used to identify the H´enon system with p = 0.2, q = 0.1, then a 3 × N × 1 neural network with inputs y (k) , y (k − 1) , y (k − 2) and −1 was chosen. The results are shown in Figs. 2.8(b) and 2.9(a) with 1000 training points taken arbitrarily from the output of the unknown H´enon attractor. From Fig. 2.9, one can remark that the identified H´enon attractor based on the Hammerstein model is closer to the true attractor than the one obtained from the Wiener model. Another interesting phenomenon is that 70 training points rather than 1000 are sufficient for the Hammerstein model to learn the H´enon attractor as shown in Fig. 2.9(b). 3 The

input −1 is the bias input emulating the threshold of neurons, which enables the system to approximate the zero-to-nonzero mapping in the H´enon attractor.

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Fig. 2.8 (a) The phase plot of the H´ enon attractor. (b) The phase plot of the plant after learning using 1000 points based on the Wiener model. Adapted from [Xu et al. (2001)].

Fig. 2.9 (a) The phase plot of the plant after learning using 1000 points based on the Hammerstein model. (b) The phase plot of the plant after learning using 70 points based on the Hammerstein model. Adapted from [Xu et al. (2001)].

2.5.5

Methods based on time series analysis

This method was introduced in Sec. 1.12. For the H´enon map (2.23) with a = 1.4, b = 0.3, the scaled distribution i2 σ m (i) for the original time series and its surrogate are almost indistinguishable as shown in Fig. 2.10 [Bhattacharya and Kanjilal (1999)]. If the algorithm for this method is repeated

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1500

{x surr (k)} {x (k)}

1000

i 2 σ m (i) 500

0 5

10

15

20

25

i Fig. 2.10 The scaled distribution i2 σ m (i) for the original data {x (k)} and for the corresponding surrogate data {xsurr (k)} sets. Adapted from [Bhattacharya and Kanjilal (1999)].

after bidirectionally filtering [Kanjilal (1995)], then the data sequence with a low-pass filter has a single pole at 0.4. From Fig. 2.10, one can see that the i2 σ m (i) profile droops while that for the respective surrogate series remains at a higher value with increasing i, Z = 2.0111, capturing the determinism with 95% confidence. Because chaos in the data cannot be induced by linear filtering, the detection of chaos in the series is therefore confirmed.

2.5.6

The validated shadowing

This method was introduced in Sec. 1.9, and it is a computer-assisted method with the principle that the given dynamical system admits the shift map as a subsystem. This method was employed in [Stoffer and Palmer (1999)] to prove the existence of a chaotic attractor for the H´enon map (2.23) in some interval including the classical parameter values, i.e., the following theorem has been proved: Theorem 2.13. For (a, b) ∈ [1.3996, 1.4004] × [0.2996, 0.3004] the H´enon map (2.23) is chaotic in the sense of Smale. The 25th iterate of the H´enon map (2.23) admits the Bernoulli shift as a subsystem.

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The method of fixed point index

This method was introduced in Sec. 1.4. In [Zgliczynski (1997b)], the following result was proved for the H´enon map (2.23) for a = 1.4, b = 0.3: Theorem 2.14. For all parameter values in a sufficiently small neighborhood of (a, b) =
(1.4, 0.3), there exists a set N and a continuous surjection 7 π : Inv N, h → Σ2 such that π ◦ h = σ ◦ π. Moreover, the preimage of any periodic sequence from Σ2 contains periodic points of h7 . Proof. The proof of Theorem 2.14 is based on a consideration of the values K = 2, d = 0.01, u = 0.28, a0 = 0.455, a1 = 0.551, a2 = 0.583, and  11 a3 = 0.615. Then the transition matrix is A = , and the set Ei′ are 11 given in the new coordinates  x1 , y1 defined by x1 = x − 0.5y (2.37) y1 = y by the following  formulas: ′ E0 = {x1 < a0 and y1 > d}  (2.38) E1′ = E1  ′ E2 = {y1 < −0.01} ∪ {x1 > a3 and y1 ≤ u} . Then using theorems given in Sec. 1.4, one has that for all parameter values in a sufficiently small neighborhood of (a, b) = (1.4, 0.3), the following conditions are satisfied4  h7 (N ) ⊂ E ′ ∪ N    7 ′   h (L (N0 )) ⊂ E2 7 ′ (2.39) h (L (N1 )) ⊂ E0  7 ′   h (R (N )) ⊂ E 0  0  7 h (R (N1 )) ⊂ E2′  2.5.8

A new chaos detector

This term was introduced in [Mcdonough et al. (1995)], and it means a new criterion for detecting chaos in dynamical systems just like the largest Lyapunov exponent and the different types of dimensions and entropies. An example of its application to the H´enon map (2.23) is shown in this section. Indeed, Fig. 2.11 shows the MESAH histogram of the x coordinate of the 4 In this case, rigorous numerical verification requires the calculation of h7 for around 60000 points.

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Fig. 2.11 The MESAH histogram of the x coordinate of H´ enon map (2.23). Adapted from [Mcdonough et al. (1995)].

H´enon map (2.23) for a = 1.4, b = 0.3, and x ∈ [−1.2, 1.2] , y ∈ [−0.4, 0.4] with an initial starting point at x (1) = 0.7511902183349609 and y (1) = 0.2728882217590332. It is clear from Fig. 2.11 that the H´enon map (2.23) has at least five different peaks that indicate a very complex dynamical behavior with many repeated sequences. Note that this histogram does not depend on the choice of initial conditions, and the broad hump between 1.8 and 3.8 falls quickly above 4.0.

2.6

Bifurcation analysis

In this section, we discuss different bifurcation phenomena observed in the H´enon map in all its simplest forms. 2.6.1

Existence and bifurcations of periodic orbits

Before discussing the different type of bifurcations observed in the H´enon map, we must first establish their existence. The H´enon map in all its simplest forms has at most 2n periodic points of period-n [Moser (1960)]. The existence of periodic orbits is given in [Galias (1998)], where a rigorous study of the H´enon map (2.23) was performed, and a computer-assisted proof given in Sec. 1.3.3 confirms the existence of symbolic dynamics for h2 and h7 and the existence of periodic orbits of all periods, where hi is ith composition of the H´enon map h. On the one hand, all periodic orbits with period n ≤ 26 for the H´enon map (2.23) are given in [Galias (1999)].

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In [Hitzl and Zele (1985)], sufficient conditions for the existence of periods 1 to 6 are also given analytically for the region −2 ≤ a ≤ 6, −1 ≤ b ≤ 1, with the determination of the associated bifurcations to stable cycles of twice the period from 2 to 12. The existence and uniqueness of low-period cycles within the H´enon attractor and estimation of topological entropy for the H´enon map (2.23) was studied in [Galias (1998b)]. On the other hand, the so-called rotation numbers defined for dynamical systems are employed in numerical studies of the accessible periodic orbits for the H´enon map (2.23) with some theoretical results on rotary homoclinic tangencies which describe some properties such as the appearance of the accessible saddles and their organization [Alligood and Sauer (1988)]. Another work focused on the global picture of bifurcations is given in [Mosekilde et al. (2000)] where division of the parameter space for the H´enon map into domains of periodic and chaotic oscillations with some coexisting phenomena is studied numerically and analytically. 2.6.2

Recent bifurcation phenomena

This section describes some important and unusual bifurcation phenomena observed in the H´enon map as follows: (1) New types of transients of high and varying periodicity for low-periodic orbits [Michelitsch and R¨ ossler (1998)]. (2) The landing phenomenon: A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the H´enon map in parameter space by numerical methods, i.e., a narrow domain of period-m solutions firstly co-exists with (lies on) a large period-n (m < n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The disappearance of the co-existing attractors is called the landing phenomenon. There is an interaction between the two domains in the course of landing. The chaotic area in the large domain is enlarged, and there is a crisis step near the landing area. This type of bifurcation was given in [Pei-Min and Bang-Chun (2004)]. (3) Sequence of global period-doubling bifurcations: This sequence has some properties such as the existence of a region of bifurcated area in the phase space that expands gradually when the control parameter increases over the critical threshold, and the relative prevalence of the

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bifurcated orbit is measured in terms of the relative ratio of the area covered by the bifurcated daughter orbit to that of the mother orbit [Murakami et al. (2002)]. (4) Noisy parametric sweep through a period-doubling bifurcation: This is done in [Davies and Rangavajhula (2002)] and means that an explicit noise model is included, and the matched asymptotic expansions together with a center-manifold reduction in the vicinity of the bifurcation are used to describe trajectories sweeping up or down. (5) Entry and exit sets5 , i.e., regions through which any forward or backward unbounded orbit escapes to infinity: In this case, the H´enon map is considered as a discrete version of an open Hamiltonian system that can exhibit chaotic scattering. The method of analysis is the proof that the right branch of the unstable manifold of the hyperbolic fixed point is the graph of a function, which is the uniform limit of a sequence of functions whose graphs are arcs of the symmetry lines of the H´enon map as a reversible map [Petrisor (2003)]. Namely, we have the following results: Proposition 2.1. (a) The H´enon map   −y − µx + x2 hµ (x, y) = x

(2.40)

 For µ 2∈ (−2, 2) has an unbounded forward invariant set E0 ⊂ (x, y) ∈ R : x > 0, y > 0 , i.e., hµ (E0 ) ⊂ E0 . Its boundary ∂E0 is Ca ∪ Γh0 where Ca is the curve parameterized by (X (a) , Y (a)) , a ∈ a2 −aµ+1 , Y (a) = aX (a) , and Γh0 = (0, X (a) =  1] , with a 2 (x, y) ∈ R : y = x >, x > xh . (b) The forward orbit (xn , yn ) = hnµ (x0 , y0 ) , n ∈ N of a point (x0 , y0 ) ∈ E0 has the property that both sequences (xn ) and (yn ) are increasing and unbounded. (c) The right branch W+u (zh ) of the unstable manifold of the hyperbolic fixed point zh is
included in the set E0 , where the branch of the unstable manifold W u zhµ having the the sense of v = (λu , 1) for the tangent semi-line l =
 direction and z = (x, y) ∈ R2 : z = zhµ + tv, 0 < t ≤ 1 at zh is denoted by W+u zhµ , and λu > 1 is the expanding
multiplier of the hyperbolic fixed point zh . (d) is the graph of an increasThe right branch W+s zhµ of
thestable manifold
ing C 1 -function ψ : xhµ , ∞ → xhµ , ∞ , which is the uniform limit of a 5 H´ enon and Feit [H´ enon (1976), Feit (1978)] have noted that for b = 0.3 and a ∈ [−0.12, 2.67], no attractors are observed for the H´ enon map numerically. All points seem to escape to infinity.

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  sequence of increasing C 1 -functions ϕn : xhµ , ∞ → xhµ , ∞ , n ∈ N,
and (d-1) ϕn xhµ = xhµ , ∀n ∈ N. (d-2) ϕn (x) > ϕm (x) , ∀n > m The graph of the function ϕn is the partial boundand x > xhµ . (d-3)   h

ary Γh−2n = h−n Γ0 µ µ

of the forward invariant set En . (e) The sequence h

µ of forward invariant sets En having the boundary ∂En = Ca ∪ Γ−2n is an increasing sequence, and its limit E = ∪n≥0 E
n is forward invariant, and also has the boundary ∂E = Ca ∪ W+s zhµ .
The set F = R (E) is backward invariant, and ∂F = C´a ∪ W+u zhµ . The set of points  G = E ∩ F = (x, y) ∈ R2 : x > xhµ , ϕ−1 (x) < y < ϕ (x) is h-invariant, i.e., hµ (G) = G, where R is the involution R (x, y) = (y, x) and ϕ is the function constructed in (d).

2.6.3

Existence of transversal homoclinic points

The role of homoclinicity and heteroclinicity in a rigorous proof of chaos was described in detail in Sec. 1.5. Let f : Rn −→ Rn be a real function defining a discrete mapping, and let Df (x) be its Jacobian matrix. To simplify the notion of a homoclinic bifurcation, we need the following definitions: Definition 2.5. (a) A saddle point q of f is a point where Df (q) has some eigenvalues λ, such that |λ| < 1, and the remainder satisfies |λ| > 1. (b) A homoclinic point p of a map f : Rn −→ Rn lies inside the intersection of its stable and unstable separatrix (invariant manifold), i.e., limn−→+∞ f (x) = limn−→−∞ f (x) = P. (c) The map f : Rn −→ Rn has a hyperbolic period-k orbit γ if there exists a q ∈ Rn such that f k (q) = q, where q is a saddle point, and its stable and unstable manifold W s (q), W u (q) intersect transversely at a point p. (d) The point p in Def. 2.5(d) is called a transverse homoclinic point. The most important results for the existence of such a point are the existence of infinitely many periodic and homoclinic points in a small neighborhood of this point as shown in the Smale–Moser theorem [Smale (1965)]: Theorem 2.15. In a neighborhood of a transverse homoclinic point, there exists an invariant Cantor set on which the dynamics are topologically conjugate to a full shift on N symbols. Hence the definition of a homoclinic bifurcation is given by the following: Definition 2.6. A homoclinic bifurcation occurs when periodic orbits appear from homoclinic orbits to a saddle, saddle-focus, or focus-focus equi-

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librium. A good reference for homoclinic bifurcations in dynamical systems is [Kuznetsov (2004)]. For the H´enon map, some important phenomena about homoclinic and heteroclinic orbits and their bifurcations are summarized in the following: The existence of transversal homoclinic points is shown by two different methods: A numerical method in [Kan et al. (1995)] for a = 1.4, b = 0.3, and an analytic method in [Marotto (1979), Misiurewicz and Szewc (1980)]. Also, [Brown (1995)] proved that the corresponding submanifolds indeed intersect transversally for a > 0. In [Kirchgraber and Stoffer (2006)] for b = −1, a ≥ 0.265625 using the so-called shadowing techniques, this result confirms an old conjecture due to Devaney and Nitecki, claiming that the H´enon map admits a transversal homoclinic point in a region of interest. Namely, the following theorem [Devaney and Nitecki (1979)] states that the nonwandering set Ω (h) of the H´enon map has the following property:   a + by − x2 h (x, y) = (2.41) x is topologically equivalent to the shift map of 2 symbols6 , where b 6= 0, i.e., −(1+|b|2 ) −(1+|b|2 ) 7 , Ω (h) = ∅. (ii) For a > , Theorem 2.16. (i) For a < 4 4 2 Ω (h) is contained in the square S = {(x, y) ∈ R , |x| ≤ R, |x| ≤ R}, where q   1+|b|+ (1+|b|2 )+4a 2 . (iii) For a > 2 1 + |b| , Λ = ∩n∈Z hn (S) is a R = 2 topological horseshoe; for b 6= 0, there is a continuous semi-conjugacy of √ (5+2 2)(1+|b|2 ) Ω (h) ⊂ Λ onto the 2-shift. (iv) For a > , Λ = Ω (h) has a 4 hyperbolic structure and is conjugate to the 2-shift.

The proof of Theorem 2.16 can be carried out by proving the seven lemmas listed below: (1) First, the proofs of statements (i) and (iii) rely on the following technical lemmas: −(1+|b|2 ) . In this case, R is Lemma 2.4. (a) R is real if and only if a > 4 2 positive and equals the larger root of R −(|b|+1)R−a = 0. (b) a−|b| R > R   2 if and only if a > 2 1 + |b| .

6 This means that the phenomena of the H´ enon attractor are part of a bifurcation occurring in the creation of a horseshoe from nothing. 7 This value is precisely the a-value at which the first fixed point of h2 appears as H´ enon remarked [H´ enon (1976)], and statement (i) in [Devaney and Nitecki (1979)] comes from the Brouwer translation theorem [Brouwer (1912), Andrea (1965)].

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Lemma 2.5. (a) The image under h of the horizontal strip |y0 | ≤ C is the region bounded by the two parabolas a − |b| C − y12 ≤ x1 < a + |b| C − y12 . The image under h of the vertical strip |x0 | ≤ C is the horizontal strip |y1 | ≤ C. (b) The inverse image8 of the vertical strip |x0 | ≤ C is the region bounded by the two parabolas −C − a − x2−1 ≤ by−1 ≤ C − a − x2−1 . The inverse image of the horizontal strip |y0 | ≤ C is the vertical strip |x−1 | ≤ C. Lemma 2.6. (a) if x0 ≤ min (− |y0 | , −R) , then x1 ≤ x0 , with equality only for x0 = −R, y0 = ±R. (b) if x0 ≥ − |y0 | and by0 ≥ max (0, |b| R) , then by−1 ≥ by0 and |y−1 | ≥ |y0 | , with equality only for x0 = −R, y0 = ±R. (2) Second, to prove statement (i) of Theorem 2.16, we need the following two lemmas: −(1+|b|2 ) , Lemma 2.7. For a < 4 (a) F (MI ∪ M2 ) ⊂ interior M1 . (b) x is strictly decreasing along h-orbits in M1 . (c) h−1 (M2 ∪ M3 ) ⊂ interior M3 . (d) |y| is strictly increasing along h−1 -orbits in M3 .

where

 2 M = (x, y) ∈ R : x ≤ − |y| 1  M = (x, y) ∈ R2 : x ≥ − |y| and by ≤ 0  2  M3 = (x, y) ∈ R2 : x ≥ − |y| and by ≥ 0 .  

Now if R is real, then define   N1 = (x, y) ∈ R2 : x ≤ min (− |y| , R)    N2 = (x, y) ∈ R2 : x ≥ −R and |y| ≤ R  N = (x, y) ∈ R2 : x ≥ − |y| and by ≥ |b| R   3  N4 = (x, y) ∈ R2 : x ≤ − |y| and by ≥ |b| R .

(2.42)

(2.43)

Hence the following properties hold:

−(1+|b|2 ) , Lemma 2.8. For a > 4 (a) h(N1 ) ⊂ N1 . (b) F (N2 ∪ N3 ) ⊂ N1 ∪ N2 . (c) x is decreasing along h-orbits in NI (strictly decreasing except at the two points x = − |y| = −R). 8 We

use negative subscripts for preimages.

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(d) h−I (N3 ∪ N4 ) ⊂ N4 . (e) h−I (N2 ) ⊂ N2 ∪ N3 ∪ N4 . (f ) |y| is increasing along h−1 -orbits in N4 (strictly increasing except at the point (−R, R sgn(b))). (3) To prove statement (iv), let Dh and Dh−1 be Jacobian matrices of h and h−1 . Then the following sets  +  Sλ = (ξ, η) ∈ R2 : |ξ| ≥ λ |η| (2.44) Sλ− = (ξ, η) ∈ R2 : λ |ξ| ≤ |η|

are invariant under the actions of Dh and Dh−1 , and we need to prove the following three lemmas:

. Lemma 2.9. Suppose that, for some λ > 1, x satisfies |x| > λ(1+|b|) 2 Then: (a) For any vector (ξ 0 , η 0 ) ∈ Sλ+ , the vector (ξ 1 , η 1 ) = Dh (ξ 0 , η 0 ) satisfies |ξ 1 | > λ |ξ 0 |.
− = Dh−1 (ξ 0 , η 0 ) satisfies (b) For any vector (ξ , η ) ∈ S , ξ , η 0 0 −1 −1 λ η −1 ≥ λ |η 0 |. Lemma 2.10. If (x0 , y0 ) and (x1 , y1 ) = h (x0 , y0 ) both satisfy |x| > λ(1+|b|) 2 for some λ > 1, then (a) for any vector (ξ 0 , η 0 ) ∈ Sλ+ , (ξ 1 , η 1 ) = Dh (ξ 0 , η 0 ) belongs to Sλ+ , and |(ξ 1 , η 1 )| ≥ λ |(ξ 0 , η 0 )| . (b) for any vector (ξ 1 , η 1 ) ∈ Sλ− , (ξ 0 , η 0 ) = Dh−1 (ξ 1 , η 1 ) belongs to Sλ− , and λ |(ξ 1 , η 1 )| ≤ |(ξ 0 , η 0 )| . √

(5+2 2)(1+|b|2 ) , there exists a λ > 1 such that |x| > Lemma 2.11. If a > 4 λ(1+|b|) −1 for all (x, y) ∈ S ∩ h (S). 2 Using the graph transform method, Fontich proved in [Fontich (1990)] that the stable and unstable manifolds of the H´enon map9   y hc (x, y) = (2.45) −x + 2y 2 + 2cy intersect transversally for c > 1. Namely, e have the following result: Theorem 2.17. (a) For c > 1, the symmetric H´enon map (2.45) has a homoclinic point pc = (µc , µc ) 6= (0, 0) on the intersection of its graph representation hc 10 with y = x. 9 Which 10 The

is conjugate to the H´ enon map (2.23) under such circumstances. function fc is defined as the limit of a specified sequence of function (fk )k∈N .

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(b) The angle between the invariant manifolds at the homoclinic point pc = (µc , µc ) is given by the function ϕ : (1, ∞) → R defined by   Dhc (µc ) + 1 . (2.46) ϕ (c) = 2 arctan Dhc (µc ) − 1

(c) ϕ is an analytic function on (1, ∞) . (d) For c > 1.78, 

ϕ (c) > 2 arctan 1 − q 1+

2 16(c−1)2 2c−1



 > 0.

(2.47)

The proof of Theorem 2.17 is based on the global graph transformation used to obtain estimates for certain invariant manifolds of the symmetric map (2.45). Tovbis et al. [Tovbis et al. (1998)] and Gelfreich and Sauzin [Gelfreich and Sauzin (2001)] investigate exponentially small phenomena for −1 < ρ << 0, especially for the H´enon map given by   x + ρy hǫ (x, y) = . (2.48) y + ρx (1 − x) Due to the symmetry of the map (2.48) one has the following definition: Definition 2.7. The intersection of separatrices with the horizontal axis is a homoclinic point. Hence the value of the angle11 between the stable and unstable separatrix at the first intersection of the separatrices with the horizontal axis was determined in [Gelfreich (1991)], namely, we have the following result: Theorem 2.18. The angle between the stable and unstable separatrix at the first intersection of the separatrices with the horizontal axis is given asymptotically by −2π 2

64πe ρ α= 9ρ7

(|Θ| + O (ρ)) , Θ ∈ C

(2.49)

Formula (2.49) implies the exponentially small transversality of the homoclinic point for all small ρ > 0 provided the factor |Θ| does not vanish. Hence the H´enon map (2.48) has a homoclinic point for all small ρ > 0 as shown in [Gelfreich and Sauzin (2001)]: 11 Called

also the splitting constant, and for the map Fǫ , one has that |Θ| ≈ 2.474 × 106 [Gelfreich and Sauzin (2001)].

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Theorem 2.19. In the case of the H´enon map (2.48) the splitting constant |Θ| does not vanish. More precisely, Θ ∈ iR and Im Θ < 0. The proof of Theorem 2.19 is based on the detailed study of the Borel transform12 of the formal separatrix of the parabolic fixed point of the map (2.48). The classical case where a = 1.4, b = 0.3 for the H´enon map (2.23) was investigated with computer-assisted methods by several authors. For example, [Coomes et al. (2005)] proved the existence of orbits homoclinic to a periodic orbit using theorems given in Sec. 1.9. For ǫ = 1.95 × 10−15 , a short and long pseudo-homoclinic orbit corresponding to the hyperbolic fixed point of the saddle type are shown in Fig. 2.12. Using Theorem 1.23, one can prove that these short and long pseudo-homoclinic orbits are shadowed by unique true transversal homoclinic orbits within ǫz = 3.23 × 10−14 and ǫz = 1.22 × 10−11 , respectively. For the period-23 orbit obtained using the global Newton’s method in [Coomes et al. (1997)], an ǫ-pseudo-periodic orbit {yk , k ∈ Z} of period-23 with ǫ near the machine precision is shown in Fig. 2.13(b). An ǫ-pseudo-connecting homoclinic orbit {zk , k ∈ Z} connecting {xk , k ∈ Z} to {yk , k ∈ Z} with τ = 10 and ǫ = 3.34 × 10−15 is shown in Fig. 2.14. Using Theorem 1.24, one has that the pseudo-periodic orbit {yk , k ∈ Z} is ǫz -shadowed, with ǫz = 1.64 × 10−12 by a unique true hyperbolic periodic orbit {wk , k ∈ Z} of minimal period-23 and that the pseudo-connecting orbit {zk , k ∈ Z} is ǫz -shadowed by a unique true orbit {ek , k ∈ Z} such that e0 is a transversal homoclinic point to the periodic orbit {vk , k ∈ Z} with phase shift τ = 10. Figure 2.14 shows the pseudoperiodic and pseudo-homoclinic orbits for the 23rd power of the H´enon map (2.23). We see that the homoclinic orbit, which is a heteroclinic orbit for the iterated map, connects e0 to e10 . Thus the point e0 is a transversal homoclinic point to the periodic orbit {vk , k ∈ Z} with phase shift τ . For b = −1 and a = 1, Fig. 2.15(a) shows the stable and √
man√ unstable ifolds of the hyperbolic saddle fixed point Q = −1 − 2, 1 + 2 of the H´enon map (2.23), and Fig. 2.15(b) shows a pseudo-homoclinic orbit with ǫ = 5.98 × 10−15 to this fixed point. Using Theorem 1.23, one can confirm the existence of a true transversal homoclinic orbit within ǫz = 1.45×10−13 of this pseudo one. 12 A Borel summation is given by the Laplace transform, and it is a generalization of the usual notion of summation of a series [Andrianov and Manevitch (2003)].

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Fig. 2.12 (a) A pseudo-homoclinic orbit (marked with diamonds z±k ) to the hyperbolic fixed point (marked with a white triangle) of the H´ enon map (2.23). There exists a true transversal homoclinic orbit within ǫz = 3.23 × 10−14 of this pseudo one. (b) Another long pseudo-homoclinic orbit (marked with diamonds) to the same hyperbolic fixed point. There exists a true transversal homoclinic orbit within ǫz = 1.22 × 10−11 of this pseudo one. Adapted from [Coomes et al. (2005)].

For finding a transversal homoclinic point for the H´enon map (2.23), the following results were proved in [Pilyugin (1999), Kirchgraber and Stoffer (2004-2006)] using the so-called shadowing lemma introduced in Sec. 1.9. Let h : R2 → R2 be a map of class C 1 . Using the introduction given in Sec. 1.9, we obtain the following definitions: Definition 2.8. A sequence w = (wn )n ∈ Z ∈ l∞ (Z, R2 ) is called a pseudoorbit of h with error d = (dn )n ∈ Z ∈ l∞ (Z, R2 ) if wn+1 − h(wn ) = dn holds for n ∈ Z.

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(a) ∆y5

∆y11

∆y20

∆y16

∆y18∆y3

∆y9

∆y ∆y14 1 ∆y22

∆y0

∆y2 ∆y15 ∆y10

∆y13 ∆y8

∆y4 ∆y19 ∆y17

∆y7

∆y21

∆y12 ∆y6

(b)

y0

y10

Fig. 2.13 (a) A pseudo-periodic orbit marked with triangles yk of period-23 of the H´ enon map (2.23). (b) A pseudo-homoclinic orbit marked with diamonds associated with the period-23 orbit. There exists a true transversal homoclinic orbit near ǫz = 1.64 × 10−12 of this pseudo-orbit. Adapted from [Coomes et al. (2005)].

For n ∈ Z, let Ωn be an open and convex set containing wn . Definition 2.9. An R-shadowing orbit z of w is a sequence z = (zn )n ∈ Z ∈ l∞ (Z, R2 ) with zn ∈ Ωn , zn+1 = h(zn ) and |zn − wn | ≤ R for n ∈ Z. Assume that there are (constant) 2 × 2-matrices An , with supn∈Z |An | < ∞ and such that |Dh − AnY | is small enough13 . We introduce the linear operator L and for z ∈ Ω = Ωn the linear operator ∆z in l∞ (Z, R2 ) n∈Z

13 This

mean that the Jacobian Dh does not vary too much in the set Ωn .

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y0 w 0

w69 y10

w23

w115

w46

w92

Fig. 2.14 The pseudo-periodic (triangles) and pseudo-homoclinic orbits (diamonds) for the 23rd power of the H´ enon map (2.23). There is a phase shift (y0 → y10 ) of τ = 10 in this homoclinic orbit. Adapted from [Coomes et al. (2005)].

as follows:



L : (Lζ)n = ζ n+1 − An ζ n ∆z : (∆z ζ)n = (Dh (zn ) − An ) ζ n .

(2.50)

Hence the following theorem was proved in [Pilyugin (1999), Kirchgraber and Stoffer (2004)]:
Theorem 2.20. (Shadowing theorem) Let h ∈ C 1 R2 , R2 be a map of class C 1 , and let δ 0 , δ 1 be positive constants. For n ∈ Z let Ωn ⊂ R2 be open and convex. Let w = (wn )n ∈ Z ∈ l∞ (Z, R2 ) with wn ∈ Ωn be a pseudo-orbit of h with error d ∈ l∞ (Z, R2 ). Let the operators L and ∆z be defined as above. Assume that L is invertible and that the following estimates are satisfied:

−1

L d ≤ δ 0 (2.51) Y

−1

L ∆z ≤ δ 1 < 1, for all z ∈ Ω = Ωn (2.52) n∈Z

δ0 , n ∈ Z. (2.53) 1 − δ1 Then there is an R-shadowing orbit z ∗ ∈ Ω of w, and z ∗ is the only orbit in Ω. Moreover, z ∗ is hyperbolic. ¯R (wn ) ⊂ Ωn , for R = B

Theorem 2.21. (a) The H´enon map hp,q given by  ! 1 2 (1 + pq) x − x − py hp,q (x, y) = q x

(2.54)

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admits a transversal homoclinic point for all parameters p, q with 0 < |p| 1 . In terms of the classical parameters a, b this means: |b| ∈ (0, 1] ≤ q ≤ 10 91 19 2 and a ≥ 20 + 20 b − 400 b . (b) The area and orientation-preserving H´enon map hp,q given by (2.54), admits a transversal homoclinic point for all parameters p, q with p = q ∈ (0, 14 ]. In terms of the classical parameters 17 = 0.265625 14 a, b, this corresponds to b = −1 and a ≥ a0 = 64 Theorem 2.21 is obtained using the properties of some well-defined linear operator K : ξ ∈ l∞ (Z, R2 ) → Kξ ∈ l∞ (Z, R2 ) and some computer-assisted proofs. In the sequel, we describe in detail the method for finding homoclinic points for the H´enon map (2.23) given in [Marotto (1979), Misiurewicz and Szewc (1980)] as follows: First, note that for some values of a and b (not for the usual parameters a = 1.4, b = 0.3) the existence of a transversal homoclinic point was proved analytically by the Marotto theorem [Marotto (1979)] given in Sec. 1.7. Theorem 2.22. The H´enon map (2.23) has transversal homoclinic points for all a > 1.55 and |b| < ǫ, for some ǫ > 0. Proof.

Indeed, the H´enon map (2.23) can be equivalently written as   1 − ax2 + by h (x, y) = . (2.55) x

For each value of a, the map (2.55) is of the form (1.46) with f (x, y) = 1 − ax2 + y. Let us consider the map: g (x) = 1 − ax2 . (2.56)
√ The fixed point x∗ = a1 12 4a + 1 − 21 is an unstable fixed point of g, and √ g (x∗ ) = 1 − 4a + 1 < −1 for all a > 0.75. If one can find a solution (xk )k∈Z of the map (2.55) with not all xk = x∗ satisfying (i) xk = x∗ for all k ≥ m for some m, (ii) limk→−∞ xk = x∗ , and (iii) g´ (xk ) 6= 015 for all k, then x∗ is a snap-back repellor. Let us define the following orbit (xk )k∈Z : x0 = x∗ . Since x∗ is a fixed point of the map (2.56), then xk = x∗ for 14 Theorem 2.21 generalizes one of the results of Coomes et al. [Coomes et al. (2001), Devaney and Nitecki (1979)] where they prove with computer assistance the existence of a transversal homoclinic point for b = −1 and a = 1 and for b = −1, a ≥ −0.866360, respectively. Also, using the graph transforms method Fontich [Fontich (1990)] has shown that the stable and unstable manifolds of the fixed point 0 intersect transversally for a ≥ −0.3916. 15 The condition g ´ (xk ) 6= 0 is the Jacobian condition required of the xk ’s in the definition of a snap-back repellor.

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(a)

(b)

w8

w4 w5

w7 w6

w9

y0

w10 w1

w3 w2

Fig. 2.15 (a) Nonrigorous pictures of the stable and unstable manifolds W s (Q) and W u (Q) of the hyperbolic saddle fixed point Q of the area-preserving H´ enon map (2.23) for a = 1, b = −1. The fixed point is marked with an encircled square. Many transversal intersections of the invariant manifolds are visible. (b) A pseudo-homoclinic orbit (marked with blue diamonds zk ) to the fixed point (marked with the triangle y0 ). There exists a true transversal homoclinic orbit within ǫz = 1.45 × 10−13 of this pseudo one. Adapted from [Coomes et al. (2005)].

all k ≥ 0. The values of xk for k < 0 can be constructed by iterating the multi-inverse of the map g, i.e., r 1 − xk −1 xk−1 = ± = g± (xk ) . (2.57) a Provided xk ≤ 1, with x0 = x∗ , one has two choices for x−1 according to (2.57). The choice of the plus sign in (2.57) does not yield an appropriate se−1 −1 quence because one has x−1 = g+ (x0 ) = g+ (x∗ ) = x∗ . Therefore, define −1 −1 the sequence: x−1 = g− (x0 ) = −x∗ < x∗ and xk−1 = g+ (xk ) in (2.57)

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0

for all k ≤ −1. Hence the sequence (xk )k=−∞ satisfies limk→−∞ xk = x∗ −1 −1 for appropriate values of a, x−2 = g+ (x−1 ) ∈ g+ ((−∞, x∗ )) ⊂ (x∗ , ∞). ∗ Indeed, let us find those values of a for which xq −2 < 1. Since x−1 = −x = √ 1− 4a+1 , 2a

−1 then by (2.57), x−2 = g+ (x−1 ) = 1+x a . Therefore, x−2 < 1
√ 1 1 1 ∗ implies x < a − 1, or a 2 4a + 1 − 2 < a − 1. This is equivalent to a3 − 2a2 + 2a − 2 > 0. Hence all values of a > 1.55 satisfy this equation, and x−2 ∈ (x∗ , 1) for these values of a. Because x−2 ∈ (x∗ , 1) for a > 1.55, then −1 −1 x−3 = g+ (x−2 ) ∈ g+ ((x∗ , 1)) ⊂ (0, x∗ ) , and consequently, x−3 ∈ (0, x∗ ) . −1 −1 −1 Also, x−4 = g+ (x−3 ) ∈ g+ ((0, x∗ )) ⊂ g+ ((x−1 , x∗ )) ⊂ (x∗ , x−2 ) , and −1 −1 ∗ so x−4 ∈ (x , x−2 ) . This implies that x−5 = g+ (x−4 ) ∈ g+ ((x∗ , x−2 )) ⊂ (x−3 , x∗ ), and hence x−5 ∈ (x−3 , x∗ ) . Repeating this procedure gives the 0 result that the constructed sequence (xk )k=−∞ satisfies the following: x−2k is a decreasing sequence bounded below by x∗ , and x−2k−1 is an increasing sequence bounded above by x∗ . There must therefore exist a point α ∈ (0, x∗ ] which is the limit of x−2k−1 , and a point β ∈ (x∗ , 1] which is the limit of x−2k as k → ∞. In the sequel, it is possible to show that α = β = x∗ . Indeed, since g (x−2k−1 ) = x−2k and g (x−2k ) = x−2k+1 , it must be that g (α) = β and g (β) = α. Consequently, g (g (α)) = α, and α is thus a fixed point of the function g ◦ g. But for a > 1.55, there are precisely four fixed
points of g ◦g 16 , each of which can be computed exactly:
√ √ 1 1 1 1 1 1 ± and ± . It is clear that, 4a + 1 − 4a − 3 + a 2 2 √ a 2
2 √
for a > 1.55, one has that a1 − 12 4a + 1 − 21 and a1 − 21 4a − 3 + 21 are both nega∗ tive, and thus neither of these
can∗ equal α ∈ (0, x ]. Also, for a > 1.55, one √ 1 1 1 has that a + 2 4a − 3 + 2 > x . Therefore, it must be that α = β = x∗ . Hence limk→−∞ xk = x∗ . Finally, the constructed sequence (xk )k∈Z satisfies (i), (ii), and (iii). It can easily be shown that (iii) is satisfied. Since g ′ (x) = −2ax, the only possible way for g ′ (xk ) to be zero is if xk = 0 for some k. But from the manner in which the sequence (xk )k∈Z is constructed, xk = x∗ for k ≥ 0, x−1 = −x∗ , x−2 > x∗ > 0, x−3 > 0, and xk ∈ (x−3 , x−2 ) ⊂ (0, 1) for all k < −3. Hence x∗ is a snap-back repellor of (2.56).  ∗

However, for a = 1.4, b = 0.3, a rigorous proof of chaos in the H´enon map (2.23) was given in [Misiurewicz and Szewc (1980)], and it is divided into nine steps, which we consider here as elementary lemmas for simplifying the proof of this method. The H´enon map (2.23) can be rewritten for the 16 Because

this function is a quartic polynomial.

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Fig. 2.16 The quadrilateral ABCD containing the trapping region Ω of the H´ enon map (2.23).

usual values as h (x, y) =



y 1 − 1.4y 2 + 0.3x



.

(2.58)

In this case, the map (2.58) has a hyperbolic fixed point P given by ! √ √ −0.7 + 6.09 −0.7 + 6.09 . (2.59) , P = (x0 , x0 ) = 2.8 2.8 Now consider the quadrilateral ABCD shown in Fig.
2.16 and defined by

1.33 −5 A = (1.4, −1.33), B = 3 , 1.32 , C = −1.4 3 , 1.245 , D = 3 , −1.06 . The image of ABCD under h is a region bounded by four arcs of a parabola, and it can be shown by elementary algebra that this image lies inside ABCD [H´enon (1976)]. Then one has the following theorem [Misiurewicz and Szewc (1980)]: Theorem 2.23. There exists a transversal homoclinic point for the fixed point P of the H´enon map (2.58). If we denote by Ω the quadrilateral ABCD, then one has Lemma 2.12. h(Ω) ⊂ Ω. Proof.

The lines AB, BC, CD, DA are given by the equations  AB : 2.87y + 7.95x = 7.3129,    BC : 2.73y − 0.225x = 3.50385,  CD : 3.6y − 6.915x = 7.709,   DA : 9.2y + 0.81x = −11.102.

(2.60)

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The images of the points A, B, C, D under h are:  h(A) = (−1.33, −1.05646),    h(B) = (1.32, −1.30636),  h(C) = (1.245, −1.310035),   h(D) = (−1.06, −1.07304).

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(2.61)

The image of the line AB is a parabola given by the equation 7.95y = −1.4x2 − 0.861x + 10.14387.

(2.62)

It is easy to check that this parabola lies below the lines AB, BC, and CD, and that h(A), h(B), h(C), and h(D) lie above the line DA. Also, since the horizontal lines are mapped onto vertical ones under h, the images of the sides BC, CD, and DA of Ω are contained within parabolas pointing upwards, and they lie below the parabola h(AB). Hence h(Ω) ⊂ Ω  Lemma 2.13. We have  

1 8

0.631 < x0 < 0.632 λ1 < −1.915 < −1.12 < 0.15 < λ2 < 0.17 < 12 .

(2.63)

Proof. The aim of this lemma is to make estimates for the fixed point P and its eigenvectors. Indeed, from (2.59), we clearly have 0.631 < x0 < 0.632, and it is easy to check that P ∈ lnt (Ω) , and hence the whole unstable manifold of P is contained  in Ω. The derivative of h at a point 0 1 (x, y) is Dh(x, y) = , and the characteristic polynomial of 0.3 −2.8y Dh(P ) p is λ2 + 2.8x0 λ − 0.3 = 0. We have 1.76 < 2.8x0 < 1.77. Thus 2.07 < (2.8x0 )2 + 4 (0.3) < 2.1, and we get the following estimates for the eigenvalues of Dh(P ): λ1 < −1.915 < −1.12, 81 < 0.15 < λ2 < 0.17 < 21 .     1 1 The corresponding eigenvectors are and respectively.  λ1 λ2 s Here, we denote the stable and unstable manifolds of P by W and  a W u , respectively. Denote Φ = {(x, y) : y ≥ 0.4}, Γ = , −b ≥ 1.12 . a b Then one has the following:

Lemma 2.14. The unstable manifold W u has the following properties: (a) A small piece of W u containing P is contained in Φ, and vectors tangent to it belong to Γ. (b)There exists a point of the form (c, 0.4) belonging to W u with c > x0 > 0.6. (c) There exists a point of the form Q = (q, 0) belonging between P and h2 (c, 0.4).

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      a a a1 (a) Let (x, y) ∈ Φ, = Dh(x, y) ∈ Γ, . We have b b1 b a 1 a1 = b, b1 = 0.3a − 2.8by. Hence −b a1 = −0.3 b + 2.8y ≥ 2.8y ≥ 1.12, and   a1 ∈ Γ. Besides, we get |a1 | ≥ 1.12 |a| . Notice that P ∈ lnt (Φ) therefore b   1 1 and ∈ Γ. Therefore, a small piece of W u containing P is contained λ1 in Φ, and vectors tangent to it belong to Γ. (b) As long as images of this piece are contained in Φ, vectors tangent to them belong to Γ, and the absolute value of the difference of the first coordinates of their endpoints grows at least by a factor of 1.12 at each step. Since the whole of W u is contained in Ω, this cannot continue forever. Hence there exists a point (c, 0.4) belonging to W u . We take such a point closest (along W u ) to P . (c) It is easy to verify that the point (d1 , d2 ) = h2 (c, 0.4) also belongs to W u . 2 We get d2 = 1.12  − 1.4 (0.776 + 0.3c) < 1.4 (1.676 + 0.3c) (0.124 − 0.3c) . x0 − c Since ∈ Γ, we have c > x0 > 0.6, and consequently, d2 < 0. x0 − 0.4 u Moreover, since λ 1 < 0, for  all points (x, y) lying on W between P and x0 − x (d1 , d2 ), we have ∈ Γ and y ≤ x0 . Hence x ≥ x0 . Since d2 < 0, x0 − y among those points, there is one with the second coordinate equal to zero. We denote it by Q = (q, 0), and the piece of W u between Q and P by Σu. Proof.

The mapping h−1 is given by the formula h −1

−1

(x, y) =

 2.8

1 0.3 x 0.3

We have Dh (x, y) = 1 0    a a 1.9}, ∆ = ,2 ≤ b ≤ 8 . b



−1+1.4x2 +y 0.3

x

!

.

(2.64)

. Denote : Ψ = {(x, y) : 0.475 ≤ 2.8x ≤

Lemma 2.15. The stable manifold W s has the following properties: (a) A small piece of W s containing P is contained in Ψ, and vectors tangent
1.9 to it belong to ∆. (b) There exist two points of the form R = , r and 2.8
0.475 s s s S = 2.8 , s on W . (c) There exists a piece Σ of W such that for all (x, y) ∈ Σs , the signs of x − x0 and y − x0 are the same.       a a a1 Proof. (a) Let (x, y) ∈ Ψ, = Dh−1 (x, y) ∈ ∆, and . We b b1 b a1 1 b 0.475 1 1 1.9 have ab11 = 2.8 0.3 x+ 0.3 a , and therefore, 2 = 0.3 + 8(0.3) ≤ b1 ≤ 0.3 + 2(0.3) =

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a1 b 1

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∈ ∆. Besides, we get |b1 | ≥ 1.12 |a| . We remark that P ∈ lnt (Ψ)  1 and ∈ ∆. λ2 Therefore, a small piece of W s containing P is contained in Ψ, and vectors tangent to it belong to ∆. (b) Look at the images of this piece (under h−n , n = l, 2, ...). As long as they are contained in Ψ, vectors tangent to them belong to ∆, and the absolute values of the differences between the first coordinates of their endpoints and x0 grow at least by the factor two at each step. This time, λ2 > 0, and we may look separately at what happens left of P . Hence we get the points
to the right and
to the 1.9 s , r and S = 0.475 , s on W closest to P (along W s ) to the right R = 2.8 2.8 and to the left of P , respectively. (c) We denote the pieces of W s between S and P and between P and s s s R by Σs1 and Σs2 , respectively.  We denote also Σ = Σ1 ∪ Σ2 . For all  x − x0 ∈ ∆. In particular, it follows that x − x0 (x, y) ∈ Σs , we have y − x0 and y − x0 have the same sign.  8,

Lemma 2.16. The piece Σs divides h(Ω) into two parts. Proof. We will show that Σs divides h(Ω) into two parts, and one of them is contained in the half-plane Π = {(x, y) : 2.8x ≥ 0.475}. The 1.08 x + 4.6023 parabola h(CD) is given by the equation y = −1.4x2 + 6.915 6.915 . 0.475 If x = 2.8 , then using the estimates 0.1 < x < 0.17, we get y > x0 . Therefore, Σs1 intersects h(CD). The parabola h(AB) is given by the equation 10.14387 1.9 y = −1.4x2 − 0.861 7.95 x + 7.95 . If x = 2.8 , then using the estimate x > 0.65, s we get y < x0 . Therefore, Σ2 intersects h(AB). Those points of intersection are unique because vectors tangent to Σs lie in the first and the third quadrants, and the tops of the parabolas h(CD) and h(AE) lie to the left s s of the line x = 0.475 2.8 . Hence Σ divides h(Ω) into two parts. Since Σ is contained in Π, one of the parts also must be contained in Π. We denote this part by Λ1 and the other by Λ2 .  Lemma 2.17. The subset h4 (Σu ) intersects Σs at some point H different from P . Proof. Suppose that h4 (Σu ) intersects Σs only at P . Then hi (Σu ) for i = 0, l, 2, 3 also intersects Σs only at P . Since λ1 < 0 and W u ⊂ Λ1 ∪ Λ2 , we get hi (Σu ) ⊂ Λ1 for i = 0, 2, 4 and hi (Σu ) ⊂ Λ2 for i = l, 3. In

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particular, it follows that h4 (Q) ∈ Λ1 ⊂ Π. The line AB intersects the xaxis at the point with the first coordinate 7.3129 7.95 < 0.92, and consequently one gets 0.63 < x0 < q < 0.92. For the points h(Q) = (0, q1 ), h2 (Q) = (q1 , q2 ), h3 (Q) = (q2 , q3 ), and h4 (Q) = (q3 , q4 ), one obtains consecutively the following estimates: 1.18 < q1 < 1.28, q2 < −0.94, and q3 < 0.15 Since 2.8 (0.15) < 0.475, one gets h4 (Q) ∈ / Π, a contradiction. Hence the subset h4 (Σu ) intersects Σs at some point different from P . Therefore, h2 (Σu ) also intersects h−2 (Σs ) at some point different from P .    au u tangent to W u at H, one has −b Lemma 2.18. For the vector au > bu 1.9. By Step 3, Σu ⊂ {(x, y) : x ≥ 0 , 0 ≤ y≤ x 0 }, and the vectors  x   a a a 1 u u tangent to Σ belong to Γ, (x, y) ∈ Σ , = Dh(x, y) ∈ Γ, , b b1 b     a a2 = Dh2 (x, y) . We have a1 = b, b1 = 0.3a − 2.8by, a1 = b1 , b2 b
a −b2 1 b2 = 0.3a1 − 2.8b1 1 − 1.4y 2 + 0.3x . Hence −b a1 = 2.8y − 0.3 b , a2 =
0.3 1 2.8 1 − 1.4y 2 + 0.3x − 0.3 ab11 . We get 0 < −b a1 ≤ 2.8x0 − 1.12 < 1.8 − 0.2 =
−b2 0.3 > 2.8x0 + 0.18 > 1.9. Thus for 1.6, and a2 > 2.8 1 − 1.4x20 + 0.3x0 + 1.6   au u tangent to W u at H, we get −b the vector au > 1.9.  bu   as tangent to W s at H, one has −1 Lemma 2.19. For the vector 2 ≤ bs 1 − as bs ≤ 0.6 < 1.7. Proof.

Proof. By Step 4, ΣS ⊂ Ψ, and vectors tangent to Σs belong to ∆. s Therefore, the vectors tangent to h−1 (Σ   belong   to ∆. DenoteH =  ) also a1 a1 as . The vector = (x2 , y2 ) , h (H) = (x1 , y1 ) , and Dh(H) b1 b1 bs is tangent to h−1 (Σs ), and therefore it belongs to ∆. We consider two cases:   as s ∈ ∆. Case 1. h(H) ∈ Σ . Then bs Case 2. h(H) ∈ / Σs . Since for x ≥ x0 the first coordinate of the inverse images of a vector from the first quadrant grows at least by the s u factor 2.8 0.3 x0 > 5 at each step, no point of Σ2 − {P } can belong to W . 2 2 s Therefore, h (H) ∈ Σ1 . Hence, if h (H) = (x, y), then 0 < x < y ≤ x0 . Using the formula for h−1 and the inequality x2 ≥ 0.8x − 0.16, we 1 −1.224 1 +1.224 get (notice that y1 = x, y2 = x1 ). x1 ≥ 2.12y0.3 , y1 ≤ 0.3x2.12

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1.4 2 1 and x2 ≤ 0.3 x1 + 2.12 x1 − 0.896 0.626 . Since H ∈ Ω, H lies to the right of 3.6y2 −7.709 . Since y2 = x1 , we get 1.4x21 + the line CD. Therefore, x ≥ 2 6.915  
0.3(3.6) 7.709 0.3 x1 + − 0.896 2.12 − 6.915 0.626 + 6.915 (0.3) ≥ 0. The second coordinate of the point of intersection of the y-axis and h(CD) is 4.6023 the estimates from Step 5, 6.915 > 0.66 >  x0 . Along with is disjoint from , y ≤ x this proves that the set (x, y) : 0 ≤ x ≤ 0.475 0 2.8 h(Ω). Thus x1 < 0. Hence we get 0 ≤ 1.4x2 + (0.14 − 0.16)x1 + (−1.42 + 2 1.12) (0.3) = 1.4x2 − 0.02x1 − 0.09. Since (0.02) + (4.1) × (4) (0.09) > 0.5, 2.8 1 we have x1 ≤ 0.02−0.7 = − 0.68 2.8 2.8 . Since as = 0.3 x1 a1 + 0.3 b1 , bs = a1 , and a1 as −0.68+0.5 = −0.6. Finally, in both cases we obtain b1 ≥ 2, we get bs ≤ 0.3 as 1 −1 ≤ − < 1.7.  ≤ 2 bs 0.6     as au tangent to W u and Lemma 2.20. For all (nonzero) vectors bs bu     as au s . 6= and W , respectively, at H, one has bs bu

Proof. Use Lemma 2.18 and 2.19, and consequently, W u intersects W s at H transversally.  On the other hand, numerical analysis of the H´enon map fa,b (x, y) = (a − x2 + by, x)

(2.65)

in [Hansen and Cvitanovic (1998)] showed that it is not structurally stable in some parameter intervals such as b = 0.3 and a ∈ [1.270, 1.420]. This means that for a dense set of parameter values in this interval, the H´enon family has a homoclinic tangency. Another rigorous computational method for finding homoclinic tangency and structurally unstable connecting orbits is proposed in [Arai and Mischaikow (2006)]. This method is a combination of several tools and algorithms, including the interval arithmetic introduced in Sec. 1.3.3, the subdivision algorithm, the Conley index theory [Mischaikow (2002)], and the computational homology theory. Indeed, one has the following theorem [Arai and Mischaikow (2006)]: Theorem 2.24. (a) Fix any b0 sufficiently close to 0.3. Then there exists an a ∈ [1.392419807915, 1.392419807931] such that the one-parameter family fa,b0 has a generic17 homoclinic tangency with respect to the saddle fixed point on the first quadrant. 17 The

tangency in a one-parameter family is generic if the intersection of unstable and stable manifolds is quadratic and the intersection is unfolded generically in the family.

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(b) Fix any b0 sufficiently close to −0.3. Then there exists an a ∈ [1.31452..., 1.31452...] such that the one-parameter family fa,b0 has a generic homoclinic tangency with respect to the saddle fixed point on the third quadrant. Similar results can also be obtained by the methods of Fornæss and Gavosto [Fornæss and Gavosto (1992-1999)] which depends on the analyticity of maps. By comparison, the method of [Arai and Mischaikow (2006)] is rather geometric and topological, and it can be applied to a wider class of maps. Essentially, a continuous family of C 2 diffeomorphisms is required for which one can compute the image of the maps using interval arithmetic. A good reference about homoclinic tangency for diffeomorphisms can be found in [Gonchenko et al. (2005)]. 2.6.4

Classification of homoclinic bifurcations

A classification of homoclinic bifurcations and some relations between them, especially those for the area-preserving case in the H´enon map (2.23), was given in [Sterling et al. (1999)] where the so-called anti-integrable limit was employed and it was shown that there exists a bound on the parameter range for which the H´enon map   y − k + x2 hk,b (x, y) = (2.66) −bx

exhibits a complete binary horseshoe and a subshift of finite type. Namely, we have the following results: Let xt , t ∈ Z be a sequence of points on an orbit of the H´enon map (2.66), which can be rewritten as a second order difference equation: xt+1 + bxt−1 + k − x2t = 0.

(2.67)

− 12

The scaled coordinate z = ǫx = k x (where 0 < k < ∞) gives an implicit map in the variable z with parameter ǫ: ǫ (zt+1 + bzt−1 ) + 1 − zt2 = 0.

(2.68)

Hence a period-n orbit of the H´enon map (2.66) is given by a sequence z0 , z1 , ..., zn−1 that satisfies (2.68) with the condition zt+n = zt . The so-called anti-integrable limit technique [Aubry (1992), MacKay and Meiss (1992), Aubry (1995)], shows that the dynamics in discrete time can be represented by a relation F (x, x′ ) = 0

(2.69)

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where x and x′ are points in some manifold. Eq.(2.69) can be explicitly solved for x′ = hk,b (x), giving a map xt = hk,b (xt−1 ) on the manifold. Suppose that F depends on a parameter18 ǫ, for example: F (x, x′ ) = ǫG(x, x′ ) + H(x).

(2.70)

In this case, the solutions of the implicit equation F = 0 when ǫ = 0 correspond to arbitrary sequences of points xt that are zeros of H 19 , and one has the following definition [Aubry (1995)]: Definition 2.10. (a) The case ǫ = 0 corresponds to an anti-integrable limit (AI) of the map hk,b . (b) If the derivative of H is nonsingular, then the AI orbits can be continued for ǫ 6= 0 to orbits of the map hk,b [Aubry (1992), MacKay and Meiss (1992)] 20 . An AI limit with this property is called nondegenerate. At the anti-integral limit, Eq. (2.68) reduces to zt2 = 1.

(2.71)

Thus orbits of this limit are arbitrary sequences in the following infinite set: n o n o Z Z Σ = (−1, +1) = (−, +) = {s : st ∈ {−1, +1} , t ∈ Z} . (2.72) Let us introduce the following subset of Σ defined by:

Σ̥ = Σ − {s ∈ Σ : ∃t ∈ Z, such that st−1 = st+1 = −1} .

(2.73)

We need the following definitions: Definition 2.11. (a) Orbits in the anti-integrable limit are bi-infinite sequences s ∈ Σ. (b) The dynamics on s ∈ Σ are given by the shift map, σ : Σ → Σ defined as σ (...s−1 .s0 s1 s2 ...) = ...s−1 s0 .s1 s2 ...

(2.74)

(c) An orbit of the symbolic dynamics is periodic if the sequence s is periodic. (d) A saddle-node bifurcation is a local bifurcation in which two fixed points collide and annihilate each other. 18 This

is not always possible. this case, the dynamics are not deterministic. 20 This can be done using a straightforward implicit function argument. 19 In

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(e) A pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations, are of two types–supercritical and subcritical. (f) A period-doubling bifurcation is a bifurcation in which the map switches to a new behavior with twice the period of the original map, i.e., the Floquet multiplier is −1. (g) A rotational bifurcation occurs when the winding number of an elliptic orbit becomes ω = m/n21 . An orbit of least period-n is denoted by the string of n symbols and a superscript ∞ to represent repetition22 : ∞

(s0 s1 s2 sn−1 )

= ...sn−2 sn−1 .s0 s1 ...sn−1 s0 ...

(2.75)

Lemma 2.21. The map σ has the following properties: (a) σ has two fixed ∞ ∞ points, (+) and (−) , corresponding to the two fixed points of the H´enon map (2.66). (b) The two fixed points are born in a saddle-node bifurcation at
− 1 + b2 , (2.76) k= 4 ∞ ∞ which we denote by sn {(+) , (−) } . In order to classify different homoclinic bifurcations for the H´enon map (2.66), we need to state the following definitions: Definition 2.12. (a) A parent refers to the orbit that is undergoing the bifurcation, if any. (b) The type is one of sn, pf, pd, or m/n, corresponding to a saddlenode, pitchfork, period-doubling, or rotational bifurcation, respectively. (c) The set of orbits created in the bifurcation is listed as the children23 . The bifurcations are denoted here with the general template: parent → type (children) .

(2.77)

Hence a classification of the periodic orbits up to period-6 and their bifurcations is shown in (2.78) [Sterling et al. (1999)], where if there are 21 For ω = 1/3, the child is not created in the bifurcation, but it exists before and after the bifurcation. 22 Note that any cyclic permutation of a periodic orbit gives another point on the same orbit. 23 We adopt the convention that the unstable child is listed first, and the stable one second.

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two children, the one listed in the first column has negative residue just after birth (except for the pf case). The real roots of the polynomials 16k 5 − 108k 4 + 105k 3 + 27k 2 − 97k − 47 and 16k 6 − 136k 5 + 213k 4 + 220k 3 + 126k 2 + 108k + 81 give exact bifurcation values for the three approximations shown in (2.78) labelled by ⊗ and ⊠, respectively. A schematic representation of the dynamics of the H´enon map (2.66) is shown in Fig. 2.17. Parent ∞

(−)

(−)∞ (−)∞ (+−)∞ (−)∞ (−)∞ ∞

(−) (+ − +)∞
∞ + −4 + ∞

(+−)

Type sn pd sn 1/3 1/4 pd 1/5 2/5 sn 1/6 pd pf sn 1/3 sn

Child ∞ (−)

Child ∞ (+) ∞ (+−) ∞ (+ − +)

∞

(− + −) (− + −)∞ (+ − −+)∞

k-values −1 3 1

∞

(− + + + −)∞ (− − + − −)∞ (+ − + −
+)∞ ∞ − +4 −

∞

(+ + − + −−) ∞ (− − + − +−) ∞ (− − + − +−)
∞ + − +3 −

(+ − ++) (− + −−)∞ (+ + − + +)∞ (− + − + −)∞ ∞ (+ − − − +)
3 ∞ + + −+
∞ + −4 + ∞ (− − + − ++)
∞ − − +−3
∞

5 4

0 4

√ 7−5 5 8√ 7+5 5 8

5.5517014⊗ −3 4 5 4

3 3.7016569⊠ 15 4

5.6793695⊠ (2.78) Let α, ζ be two homoclinic orbits of the H´enon map (2.66), and let W s , W u be the stable and unstable manifolds of the H´enon map, respectively. 3

−−+ −

Definition 2.13. (a) A segment of a manifold from a point to its iterate W s (β, h (β)) is called a fundamental segment. (b) The transition time is the number of iterates required for β ∈
u −1 W h (ζ) , ζ to reach the stable segment:
(2.79) ttrans (β) = m, if hm (β) ∈ W s h−1 (ζ) , ζ . (c) The type of a homoclinic point is the number of iterates for which the stable initial segment hj (β) intersects with the unstable initial segment β, i.e., 
type (β) = sup j ≥ 0, : W s p, hj (β) ∩ W u (p, β) 6= ∅ . (2.80)

(d) Primary homoclinic points have type 0. (e) Horseshoes have type 1.

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Note that the union of the iterates of a fundamental segment is the entire branch of the manifold that contains β, and the value of the transition time depends on the choice of the fundamental segments, and so it is not as basic a property as the type. The type of a homoclinic point is invariant along its orbit. These notions are defined for the classification of homoclinic points. For more details see [Easton (1998)]. Define the following sets shown in Fig. 2.18:  U = W u (α, ζ) ,
(2.81) S = W s f −1 (ζ) , α .

Hence the following results with their detailed proofs in addition to other results can be found in [Sterling et al. (1999)]:

Theorem 2.25. (a) For every symbol sequence s ∈ Σ, there exists a unique orbit z (ǫ) of the H´enon map (2.68) such that z (0) = s if s 2 (2.82) |ǫ| (1 + |b|) < 2 1 − √ ≈ 0.649839. 5 (b) There are no bifurcations in the H´enon map (2.68) when ǫ and b are in the range (2.82). Conjecture 2.1. (a) (No bubbles): For the area-preserving H´enon map, every orbit is (at least) continuously connected to the anti-integral limit. (b) (Homoclinic saddle-node bifurcation): For positive b, the first bifurcation as ǫ increases from 0 corresponds to the homoclinic saddle-node bifuraction24 sn {+∞ − (+) − +∞ , +∞ − (−) − +∞ } .

(2.83)

(c) (Heteroclinic saddle-node bifurcation): For negative b, the first bifurcation as ǫ increases from 0 corresponds to the heteroclinic saddle-node bifurcation sn {−∞ + (−) − +∞ , −∞ + (+) − +∞ } .

(2.84)

(d) Every orbit of the H´enon map (2.68) can be continuously connected to the anti-integrable limit. 24 A theorem of Smillie [Smillie (1997)] implies that the first bifurcation destroying the H´ enon horseshoe must be a quadratic homoclinic tangency for some orbit.

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Theorem 2.26. (Existence and uniqueness of Σ̥ orbits): Suppose 0 ≤ b ≤ 1. Then for every symbol s ∈ Σ̥ , there exists a unique orbit z (ǫ) of the H´enon map such that z (0) = s if 0 ≤ ǫ ≤ ǫmax , where s √ −b2 + 2b + 5 − 2 5 + 4b 2 . (2.85) ǫmax = 1+b (1 − b) (5 − b) Lemma 2.22. (a) Assume there are exactly two primary homoclinic orbits, α and ζ, and the segments S and U defined in Eq. (2.81) contain all of the homoclinic orbits. Then for each homoclinic point in β ∈ U, one has ttrans (µ) = type (β) . (b) Two homoclinic orbits β and γ cannot bifurcate unless they are double neighbors. (c) If two homoclinic orbits β and γ bifurcate, then they must have the same transition time ttrans . (d) The transition time of a homoclinic orbit never changes. (e) Two homoclinic orbits on U are neighbors in the complete horseshoe if and only if they are of the form +∞ −· (s+) −+∞ and +∞ −· (s−) −+∞ . (f ) The first homoclinic bifurcation of the invariant manifolds of the fixed point (+)∞ is sn {+∞ − (+) − +∞ , +∞ − (−) − +∞ } .

(2.86)

For more detail, see [Devaney (1984), Grassberger et al. (1989), Davis et al. (1991), Easton (1998), Sterling et al. (1999)], and references therein. 2.6.5

Basins of attraction

In this section, we discuss some relevant results about the basins of attraction of the H´enon map. It is well known that basin boundaries arise in dissipative dynamical systems when two or more attractors are present. In such situations, each attractor has a basin of initial conditions that lead asymptotically to that attractor. The sets that separate different basins are called the basin boundaries [Nusse and Yorke (1996)]. The basin of the H´enon attractor with the usual parameters is shown in Fig. 2.6(a). In some cases, the basin boundaries can have very complicated fractal structure and hence pose an additional impediment to predicting long-term behavior. For the H´enon map, this phenomenon can be seen in [Endler and Gallas (2001)], where period-4 stability and multistability domains are analytically derived. An exact description of the basin and the nonwandering set of some H´enon

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Fig. 2.17 First bifurcation for the H´ enon map (2.66). The dark shaded region represents Theorem 2.25, and the lighter that of Theorem 2.27. The curves represent the numerical results for the first orbits destroyed up to period-24. Bounds for the subshift Σ̥ are indicated with a triangle symbol. Adapted from [Sterling et al. (1999)].

strange attractors obtained for small |b| and b < 0 were given in [Cao and Mao (2000)]. 2.6.6

Structure of the parameter space

In this section, we discuss the full image of the dynamical behaviors for the H´enon map (2.23). However, most of the results on this topic are done numerically, but there are some works that focus on the analytical investigation of such dynamics. For example, the paper [Benedicks and Carleson (1991)] gives an excellent global characterization of such dynamics, especially the question of whether an orbit of a point will generically go into an attractive cycle. The main theorem for answering this question is stated as follows: Theorem 2.27. Let W u be the unstable manifold of the map h given by (2.23) at its fixed point in x, y > 0. Then for all c < log 2, there is a b0 > 0 such that for all b ∈ (0, b0 ), there is a set E (b) of positive one-dimensional Lebesgue measure such that for all a ∈ E (b) : (i) There is an open set U = U (a, b) such that for all z ∈ U ,
(2.87) dist hn (z) , W u → 0, as n → ∞.

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Fig. 2.18 Stable and unstable manifolds for the H´ enon map (2.66) at k = 5 and b = 1 shown in the (z, z ′ ) coordinates [Sterling et al. (1999)].

(ii) There is a point z0 = z0 (a, b) ∈ W u such that: u (a) The sequence (hn (z0 ))∞ n=0 is dense in W , n cn (b) kDh (z0 ) (0, 1)k ≥ e . It is important to note that if a ∈ E (b) , then the following characterizations of the dynamics of the H´enon map (2.23) hold: a The forward orbit is dense on the unstable manifold W u for almost all starting points in a neighborhood of the origin. b The closure of the parameter values a for which h has an attractive cycle contains E (b) . c There exist Sinai-Bowen-Ruelle measures which have absolutely continuous conditional measures with respect to the unstable foliation. d A partial theory of kneading sequences can be developed for a ∈ E (b) . More specifically, if a is approximately equal to 2 and b is sufficiently small (b = 0 corresponds to the logistic map), the H´enon map (2.23) admits a strange attractor. The proof of Theorem 2.28 is based on the analysis of the one-dimensional map x → 1−ax2 using a very long treatment. Fig.2.17 shows bounds for the Benedicks and Carleson results as a black curve. A more general theory can be found in [Mora and Vianna (1993)]: Theorem 2.28. Let (fµ )µ be a C ∞ one-parameter family of diffeomorphisms on a surface, and suppose that f0 has a homoclinic tangency associated with some periodic point P0 . Then under generic (even open and

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Fig. 2.19 Regions of dynamical behaviors in the ab-plane for the H´ enon map (2.23) [Sprott (1996)].

dense) assumptions, there is a positive Lebesgue measure set E of parameter values near µ = 0 such that for µ ∈ E, the diffeomorphism fµ exhibits a strange attractor or repellor near the orbit of tangency. The full structure of the parameter space of the H´enon map (2.23) was the subject of several works, especially, in [Gallas and Jason (1993), Sprott (1996), Cao and Kiriki (2000)]. Fig.2.19 shows regions of unbounded, fixed point, periodic, and chaotic solutions in the ab-plane for the H´enon map (2.23), where we use |LE| < 0.0001 as the criterion for quasi-periodic orbits with 106 iterations for each point. Some things to note if this structure is denoted by S and the H´enon map is given by   a − x2 + by h (x, y) = (2.88) x include the following:

(1) S contains a regular structure-parallel-to-structure sequence of shrimpshaped robust isoperiodic domains. (2) The orientation-preserving and reversing isoperiodic domains are densely concentrated in a neighborhood of the parameter a. Thus the a direction25 defined by b = −0.583 + 1.025 is rich in new phenomena, and it is conjectured to be typical of bimodal maps. This range is valid for 0.94 ≤ a ≤ 1.86, which covers the range −0.059 ≤ b ≤ 0.48. 25 This

direction was obtained from a least-square fit to points located roughly at the center of the main body of many shrimps.

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(3) There is a secondary b direction which contains the range 1.382 ≤ b ≤ 2.056 perpendicular to a very dense “foliation of legs” emanating from the isoperiodic domains. (4) Familiar bifurcation phenomena observed in unimodal maps correspond to particular cuts along the b direction. (5) The H´enon map (2.23) is topologically equivalent to a 2-D section through the infinite-dimensional parameter space characterizing a generic map of the H´enon type. Note that the properties (1), (2), (3), and (4) are given in [Gallas and Jason (1993)], while the property (5) is observed in [Hansen and Cvitanovic (1998)], and it is obtained using a series of n-unimodal approximations to maps of the H´enon type and their associated symbolic dynamics. Some other properties of the H´enon map (2.23) are listed below: (1) The uniform hyperbolicity, i.e., the existence of many regions of hyperbolic parameters in the parameter plane. This was done for the H´enon maps in [Arai (2007)] using a rigorous computational method. (2) The complex susceptibility, i.e., a complex function associated with a perturbation of the considered map and to an observable function. The method of analysis is based on the spectral description of transfer operators. In [Cessac (2007)], it is shown that the complex susceptibility of the H´enon map (2.23) has a pole in the upper-half complex plane obtained by a numerical procedure using time-series analysis. Some open problems about the H´enon map (2.23) can be found in [Palis (2008), Benedicks (2002)].

2.7

Exercises

(1) Show that the polynomials in the H´enon maps (2.23) and the elementary transformations τ ω , e1 , e2 and t given in Theorem 2.6 satisfy a common scaling condition. (2) Show that if a < −(1 − b)2 /4 in the H´enon map (2.23), then all orbits tend to infinity, and therefore there are no interesting dynamics in bounded domains (Lemma 2.1). (3) Draw the mapping h7 where h is given by (2.23), and give the problem details such as the transformed sides of the parallelograms, h7 (A), h7 (B), h7 (C), and h7 (D) of Theorem 2.8.

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(4) Draw the partitions Mi , i = 1, 2, 3 of Lemma 2.7, and determine their images under h and h−1 . (5) Draw the partitions Ni , i = 1, 2, 3, 4 of Lemma 2.8, and determine their images under h and h−1 . (6) Prove that the H´enon map (2.23) after scaling is equivalent to (2.54) using the transformation x = x1 + uq , y = x1 + vq , where p = λ1 and q = λ12 , and λ1 , λ2 , with |λ1 | denoting the eigenvalues of the Jacobian √ (b−1− (b−1)2 +4a) Dh at (x1 , x1 ), and x1 = . 2 (7) Show that the H´enon map (2.23) is topologically equivalent to a 2-D section through the infinite-dimensional parameter space characterizing a generic map of the H´enon type.

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Chapter 3

Classification of chaotic orbits of the general 2-D quadratic map

3.1

Analytical prediction of system orbits

In this section, we investigate domains for the parameters (ai , bi )0≤i≤5 ∈ R12 in which the map (2.1) has unbounded or bounded orbits. We use the idea of the nonexistence of fixed points, because if there is no fixed point, then there is no chaos in the map (2.1) as shown in the next section. Note that in this chapter we use the following simple result available in most textbooks on linear algebra: Theorem 3.1. The polynomial Ax2 + Bx + C has no real zeros if and only if A > 0 and B 2 − 4AC < 0, or A < 0 and B 2 − 4AC < 0. 3.1.1

Existence of unbounded orbits

First, a fixed point (x, y) of the map (2.1) must simultaneously satisfy the following two equalities:  a0 + a1 x + a2 y + a3 x2 + a4 y 2 + a5 xy = x (3.1) C0 : b0 + b1 x + b2 y + b3 x2 + b4 y 2 + b5 xy = y. Second, note that most special cases of the map (2.1) are unbounded. In this subsection, we give sufficient conditions for the existence of these unbounded orbits. Indeed, if there are no fixed points, then one has from (3.1) that the polynomials f (x, y) − x or g (x, y) − y are either positive or negative for all (x, y) ∈ R2 . Assume, for example, that f (x, y) − x is positive, and let x0 ≥ 0. Then one has for all integer k, that f (xk , yk ) > xk , i.e., xk+1 > xk > xk−1 > .... > x0 ≥ 0. Consider the Euclidean distance d (xk , 0) = xk that measures the distance between the first component xk of the map (2.1) and the origin 0 on the real line. Then we have 105

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d (xk+1 , 0) > d (xk , 0) > d (xk−1 , 0) > .... > d (x0 , 0) ≥ 0. Hence there is a real number △ > 0 such that d (xk , 0) = d (xk−1 , 0) + △, which implies that d (xk , 0) = d (x0 , 0) + (k + 1) △. Finally, one has limk→+∞ d (xk , 0) = +∞. If x0 < 0, the same logic applies. When there are fixed points, there are domains that contain all bounded orbits, i.e., possibly chaotic attractors. On the other hand, chaotic attractors in a system without fixed points are probably rare, but it is possible for them to exist. There is at least one case of a conservative system that is chaotic without any fixed points [Sprott (1994)]. The idea of the nonexistence of fixed points for the map (2.1) is used to prove the following theorem: 12 Theorem 3.2. If (ai , bi )0≤i≤5 ∈ Ω = ∪i=4 i=1 Ci ⊂ R , then all orbits of the map (2.1) are unbounded, where  a2  a3 > 4a54 , a4 > 0 2 2 (3.2) C1 :  a0 > (2a1 −1−a1 )a24 +(a1 −1)a2 a5 −a2 a3 a5 −4a3 a4  a2  a3 < 4a54 , a4 < 0 2 2 C2 : (3.3)  a0 > (2a1 −1−a1 )a24 +(a1 −1)a2 a5 −a2 a3 a5 −4a3 a4  b2  b3 > 4b54 , b4 > 0 2 2 (3.4) C3 :  b0 > b1 b4 −(2b2 −b2 −1)b32−(b2 −1)b1 b5 4b3 b4 −b5  b2  b3 < 4b54 , b4 < 0 2 2 C4 : (3.5)  b0 > b1 b4 −(2b2 −b2 −1)b32−(b2 −1)b1 b5 . 4b3 b4 −b 5

Proof. The map (2.1) has no fixed points if one of the following inequalities holds for all (x, y) ∈ R2 : (1) a3 x2 + (a1 − 1 + a5 y) x + a4 y 2 + a2 y + a0 > 0 (2) a3 x2 + (a1 − 1 + a5 y) x + a4 y 2 + a2 y + a0 < 0 (3) b3 x2 + (b1 + b5 y) x + b4 y 2 + (b2 − 1) y + b0 > 0 (4) b3 x2 + (b1 + b5 y) x + b4 y 2 + (b2 − 1) y + b0 < 0.
The discriminant of the first case is given by d = a25 − 4a3 a4 y 2 + (2a1 a5 − 4a2 a3 − 2a5 ) y − 4a0a3 + a21 − 2a1 + 1, and so the inequality a3 x2 + 2 (a1 − 1 + a5 y) x + a4 y 2 + a2 y + a0 > 0 holds for all (x,
y) ∈ R if and only 2 if d < 0 for
all y ∈ R, and a3 > 0, i.e., 1 + a1 − 2a1 a4 + (1 − a1 ) a2 a5 + a25 − 4a3 a4 a0 + a22 a3 < 0 and a25 − 4a3 a4 < 0, and this is possible if (3.2) holds. The other cases are obtained using the same logic. 

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Note that conditions (3.2) to (3.5) do not contain all domains for unbounded orbits because they do not determine domains where zero sets of the two quadratic forms of (3.1) do not intersect. We will determine these domains when discussing the bifurcation phenomena in the following section. 3.1.2

Existence of bounded orbits

Define the following subsets of R12 :  a2  C˜11 : a3 < 4a54   C˜12 : a4 < 0 2 2    C˜13 : a0 < (2a1 −1−a1 )a24 +(a1 −1)a2 a5 −a2 a3

(3.6)

a5 −4a3 a4

   

2

   C˜23 : a0 <

b C˜31 : b3 < 4b54 C˜32 : b4 < 0 b21 b4 −(2b2 −b22 −1)b3 −(b2 −1)b1 b5

   

b C˜41 : b3 > 4b54 C˜42 : b4 > 0 2 b1 b4 −(2b2 −b22 −1)b3 −(b2 −1)b1 b5

   C˜43 : b0 <


(3.7)

a25 −4a3 a4

   

   C˜33 : b0 <

Then one has

a C˜21 : a3 > 4a54 C˜22 : a4 > 0 (2a1 −1−a21 )a4 +(a1 −1)a2 a5 −a22 a3

2

(3.8)

4b3 b4 −b25 2

4b3 b4 −b25

˜ C¯i = ∪j=3 j=1 Cij , i = 1, 2, 3, 4.

(3.9) .

(3.10)

Here, the subsets C¯i 1≤i≤4 are the complements in R12 of the subsets (Ci )1≤i≤4 given in (3.2), (3.3), (3.4), and (3.5). As a result, we have the following theorem: ¯ = ∩i=4 ∪j=3 C˜ij ⊂ R12 , then the Theorem 3.3. If (ai , bi )0≤i≤5 ∈ Ω i=1 j=1 ¯ is the complement in R12 map (2.1) has possible bounded orbits, where Ω of the set Ω.

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Because both equations of the map (2.1) are quadratic, the possible number of fixed points is from 0 to 8, and thus one has the following theorem: Theorem 3.4. For the map (2.1), there are at least eight possible topologically different bounded attractors. This section includes some theorems that determine rigorously the domains for the parameters (ai , bi )1≤i≤6 ∈ R12 in which the orbits of the map (2.1) are asymptotically stable and possibly chaotic. Generally, chaos can occur in the map (2.1) if it has at least one fixed point that is not asymptotically stable, i.e., it must  be asaddle or unstable fixed point. ab First, consider a matrix A = whose determinant is δ = ad − bc cd and whose trace is τ = a+d as a Jacobian matrix of the map (2.1) evaluated at a fixed point. Then the eigenvalues ω 1,2 of A can be expressed in terms of δ and τ as follows: If then one has ω1,2 = and if then one has

τ 2 − 4δ ≥ 0,

(3.11)

 p 1 τ ± τ 2 − 4δ , 2

(3.12)

τ 2 − 4δ < 0,

(3.13)

 p 1 τ ± i 4δ − τ 2 . 2 Thus one has the following theorems: ω 1,2 =

(3.14)

Theorem 3.5. The fixed points of the map (2.1) with the Jacobian matrix A are not all asymptotically stable if one or more than one of the following conditions hold: τ 2 − 4δ ≥ 0, |τ | > 2

(3.15)

τ 2 − 4δ ≥ 0, δ + 1 < τ < 2, δ < 1

(3.16)

τ − 4δ ≥ 0, − (δ + 1) < τ < −2, δ > 1

(3.17)

2

τ 2 − 4δ ≥ 0, 2 < τ < δ + 1, δ > 1

(3.18)

τ − 4δ ≥ 0, −2 < τ < − (δ + 1) , δ < 1 √ |τ | < 2 δ, δ > 1

(3.19)

2

(3.20)

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Theorem 3.6. The fixed points of the map (2.1) with the Jacobian matrix A are all asymptotically stable if one of the following conditions holds: ( 2 τ − 1 < δ < τ4 (3.21) 0≤τ <2 or

or

(

2

−τ − 1, < δ < τ4 −2 < τ ≤ 0

(3.22)

√ 0 < δ < 1, |τ | < 2 δ

(3.23)

For the map (2.1), one has  δ = ξ 1 x2 + ξ 2 y 2 + ξ 3 xy + ξ 4 x + ξ 5 y + ξ 6 τ = ξ7x + ξ8y + ξ9, where (ξ i )1≤i≤6 are given in (2.20), and   ξ 7 = 2a3 + b5 ξ = a5 + 2b4  8 ξ 9 = a1 + b 2 . 3.2

(3.24)

(3.25)

A zone of possible chaotic orbits

From some special cases of the map (2.1), a condition for the existence of chaos is that the map has at least one saddle or unstable fixed point, and thus one has the following theorem: Theorem 3.7. A possible range for chaos in the 2-D quadratic map (2.1) 12 is the set Ω1 = ∪i=10 i=5 Ci ⊂ R . where (Ci )5≤i≤10 are the following subsets of R12 :  ξ ξ 2 −ξ 12 ξ 13 ξ 14 +ξ 11 ξ 213 2   ξ 10 > 0, ξ 12 − 4ξ 10 ξ 11 < 0, ξ 15 > 10 14 4ξ10 ξ 11 −ξ 212 2 C5 : b3 ξ 8 > 0, ξ 16 < 0, ξ 17 − 4ξ 16 ξ 18 < 0, or   b3 ξ 8 < 0, ξ 19 < 0, ξ 220 − 4ξ 19 ξ 21 < 0

 ξ 10 ξ 214 −ξ 12 ξ 13 ξ 14 +ξ 11 ξ 213 2  ξ > 0, ξ − 4ξ ξ < 0, ξ >  10 12 10 11 15 4ξ 10 ξ 11 −ξ 212   2 ξ < 0, ξ < 0, ξ − 4ξ ξ 1 22 23 22 24 < 0 C6 : 2  b ξ < 0, ξ < 0, ξ − 4ξ  3 8 16 17 16 ξ 18 < 0   2 ξ 1 < 0, ξ 25 < 0, ξ 26 − 4ξ 25 ξ 27

(3.26)

(3.27)

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 ξ 10 ξ 214 −ξ 12 ξ 13 ξ 14 +ξ 11 ξ 213 2  ξ > 0, ξ − 4ξ ξ < 0, ξ >  10 12 10 11 15 4ξ 10 ξ 11 −ξ 212   2 ξ > 0, ξ < 0, ξ − 4ξ ξ 1 28 29 28 30 < 0 C7 : 2  b ξ < 0, ξ < 0, ξ − 4ξ  3 8 19 20 19 ξ 21 < 0   2 ξ 1 > 0, ξ 25 < 0, ξ 26 − 4ξ 25 ξ 27  ξ 10 ξ 214 −ξ 12 ξ 13 ξ 14 +ξ 11 ξ 213 2   ξ 10 > 0, ξ 12 − 4ξ 10 ξ 11 < 0, ξ 15 > 4ξ 10 ξ 11 −ξ 212   2 b ξ > 0, ξ < 0, ξ − 4ξ ξ 3 8 16 17 16 18 < 0 C8 : 2  − 4ξ ξ > 0, ξ < 0, ξ  23 22 ξ 24 < 0 1 22   2 ξ 1 > 0, ξ 25 < 0, ξ 26 − 4ξ 25 ξ 27  ξ ξ 2 −ξ ξ ξ +ξ ξ 2 2  ξ 10 > 0, ξ 12 − 4ξ 10 ξ 11 < 0, ξ 15 > 10 14 4ξ 12ξ 13−ξ142 11 13   10 11 12  b3 ξ 8 > 0, ξ 19 < 0, ξ 220 − 4ξ 19 ξ 21 < 0 C9 :  ξ 1 < 0, ξ 28 < 0, ξ 229 − 4ξ 28 ξ 30 < 0    ξ 1 < 0, ξ 25 < 0, ξ 226 − 4ξ 25 ξ 27 ( ξ ξ 2 −ξ ξ ξ +ξ ξ 2 ξ 10 < 0, ξ 212 − 4ξ 10 ξ 11 < 0, ξ 15 > 10 14 4ξ 12ξ 13−ξ142 11 13 10 11 12 C10 : ξ 1 > 0, ξ 25 < 0, ξ 226 − 4ξ 25 ξ 27 , where  ξ 10 = ξ 27 − 4ξ 1     ξ 11 = ξ 28 − 4ξ 2     ξ 12 = 2ξ 7 ξ 8 − 4ξ 3      ξ 13 = 2ξ 7 ξ 9 − 4ξ 4     ξ 14 = 2ξ 8 ξ 9 − 4ξ 5  ξ 15 = ξ 29 − 4ξ 6    ξ 16 = b25 ξ 28 − 4b3 b4 ξ 28     ξ 17 = 2b5 ξ 7 ξ 8 + 2b1 b5 ξ 28 − 4b2 b3 ξ 28 ξ 9    2 2 2 2    ξ 18 = 8b3 ξ 8 + 2b1 ξ 7 ξ 8 + ξ 7 − 4b0 b3 ξ 8 + b1 ξ 8  2 2 2   ξ 19 = b5 ξ 8 − 4b3 b4 ξ 8   ξ 20 = 2b5 ξ 7 ξ 8 − 4b2 b3 ξ 28 + 2b1 b5 ξ 28 and  ξ 21 = 2b1 ξ 7 ξ 8 − 8b3 ξ 8 − 4b3 ξ 8 ξ 9 + ξ 27 − 4b0 b3 ξ 28 + b21 ξ 28     ξ 22 = ξ 23 − 4ξ 1 ξ 2      ξ 23 = 4ξ 1 ξ 8 − 2ξ 3 ξ 7 − 4ξ 1 ξ 5 + 2ξ 3 ξ 4     ξ = 4ξ 1 ξ 9 − 4ξ 1 ξ 6 − 4ξ 1 − 2ξ 4 ξ 7 + ξ 24 + ξ 27  24   ξ 25 = ξ 23 − 4ξ 1 ξ 2  ξ 26 = 2ξ 3 ξ 4 − 4ξ 1 ξ 5   2   ξ  27 = 4ξ 1 − 4ξ 1 ξ 6 + ξ 4   2  ξ 28 = ξ 3 − 4ξ 1 ξ 2      ξ = 2ξ ξ 29 3 4 − 4ξ 1 ξ 8 + 2ξ 3 ξ 7 − 4ξ 1 ξ 5   ξ 30 = 2ξ 4 ξ 7 − 4ξ 1 ξ 6 − 4ξ 1 ξ 9 − 4ξ 1 + ξ 24 + ξ 27 .

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

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The proof of Theorem 3.7 is given using Theorem 3.1 as follows: Proof. From (3.24), one has that τ 2 − 4δ = ξ 10 x2 + (ξ 12 y + ξ 13
) x + ξ 11 y 2 + ξ 14 y + ξ 15 ≥ 0 if and only if ξ 10 > 0 and ξ 212 − 4ξ 10 ξ 11 y 2 + (2ξ 12 ξ 13 − 4ξ 10 ξ 14 ) y − 4ξ 10 ξ 15 + ξ 213 < 0 for all y ∈ R, i.e., ξ 10 > 0, ξ 212 − ξ ξ 2 −ξ ξ ξ +ξ ξ 2 4ξ 10 ξ 11 < 0, and ξ 15 > 10 14 4ξ 12ξ 13−ξ142 11 13 . On the other hand, we have 10 11 12 |τ | = |ξ 7 x + ξ 8 y + ξ 9 | > 2 if and only if ξ 7 x + ξ 8 y + ξ 9 > 2 or ξ 7 x + ξ 8 y + ξ 9 < −2 using the curve C0 given in (3.1). Then one has b3 ξ 8 x2 + (b5 ξ 8 y + ξ 7 + b1 ξ 8 ) x + b4 ξ 8 y 2 + b2 ξ 8 yξ 9 + b0 ξ 8 − 2 > 0 if b3 ξ 8 > 0, ξ 16 < 0, ξ 217 − 4ξ 16 ξ 18 < 0, or b3 ξ 8 x2 + (ξ 7 + b1 ξ 8 + b5 ξ 8 y) x + b4 ξ 8 y 2 + b2 ξ 8 y + b0 ξ 8 + ξ 9 + 2 < 0 if b3 ξ 8 < 0, ξ 19 < 0, ξ 220 − 4ξ 19 ξ 21 < 0. The other cases given in (3.27) to (3.30) can be obtained from the conditions (3.16) to (3.20) using the same logic.  ˜ 1 = ∪i=10 Ci ⊂ R12 is not empty. Theorem 3.8. The set Ω i=5 Proof. The H´enon map [H´enon (1976)] has a chaotic attractor with a saddle fixed point for a0 = b1 = 1, a2 = 0.3, a3 = −1.4, and a1 = a4 = a5 = b0 = b2 = b3 = b4 = b5 = 0.  3.2.1

Zones of stable fixed points

Generally, the geometric structure of an orbit of the map (2.1) depends on the number of fixed points. It can be verified that map (2.1) has at most eight fixed points. Then if all its fixed points are stable1 , then the map (2.1) convergesto a fixed point, i.e., for the following subsets of R12 defined by: ξ 1 < 0, ξ 28 < 0, ξ 229 − 4ξ 28 ξ 30 < 0     ξ ξ 2 −ξ ξ ξ +ξ ξ 2 ξ 10 > 0, ξ 212 − 4ξ 10 ξ 11 < 0, ξ 15 > 10 14 4ξ 12ξ 13−ξ142 11 13 10 11 12 (3.34) C11 : 2  b ξ > 0, ξ < 0, ξ − 4ξ ξ < 0  3 8 31 32 31 33   b3 ξ 8 < 0, ξ 16 < 0, ξ 217 − 4ξ 16 ξ 18 < 0 or  b3 ξ 8 > 0, ξ 16 < 0, ξ 217 − 4ξ 16 ξ 18 < 0     ξ ξ 2 −ξ ξ ξ +ξ ξ 2 ξ 10 > 0, ξ 212 − 4ξ 10 ξ 11 < 0, ξ 15 > 10 14 4ξ 12ξ 13−ξ142 11 13 10 11 12 C12 : (3.35)  b3 ξ 8 > 0, ξ 19 < 0, ξ 220 − 4ξ 19 ξ 21 < 0    b3 ξ 8 < 0, ξ 31 < 0, ξ 232 − 4ξ 31 ξ 33 < 0 or  ξ 1 > 0, ξ 36 < 0, ξ 237 − 4ξ 36 ξ 38 < 0   ξ 1 < 0, ξ 25 < 0, ξ 226 − 4ξ 25 ξ 27 (3.36) C13 :   ξ < 0, ξ 2 − 4ξ ξ < 0, ξ > ξ10 ξ214 −ξ12 ξ 13 ξ14 +ξ11 ξ213 2 10 10 11 15 12 4ξ ξ −ξ 10 11

1 The

absolute value of their eigenvalues does not exceed 1.

12

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where

  ξ 31 = b25 ξ 28 − 4b3 b4 ξ 28     ξ 32 = 2b5 ξ 7 ξ 8 + ξ 27 − 4b2 b3 ξ 28 + 2b1 b5 ξ 28    ξ 33 = 2b1 ξ 7 ξ 8 − 4b3 ξ 8 ξ 9 − 4b0 b3 ξ 28 + b21 ξ 28  ξ 35 = ξ 23 − 4ξ 1 ξ 2     ξ 36 = 2ξ 3 ξ 4 − 4ξ 1 ξ 5    ξ 37 = −4ξ 1 ξ 6 + ξ 24 .

(3.37)

Hence the following theorem is proved:

Theorem 3.9. If (ai , bi )1≤i≤6 ∈ Ω2 = ∪i=13 i=11 Ci , then the 2-D quadratic map (2.1) converges to a fixed point. 12 Theorem 3.10. The set Ω2 = ∪i=13 is not empty. i=11 Ci ⊂ R

Proof. The delayed logistic map [Aronson (1982)] is asymptotically stable for all −7 ≤ a5 ≤ 7 and −6 ≤ b1 ≤ 6, with a2 = −a5 and a0 = a1 = a3 = a4 = b0 = b2 = b3 = b4 = b5 = 0.  The above analysis is not valid in some cases where at least one fixed point is not asymptotically stable. For example, in [Zeraoulia and Sprott (2008a)], there are some regions in a4 -a1 space where two coexisting attractors occur as shown in the black region of Fig. 3.1. For example, with a4 = 1 and a1 = −0.8, a fixed point (at x = y = 0.4329311) coexists with a period-3 orbit, and with a4 = 1 and a1 = −0.8, a fixed point (at x = y = 0.445362) coexists with a quasi-periodic orbit. Another example can be found in [Zeraoulia and Sprott (2008a)]. 3.3

Boundary between different attractors

In this section, we use the fact that attractors with different numbers of fixed points are not topologically equivalent to derive some explicit formulas for the subsets of (ai , bi )0≤i≤5 ∈ R12 where the map (2.1) has different chaotic attractors. For this purpose, we investigate the number of fixed points of the map (2.1). Indeed, Eq. (3.1) can be rewritten as
a3 x2 + (a1 + a5 y − 1) x + a4 y 2 + a2 y + a0 = 0 (3.38) and

b3 x2 + (b1 + b5 y) x + b4 y 2 + (b2 − 1) y + b0 = 0.

(3.39)

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Fig. 3.1 The regions of a4 -a1 space with multiple attractors for the map (2.15). [Zeraoulia and Sprott (2008a)].

We have two cases, the first of which is when a3 = 0, and the second of which is when a3 6= 0. Let us begin with the first case, i.e., a3 = 0.

(3.40)

Then one has x= with the condition

−a0 − a2 y − a4 y 2 a1 + a5 y − 1

Π1 : a5 y 6= 1 − a1 .

(3.41)

(3.42)

Substituting (3.38) into Eq. (3.39) gives ψ 8 y 4 + (ψ 6 + ψ 7 ) y 3 + (ψ 4 + ψ 5 ) y 2 + (ψ 2 + ψ 3 ) y + ψ 1 = 0 where  ψ 1 = b0 + a0 b1 − 2a1 b0 − a0 a1 b1 + a21 b0 + a20 b3     ψ 2 = 2a0 a2 b3 − a2 b1 (a1 − 1) + 2b0 a5 (a1 − 1)    2   ψ  3 = −a0 (b1 a5 + b5 (a1 − 1)) + (a1 − 1) (b2 − 1)   ψ 4 = b0 a25 − a0 a5 b5 − b1 a4 (a1 − 1) − a2 (b1 a5 + b5 (a1 − 1))
2 2   ψ 5 = b4 (a1 − 1) + 2a5 (a1 − 1) (b2 − 1) + b3 2a0 a4 + a2    ψ 6 = a4 b5 − 2a5 b4 + 2a2 a4 b3 − a1 a4 b5    2 2  ψ  7 = 2a1 a5 b4 − b1 a4 a5 − a2 a5 b5 − a5 + b2 a5   2 2 ψ 8 = a4 b 3 − a4 a5 b 5 + a5 b 4 .

(3.43)

(3.44)

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According to the position of the coefficients (ψ i )1≤i≤8 , we have the following cases: (1-2) If ψ 8 = 0, ψ6 + ψ 7 6= 0,

(3.45)

then there are one or three fixed points because we have a third-order polynomial equation in y given by y3 +

ψ1 (ψ 4 + ψ 5 ) 2 (ψ 2 + ψ 3 ) y + y+ = 0. ψ 6 + ψ7 ψ6 + ψ7 ψ6 + ψ7

(3.46)

If ψ 12 > 0, then there is a unique fixed point   −a0 − a2 y1 − a4 y12 , y1 P1 = a1 + a5 y 1 − 1 where y1 is given by

(3.47)

(3.48)

  31   31 ψ10 + ψ 11 −ψ 4 − ψ 5 ψ 10 + ψ 11 + − y1 = + − + ψ 13 − ψ 13 3 (ψ 6 + ψ 7 ) 2 2 (3.49) with the condition a1 + a5 y1 − 1 6= 0

(3.50)

where  (3ψ2 ψ6 +3ψ2 ψ7 +3ψ3 ψ6 −2ψ4 ψ5 +3ψ 3 ψ7 −ψ24 −ψ25 )   ψ =  9 3(ψ 6 +ψ 7 )2   54ψ 1 ψ 6 ψ 7 −9ψ 2 ψ 4 ψ 7 −9ψ 2 ψ 5 ψ 6 −9ψ 3 ψ 4 ψ 6 −9ψ 2 ψ 4 ψ 6 −9ψ 2 ψ 5 ψ 7 −9ψ 3 ψ 4 ψ 7     ψ 10 = 27(ψ 6 +ψ 7 )3 −9ψ 3 ψ 5 ψ 6 −9ψ 3 ψ 5 ψ 7 +2ψ 34 +2ψ 35 +27ψ 1 ψ 26 +27ψ 1 ψ 27 +6ψ 4 ψ 25 +6ψ 24 ψ 5 ψ 11 = 27(ψ 6 +ψ 7 )3   2  3  ψ 12 = 4ψq  9 + 27 (ψ 10 + ψ 11 )   3 2  ψ  11 ) + 279 , ψ 13 = (ψ10 +ψ 2 (3.51) and if ψ 12 < 0, then there are three fixed points   −a0 − a2 yi − a4 yi2 , yi (Pi )2≤i≤4 = a1 + a5 y i − 1 2≤i≤4

(3.52)

(3.53)

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with the conditions a1 + a5 yi − 1 6= 0, 2 ≤ i ≤ 4 where

q 
−ψ 4 −ψ 5  + 2 − ψ39 sin 3θ  y2 = 3(ψ  6 +ψ 7 )  q    y3 = −ψ4 −ψ5 + 2 − ψ9 sin θ+2π
3(ψ 6 +ψ 7 ) 3 q 3
ψ9 −ψ 4 −ψ 5 θ+4π   y4 = 3(ψ +ψ ) + 2 − 3 sin 3   6 q 7    −27(ψ 10 +ψ11 )2  ∈ [0, π] . θ = arcsin 4ψ 3

(3.54)

(3.55)

9

(1-3) If

ψ 8 = 0, ψ6 + ψ 7 = 0, ψ 4 + ψ 5 6= 0,

(3.56)

then there are none, one, or two fixed points because we have a second-order polynomial equation in y. We have the following cases: (1-3-1) If 2

ψ 8 = 0, ψ6 + ψ 7 = 0, (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 < 0, ψ 4 + ψ 5 < 0, (3.57) then there are no fixed points for the map (2.1). (1-3-2) If 2

ψ 8 = 0, ψ6 + ψ 7 = 0, ψ4 + ψ 5 6= 0, (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 = 0, (3.58) then there is one fixed point given by   −a0 − a2 y5 − a4 y52 , y5 P5 = a1 + a5 y 5 − 1

(3.59)

where

y5 =

−ψ 2 − ψ 3 2 (ψ 4 + ψ 5 )

(3.60)

with the condition 2ψ 4 + 2ψ5 − 2a1 ψ 4 − 2a1 ψ 5 + a5 ψ 2 + a5 ψ 3 6= 0.

(3.61)

(1-3-3) If 2

ψ 8 = 0, ψ6 + ψ 7 = 0, ψ4 + ψ 5 6= 0, (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 > 0, (3.62) then there two fixed points given by    2  P6 = −a0 −a2 y6 −a4 y6 , y6 a +a y −1   1 5 6 2  P7 = −a0 −a2 y7 −a4 y7 , y7 a1 +a5 y7 −1

(3.63)

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where y6,7 =

− (ψ 2 + ψ 3 ) ±

q 2 (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 2 (ψ 4 + ψ 5 )

(3.64)

with the condition a1 + a5 y6,7 − 1 6= 0.

(1-4) If

(3.65)

ψ 8 = 0, ψ 4 + ψ 5 = 0, ψ6 + ψ 7 = 0, ψ2 + ψ 3 6= 0, (3.66) then there is only one fixed point given by   −a0 − a2 y8 − a4 y82 P8 = (3.67) , y8 a1 + a5 y 8 − 1 where −ψ 1 y8 = (3.68) ψ2 + ψ3 with the condition a1 + a5 y8 − 1 6= 0. (3.69) (1-5) If ψ 8 = 0, ψ4 + ψ 5 = 0, ψ6 + ψ 7 = 0, ψ2 + ψ 3 = 0, ψ 1 6= 0, (3.70) then there are no fixed points for the map (2.1). (1-6) If ψ 8 = 0, ψ4 + ψ 5 = 0, ψ6 + ψ 7 = 0, ψ2 + ψ 3 = 0, ψ 1 = 0, (3.71) then there are infinitely many nonisolated fixed points of the form  −a0 −a2 y−a4 y 2 ,y for the map (2.1). (1-7)If a1 +a5 y−1 y∈R

ψ 8 6= 0, (3.72) then it is possible to have four fixed points for the map (2.1). Now, define the following subsets in R12 :  D1 : a3 = 0, ψ8 = 0, ψ 6 + ψ 7 = 0, (ψ 2 + ψ 3 )2 − 4 (ψ 4 + ψ 5 ) ψ 1 < 0      ψ 4 + ψ 5 < 0.     D  2 : a3 = 0, ψ 8 = 0, ψ 4 + ψ 5 = 0, ψ 6 + ψ 7 = 0, ψ 2 + ψ 3 = 0, ψ 1 6= 0.    D  3 : a3 = 0, ψ 8 = 0, ψ 6 + ψ 7 6= 0, ψ 12 > 0     D4 : a3 = 0, ψ8 = 0, ψ 6 + ψ 7 = 0, ψ4 + ψ 5 6= 0    2 (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 = 0  D5 : a3 = 0, ψ8 = 0, ψ 4 + ψ 5 = 0, ψ6 + ψ 7 = 0, ψ2 + ψ 3 6= 0     D6 : a3 = 0, ψ8 = 0, ψ 6 + ψ 7 = 0, ψ4 + ψ 5 6= 0    2    (ψ 2 + ψ 3 ) − 4 (ψ 4 + ψ 5 ) ψ 1 > 0     D7 : a3 = 0, ψ8 = 0, ψ 6 + ψ 7 6= 0, ψ12 < 0    D8 : a3 = 0, ψ8 6= 0    D9 : ψ 8 = 0, ψ 4 + ψ 5 = 0, ψ 6 + ψ 7 = 0, ψ 2 + ψ 3 = 0, ψ1 = 0. (3.73)

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(2) If a3 6= 0,

(3.74)

then the discriminant of Eq. (3.38) is given by d = ψ 14 y 2 + ψ 15 y + ψ 16 where

(3.75)

 

ψ 14 = a25 − 4a3 a4 ψ = 2a5 (a1 − 1) − 4a2 a3  15 ψ 16 = (a1 − 1)2 − 4a0 a3 .

First, if



(3.76)

ψ 215 − 4ψ 14 ψ 16 < 0 ψ 14 < 0,

(3.77)

then d < 0 for all y ∈ R, and there are no fixed points for map (2.1), and if ψ 14 = 0, ψ 15 = 0, ψ16 < 0,

(3.78)

then there are also no fixed points for the map (2.1). Second, Eq. (3.38) has real solutions if and only if d ≥ 0, i.e., in the following cases: Π2 : ψ 14 = 0, y ≥

−ψ16 , ψ 15 > 0 ψ 17

(3.79)

Π3 : ψ 14 = 0, y ≤

−ψ16 , ψ 15 < 0 ψ 15

(3.80)

or

or Π4 :

(

or Π5 :

2 √ψ215 − 4ψ14 ψ 16 ≥ 0, ψ14 > 0√ 2 −ψ 15 − ψ 15 −4ψ 14 ψ 16 −ψ 15 + ψ 15 −4ψ 14 ψ 16 , or y ≤ y≥ 2ψ 2ψ 14

(

2 <0 √ 2ψ 16 − 4ψ14 ψ 17 ≥ 0, ψ14√ −ψ15 − ψ 15 −4ψ 14 ψ 16 −ψ 15 + ψ 215 −4ψ 14 ψ 16 ≤y≤ . 2ψ 2ψ 14

(3.81)

14

(3.82)

14

If (3.79) or (3.80) or (3.81) or (3.82) holds, then there are two real solutions for Eq. (3.43), i.e., two fixed points of the map (2.1): √ ( 5 y−1)+ d x9 = −(a1 +a2a 3 √ (3.83) 5 y−1)− d x10 = −(a1 +a2a . 3 (2-1) For the case of x9 given by (3.83), one has

√ ψ 17 + ψ 18 y + ψ 19 y 2 + ψ 20 d + (ψ 21 + ψ 22 y) d = 0

(3.84)

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where

 b −2a1 b3 +2b1 a3 −2a1 b1 a3 +4b0 a23 +a21 b3  ψ 17 = 3  4a23    a3 b5 −b3 a5 −a1 a3 b5 +a1 b3 a5 −b1 a3 a5 −2a23 +2a23 b2   ψ 18 =  2a23    4a23 b4 −2a3 a5 b5 +b3 a25 ψ 19 = 4a23 b3   ψ 20 = 4a 2   3  b3 −a1 b3 +b1 a3   ψ 21 =  2a23    ψ = a3 b5 −b2 3 a5 , 22

(3.85)

2a3

then one has the following quartic equation in the variable y: ψ 28 y 4 + ψ 27 y 3 + (ψ 25 + ψ 26 ) y 2 + ψ 24 y + ψ 23 = 0 where  ψ 23 = 2ψ 20 ψ 16 ψ 17 + ψ 217 − ψ 221 ψ 16 + ψ 220 ψ 216     ψ 24 = 2ψ17 ψ 18 + 2ψ 16 (ψ 20 ψ 18 − ψ 21 ψ 22 )
+ γ 0      γ 0 = ψ 15 2ψ20 ψ 17 − ψ 221 + 2ψ 220 ψ 16      ψ 25 = 2 (ψ 17 ψ 19 − ψ 21 ψ 22 ψ 15 + ψ 20 ψ 14 ψ 17 + ψ 20 ψ 15 ψ 18 ) ψ 26 = 2ψ 20 ψ 16 ψ 19 + ψ 218 − ψ 221 ψ 14 −
ψ 222 ψ 16 + γ 1    γ 1 = ψ 220 2ψ14 ψ 16 + ψ 215     ψ 27 = 2ψ18 ψ 19 + 2ψ 14 (ψ 20 ψ 18 − ψ 21 ψ 22 )
+ γ 2     γ 2 = ψ 15 2ψ20 ψ 19 − ψ 222 + 2ψ 220 ψ 14    ψ 28 = 2ψ20 ψ 14 ψ 19 + ψ 219 − ψ 222 ψ 14 + ψ 220 ψ 214 ,

(3.86)

(3.87)

then we have the following cases: (2-1-1) If

ψ 28 = 0, ψ27 6= 0,

(3.88)

then Eq. (3.86) is third-order and can be solved using the Cardan method. Especially, one has that if ψ 29 = 4ψ330 + 27ψ231 > 0 where  

 ψ 31 =

(3.89)

3ψ 24 ψ 27 −2ψ 25 ψ 26 −ψ 225 −ψ 226 3ψ 227 2ψ 325 −9ψ 24 ψ 26 ψ 27 −9ψ 24 ψ 25 ψ 27 +2ψ 326 +27ψ 23 ψ 227 +6ψ 25 ψ 226 +6ψ 225 ψ 26 , 27ψ 327

ψ 30 =

(3.90) then there is a unique real solution of Eq. (3.86) if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds, while if ψ 29 < 0,

(3.91)

then there are three real solutions for Eq. (3.86) with only one of the conditions (3.79), (3.80), (3.81), and (3.82). The case ψ 29 = 0 corresponds to a

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measure-zero set of parameters. Therefore, by a slight perturbation of parameters, without changing the behavior of the system, a system belonging to one of the two cases is obtained. (2-1-2) If ψ 28 = 0, ψ27 = 0, ψ 25 + ψ 26 6= 0,

(3.92)

then one has that if ψ 28 = 0, ψ 27 = 0, ψ224 − 4 (ψ 25 + ψ 26 ) ψ 23 < 0, ψ25 + ψ 26 < 0,

(3.93)

then there are no real solutions for Eq. (3.86). However, if ψ 28 = 0, ψ 27 = 0, ψ25 + ψ 26 6= 0, ψ224 − 4 (ψ 25 + ψ 26 ) ψ 23 ≥ 0,

(3.94)

then there are two real solutions if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. (2-1-3) If ψ 28 = 0, ψ27 = 0, ψ 25 + ψ 26 = 0, ψ 24 6= 0,

(3.95)

ψ 28 = 0, ψ 27 = 0, ψ25 + ψ 26 = 0, ψ 24 = 0, ψ23 6= 0,

(3.96)

23 then Eq. (3.86) has one real solution given by y1 = −ψ ψ 24 if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. (2-1-4) If

then there are no real solutions for Eq. (3.86). (2-1-5) If ψ 28 = 0, ψ 27 = 0, ψ25 + ψ 26 = 0, ψ 24 = 0, ψ23 = 0,

(3.97)

then there are infinitely many nonisolated real solutions for Eq. (3.86). (2-1-6) If ψ 28 6= 0,

(3.98)

then there are possibly four fixed points for the map (2.1) if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. (2-2) For the case of x10 given by (3.83), one has √ (3.99) ψ 32 + ψ 33 y + ψ 34 y 2 + ψ 35 d + (ψ 36 + ψ 37 y) d = 0 where

 b −2a1 b3 +2b1 a3 −2a1 b1 a3 +4b0 a23 +a21 b3 ψ 32 = 3   4a23    a b −b a −a a b +a b3 a5 −b1 a3 a5 −2a23 +2a23 b2 3 5 3 5 1 3 5 1  ψ 33 =  2 2a  3   4a23 b4 −2a3 a5 b5 +b3 a25 ψ 34 = 4a23   ψ 35 = 41 a−2  3 b3   a1 b3 −b3 −b1 a3  ψ =  36  2a23   −a3 b5 . ψ 37 = b3 a52a 2 3

(3.100)

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Then one has ψ 38 y 4 + ψ 39 y 3 + (ψ 40 + ψ 41 ) y 2 + ψ 42 y + ψ 43 = 0 where  ψ 38 = 2ψ14 ψ 34 ψ 35 + ψ 234 − ψ 14 ψ 237 + ψ 214 ψ 235     ψ 39 = 2ψ33 ψ 34 − 2ψ14 (ψ 36 ψ 37 − ψ 33 ψ 35 )
+ γ 3      γ 3 = ψ 15 2ψ 34 ψ 35 − ψ 237 + 2ψ14 ψ 235      ψ 40 = 2ψ 32 ψ 34 + 2ψ 14 ψ 32 ψ 35 + 2ψ 15 ψ 33 ψ 35 + 2ψ 16 ψ 34 ψ 35 ψ 41 = −2ψ15 ψ 36 ψ 37 + ψ 233 − ψ 14 ψ 236 −
ψ 16 ψ 237 + γ 4    γ 4 = ψ 235 2ψ14 ψ 16 + ψ 215     ψ 42 = 2ψ32 ψ 33 − 2ψ16 (ψ 36 ψ 37 − ψ 33 ψ 35 )
+ γ 4     γ 4 = ψ 15 2ψ 32 ψ 35 − ψ 236 + 2ψ16 ψ 235    ψ 43 = 2ψ 32 ψ 16 ψ 35 + ψ 232 − ψ 16 ψ 236 + ψ 216 ψ 235 .

(3.101)

(3.102)

Since Eq. (3.101) is a quartic equation in the variable y, we have the following cases: (2-2-1) If ψ 38 = 0, ψ39 6= 0,

(3.103)

then Eq. (3.101) is a third-order equation and can be solved using the Cardan method. Especially, one has that if ψ 44 = 4ψ345 + 27ψ246 > 0 where  

 ψ 46 =

(3.104)

3ψ 42 ψ 39 −2ψ 40 ψ 41 −ψ 240 −ψ 241 3ψ 239 2ψ 340 −9ψ 41 ψ 42 ψ 39 −9ψ 40 ψ 42 ψ 39 +2ψ 341 +6ψ 40 ψ 241 +6ψ 240 ψ 41 +27ψ 43 ψ 239 , 27ψ 339

ψ 45 =

(3.105) then there is a unique real solution of Eq. (3.101) if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds, while if ψ 44 < 0,

(3.106)

then there are three real solutions of Eq. (3.101) in which only one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. The case ψ 44 = 0 corresponds to a measure-zero set of parameters. Therefore, by a slight perturbation of parameters, without changing the behavior of the system, a system belonging to one of the two cases is obtained. (2-2-2) If ψ 38 = 0, ψ39 = 0, ψ 40 + ψ 41 6= 0,

(3.107)

then one has that if ψ 38 = 0, ψ 39 = 0, ψ242 − 4 (ψ 40 + ψ 41 ) ψ 43 < 0, ψ 40 + ψ 41 < 0,

(3.108)

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then there are no real solutions of Eq. (3.101). However, if ψ 38 = 0, ψ 39 = 0, ψ40 + ψ 41 6= 0, ψ 242 − 4 (ψ 40 + ψ 41 ) ψ 43 ≥ 0, (3.109) then there are two real solutions if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. (2-2-3) If ψ 38 = 0, ψ39 = 0, ψ 40 + ψ 41 = 0, ψ 42 6= 0, (3.110) 43 if one of the then Eq. (3.101) has one real solution given by y1 = −ψ ψ 42 conditions ((3.79), (3.80), (3.81), and (3.82) holds. (2-2-4) If ψ 38 = 0, ψ 39 = 0, ψ40 + ψ 41 = 0, ψ 42 = 0, ψ43 6= 0, (3.111) then there are no real solutions of Eq. (3.101). (2-2-5) If ψ 38 = 0, ψ 39 = 0, ψ40 + ψ 41 = 0, ψ 42 = 0, ψ43 = 0, (3.112) then there are infinitely many nonisolated real solutions of (3.101) if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. (2-2-6) If ψ 38 6= 0, (3.113) then there are possibly four fixed points for the map (2.1) if one of the conditions (3.79), (3.80), (3.81), and (3.82) holds. Now, let us define the following subsets in R12 :   D10 : a3 6= 0, ψ215 − 4ψ 14 ψ 16 < 0, ψ14 < 0.     D11 : a3 6= 0, ψ14 = 0, ψ 15 = 0, ψ16 < 0     D12 : a3 6= 0, ψ28 = 0, ψ 27 = 0, ψ224 − 4 (ψ 25 + ψ 26 ) ψ 23 < 0       and ψ 25 + ψ 26 < 0     D13 : a3 6= 0, ψ28 = 0, ψ 27 = 0, ψ25 + ψ 26 = 0, ψ 24 = 0, ψ23 6= 0   D14 : a3 6= 0, ψ28 = 0, ψ 27 6= 0, ψ29 > 0 (3.114)  D15 : .a3 6= 0, ψ28 = 0, ψ 27 = 0, ψ25 + ψ 26 = 0, ψ 24 6= 0     D16 : a3 6= 0, ψ28 = 0, ψ 27 = 0, ψ25 + ψ 26 6= 0     and ψ 224 − 4 (ψ 25 + ψ 26 ) ψ 23 ≥ 0      D17 : a3 6= 0, ψ28 = 0, ψ 27 6= 0, ψ29 < 0     D18 : a3 6= 0, ψ28 6= 0    D19 : a3 6= 0, ψ28 = 0, ψ 27 = 0, ψ25 + ψ 26 = 0, ψ 24 = 0, ψ23 = 0 and  D20 : a3 6= 0, ψ38 = 0, ψ 39 = 0, ψ40 + ψ 41 = 0, ψ 42 = 0, ψ43 6= 0     D21 : a3 6= 0, ψ38 = 0, ψ 39 = 0, ψ242 − 4 (ψ 40 + ψ 41 ) ψ 43 < 0       and ψ 40 + ψ 41 < 0  3 2    D22 : a3 6= 0, ψ38 = 0, ψ 39 6= 0, ψ44 = 4ψ 45 + 27ψ46 > 0   D23 : a3 6= 0, ψ38 = 0, ψ 39 = 0, ψ40 + ψ 41 = 0, ψ 42 6= 0 (3.115)  D24 : a3 6= 0, ψ38 = 0, ψ 39 = 0, ψ40 + ψ 41 6= 0     and ψ 242 − 4 (ψ 40 + ψ 41 ) ψ 43 ≥ 0     D25 : a3 6= 0, ψ38 = 0, ψ 39 6= 0, ψ44 < 0      D : a 6= 0, ψ38 6= 0   26 3 D27 : a3 6= 0, ψ38 = 0, ψ 39 = 0, ψ40 + ψ 41 = 0, ψ 42 = 0, ψ43 = 0

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and

and

 E1 = (D12 ∪ D13 ) ∩ (D20 ∪ D21 ) ∪ (D10 ∪ D11 )     E = (D12 ∪ D13 ) ∩ (D22 ∪ D23 ) ∪ ((D14 ∪ D15 ) ∩ (D20 ∪ D21 ))    2   E = ((D14 ∪ D15 ) ∩ (D22 ∪ D23 ))   3   E4 = ((D12 ∪ D13 ) ∩ D24 ) ∪ (D16 ∩ (D20 ∪ D21 ))     E5 = E3 ∪ E4      E6 = ((D12 ∪ D13 ) ∩ D25 ) ∪ ((D14 ∪ D15 ) ∩ D24 )     E7 = (D16 ∩ (D22 ∪ D23 )) ∪ (D17 ∩ (D20 ∪ D21 ))     E8 = E6 ∪ E7      E9 = ((D14 ∪ D15 ) ∩ D25 ) ∪ ((D12 ∪ D13 ) ∩ D26 ) ∪ (D16 ∩ D24 )     E10 = (D17 ∩ (D22 ∪ D23 )) ∪ (D18 ∩ (D20 ∪ D21 ))  E11 = E9 ∪ E10    E12 = ((D14 ∪ D15 ) ∩ D26 ) ∪ (D16 ∩ D25 )     E13 = (D17 ∩ D24 ) ∪ (D18 ∩ (D22 ∪ D23 ))     E14 = E12 ∪ E13      E15 = D16 ∩ D26 ∪ (D17 ∩ D25 ) ∪ (D18 ∩ D24 )     E16 = D17 ∩ D26 ∪ (D18 ∩ D25 )     E17 = D18 ∩ D26      E18 = (D12 ∪ D13 ) ∩ D27 ∪ ((D14 ∪ D15 ) ∩ D27 )     E19 = (D17 ∩ D27 ) ∪ (D16 ∩ D27 ) ∪ (D18 ∩ D27 )     Π : a y 6= 1 − a1 , y is a solution of (3.42)    1 5i=5 Π = ∪i=2 Πi (3.116)    E20   E21      E22     E23    E24  E25     E26     E  27     E28   E29

= (E1 ∩ Π) ∪ ((D1 ∪ D2 ) ∩ Π1 ) = (E2 ∩ Π) ∪ ((D3 ∪ D4 ∪ D5 ) ∩ Π1 ) = (E5 ∩ Π) ∪ (D6 ∩ Π1 ) = (E8 ∩ Π) ∪ (D7 ∩ Π1 ) = (E11 ∩ Π) ∪ (D8 ∩ Π1 ) = E14 ∩ Π = E15 ∩ Π = E16 ∩ Π = E17 ∩ Π = (E20 ∩ Π) ∪ (D9 ∩ Π1 ) .

(3.117)

Thus, the following theorems have been proved: Theorem 3.11. (a) If (ai , bi )0≤i≤5 ∈ E20 , then the map (2.1) has no fixed points. (b) If (ai , bi )0≤i≤5 ∈ E20 , then all orbits of the map (2.1) are

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unbounded. (c) All the bounded orbits of the map (2.1) are confined in the set (ai , bi )0≤i≤5 ∈ R12 − E20 . Theorem 3.12. (a) If (ai , bi )0≤i≤5 ∈ E2j , then the map (2.1) has j fixed points, where j = 1, 8. (b) If (ai , bi )0≤i≤5 ∈ E29 , then the map (2.1) has infinitely many nonisolated fixed points. 3.4

Finding chaotic and nonchaotic attractors

The paper [Zeraoulia and Sprott (2009)] gives some criteria for the existence and the nonexistence of chaotic attractors in the general 2-D quadratic map of the form (2.1). This section offers a rigorous proof for the chaoticity and the nonchaoticity of the general 2-D quadratic map (2.1) using the so-called second-derivative test defined for real functions introduced in Sec. 1.10 and in Theorem 1.26. For the map (2.1), assume that the following conditions holds:  a3 > 0, a4 > 0, 4a3 a4 > a25 Ω1 : (3.118) b3 < 0, b4 < 0, 4b3 b4 > b25 , where Ω1 defines a subset of the elements (ai , bi )0≤i≤5 ∈ R12 . Let the two functions za (x, y) and zb (x, y) be the first and second components of the map (2.1), i.e.,  za (x, y) = a0 + a1 x + a2 y + a3 x2 + a4 y 2 + a5 xy (3.119) zb (x, y) = b0 + b1 x + b2 y + b3 x2 + b4 y 2 + b5 xy. If the second-derivative test for both za (x, y) and zb (x, y) is used separately, then one has for all (x, y) ∈ R2 that  2 2 0 a3 a4 +a1 a4 +a2 a3  za (x, y) ≥ a0 a25 −a1 a2 a5 −4a = La a25 −4a3 a4 (3.120) 2 2 2 0 b3 b4 +b1 b4 +b2 b3  zb (x, y) ≤ b0 b5 −b1 b2 b5 −4b = Lb , 2 b5 −4b3 b4

i.e., for all iterations (x, y) ∈ R2 of the map (2.1), one has x ≥ La and y ≤ Lb

(3.121)

using only the inequality (1.64) of Theorem 1.26 and searching for some real N such that 0 < N ≤ 1 for which the map (2.1) has no chaotic attractors. This result permits us to use the notion of complement defined for ensembles to determine rigorously all regions of the parameters (ai , bi )0≤i≤5 ∈ R12 for the occurrence of chaos in the quadratic map of the plane (2.1).

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For the map (2.1), one has JT J =



J11 J12 J12 J22



(3.122)

where J12 = J21 because J T J is symmetric and  2 2  J11 = ((a1 + 2a3 ) x + a5 y) + ((b1 + 2b3 ) x + b5 y)     J12 = o1 + o2  o1 = ((a1 + 2a3 ) x + a5 y) (a5 x + (a2 + 2a4 ) y)    o2 = ((b1 + 2b3 ) x + b5 y) (b5 x + (b2 + 2b4 ) y)   2 2  J22 = (a5 x + (a2 + 2a4 ) y) + (b5 x + (b2 + 2b4 ) y) .

(3.123)

Because J T J is at least a positive
semi-definite
matrix, all its eigenvalue are real and positive, i.e., λmax J T J ≥ λmin J T J ≥ 0. Hence the eigenvalues of J T J are given by  2 +J 2
J +J +√J 2 −2J J +4J12  22 λmax J T J = 11 22 √ 11 2 11 22 (3.124) 2 2 2
J11 +J22 − J11 −2J11 J22 +4J12 +J22 λ . JT J = min

2

We have

where         

  J11 = C1 x2 + C2 y 2 + C3 xy J = 12 C3 x2 + C4 y 2 + C5 xy  12 J22 = C2 x2 + C6 y 2 + 2C4 xy 2

(3.125)

2

C1 = (2a3 + a1 ) + (2b3 + b1 ) ≥ 0 C2 = a25 + b25 ≥ 0 C3 = 2 ((a1 + 2a3 ) a5 + (b1 + 2b3 ) b5 )  C4 = (a2 + 2a4 ) a5 + (b2 + 2b4 ) b5    2 2  C = (a + 2a 5 1 3 ) (a2 + 2a4 ) + (b1 + 2b3 ) (b2 + 2b4 ) + a5 + b5    2 2 C6 = (2a4 + a2 ) + (2b4 + b2 ) ≥ 0.

(3.126)

The 2-D quadratic map (2.1) is nonchaotic if there exists a real N satisfying the following inequalities:  0
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where

            

α1 = C1 + C2 ≥ 0 α2 = C2 + C6 ≥ 0 α3 = C3 + 2C4 α4 = 14 C32 − C1 C2  α5 = C42 − C2 C6     α6 = C3 C5 − C2 C3 − 2C1 C4     α7 = 2C4 C5 − C3 C6 − 2C2 C4    α8 = C52 − C1 C6 − C3 C4 − C22 .

Assume first that

125

(3.128)

Ω2 : α3 < 0.

(3.129)

The aim of the following investigation is to determine an interval for the quantity 0 < N ≤ 1 such that (3.127) holds for all x ≥ La and y ≤ Lb . For this purpose, begin with the second condition of (3.127) and consider the function m (x, y) = α1 x2 + α2 y 2 + α3 xy − 2N, assuming that Ω3 : a1 < 0.

(3.130)

Then from (3.118) and (3.120), one has La ≤ x ≤ −

a4 y 2 a5 xy a2 y La a0 a3 x2 − − − + − . a1 a1 a1 a1 a1 a1

(3.131)

a3 x2 a4 y 2 a5 xy a2 y L a a0 − − − + − ≤ x2 a1 a1 a1 a1 a1 a1

(3.132)

Thus we can choose x1 ≤ La ≤ x ≤ −

where x1 and x2 are the roots of the equation m
(x, y) = 0 with respect to x, i.e., its discriminant is 8N α1 + α23 − 4α1 α2 y 2 > 0 for all y ∈ R. Then one has  q 2   x = −α3 y− 8N α1 +(α3 −4α1 α2 )y2 1 2α1 q (3.133) 2 2   x = −α3 y+ 8N α1 +(α3 −4α1 α2 )y . 2 2α1 The inequality x1 ≤ La holds for all y ≤ Lb if Ω4 : L b ≤ 2

and the inequality − aa3 x1 − y ≤ Lb if

a4 y 2 a1

−

−2α1 La , α3

a5 xy a1

−

a2 y a1

+

(3.134) La a1

−

w1 (x, y) + w2 (x, y) + α21 ≤ 0

a0 a1

≤ x2 holds for all (3.135)

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w1 (x, y) = α9 x4 + α10 y 4 + α11 x2 y 2 + α12 x3 y + α13 xy 3 + α14 y 3 w2 (x, y) = α15 x2 y + α16 xy 2 + α17 x2 + α18 y 2 + α19 xy + α20 y (3.136)

and                

4a2 α2

α11

4a2 α2

α9 = a32 1 , α10 = a42 1 1 1 4(2a3 a4 +a25 )α21 8a3 a5 α21 = , α = 2 12 a a2 1

1

8a4 a5 α21 2 α1 )a4 α1 , α14 = −4(a1 α3 −2a a21 a21 2 α1 )a3 α1 2 α1 )a5 α1 α15 = −4(a1 α3 −2a , α16 = −4(a1 α3 −2a a21 a21 −4(a1 a2 α3 −2a0 a4 α1 +2a4 α1 La −a21 α2 −a22 α1 )α1 −8(La −a0 )a3 α21 , α18 = a21 a21 −8(La −a0 )a5 α21 4(La −a0 )(a1 α3 −2a2 α1 )α1 , α20 = α19 = a21 a21 4(a20 α1 −2N a21 −2a0 α1 La +α1 L2a )α1 α21 = . a21

α13 =

    α17 =           

(3.137) Now consider the function w (x, y) = w1 (x, y) + w2 (x, y) + α21 . The critical points of w are the solutions of the system
 3 2 2 x +
t0 = 0 4α x + 3α yx + 2α + 2α y + 2y α 9 12 17 15 11    3 2 4α10 y + (3α14 + 3α13 x) y + 2α18 + 2α16 x + 2α11 x2 y + t1 = 0  t0 (x, y) = α13 y 3 + α16 y 2 + α19 y   t1 (x, y) = α12 x3 + α15 x2 + α19 x + α20 (3.138) Assume that Ω5 : α9 6= 0, α10 6= 0. (3.139) Then both equations in (3.138) are cubic, and its first equation  has at  least (1)

(i)

one real solution sc for all values of y, and at most three roots sc

1≤i≤3

for all values of y. The second equation of (3.138) has at least one real  (1) (i) solution qc for all values of x, and at most three roots qc for all 1≤i≤3   (i) (i) values of x. Thus there are still solutions sc , qc of Eq. (3.138) that are critical ( points of the function h. On the other hand, one has d2 w dx2

(x, y) = 12α9 x2 + 2α11 y 2 + 6α12 xy + 2α15 y + 2α17 dw (x, y) = d1 (x, y) + d2 (x, y)

(3.140)

where d1 = α22 x4 + α23 y 4 + α24 x2 y 2 + α25 x3 y + α26 xy 3 + t′2    t′2 (x, y) = α27 y 3 + α28 x2 y  d = α29 xy 2 + α30 y 3 + α31 x2 + α32 y 2 + α33 xy + t′3   2 t′3 (x, y) = α34 x + α35 y + α36

(3.141)

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and

 α22 = 24α9 α11      α 23 = 24α10 α11     α24 = 144α9 α10 + 36α12 α13 + 4α211    α25 = 72α9 α13 + 12α11 α12  α 26 = 12α11 α13 + 72α10 α12     α 27 = 24α10 α15 + 12α11 α14       α28 = 12α16 α12 + 4α11 α15 + 72α9 α14  α29 = 4α16 α11 + 36α12 α14 + 12α13 α15

and

127

(3.142)

 α30 = 24α10 α15 + 12α11 α14      α31 = 24α9 α18 + 4α17 α11 − 16α211      α32 = 24α17 α10 + 4α18 α11 + 12α14 α15 − 36α213 (3.143) α33 = 12α17 α13 + 12α18 α12 + 4α16 α15 − 48α11 α13    α = 4α α − 16α α 34 16 17 16 11     α = 12α α − 24α α + 4α18 α15  35 17 14 16 13   2 α36 = 4α17 α18 − 4α16 + 24α16 α9 .   (1) (1) If one root sc , qc exists for Eq. (3.138), then assume that     2 (1) (1) (1) (1) d w s , q > 0, i.e., s , q < 0 and d c c c c w 2 dx   2  2 (1) (1) (1) (1)  12α9 s(1) + 6α12 sc qc + 2α15 qc + 2α17 < 0 + 2α11 qc c     Ω6 : (1) (1) (1) (1)  > 0. + d sc , qc d sc , qc 1

2



(1) (1) sc , qc



(3.144)

, i.e., w (x, y) ≤ Hence the function w has a relative maximum at     (1) (1) (1) (1) 2 w sc , qc for all (x, y) ∈ R , and in this case we choose w sc , qc < 0, i.e.,     (1) (1) (1) s , q + α21 < 0 (3.145) , q + w w1 s(1) 2 c c c c

or

Ω7 :

 

(w1 +w2 )a21 +(4a20 +4L2a −8a0 La )α21 = N1 < N 8α1 a21    (1) (1) (1)  w1 + w2 = w1 s(1) + w2 sc , qc c , qc

(3.146)

because only the coefficient α21 depends on N. If Eq. (3.138) than one root, then one calculates  has more    (i) (i) (i) (i) (i) (i) d2 w and determines the type , dw sc , qc , and w sc , qc dx2 sc , qc

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of each point by imposing  some conditions as above, and according to (i) (i) the values of w sc , qc , one can determine the global maximum of   (i) (i) the function h, and finally make the quantity w sc , qc strictly nega-

tive. For the third condition of (3.127), consider the function v (x, y) = v1 (x, y) + v2 (x, y) , where  2 2 v1 (x, y) = α4 x4 + α5 y 4 + α6 x3 y + α7 xy 3 + α8 x

y +1 (3.147) 2 2 2 2 v2 (x, y) = N N − α1 x + α2 y + α3 xy .

The critical points of the function v are the solutions of the system
 4α4 x3 + (3α6 y) x2 + 2α8 y 2 − 2N 2 α1
x + α7 y 3 − N 2 α3 y = 0 (3.148) 4α5 y 3 + (3α7 x) y 2 + 2α8 x2 − 2N 2 α2 y + α6 x3 − N 2 α3 x = 0.   (i) (i) of With the same analysis as above, there are still solutions kc , lc Eq. (3.148) that are critical points for the function v. On the other hand, one has ( 2 d v 2 2 2 dx2 (x, y) = 12α4 x + 2α8 y + 6α6 xy − 2N α1 (3.149) dv (x, y) = p1 (x, y) + p2 (x, y) where 

p1 (x, y) = α51 x4 + α52 y 4 + α53 x3 y + α54 xy 3 + α55 x2 y 2 p2 (x, y) = N 2 h1 (x, y) + h2 (x, y) + 4α1 α2 N 4

and  2  h1 (x, y) = 12 (α2 α6 − α1 α7 ) xy − 4 (6α2 α4 + α1 α8 ) x + h0 2 h0 (y) = −4 (6α1 α5 + α2 α8 ) y
 h2 (x, y) = − 16α28 x2 + 36α27 y 2 + 48α7 α8 xy

(3.150)

(3.151)

and

              

α37 = 24α4 α8 α38 = 24α5 α8 α39 = 72α4 α7 + 12α6 α8 α40 = 12α7 α8 + 72α5 α6 α41 = 144α4 α5 + 36α6 α7 + 4α28    α42 = −16α28 − 4N 2 (6α2 α4 + α1 α8 )     α43 = −36α27 − 4N 2 (6α1 α5 + α2 α8 )     α = 12N 2 (α2 α6 − α1 α7 ) − 48α7 α8    44 α45 = 4N 4 α1 α2 .

(3.152)

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  (1) (1) If one root kc , lc exists for Eq. (3.148), then assume that     2 (1) (1) (1) (1) d v k , l > 0, i.e., k , l > 0 and d c c c c v 2 dx  r     (1) (1) (1) 2 (1) 2  +6α6 kc lc +2α8 lc 12α4 kc  = N2 N < 1 Ω8 :  i    h 2α  (1) (1) (1) (1) (1) (1)  4α1 α2 N 4 + h1 kc , lc N 2 + p1 kc , lc > 0. + h2 kc , lc (3.153) The first condition of (3.153) is possible if  2 2  (3.154) Ω9 : 12α4 kc(1) + 2α8 lc(1) + 6α6 kc(1) lc(1) > 0,

and the second condition of (3.153) is possible for all N ∈ R if       Ω10 : h21 kc(1) , lc(1) − 16p1 kc(1) , lc(1) α1 α2 − 16h2 kc(1) , lc(1) α1 α2 < 0 (3.155) because α1 α2 > 0, and from the first condition of (3.127) and conditions (3.146) and (3.153) one has that Ni , i = 1, 2 must satisfy the inequalities max (0, N1 ) < N < min (1, N2 ) .

(3.156)

We have the following cases: (a) If N1 ≤ 0 and N2 ≥ 1, i.e.,      
(1) (1) (1)  Ω11 : w1 s(1) a21 + 4a20 + 4L2a − 8a0 La α21 ≤ 0 + w2 sc , qc c , qc  2 2  (1) (1) (1) (1)  + 6α6 kc lc − 2α1 ≥ 0, + 2α8 lc Ω12 : 12α4 kc (3.157) then one has 0 < N < 1. (b) If N1 ≤ 0 and N2 ≤ 1, i.e.,      
(1) (1) (1)  Ω11 : w1 s(1) a21 + 4a20 + 4L2a − 8a0 La α21 ≤ 0 + w2 sc , qc c , qc   2 2  ¯ 12 : 12α4 kc(1) + 2α8 lc(1) + 6α6 kc(1) lc(1) − 2α1 ≤ 0 Ω (3.158) ¯ 12 is the complement of the subset Ω12 , then there exists an N such where Ω that 0 < N < N2 ≤ 1. (c) If N1 ≥ 0 and N2 ≥ 1, i.e.,      
(1) (1) (1)  Ω13 : w1 s(1) a21 + 4a20 + 4L2a − 8a0 La α21 ≥ 0 + w2 sc , qc c , qc  2  2 (1) (1) (1) (1)  + 6α6 kc lc − 2α1 ≥ 0, + 2α8 lc Ω12 : 12α4 kc (3.159)

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then there exists an N such that 0 ≤ N1 < N ≤ 1, with the condition N1 < 1, i.e.,    
(1) Ω14 : (w1 + w2 ) s(1) − 8α1 a21 + 4a20 + 4L2a − 8a0 La α21 < 0. c , qc (3.160) (d) If N1 ≥ 0 and N2 ≤ 1, i.e.,   2  2   Ω15 : 12α4 kc(1) + 2α8 lc(1) + 6α6 kc(1) lc(1) − 2α1 ≤ 0  2 2  (3.161)  ¯ 12 : 12α4 kc(1) + 2α8 lc(1) + 6α6 kc(1) lc(1) − 2α1 ≤ 0, Ω

then one has 0 ≤ N1 < N < N2 ≤ 1, with the condition N1 < N2 , i.e.,  r     (1) 2 (1) 2 (1) (1)  12α4 kc +2α8 lc +6α6 kc lc  (w1 +w2 )a21 +(4a20 +4L2a −8a0 La )α21 < 2 2α 8α a 1 1 1 Ω16 :    (1) (1)  w1 + w2 = (w1 + w2 ) sc , qc . (3.162) Therefore, for all the above cases, there exists an N such that 0 < N ≤ 1 in which inequality (3.127) holds for all x ≥ La and y ≤ Lb . Finally, the general 2-D quadratic map (2.1) has no chaotic attractors if all the above inequalities hold. Hence we have proved the following theorem: i=11 i=14 ¯ Theorem 3.13. If ∩i=12 i=1 Ωi 6= ∅, or ∩i=1 Ωi ∩ Ω12 6= ∅, or ∩i=1,i6=11 Ωi 6= ∅, ¯ or ∩i=10 i=1 Ωi ∩ Ω12 ∩ Ω15 ∩ Ω16 6= ∅, then the general quadratic map of the plane given by Eq. (2.1) has no chaotic attractors (x, y) with the condition x ≥ La and y ≤ Lb , where La and Lb are given by (3.120).

An immediate and fundamental result of Theorem 3.13 is given by: i=11 ¯ ¯ Theorem 3.14. If (ai , bi )0≤i≤5 ∈ ∪i=12 i=1 Ωi , or (ai , bi )0≤i≤5 ∈ ∪i=1 Ωi ∪ i=14 i=10 ¯ ¯ ¯ i , or (ai , bi ) Ω12 , or (ai , bi )0≤i≤5 ∈ ∪i=1,i6=11 Ω 0≤i≤5 ∈ ∪i=1 Ωi ∪ Ω12 ∪ Ω15 ∪ ¯ 16 , then the general quadratic map of the plane given by Eq. (2.1) has Ω chaotic attractors (x, y) with the condition x ≥ La and y ≤ Lb , where La and Lb are given by (3.120).

We conclude with the following remarks: (1) The above inequalities do not guarantee the boundedness of the attractors. (2) Not all chaotic or nonchaotic attractors are obtained from the above conditions. (3) Finding a specific example is not simple because at each step the solution of third-degree equations and very complicated inequalities with twelve unknown variables are required.

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(4) It may be possible to convert the proof to a numerical algorithm. (5) Some of the above chaotic or nonchaotic attractors can be infinitely large or very large. 3.5

Finding hyperchaotic attractors

In this section we derive sufficient conditions for the existence of hyperchaotic attractors in the general 2-D quadratic map (2.1) [Zeraoulia and Sprott (2008d)]. Indeed, if two or more Lyapunov exponents of a dynamical system are positive throughout a range in the parameters space, then the resulting attractors are called “hyperchaotic”. The importance of these attractors is that they are less regular and the iteration points seemingly “almost” fill the space, which suggests one of the applications of chaos in fluid mixing [Ottino (1989)]. This section offers a similar rigorous proof of the hyperchaoticity of the general map (2.1) using the so-called second-derivative test defined for real functions in Sec. 1.10 and Theorem 1.26. We reconsider all the conditions from (3.118) through (3.126) of Sec.
3.4. The 2-D quadratic map (2.1) has hyperchaotic attractors if λmin J T J > 1, i.e.,  J11 + J22 > 2  2 (3.163) +1 J11 + J22 < J11 J22 − J12  2 J11 J22 − J12 > 1. The first condition of (3.163) gives

µ1 x2 + (µ3 y) x + µ2 y 2 − 2 > 0 where

  µ1 = C1 + C2 ≥ 0 µ = C2 + C6 ≥ 0  2 µ3 = C3 + 2C4 .

(3.164)

(3.165)

The second-derivative test is inconclusive for the function µ1 x2 + (µ3 y) x + µ2 y 2 − 2 because its minimum is −2 and does not depend on bifurcation parameters. Hence it is not possible to choose a lower bound for it. Thus inequality (3.164) has a solution if µ23 − 4µ1 µ2 > 0 and q −µ3 y+ (µ23 −4µ1 µ2 )y 2 +8µ1 for all y ≤ Lb . We choose y ≤ Lb such that x> 2 µ x ≥ La ≥

−µ3 y+

q1

(µ23 −4µ1 µ2 )y2 +8µ1 2 µ1

, i.e.,

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    

µ23 − 4µ1 µ2 > 0 −2 µ1 La Lb ≤ µ3 ̥2 :  µ < 0  3   8µ2 − 4µ1 µ2 L2a + L2a µ23 < 0.

(3.166)

The second condition of (3.163) gives

µ4 x4 + µ5 y 4 +µ6 x3 y +µ7 xy 3 +µ8 x2 y 2 +µ1 x2 +µ2 y 2 +µ3 xy −1 < 0 (3.167) where      

µ4 = 14 C32 − C1 C2 µ5 = C42 − C2 C6 µ6 = C3 C5 − C2 C3 − 2C1 C4    µ = 2C4 C5 − C3 C6 − 2C2 C4   7 µ8 = C52 − C1 C6 − C3 C4 − C22 .

(3.168)

˜ (x, y) = µ x4 + µ y 4 +µ x3 y+µ xy 3 +µ x2 y 2 + Now consider the function h 4 5 6 7 8 2 2 ˜ are the solutions of the µ1 x + µ2 y + µ3 xy − 1. The critical points of h system 


(4µ4 ) x3 + (3µ6 y) x2 + 2µ1 + 2µ8 y 2
x + µ7 y 3 + µ3 y = 0 (4µ5 ) y 3 + (3xµ7 ) y 2 + 2µ8 x2 + 2µ2 y + µ6 x3 + xµ3 = 0.

(3.169)

Assume first that

̥3 : µ4 6= 0, µ5 6= 0.

(3.170)

Then both equations in (3.169) are cubic, and the first equation of (3.169) (1) has atleast  one real solution z˜c for all values of y, and at most three (i) roots z˜c for all values of y. The second equation of (3.169) has 1≤i≤3

(1)

at least  one real solution u˜c for all values of x, and at most three roots  (i) u ˜c for all values of x. Thus there are still solutions of Eq. (3.169) 1≤i≤3

˜ On the other hand, one has that are critical points of the function h.  ˜ d2 h 2 2   dx2 (x, y) = 12x µ4 + 2µ8 y + 6µ6 xy + 2µ1 4 4 3 dh˜ (x, y) = µ9 x + µ10 y + µ11 xy + µ12 x3 y + µ14 x2 y 2 + t2 (x, y)   t2 (x, y) = µ15 x2 + µ16 y 2 + µ17 xy + µ18 (3.171)

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                

µ9 = 24µ4 µ8 − 9µ26 µ10 = 24µ5 µ8 − 9µ27 µ11 = 72µ5 µ6 − 12µ7 µ8 µ12 = 72µ4 µ7 − 12µ6 µ8 µ13 = 144µ4 µ5 + 18µ6 µ7 − 12µ28 (3.172)  µ14 = 144µ4 µ5 + 18µ6 µ7 − 12µ28     µ15 = 24µ2 µ4 + 4µ1 µ8 − 6µ3 µ6     µ  16 = 24µ1 µ5 + 4µ2 µ8 − 6µ3 µ7     µ = 12µ1 µ7 + 12µ2 µ6 − 8µ3 µ8   17 µ18 = 4µ1 µ2 − µ23 .   (1) (1) If one root z˜c , u ˜c exists for Eq. (3.169), then assume that     2˜ (1) (1) (1) (1) d h ˜c > 0, i.e., ˜c , u ˜c < 0 and dh˜ z˜c , u dx2 z (  2  2 (1) (1) (1) (1) 12 z˜c µ4 + 2µ8 u ˜c + 6µ6 z˜c u ˜c + 2µ1 < 0 ̥4 : (3.173) µ19 + µ20 > 0 where  3   4  3  4  (1) (1) (1) (1) (1) (1)  u˜c + µ11 z˜c uc + µ12 z˜c + µ10 u˜c µ19 = µ9 z˜c 2 2 2 2     (1) (1) (1) (1) (1)   µ20 = µ14 z˜c(1) u ˜c + µ15 z˜c + µ16 u ˜c ˜c + µ18 . + µ17 z˜c u (3.174)   ˜ ˜ has a relative maximum at z˜c(1) , u˜(1) , i.e., h (x, y) ≤ Hence the function h c   (1) ˜ z˜c(1) , u h ˜c for all (x, y) ∈ R2 , especially for x ≥ La and y ≤ Lb . In this   (1) ˜ z˜c(1) , u case, we choose h ˜c < 0, i.e., ̥5 : µ21 + µ22 < 0

(3.175)

where  3  4 4 3     (1) (1) (1) (1) (1)  µ21 = µ4 z˜c(1) + µ5 u u ˜c ˜c + µ6 z˜c u ˜c + µ7 z˜c  2  2  2  2 (1) (1) (1) (1) (1)   µ22 = µ8 z˜c(1) u˜c + µ1 z˜c + µ2 u˜c + µ3 z˜c u ˜c − 1.

(3.176) Therefore, the second inequality (3.163) holds for all x ≥ La and y ≤ Lb . If Eq. (3.169) than one root, then one calculates  has more    ˜ (i) (i) (i) (i) (i) (i) d2 h ˜ ˜c , and h z˜c , u ˜c and determines the type z˜c , u ˜c , d˜ z˜c , u 2 dx

h

of each point by imposing some conditions as above, and according to the

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  (i) ˜ z˜c(i) , u values of h ˜c , one can determine the global maximum of the func  (i) ˜ and finally make the quantity h ˜ z˜c(i) , u tion h, ˜c strictly negative. The third condition of (3.163) gives µ4 x4 + µ5 y 4 + µ6 x3 y + µ7 xy 3 + µ8 x2 y 2 + 1 < 0.

(3.177)
Now consider the function g˜ (x, y) = µ4 x4 +(µ6 y) x3 + µ8 y x2 + µ7 y 3 x+ µ5 y 4 +1. The critical points of the function g˜ are the solutions of the system
 (4µ4 ) x3 + (3yµ6 ) x2 + 2y 2 µ8
x + µ7 y 3 = 0 (3.178) (4µ5 ) y 3 + (3xµ7 ) y 2 + 2x2 µ8 y + µ6 x3 = 0.
2

Using the same analysis as above, one has that the first equation   of (3.178) (i) (1) for all has at least one real solution x ˜c and at most three roots x˜c 1≤i≤3

values of y, and the second equation   of (3.178) has at least one real solution (1)

y˜c

(i)

and at most three roots y˜c

1≤i≤3

for all values of x. Thus there are

still solutions of Eq. (3.178) that are critical points for the function g˜. On the other hand, one has ( d2 g ˜ 2 2 dx2 (x, y) = 12µ4 x + 2µ8 y + 6µ6 xy (3.179) 4 4 3 dg˜ (x, y) = µ9 x + µ10 y + µ11 xy + µ12 x3 y + µ13 x2 y 2 .   (1) (1) exists for Eq. (3.178), then assume that If one root x ˜c , y˜c     (1) (1) (1) (1) d2 g ˜ > 0, i.e., ˜c , y˜c < 0 and dg˜ x˜c , y˜c dx2 x   2  2 (1) (1) (1) (1)   y + 6µ6 x ˜c y˜c < 0 x ˜ + 2µ 12µ c c  8 4    3 4 4   (1) (1) (1) (1) (3.180) ̥6 : µ9 x + ζ 1 > 0. ˜c + µ10 y˜c + µ11 x ˜c y˜c    3  2  2   (1) (1) (1) (1)  ζ 1 = µ12 x˜c y˜c + µ13 x ˜c y˜c .   (1) (1) Hence the function g˜ has a relative maximum at x˜c , y˜c , i.e., g˜ (x, y) ≤     (1) (1) (1) (1) g˜ x ˜c , y˜c for all (x, y) ∈ R2 , and in this case we choose g˜ x˜c , y˜c < 0, i.e.,    4  4  3  (1) (1) (1) (1)  µ4 x ˜c + µ5 y˜c + µ6 y˜c x ˜c + ζ2 < 0 ̥7 :  3    2  2 (3.181) (1) (1) (1)   ζ 2 = µ7 y˜c(1) x ˜c + µ8 y˜c x ˜c + 1.

Therefore, inequality (3.177) holds for all x ≥ Laand y ≤  Lb . If Eq. (3.178)  (i) (i) (i) (i) d2 g ˜ ˜c , y˜c , dg˜ x˜c , y˜c , has more than one root, then one calculates dx2 x

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  (i) (i) and g˜ x ˜c , y˜c , and determines the type of each point by imposing some   (i) (i) conditions as above, and according to the values g˜ x˜c , y˜c , one can determine theglobal maximum of the function g˜ and finally make the quantity  (i) (i) strictly negative. g˜ x ˜c , y˜c Condition (1.65) of Theorem 1.26 gives  ˜ ≤0 µ x2 + µ y 2 + µ xy − 2N  1

2

3

µ x4 + µ5 y 4 + µ6 x3 y + µ7 xy 3 + µ8 x2 y 2 + t3 (x, y) ≥ 0  4 ˜ 2 µ1 x2 − N ˜ 2 µ2 y 2 − N ˜ 2 µ3 xy + 1 + N ˜ 4. t3 (x, y) = −N

(3.182)

The aim of the following investigation is to determine an interval for the ˜ > 1. For this purpose, begin with the first condition of (3.182) quantity N ˜ assuming and consider the function m1 (x, y) = µ1 x2 + µ2 y 2 + µ3 xy − 2N, that ̥8 : a1 < 0.

(3.183)

Then from (3.120) and (3.121), one has La ≤ x ≤ −

a4 y 2 a5 xy a2 y La a0 a3 x2 − − − + − . a1 a1 a1 a1 a1 a1

(3.184)

Thus we can choose a3 x2 a4 y 2 a5 xy a2 y L a a0 − − − + − ≤x ˜2 (3.185) a1 a1 a1 a1 a1 a1 where x ˜1 and x ˜2 are the roots of the equation m
1 (x, y) = 0 with respect to ˜ 1 + µ2 − 4µ1 µ2 y 2 > 0 for all y ∈ R. Then x, i.e., its discriminant is 8Nµ 3 one has  q 2 2 ˜   x˜ = −µ3 y− 8N µ1 +(µ3 −4µ1 µ2 )y 1 2µ 1 q (3.186) 2 2 ˜   x˜ = −µ3 y+ 8N µ1 +(µ3 −4µ1 µ2 )y . 2 2µ x ˜1 ≤ La ≤ x ≤ −

1

The inequality x ˜1 ≤ La holds for all y ≤ Lb if −2µ1 La ̥9 : Lb ≤ , µ3 2

and the inequality − aa3 x1 − y ≤ Lb if

a4 y 2 a1

−

a5 xy a1

−

a2 y a1

+

(3.187) La a1

−

w ˜1 (x, y) + w ˜2 (x, y) + µ35 ≤ 0

a0 a1

≤x ˜2 holds for all (3.188)

where  w ˜1 (x, y) = µ23 x4 + µ24 y 4 + µ25 x2 y 2 + µ26 x3 y + µ27 xy 3 + µ28 y 3 w ˜2 (x, y) = µ29 x2 y + µ30 xy 2 + µ31 x2 + µ32 y 2 + µ33 xy + µ34 y (3.189)

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and                                 

4a23 µ21 a21 4a24 µ21 µ24 = a2 1 4(2a3 a4 +a25 )µ21 µ25 = a21 8a3 a5 µ21 µ26 = a2 1 8a a µ2 µ27 = 4a25 1 1 −4(a1 µ3 −2a2 µ1 )a4 µ1 µ28 = a21 2 µ1 )a3 µ1 µ29 = −4(a1 µ3 −2a a21   2 µ1 )a5 µ1  µ30 = −4(a1 µ3 −2a   a21    −8(La −a0 )a3 µ21  µ31 =  a21    −4(a1 a2 µ3 −2a0 a4 µ1 +2a4 µ1 La −a21 µ2 −a22 µ1 )µ1    µ32 =  a21    −8(La −a0 )a5 µ21  µ33 =   a21   4(La −a0 )(a1 µ3 −2a2 µ1 )µ1   µ34 =  a21   ˜ a2 −2a0 µ La +µ L2 )µ  4(a20 µ1 −2N  1 1 1 a 1 . µ35 = a21

µ23 =

(3.190)

Now consider the function w ˜ (x, y) = w ˜1 (x, y) + w ˜2 (x, y) + µ35 . The critical points of w ˜ are the solutions of the system  4µ23 x3 + 3µ26 yx2 + z1 x + µ27 y 3 + µ30 y 2 + µ33 y = 0    z1 = 2µ31 + 2yµ29 + 2y 2 µ25
3 2 2   4µ24 y + (3µ28 + 3µ27 x) y + 2µ32 + 2µ30 x + 2µ25 x y + t4 (x, y) = 0  t4 (x, y) = µ26 x3 + µ29 x2 + µ33 x + µ34 . (3.191)   (i) (i) With the same analysis as above, there are still solutions s˜c , q˜c of Eq. (3.191) that are critical points for the function w. ˜ On the other hand, one has ( 2 d w ˜ 2 2 dx2 (x, y) = 12µ23 x + 2µ25 y + 6µ26 xy + 2µ29 y + 2µ31 (3.192) dw˜ (x, y) = d1 (x, y) + d2 (x, y)

where ˜ d1 (x, y) = µ36 x4 + µ37 y 4 + µ38 x2 y 2 + µ39 x3 y + t′3    t′3 (x, y) = µ40 xy 3 + µ41 y 3 + µ42 x2 y  d˜ (x, y) = µ43 xy 2 + µ44 y 3 + µ45 x2 + µ46 y 2 + t′4   2 t′4 (x, y) = µ47 xy + µ48 x + µ49 y + µ50

(3.193)

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and

 µ36 = 24µ23 µ25      µ37 = 24µ24 µ25     µ38 = 144µ23 µ24 + 36µ26 µ27 + 4µ225    µ39 = 72µ23 µ27 + 12µ25 µ26  µ40 = 12µ25 µ27 + 72µ24 µ26     µ41 = 24µ24 µ29 + 12µ25 µ28       µ42 = 12µ30 µ26 + 4µ25 µ29 + 72µ23 µ28  µ43 = 4µ30 µ25 + 36µ26 µ28 + 12µ27 µ29

and

137

(3.194)

 µ44 = 24µ24 µ29 + 12µ25 µ28     2  µ  45 = 24µ23 µ32 + 4µ31 µ25 − 16µ25     µ46 = 24µ31 µ24 + 4µ32 µ25 + 12µ28 µ29 − 36µ227 (3.195) µ47 = 12µ31 µ27 + 12µ32 µ26 + 4µ30 µ29 − 48µ25 µ27    µ = 4µ µ − 16µ µ  48 30 31 30 25    µ49 = 12µ31 µ28 − 24µ30 µ27 + 4µ32 µ29    µ50 = 4µ31 µ32 − 4µ230 + 24µ30 µ23 .   (1) (1) If one root s˜c , q˜c exists for Eq. (3.191), then assume that     2 (1) (1) (1) (1) d w ˜ s ˜ , q ˜ > 0, i.e., s ˜ , q ˜ < 0 and d c c c c w ˜ 2 dx   2 2  (1) (1) (1) (1) (1)  12µ + 6µ26 s˜c q˜c + 2µ29 q˜c + 2µ31 < 0 ˜c + 2µ25 q˜c 23 s     ̥10 : (1) (1) (1) (1)  > 0. + d˜ s˜c , q˜c d˜ s˜c , q˜c 1

2



(1) (1) s˜c , q˜c



(3.196)

, i.e., w ˜ (x, y) ≤ Hence the function w ˜ has a relative maximum at     (1) (1) (1) (1) w ˜ s˜c , q˜c for all (x, y) ∈ R2 , and in this case we choose w ˜ s˜c , q˜c < 0, i.e.,     ˜c(1) + µ35 < 0 (3.197) ˜c(1) + w ˜2 s˜(1) w ˜1 s˜(1) c ,q c ,q or

̥11

 2 2 2 2 a −8a0 La )µ1  N ˜ > (w˜1 +w˜2 )a1 +(4a0 +4L ˜1 =N 8µ1 a21     : (1) (1) (1) (1) w +w ˜2 s˜c , q˜c ˜1 + w ˜2 = w ˜1 s˜c , q˜c

(3.198)

˜ , and because we want N ˜ > 1, because only the coefficient µ35 depends on N ˜ then we must assume N1 > 1, i.e.,
˜1 + w ˜2 − 8µ1 ) a21 > 0 (3.199) ̥12 : 4a20 + 4L2a − 8a0 La µ21 + (w

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    (1) (1) (1) (1) . Therefore, the first in+w ˜2 s˜c , q˜c where w ˜1 + w ˜2 = w ˜1 s˜c , q˜c

equality of (3.182) holds for all x ≥ La and y ≤ Lb . For the second condition of (3.182), consider the function v˜ (x, y) = v˜1 (x, y) + v˜2 (x, y) , where ( 4 3 3 2 2 v˜1 (x, y) = µ4 x4 + µ 5 y + µ6 x y + µ7 xy + µ8 x
y + 1 (3.200) ˜2 N ˜ 2 − µ x2 + µ y 2 + µ xy . v˜2 (x, y) = N 1 2 3 The critical points of the function v˜ are the solutions of the system    ˜ 2 µ1 x + µ7 y 3 − N ˜ 2 µ3 y = 0  4µ4 x3 + (3µ6 y) x2 + 2µ8 y 2 − 2N   ˜ 2 µ2 y + µ6 x3 − N ˜ 2 µ3 x = 0.  4µ5 y 3 + (3µ7 x) y 2 + 2µ8 x2 − 2N (3.201)   (i) ˜(i) ˜ With the same analysis as above, there are still solutions kc , lc of

Eq. (3.201) that are critical points for the function v˜. On the other hand, one has ( 2 d v ˜ 2 2 ˜2 dx2 (x, y) = 12µ4 x + 2µ8 y + 6µ6 xy − 2N µ1 (3.202) dv˜ (x, y) = p1 (x, y) + p2 (x, y) where 

p˜1 (x, y) = µ51 x4 + µ52 y 4 + µ53 x3 y + µ54 xy 3 + µ55 x2 y 2 ˜ 2 h1 (x, y) + h2 (x, y) + 4µ µ N ˜4 p˜2 (x, y) = N 1 2

and ˜ 2 2  h1 (x, y) = 12 (µ2 µ6 − µ1 µ7 ) xy + z2 x − 4 (6µ1 µ5 + µ2 µ8 ) y z2 = −4 (6µ2 µ4 + µ1 µ8 )
 ˜ 2 (x, y) = − 16µ2 x2 + 36µ2 y 2 + 48µ µ xy h 8 7 7 8

(3.203)

(3.204)

and

              

µ51 = 24µ4 µ8 µ52 = 24µ5 µ8 µ53 = 72µ4 µ7 + 12µ6 µ8 µ54 = 12µ7 µ8 + 72µ5 µ6 µ55 = 144µ4 µ5 + 36µ6 µ7 + 4µ28   ˜ 2 (6µ µ + µ µ )  µ56 = −16µ28 − 4N 2 4 1 8    2 2 ˜  µ = −36µ − 4 N (6µ µ + µ  57 1 5 2 µ8 ) 7   2  ˜ µ = 12 N (µ µ − µ µ ) − 48µ  2 6 1 7 7 µ8   58 4 ˜ µ59 = 4N µ1 µ2 .

(3.205)

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139

  (1) (1) If one root k˜c , ˜lc exists for Eq. (3.201), then assume that     2 d v ˜ ˜ (1) ˜(1) ˜c(1) , ˜lc(1) > 0, i.e., k k , l > 0 and d c c v ˜ 2 dx  r  2   (1) 2 (1) ˜c(1) +2µ ˜ ˜c(1) ˜  12µ4 k +6µ6 k lc  8 lc ˜ ˜ =N N< ̥13 : 1  h 2µ    2 i   4µ µ N ˜ 1 k˜c(1) , ˜lc(1) N ˜ 2 k˜c(1) , ˜lc(1) > 0. ˜4 + h ˜ 2 + p˜1 k˜c(1) , ˜lc(1) + h 1 2 (3.206) The first condition of (3.206) is possible if  2  2 (1) (1) (1) (1) 12µ4 k˜c + 2µ8 ˜lc + 6µ6 k˜c ˜lc > 0, (3.207) ̥14 : 2µ1 ˜ ∈ R if and the second condition of (3.206) is possible for all N i   h    ˜ 2 k˜(1) , ˜l(1) < 0 (3.208) ̥15 : ˜ h2 k˜(1) , ˜l(1) − 16µ µ p˜1 k˜(1) , ˜l(1) + h 1

c

c

1 2

c

c

c

c

˜i , i = 1, 2 because µ1 µ2 > 0, and from (3.198) and (3.206), one has that N ˜ ˜ must satisfy the inequalities 1 < N1 < N2 , i.e., 
4a20 + 4L2a − 8a0 La µ21 r + (w ˜1 + w ˜2 − 8µ1 ) a21 > 0      2   (1) 2 (1) ˜c(1) +2µ ˜ ˜c(1) ˜ 12µ4 k +6µ6 k lc (w ˜ 1 +w ˜2 )a21 +(4a20 +4L2a −8a0 La )µ21 8 lc ̥16 : < 2 2µ1  8µ1 a1       (1) (1) (1) (1)  . +w ˜2 s˜c , q˜c w ˜1 + w ˜2 = w ˜1 s˜c , q˜c

(3.209) ˜ such that 1 < N ˜1 < N ˜
Theorem 3.15. If ∩i=16 i=2 ̥i ∩Ω1 6= ∅, then the general quadratic map of the plane given by Eq. (2.1) has hyperchaotic attractors (x, y) with the condition x ≥ La and y ≤ Lb , where La and Lb are given by (3.120). The same conclusion given at the end of Sec. 3.4 is still valid for this case.

3.6

Some criteria for finding chaotic orbits

Roughly speaking, chaotic behavior implies sensitive dependence on initial conditions, with at least one positive Lyapunov exponent. From some well known special cases of the 2-D quadratic map (2.1), several remarks are in order:

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(1) Two of the twelve coefficients can be set to 1 by renormalizing x and y. We use this criterion for each case separately. (2) Because there must be at least one nonlinearity for chaos, we classify the special maps of (2.1) according to their number of nonlinearities. We found thirty cases, some of which are topologically equivalent by the above analysis. (3) The function f must depend on y, and g must depend on x if the map (2.1) is truly a 2-D map. (4) For bounded orbits, the absolute value of the determinant (averaged along the orbit) of the Jacobian matrix is not greater than unity. (5) There must be an expanding direction for chaos. (6) We choose a classification according to the number of nonlinearities, with the simplest examples having the maximum number of vanishing coefficients. In the following sections, all possible cases of chaos in the general 2-D quadratic map (2.1) are classified according to their number of nonlinearities, and for each class, several examples are given showing the structure of the parameter space for the considered map with selected values of the vectors (ai , bi )1≤i≤6 ⊂ R12 . To simplify the search for the possible classes of chaos in the map (2.1), we introduce the following notations: A special case of the map (2.1) is of type (m − l) if it has m nonlinearities, and l indicates the number of possible cases of it. This classification is based primarily on the above theorems of equivalence between quadratic maps. As a test of the above theorems, we give detailed examples for cases with one nonlinearity including the H´enon map (2.11) and the map (2.15) given in [Zeraoulia and Sprott (2008a)]. The other cases can be studied using the same logic. 3.7

2-D quadratic maps with one nonlinearity

For the class of one nonlinearity, we have three cases:    xk+1 = a0 + a1 xk + a2 yk + a3 x2k   (1 − 1) :   y = b0 + b1 xk + b2 yk    k+1  xk+1 = a0 + a1 xk + a2 yk + a4 yk2 (1 − 2) :  y = b0 + b1 xk + b2 yk   k+1    x k+1 = a0 + a1 xk + a2 yk + a5 xk yk    (1 − 3) : yk+1 = b0 + b1 xk + b2 yk .

(3.210)

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The maps (1 − 1), including the H´enon map [H´enon (1976)] obtained for a1 = 0, b0 = 0, and b2 = 0, are shown in Fig. 2.6(a). The maps (1 − 2), including the map studied in [Zeraoulia and Sprott (2008a)] obtained for a2 = 0, b0 = 0, and b2 = 0, are shown in Fig. 2.6(b). The maps (1 − 3), including the delayed logistic map [Aronson (1982)] obtained for a0 = 0, a2 = 0, b0 = 0, b1 = 1, and b2 = 0, and the two cases obtained for a0 = 1, a1 = 0, b0 = 0, b1 = 1, and b2 = 0, or a0 = 1, a2 = 0, b0 = 0, b1 = 1, and b2 = 0, where it is easy to show that both maps are not topologically equivalent to the delayed logistic map, the H´enon map (2.11), or the map (2.15). Some simple cases of the maps (1 − i)i=1,3 that give chaotic attractors are as follows:
 (1 − 1) : (xk+1 , yk+1 ) −→ 1 − a3 x2k + a2 yk , xk
    (xk+1 , yk+1 ) −→ 1 − a4 yk2 + a1 xk , xk  (1 − 2) :  (3.211)  (xk+1 , yk+1 ) −→ (a1 xk − a1 xk yk , xk )    (1 − 3) : (x , y ) −→ (1 − a5 xk yk + a2 yk , xk )    k+1 k+1 (xk+1 , yk+1 ) −→ (1 − a5 xk yk + a1 xk , xk ) . As an example, we apply the above theorems to the map (1 − 1) as follows: The map (1 − 1) has no fixed points if (ai , bi ) ∈ Ω ⊂ R12 , where Ω is given by (3.213) or (3.213) below:  b 6= 1     22  a2 2 < 4a3 − bb20−1 + a0 a1 − 1 − b2a−1 (3.212) Ω:     b0 a2  a − +1 >0 3

or



b2 −1

b2 = 1, a2 = 0 (3.213) b0 − a1 b30 a3 + 1 6= 0. On the other hand, the map (1 − 1) has one fixed point:   2 (b0 a2 − b2 + 1) 2 (b2 − 1) + b0 (a1 − 1 − a2 + b2 − a1 b2 ) P = , a1 b 2 − a2 − b 2 − a1 + 1 (a1 b2 − a2 − b2 − a1 + 1) (b2 − 1) (3.214) ¯ ⊂ R12 , where if (ai , bi ) ∈ Ω ( 2 +b2 −1 , b2 6= b0 a2 + 1 b2 6= 1, a1 6= a(b 2 −1) ¯ Ω: (3.215) −(a1 b2 −a2 −b2 −a1 +1)2 a3 = 4(b0 a2 −b2 +1)(b2 −1)     −x2 −b0 1 −b0 and x , , where and has two fixed points x1 , −x 2 b2 −1 b2 −1    r 2   a2 a2 a2    x = − a1 −1− b2 −1 + a1 −1− b2 −1 −4a3 −b0 b2 −1 +1 1 2a3 (3.216)    r 2  a2 a2 a2  − a1 −1− b −1 −4a3 −b0 b −1 − +1 a1 −1− b −1   2 2 2 x2 = 2a3 Ω:

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¯ ⊂ R12 , where if (ai , bi ) ∈ Ω ( ¯: Ω

(a1 +a2 +b2 −a1 b2 −1)2 4

b2 6= 1 > a3 (b2 − 1) (b2 − b0 a2 − 1) .

(3.217)

We remark that the chaotic H´enon attractor (2.11) has at least one saddle fixed point. We then use this idea to identify some possible chaotic systems for the map (1 − 1). The Jacobian matrix of the map (1 − 1) is given by:   a1 + 2a3 x a2 . (3.218) J (x, y) = 1 b2 The map (1 − 1) has one fixed point, and it is of the saddle-type if ˜ 1 , where Ω ˜ 1 is the union of the following conditions: (ai , bi )0≤i≤5 ∈ Ω
a2 − 2b2 + b22 + 1 > − sgn (b2 − 1) a2 − 2b2 + b22 + 1


 a2 − 2b2 + b22 + 1 < − sgn (b2 − 1) a2 + 2b2 + b22 − 3
− sgn (b2 − 1) a2 + 2b2 + b22 − 3 > 0
 a2 − 2b2 + b22 + 1 < sgn (b2 − 1) a2 − 2b2 + b22 + 1
sgn (b2 − 1) a2 − 2b2 + b22 + 1 > 0
a2 − 2b2 + b2 + 1 > sgn (b2 − 1) a2 + 2b2 + b2 − 3 , 2 2

(3.219)

(3.220)

(3.221)

(3.222)

where sgn(.) is the standard signum function that gives the sign of its argument. On the other hand, the map (1 − 1) has two fixed points, one of ˜ 1 , where Ω ˜ 1 is the union of the which is a saddle point if (ai , bi )0≤i≤5 ∈ Ω following conditions:  2 2 2 2  4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b2 ≥ 0 −2 + a1 + b2 + 2a3 x1 > 0  (2a3 b2 − 2a3 ) x1 + a1 b2 − a2 − b2 − a1 + 1 > 0

 2 2 2 2  p4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b2 ≥ 0 2 2 2 4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b22 > r1  r1 = 2 + a1 + b2 + 2a3 x1  2 2 2 2  p4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b2 ≥ 0 4a23 x21 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a21 + b22 > r2  r2 = 2 − b2 − 2a3 x1 − a1

(3.223)

(3.224)

(3.225)

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Fig. 3.2 Regions in the a4 a1 -plane for unbounded and bounded orbits for the map (2.15).

 

 2 2 2 2  p4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b2 ≥ 0 2 2 2 4a3 x1 + (4a1 a3 − 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a1 + b22 < r3  r3 = −2 − a1 − b2 − 2a3 x1 > 0

(3.226)

4a23 x21 + (4a1 a3 − r 4a3 b2 ) x1 + 4a2 − 2a1 b2 + a21 + b22 < 0   2  a2 a2  0 < b2 − a2 + a2 b2 + b2 + 1 < 1. − 4a −b a − 1 − 3 0 1 b2 −1 b2 −1 b2 −1

(3.227) A description of the dynamics of the H´enon map is given in Sec. 2.6.6. Note that the third case is in preparation in [Zeraoulia and Sprott (2008b-2008c)]. For the second map (1 − 1) , we consider the map studied in [Zeraoulia and Sprott (2008a)] given by Eq. (2.15). Because it is proved in [Zeraoulia and Sprott (2008a)] that one of the fixed points of the map (2.15) is either ˜ 1 = Ω. ¯ The sets Ω, Ω ¯ ⊂ R12 are unstable or of a saddle type, one has Ω defined by: 2  −a1 + 1 (3.228) Ω : a4 < − 2 ¯ = Ω1 : a 4 ≥ − Ω



−a1 + 1 2

2

.

(3.229)

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Fig. 3.3 (a) A periodic orbit of the map (2.15) with its basin of attraction (white) obtained for a4 = 1 and a1 = 0.1. (b) The chaotic attractor with its basin of attraction (white) for a4 = 1 and a1 = 0.675. (c) Another chaotic attractor with its basin of attraction (white) for a4 = 0.59948 and a1 = 1. (d) A quasi-periodic orbit with its basin of attraction (white) for a4 = 1 and a1 = 0.17. [Zeraoulia and Sprott (2008a)].

A schematic representation
of these results is given in Fig. 3.2, where C is 2 the line C : a4 = − −a21 +1 . Eq.(2.15) is an interesting minimal system similar to the H´enon map, but with the time delay in the nonlinear rather than the linear term. Despite its apparent similarity and simplicity, it differs from the H´enon map in that it has a non-uniform dissipation, a more rich and varied route to chaos, and a much wider variety of attractors. Whereas the attractors for the H´enon map (2.11) have a maximum dimension of about 1.31, with all the attractors qualitatively similar, the map (2.15) has attractors covering the entire range of dimensions from 1 to 2 (as well as zero) with basins of attraction that are often much more complicated than for the H´enon map (2.11).

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Fig. 3.4 Attractors for the system (2.15) with their basins of attraction (white) when (a) a4 = 0.7, a1 = 0.9, (b) a4 = 0.8, a1 = 0.6, (c) a4 = 1.5, a1 = −0.2, (d) a4 = 0.9, a1 = 0.6, (e) a4 = 1, a1 = 0.3, (f) a4 = 1.3, a1 = −0.2, (g) a4 = 1.1, a1 = −0.9, (h) a4 = 1.1, a1 = 0.4. [Zeraoulia and Sprott (2008a)].

For the system (2.15), the values of a4 and a1 that maximize the largest Lyapunov exponent with a4 = 1 and with a1 = 1 are as follows: For a4 = 1, one has b = 0.675 and Lyapunov exponents (base-e) of 0.171496 and 0.007595, while for a1 = 1, one has a4 = 0.59948 and Lyapunov exponents of 0.091912 and −0.074313. The corresponding chaotic attractors are shown respectively in Fig. 3.3, along with their basins of attraction in white. Note that the basin boundary nearly touches the attractor for these cases and is apparently a fractal for the case in Fig. 3.3(c). The quadratic chaotic attractor obtained from the map (2.15) results from a quasi-periodic route to chaos as shown in Fig. 3.13(a). One interesting feature is that this map (2.15) is not dissipative for all combinations

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Fig. 3.5 Attractors for the system (2.15) with their basins of attraction (white) for (a) a4 = 1.6, a1 = −0.2, (b) a4 = 2.1, a1 = −1.5, (c) a4 = 1.5, a1 = −0.4, (d) a4 = 1.4, a1 = 0.1, (e) a4 = 1.5, a1 = 0, (f) a4 = 1.6, a1 = −0.3, (g) a4 = 2.6, a1 = −1.8, (h) a4 = 4, a1 = −2. [Zeraoulia and Sprott (2008a)].

of a1 and a4 . In fact, there are values for which both Lyapunov exponents are positive as shown in Fig. 3.13(b), indicating hyperchaos. Since the map (2.15) is not everywhere dissipative, its attractor can have a dimension equal to or even greater than 2.0 by virtue of the folding afforded by the quadratic nonlinearity. There are parameters such as a4 = 0.765 and a1 = 0.854 for which the two Lyapunov exponents are nearly equal and opposite (0.10710 and −0.10744), implying an attractor with a dimension of 1.9969 by the Kaplan–Yorke conjecture. Furthermore, when both Lyapunov exponents are positive, the dimension in principle could exceed 2.0, and this would be evident by examining the attractor in embeddings higher than 2. Takens’ theorem [Newhouse, et al. (1983)]

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Fig. 3.6 Correlation dimension versus embedding dimension for the map (2.15) with a4 = 1 and a1 = 0.675. [Zeraoulia and Sprott (2008a)].

states that an embedding as large as 2D + 1 might be necessary to resolve the overlaps. As a test of this prediction, the correlation dimension was calculated for various embeddings using the extrapolation method of Sprott and Rowlands [Sprott and Rowlands (2001)], and the results are plotted in Fig. 3.6 for the map (2.15) with a4 = 1 and a1 = 0.675 where the Lyapunov exponents are 0.171496 and 0.007595. The correlation dimension is approximately constant with a value of about 1.87 for all embeddings greater than 1. These results suggest that the Takens’ criterion is overly restrictive for the map (2.15) even though the map is noninvertible for all combinations of a1 and a4 , and hence there is not a one-to-one reconstruction for the map. On the other hand, it is well known that basin boundaries arise in dissipative dynamical systems when two or more attractors are present. In such situations each attractor has a basin of initial conditions that lead asymptotically to that attractor. The sets that separate different basins are called the basin boundaries. In some cases the basin boundaries can have very complicated fractal structure and hence pose an additional impediment to predicting long-term behavior. For the map (2.15), we have calculated the attractors and their basins of attraction on a grid in a4 a1 -space where the system is chaotic. There is a very wide variety of possible attractors, only some of which are shown in Figs. 3.3, 3.4, and 3.5. Also, most of the basin boundaries are smooth, but some appear to be fractal, and this is not a result of numerical errors since the structure persists as the number of iterations of each initial condition is increased. This indicate the coexistence of several attractors as shown in the black region of Fig. 3.1.

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Fig. 3.7

3.8

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Chaotic attractors of the type (2 − i)i=1,6 .

2-D quadratic maps with two nonlinearities

For the case of two nonlinearities, we have six cases given by:    xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2   (2 − 1) :   y = b0 + b1 xk + b2 yk    k+1  2  x  k+1 = a0 + a1 xk + a2 yk + a3 xk + a5 xk yk  (2 − 2) :   y = b0 + b1 xk + b2 yk    k+1    x = a0 + a1 xk + a2 yk + a3 x2k k+1    (2 − 3) : y = b0 + b1 xk + b2 yk + b3 x2k  k+1  xk+1 = a0 + a1 xk + a2 yk + a3 x2k    (2 − 4) : y  = b0 + b1 xk + b2 yk + b4 yk2    k+1   xk+1 = a0 + a1 xk + a2 yk + a3 x2k   (2 − 5) :    y = b0 + b1 xk + b2 yk + b5 xk yk   k+1    x k+1 = a0 + a1 xk + a2 yk + a5 xk yk    (2 − 6) : yk+1 = b0 + b1 xk + b2 yk + b5 xk yk .

(3.230)

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A special case of the map (2 − 1) is under preparation in [Zeraoulia and Sprott (2008c)]. Some simple cases of the maps (2 − i)i=1,6 that give chaotic attractors are given by the following: 
 (2 − 1) : (xk+1 , yk+1 ) −→ 1 − 1.4x2k − 1.7yk2 , xk
   (2 − 2) : (x , y 2  k+1 k+1 ) −→ 1 − 1.6xk + 0.7xk yk , xk
   (2 − 3) : (xk+1 , yk+1 ) −→ 1 + 0.2yk − 1.7x2k , 0.7x2k
(3.231) 2 2   (2 − 4) : (xk+1 , yk+1 ) −→ 1 + 0.1yk − 1.1xk , xk − yk
  2   (2 − 5) : (xk+1 , yk+1 ) −→ 1 − 0.8yk − 1.3xk , 1 − 1.6xk yk   (2 − 6) : (xk+1 , yk+1 ) −→ (1 + 0.3xk − 1.5xk yk , 1 − 1.9xk yk ) . The corresponding chaotic attractors are shown in Figs. 3.7.

3.9

2-D quadratic maps with three nonlinearities

For the case of three nonlinearities, we have ten cases:   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk   (3 − 1) :    y = b0 + b1 xk + b2 yk   k+1   2 2  x k+1 = a0 + a1 xk + a2 yk + a3 xk + a4 yk   (3 − 2) :   y = b0 + b1 xk + b2 yk + b3 x2k    k+1    xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2   (3 − 3) :   y = b0 + b1 xk + b2 yk + b4 yk2    k+1   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2   (3 − 4) :    y = b0 + b1 xk + b2 yk + b5 xk yk   k+1    xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a5 xk yk    (3 − 5) : y = b0 + b1 xk + b2 yk + b3 x2k  k+1  xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a5 xk yk   (3 − 6) :   y = b0 + b1 xk + b2 yk + b4 yk2    k+1   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a5 xk yk   (3 − 7) :    y = b0 + b1 xk + b2 yk + b5 xk yk   k+1    x = a0 + a1 xk + a2 yk + a4 yk2 + a5 xk yk k+1   (3 − 8) :   yk+1 = b0 + b1 xk + b2 yk + b3 x2k       xk+1 = a0 + a1 xk + a2 yk + a4 yk2 + a5 xk yk   (3 − 9) :   y = b0 + b1 xk + b2 yk + b4 yk2    k+1   xk+1 = a0 + a1 xk + a2 yk + a4 yk2 + a5 xk yk    (3 − 10) : yk+1 = b0 + b1 xk + b2 yk + b5 xk yk .

(3.232)

Some simple cases of the maps (3 − i)i=1,10 that give chaotic attractors

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Fig. 3.8

Chaotic attractors of the type (3 − i)i=1,10 .

are listed in the following:
 (3 − 1) : (xk+1 , yk+1 ) −→ 1 − 1.4x2k − 1.5yk2 − 1.8xk yk , xk
    (3 − 2) : (xk+1 , yk+1 ) −→ 1 − 1.4x2k − 1.7yk2 , −yk + 0.6x
2k      (3 − 3) : (xk+1 , yk+1 ) −→ 1 − 1.4x2k − 1.7yk2 , xk + 0.9yk2
    (3 − 4) : (xk+1 , yk+1 ) −→ 1 − 1.4x2k − 1.1yk2 , xk + 0.9xk yk  
 2 (3 − 5) : (xk+1 , yk+1 ) −→ 1 − 1.3x2k − 1.1xk yk , −0.3y k − xk
 (3 − 6) : (xk+1 , yk+1 ) −→ 1 − 1.3x2k − xk yk , xk − yk2 
   (3 − 7) : (xk+1 , yk+1 ) −→ 1 − 1.5x2k + 0.6xk yk , 1 − 1.2xk y
k     (3 − 8) : (xk+1 , yk+1 ) −→ 1 − 1.2yk2 + 0.6xk yk , 0.4yk − x2k
     (3 − 9) : (xk+1 , yk+1 ) −→ 1 − 1.5yk2 − 0.9xk yk , xk − 0.6yk2
  (3 − 10) : (xk+1 , yk+1 ) −→ 1 − 1.4yk2 + 0.4xk yk , 1 + 0.1xk yk . (3.233) The corresponding chaotic attractors are shown in Figs.3.8.

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Fig. 3.9

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Chaotic attractors of the type (4 − i)i=1,7 .

2-D quadratic maps with four nonlinearities

For the case of four nonlinearities, we have seven cases:   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk   (4 − 1) :    y = b0 + b1 xk + b2 yk + b3 x2k   k+1    xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk   (4 − 2) :   y = b0 + b1 xk + b2 yk + b5 xk yk    k+1  2 2   x k+1 = a0 + a1 xk + a2 yk + a3 xk + a4 yk    (4 − 3) : y 2 2  = b0 + b1 xk + b2 yk + b3 xk + b4 yk    k+1  xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 (4 − 4) :  y = b0 + b1 xk + b2 yk + b3 x2k + b5 xk yk   k+1   2 2   k + a4 y k  (4 − 5) : xk+1 = a0 + a1 xk + a2 yk + a3 x  2  y = b0 + b1 xk + b2 yk + b4 yk + b5 xk yk    k+1    x = a0 + a1 xk + a2 yk + a3 x2k + a5 xk yk k+1   (4 − 6) :   y = b0 + b1 xk + b2 yk + b3 x2k + b5 xk yk    k+1   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a5 xk yk    (4 − 7) : yk+1 = b0 + b1 xk + b2 yk + b4 yk2 + b5 xk yk .

(3.234)

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Fig. 3.10

Chaotic attractors of the type (5 − i)i=1,3 .

Some simple cases of the maps (4 − i)i=1,7 that give chaotic attractors are listed in the following: 
2 2 2   (4 − 1) : (xk+1 , yk+1 ) −→ 1 + 0.1xk − 0.5yk − 0.4xk yk , −1.6xk
   (4 − 2) : (xk+1 , yk+1 ) −→ 1 − 0.3x2k − 1.5yk2 − 1.2xk yk , 1 +
0.6xk yk    2 2 2 2   (4 − 3) : (xk+1 , yk+1 ) −→ 1 − 0.3xk − 0.6yk , 1 + xk − 1.4yk
(4 − 4) : (xk+1 , yk+1 ) −→ 1 − 0.3x2k − 0.8yk2 , 1 − x2k + 1.3xk yk
   (4 − 5) : (xk+1 , yk+1 ) −→ 1 − 0.4x2k − 0.7yk2 , 1 − yk2 + 1.3xk yk 
   (4 − 6) : (xk+1 , yk+1 ) −→ 1 − 0.4x2k − 1.8xk yk , 0.6x2k − 1.8xk yk  
 (4 − 7) : (xk+1 , yk+1 ) −→ 1 − 0.3x2k − 0.7xk yk , 1 − 1.1yk2 + 1.3xk yk . (3.235) The corresponding chaotic attractors are shown in Fig. 3.9.

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Fig. 3.11

3.11

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Chaotic attractors of the type (6 − i)i=1 .

2-D quadratic maps with five nonlinearities

For the case of five nonlinearities, we have three cases:   xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk    (5 − 1) :   yk+1 = b0 + b1 xk + b2 yk + b3 x2k + b4 yk2     xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk (5 − 2) : (3.236)  yk+1 = b0 + b1 xk + b2 yk + b3 x2k + b5 xk yk      xk+1 = a0 + a1 xk + a2 yk + a3 x2k + a4 yk2 + a5 xk yk    (5 − 3) : yk+1 = b0 + b1 xk + b2 yk + b4 yk2 + b5 xk yk . Some simple cases of the maps (5 − i)i=1,7 that give chaotic attractors are listed in the following:     1 − 1.8x2k + 1.4yk2 + 0.1xk yk   (5 − 1) : (x , y ) −→ k+1 k+1   −0.2x2k + 1.8yk2      1 − 0.5x2k − yk2 + xk yk (5 − 2) : (xk+1 , yk+1 ) −→ (3.237)  −1.2x2k − xk yk       1 − 0.5x2k − 1.7yk2 + 0.9xk yk    (5 − 3) : (xk+1 , yk+1 ) −→ 1 − 0.4yk2 − 1.5xk yk The corresponding chaotic attractors are shown in Figs. 3.10. 3.12

2-D quadratic maps with six nonlinearities

For the case of six nonlinearities, we have only one case given by the 2-D quadratic map (2.1) itself. A simple example of it is the following:  xk+1 = 1 + 0.1x2k − 0.5yk2 − 0.6xk yk (3.238) yk+1 = 1 − x2k + 0.4yk2 + 0.7xk yk . The corresponding chaotic attractor is shown in Fig. 3.11.

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Fig. 3.12 (a) The quasi-periodic route to chaos for the map (2.15) obtained for a1 = 0.6 and 0 < a4 ≤ 1.07. (b) Variation of the Lyapunov exponents of map (3.15) versus the parameter 0 < a4 ≤ 1.07 with a1 = 0.6. [Zeraoulia and Sprott (2008a)].

3.13

Numerical analysis

In this section, we investigate some important dynamical behaviors observed for some special cases of the map (2.1). For the case of 2-D quadratic maps with one nonlinearity, it is well known that the H´enon map (2.11) typically undergoes a period-doubling route to chaos when the parameters are varied as shown in Fig. 2.19. Furthermore, the minimal quadratic chaotic attractors considered in [Zeraoulia and Sprott (2008a)] result from a quasiperiodic route to chaos as shown in Figs. 3.12(a) and 3.13. The dynamics of other simple 2-D quadratic maps are shown in Figs. 3.14(a) and 3.14(b) [Zeraoulia and Sprott (2008(b-c)], where different regions are labelled in these figures. We use |LE| < 0.0001 as the criterion for quasi-periodic or-

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Fig. 3.13 Regions of dynamical behaviors in the a3 a1 -plane for the map (2.15) given in [Zeraoulia and Sprott (2008a)] obtained for a2 = a4 = a5 = b0 = b2 = b3 = b4 = b5 = 0.

bits with 106 iterations for each point. Thus the two chaotic systems go by different and distinguishable routes to chaos. More generally, due to the smoothness of the quadratic map of the plane given in (2.1), if a bifurcation occurs, then it is one of the generic types, namely, period-doubling, saddle-node, or Hopf. 3.13.1

Some observed catastrophic solutions in the dynamics of the map

In this section, we investigate some important dynamical behaviors observed for some special cases of the map (2.1), i.e., we discuss the occurrence of some isolated islands that make “breaks” in the dynamics of the map where such maps pass between two bounded states through unbounded orbits. Such catastrophes render the system unsuitable for many potential applications. This phenomenon is not observed for the well known simple 2-D quadratic maps [H´enon (1976), Zeraoulia and Sprott (2008a)] whose dynamics seem to be a single island of bounded orbits in a “sea” of un-

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Fig. 3.14 Regions of dynamical behaviors in the a1 a5 -plane for the map given in [Zeraoulia and Sprott (2008b)] for a2 = a3 = a4 = b0 = b2 = b3 = b4 = b5 = 0.

bounded orbits. These islands contain all the bounded state orbits of the corresponding map. For a1 = −0.2, a4 = a5 = b3 = b4 = b5 = 0, b0 = 0.2, and b2 = 0.2, in the map (2.1), i.e., a map of the type (1 − 1) with |a2 | < 1.5 and −7 ≤ a3 ≤ 4, we observe some islands of periodic and chaotic orbits

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Fig. 3.15 Regions of dynamical behaviors in the a2 a5 -plane for the map given in [Zeraoulia and Sprott (2008c)] for a1 = a3 = a4 = b0 = b2 = b3 = b4 = b5 = 0.

with a period-doubling route to chaos surrounded by unbounded orbits, or maybe these unbounded orbits are large chaotic orbits obtained by a perioddoubling route to chaos. This phenomenon can be seen in Fig. 3.15(a).On the other hand, for a4 = −1.8 and a2 = a5 = b0 = b2 = b3 = b4 = b5 = 0

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in the map (2.1), i.e., a map of the type (2 − 1) with −3 ≤ a1 ≤ 2 and −5 ≤ a3 ≤ 2, we observe some islands of chaotic orbits without any route to chaos, surrounded by unbounded orbits, or maybe these unbounded orbits are large chaotic orbits obtained by a quasi-periodic route to chaos. These phenomena can be seen in Fig. 3.15(b).

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Chapter 4

Rigorous proof of chaos in the double-scroll system

4.1

Introduction

Chua’s circuit is one of the famous 3-D dynamical systems because its equations are among the simplest piecewise linear systems, while dynamically it is very complicated and rich in bifurcations and strange attractors. This circuit can display more than 890 chaotic attractors including smooth Chua’s systems [Billota et al. (2007)], and the existence of several of them can be proved numerically [Matsumoto (1984), Zhong and Ayrom (1985)] and analytically by two independent methods [Chua et al. (1986), Matsumoto et al. (1988)] and experimentally [Matsumoto et al. (1985), Komuro et al. (1991)]. Chua’s circuit and related topics are the subject of hundreds of papers [Madan (1993)]. Chua’s circuit is easy to implement for potential applications [Chua (1993), Chua et al. (1993a-1993b), Madan (1993)], and it leads to a large number of equivalent circuits [Chua and Ying (1989), Chua and Lin (1990), Lin (1991), Altman (1993), Bohme and Schwarz (1993), Chua et al. (1993a-b), Pospisil et al. (1995), Chua et al. (1995), Wu and Chua (1996), Pospisil and Brzobohaty (1996), Pospisil, et al. (1999-2000)]. The double-scroll system is described by a third-order autonomous differential equation. In particular, we will choose a dimensionless form given by [Matsumoto et al. (1985)], which we rewrite in the equivalent form  ′  x = α (y − h (x)) (4.1) y′ = x − y + z  z ′ = −βy where 1 (4.2) h(x) = m1 x + (m0 − m1 ) (|x + 1| − |x − 1|) 2 is the canonical piecewise linear equation [Chua and Ying (1983)] describing an odd-symmetric three-segment piecewise linear curve having a breakpoint 159

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at x = ±1, a slope equal to m0 = a + 1 < 0 for the inner segment, and a slope equal to m1 = b + l > 0 for the outer segments, namely,   m1 x + m0 − m1 , x ≥ 1 h(x) = (4.3) m0 x, |x| ≤ 1  m1 x − m0 + m1 , x ≤ −1.

The double-scroll equation has a chaotic attractor [Chua et al. (1986)] for α = 9.35, β = 14.79, m0 = − 17 , and m1 = 72 as shown in Fig. 4.2. Fig.4.1 shows the complete circuit diagram for Chua’s circuit [Billota et al. 2007] described by the following system of differential equations: 
V2 −V1 dV1 1  − f (V1 )  dt = C1 R
dV2 V1 −V2 1 (4.4) + i3 dt = C2 R   di3 = − 1 (V + R i ) 2 0 3 dt L

where C1 and C2 are two capacitors, L is an inductor, R and R0 are two resistors, and NR is a nonlinear element called Chua’s diode characterized by the piecewise linear function in the iv-plane: 1 I = f (V ) = Gb V + (Ga − Gb ) (|V + E| − |V − E|) . (4.5) 2 Note that Eq. (4.4) can be transformed into the dimensionless form (4.1)– (4.2) by a linear transformation with scaled parameters as follows:  V2 Ri3 C2 V1   x = E 2, y = E , z = E , α = C1 2 (4.6) , a = RGa β = R LC2 , γ = R0 RC L   b = RG , τ = t , k = sgn (RC ) . b 2 |RC2 |

Chua’s system of equations, or specifically the piecewise linear function h(x) given by (4.2), can be modified to take other forms of smooth or nonsmooth functions. This operation is called the generalization of Chua’s circuit . For example, the original piecewise-linear function can be replaced by a discontinuous function [Mahla and Palhares (1993), Lamarque et al., (1999)], a C ∞ “sigmoid functionpolynomial h (x) = c0 x3 + c1 x [Altman (1993), Hartley and Mossayebi (1993), Khibnik and Chua (1993), Tsuneda (2005)], or an x|x| function [Tang, et al. (2003)]. These generalizations conserve some properties of the original system and display other new phenomena [Madan (1992-1993), Chua et al. (1993a-b)]. The number of equilibrium points can be extended enough to generate chaotic attractors with several scrolls [Suykens and Vandewalle (1993), Suykens et al. (1997), Yalcin et al. (2000), Yalcin (2001-2002), Tang, et al. (2001), Zhong et al. (2002), Aziz-Alaoui (2002), Ozoguz et al. (2002), Salama et al. (2003),

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Fig. 4.1 (a) A circuit diagram for Chua’s Oscillator. (b) Typical i-v characteristic of Chua’s diode. (c) Circuit realization scheme [Billota et al. (2007a)].

Yalccin et al. (2005)]. For more detail, see [Billota et al. (2007)]. For the remainder of this chapter, we concentrate on the rigorous proof of chaos in the main Eqs. (4.1)–(4.2), namely the double-scroll system. Chua’s circuit is very interesting from the viewpoint of mathematical analysis because of the infinitely many stability windows for certain parameter values, which implies a sensitive dependence of the structure of the attractor on small variation of parameters [Sil’nikov (1993b)]: (1) Chua’s circuit is a universal problem because it exhibits a number of distinct routes to chaos: period-doubling, breakdown of an invariant torus, intermittency, ... (2) The Double-scroll Chua’s attractor is formed by a pair of nonsymmetric spiral Chua’s attractors with three equilibrium states of a saddle-focus

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Fig. 4.2 The classic double-scroll attractor obtained from (4.1) for α = 9.35, β = 14.79, m0 = − 17 , and m1 = 27 [Chua et al. (1986)].

type visible in the attractor, which indicates that the attractor is multistructural, making it different from the other existing attractors of 3-D systems discussed in Sec. 1.2. (3) Chua’s Eqs. (4.1)–(4.2) are “close”in the sense that the bifurcation portraits to the equations defining a 3-D normal form for bifurcations of an equilibrium point have three zero characteristic exponents for the case with additional symmetry and a periodic orbit with three multipliers equal to −1. (4) The results given in [Ovsyannikov and Sil’nikov (1991), Gonchenko et al. (1993)] imply that the attractors that occur in Chua’s circuit (4.1)–(4.2) are new and more complicated mathematically than was earlier realized. This complication is due to the presence of structurally unstable Poincar´e homoclinic orbits in either the attractor itself or the attractor of a nearby system.1 Therefore, a “complete description” of the dynamics and bifurcations in Chua’s equations is impossible, as it is for many other models, because it was shown in [Gonchenko et al. (1993)] that systems with infinitely many structurally unstable periodic orbits of any degree of degeneracy are dense in the Newhouse regions. For a rigorous proof of chaos in the system (4.1)–(4.2), the Poincar´e map introduced in Sec. 1.3 was calculated to prove the existence of chaos in the double-scroll family in the mathematically rigorous sense. Before doing that, we need to define and state a long list of notations, lemmas, 1 These

orbits arise from the tangency of the stable and the unstable manifolds of some saddle periodic orbit (cycle).

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and theorems. Note that when we do not indicate the proof of a lemma or theorem, this means that the proof is direct, and it is listed at the end of the chapter as an exercise for the reader. Otherwise, the reader will be guided by some remarks and indications. The most essential point of the proof is the derivation of the canonical piecewise linear normal form from the real and normalized Jordan form, and in order to prove the spiral image property, the proof needs the introduction of many concepts such as the piecewise linear vector field in R3 and its spectral and geometrical properties, normalized eigenvalue parameters, the straight line tangency property, the D unit, the derivation of the real Jordan form, and hence the normalized form, by introducing affine transformations, especially the linear conjugacy and equivalence criteria. Also a change of coordinate, namely the so-called weighted sum coordinate, was used to facilitate and justify some features of the so-called double-scroll family. Also, vector dot products, normal vectors, linear independence of vectors, lines, and courbes are introduced for this proof, especially the so-called fundamental points and strategic points that are used directly for the derivation of the Poncar´e and half-return map elements and their continuity and discontinuity properties. The monotonicity and nonmonotonicity of the inverse return-time function are also studied for the half-return maps constituting the Poincar´e map of the system using local minimum and local maximum points. To prove that boundary planes of the double-scroll attractor consist of two tightly wound odd-symmetric spirals, the so-called spiral image property was used to prove the existence of homoclinic orbits in the double-scroll attractor. Note that the basis of the present proof is the papers [Chua et al. (1986), Mees and Chapman (1987), Blazquez and Tuma (1992-1993), Parker and Chua (1987), Bartissol and Chua (1988), Silva (1991), Belykh and Chua (1992), Gavrilov and Sil’nikov (1973), Newhouse (1979), Ovsyannikov and Sil’nikov (1986-1991), Matsumoto et al. (1988), Galias (1996-2003), Yang and Li (2004)], where we reformulate the proof in a consistent pedagogical way. Our approach is to present the proof by several definitions, lemmas, and theorems listed in order so that the reader can understand this long and difficult proof. In Sec. 4.5, the Sil’nikov criterion for proving chaos in Chua’s system (4.1)–(4.2) is applied when there exist homoclinic or heteroclinic orbits. The existence of these orbits is proved analytically using the piecewise linear geometry of the vector field associated with Chua’s system (4.1)–(4.2). The dynamics near these orbits is also presented and discussed. The subfamilies of the double-scroll and their properties are presented in Sec. 4.6.

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The so-called geometric model of Chua’s system (4.1)–(4.2) is presented in Sec. 4.7, and it presents a “twin” Smale’s horseshoe, each coupled to the other by its preimage. This geometric object is considered an attribute of the double-scroll attractor, namely the double-horseshoe. Some computer assisted proofs of Chua’s system (4.1)–(4.2) are presented in Sec. 4.8, where the topological entropy is calculated by two methods for both Rˆ ossler-type and double-scroll-type attractors. The final section collects some helpful exercises to enhance understanding the different proofs given in the chapter. 4.2

Piecewise linear geometry and its real Jordan form

First, let us consider the following notations: T Vectors in R3 are denoted by X = (x, y, z) . σ and ω are real and imaginary parts of a complex eigenvalue. ξ(x) : R3 −→ R3 denotes the piecewise linear vector field evaluated at x ∈ R3 . The set L indicates the generalized family of vector fields ξ, and it is called a piecewise linear vector field family. 4.2.1

Geometry of a piecewise linear vector field in R3

In this section, some geometric properties of a vector fields ξ of the generalized family L are shown in Fig. 4.3, and they are summarized in the following lemma: Lemma 4.1. Any member ξ of the family L satisfies the following properties: (P.0) ξ is a continuous piecewise linear vector field. (P.1) ξ is symmetric with respect to the origin, i.e., ξ (−x, −y, −z) = −ξ (x, y, z) for all (x, y, z) ∈ R3 . (P.2) There are two planes U1 , U−1 which are symmetric with respect to the origin, and they partition R3 into three closed regions D1 , D0 , and D−1 . (P.3) In each region Di , i = −1, 0, 1, the vector field ξ is affine, i.e., Dξ (x, y, z) = Mi for all (x, y, z) ∈ Di , where Dξ denotes the Jacobian matrix of ξ, and Mi denotes a 3 × 3 real constant matrix. (P.4) ξ has three equilibrium points, P − ∈ D−1 , O ∈ D0 , P + ∈ D1 , where O = (0, 0, 0) ∈ R3 .

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N∞

W

Er (P +) N B′

EC (P +)

D1

P+ A′∞ L0

L2 E L1

U1

D

C

F AM

B

D0

EC (0)

C−

Er (0) 0 U−1

B−| F−

A− E−

D−

D−1

Fig. 4.3 (1986)].

A = L 0 ∩ L1

L1 = U1 ∩ E C (P+)

B = L 1 ∩ L2

L2 = {x ∈U1, ξ (x) U1}

C = U1 ∩ Er (0) D = U1 ∩ Er (P +)

P−

EC (P −)

L0 = U1 ∩ E C (0)

EC (P −)

E = L 0 ∩ L2 F = {x ∈L2, ξ (x) L2}

Eigenspaces of the equilibria and their related sets. Adapted from [Chua et al.

(P.5) Each matrix Mi has a pair of complex conjugate eigenvalues and one real eigenvalue labelled respectively by σ ˜0 ± jω ˜ 0 , and γ˜ 0 for M0 and σ ˜1 ± jω ˜ 1 , and γ˜ 1 for M−1 and M1 , where ω ˜ 0 > 0, ω ˜ 1 > 0, γ˜i 6= 0, i = 0, 1, 2 and j = −1. (P.6) The eigenspace associated with either the real or the complex eigenvalue at each equilibrium point is not parallel to U1 or U−1 . Fig.4.3 shows the eigenspaces of the equilibria and related sets, where E c (0) is a 2-D eigenspace corresponding to the complex eigenvalue σ ˜0 + jω ˜ 0 at O. E r (0) is a 1-D eigenspace corresponding to the real eigenvalue γ˜0 at O. E c (P + ) is a 2-D eigenspace corresponding to the complex eigenvalue σ ˜1 + jω ˜ 1 at P + .

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E r (P + ) is a 1-D eigenspace corresponding to the real eigenvalue γ˜ 1 at P . and  L0 = U1 ∩ E C (0)  (4.7) L1 = U1 ∩ E C (P + )  L2 = {x ∈ U1 , ξ (x) kU1 } . +

Here, ξ (x) kU1 means that the vector ξ (x) lies on a plane parallel to U1 , and hence using Lemma 4.2 below, one can show that L2 is a straight line in Fig. 4. 3. 4.2.2

Straight line tangency property

Lemma 4.2. (Straight line tangency property): Let ξ be a linear vector field in R3 having a pair of complex conjugate eigenvalues σ ˜ + jω ˜ and a real eigenvalue γ˜. Let U denote any plane which is not parallel to each eigenspace and which does not pass through the origin. Then L = {x ∈ U, ξ (x) kU }

(4.8)

is a straight line. Before we prove the Lemma 4.2, we need to state the following two lemmas, the first of which is given in [Hirsch and Smale (1974)], and the second is about the so-called weighted sum coordinate used to evaluate the images of U and L in this new coordinate system. To simply the study of the Chua’s system given in (4.1)–(4.2), we must introduce the so-called Jordan canonical form [Hirsch and Smale (1974)], which is essentially a classification result for square matrices. In the actual statement, the Jordan canonical form plays a crucial role in the main proof, and some calculation leads to the following: Lemma 4.3. (Derivation of real Jordan form): If ξ is a linear vector field in R3 having a pair of complex conjugate eigenvalues σ ˜ + jω ˜ and a real eigenvalue γ˜ , then its real Jordan form is given by:   ′  σ ˜ −˜ ω0 x ′ ′ ′    ξ (x , y , z ) = ω ˜ σ ˜ 0 y′  (4.9) 0 0 γ˜ z′ where (x′ , y ′ , z ′ ) indicate a new coordinate system.

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Furthermore, it is not hard to show the following statements: Lemma 4.4. (a) The equation for U in the new coordinate system (x′ , y ′ , z ′ ) has the following simplified form:  U = (x′ , y ′ , z ′ ) ∈ R3 , x′ + z ′ = 1 . (4.10)

(b) For each x ∈ L, the vector dot product is zero, i.e., hξ (x) , hi = 0

where h = (1, 0, 1)T is a normal vector to U in view of (4.8). (c) The set L is the straight line defined by the equations:   γ˜ σ ˜ L = y ′ = x′ + (1 − x′ ) , z ′ = 1 − x′ . ω ˜ ω ˜

(4.11)

(4.12)

Let us now define the following important points in Fig. 4.3:         

A = L0 ∩ L1 B = L1 ∩ L2 C = U1 ∩ E r (0)  D = U1 ∩ E r (P + )     E = L0 ∩ L2    F = {x ∈ L2 , ξ (x) kL2 }

(4.13)

where ξ (x) kL2 means the vector ξ (x) lies on U1 and is parallel to the straight line L2 . Then one has: Definition 4.1. (Fundamental points of ξ): The four points A, B, E, and P + defined above are called the fundamental points of the vector field ξ. Lemma 4.5. The fundamental points of ξ satisfy the following geometric properties:   ξ (A) kE c (P )    c    ξ (A) kE (0)  ξ (B) kL1 (4.14)  ξ (E) kL0     ξ (C) kE r (0)    ξ (D) kE r (P )

Proof.

Use the continuity of the vector field ξ.



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4.2.3

The real Jordan form

In this section, let ψ 0 : D0 → R3 and ψ 1 : D1 → R3 denote the appropriate affine transformations which reduce M0 and M1 to the real Jordan form in (4.9). The existence of such a transformation is guaranteed by the standard theory of linear algebra [Hirsch and Smale (1974)]. Hence it is easy to prove the following lemma: Lemma 4.6. The transformations ψ i , i = 0, 1 satisfy the following equalities:  ψ i (U± ) = U       ψ i (0) = 0 3    ψ (U ) = V = (x, y, z) ∈ R , x + z = 1  1 i i  (4.15) ψ 0 (U−1 ) = V0− = (x, y, z) ∈ R3 , x + z = −1      σ −1 0 i 

 −1  1  = ξ i (X) =  1 σ i 0  X  ω ˜ i Dψ i ξ ψ i X   0 0 γi where ( σ i = ωσ˜˜ ii , i = 0, 1 (4.16) γ i = ωγ˜˜ ii , i = 0.1. However, we have the following definition: Definition 4.2. The last equations of (4.15) are called the normalized Jordan form of M0 and M1 , respectively. To simply the search and the construction of the Poincar´e maps for Chua’s system (4.1)–(4.2), we must define the so-called D0 unit and D1 unit of an element ξ in the family L, namely, we have the following definition: Definition 4.3. (D0 unit and D1 unit of ξ): The set (ξ 0 , V0 , ψ 0 ) is called the D0 unit of ξ, and the set (ξ 1 , V1 , ψ 1 ) is called the D1 unit of ξ. Geometrically, the D0 unit of ξ is simply the middle region D0 in its new reference frame (x′ , y ′ , z ′ ), which we labelled simply as (x, y, z) in Fig. 4.4. and they satisfy the following: Lemma 4.7. The D0 unit and D1 unit satisfy the following equalities:  A0 = ψ 0 (A) , B0 = ψ 0 (B) , C0 = ψ 0 (C) D0 : (4.17) D0 = ψ 0 (D) , E0 = ψ 0 (E) , F0 = ψ 0 (F ) and  A1 = ψ 1 (A) , B1 = ψ 1 (B) , C1 = ψ 1 (C) D1 : (4.18) D1 = ψ 1 (D) , E1 = ψ 1 (E) , F1 = ψ 1 (F ) .

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Unstable eigenspace E c (P)

Stable eigenvector E r (P)

D1

P+

A∞

x = −1

E∞ L2 E F A B

E1

y

B0 ψ0 (Er(0)) x = −1

Unstable eigenvector E r (0)

Φ

z

B− − A F− E−

x L21= ψ1 (L2)

A1 ψ0

C−

B1

ψ1 (Ec(P))

C 0

D0

V1 = ψ1 (U1) x=1

C1 F1

π1

D

Stable eigenspace E c (0)

y

D1

ψ1

Γ1

D1−unit ψ1 (Er(P))

z

C0

π +0

x=1

V0= ψ0 (U1) F0

A0

E0

x

P− D−1

Γ2

L20= ψ0 (L2)

π −0

ψ0 (Ec(0)) D0−unit

(a)

V− 0

V0−= ψ0 (U−1)

(b)

Fig. 4.4 Geometrical structure and typical trajectories of the original piecewise linear system and their images in the D0 unit and in the D1 unit of the transformed system (real Jordan form). (a) Original system and typical trajectories. (b) D0 and D1 units and half-return maps. Adapted from [Chua et al. (1986)].

To derive the coordinates of each of these points in their new reference frames, we must prove the following lemmas: Lemma 4.8. We have the following equalities:  ψ 0 (E c (0)) = (x, y, z) ∈ R3 , z = 0 = xy-plane  ψ 0 (E r (0)) = (x, y, z) ∈ R3 , x = y = 0 = z-axis  ψ 0 (L0 ) = (x, y, z) ∈ R3 , x = 1, z = 0  ψ 0 (L2 ) = (x, y, z) ∈ R3 , y = σ 0 x + γ 0 (1 − x) , z = 1 − x . Lemma 4.9. We have the following equalities: (a)   A0 = (1, p0 , 0) , p0 ∈ R C0 = (0, 0, 1)  E0 = (1, σ 0 , 0) .

(4.19) (4.20) (4.21) (4.22)

(4.23)

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−−−→

(b) The coordinate of B0 is determined by B0 ∈ ψ 0 (L2 ) and ξ 0 (B0 ) B0 A0 . (c)

−−−→ −−−→ F0 B0 = k0 E0 F0 .

(4.24)

(d) The coordinate of F0 must satisfy: ( y = σ 0 x + γ 0 (1 − x) , z = 1 − x γ0z σ0 x−y = σσ00 y+x 1 −γ = −1

(4.25)

0

where the arrows indicate the vector from B0 to A0 , and k0 is a scaling constant. Similarly, we can derive the coordinate of A1 , B1 , D1 , E1 , and Fl in the new reference frame for the D1 unit in Fig. 4.4 to obtain the following: −−−→ −−−→ (4.26) E1 F1 = k1 F1 B1 where k1 is a scaling constant. Definition 4.4. (a) The strategic points in the D0 unit are A0 , B0 , C0 , B0 , and F0 . (b) The strategic points in the D1 unit are A1 , B1 , D1 , E1 , and F1 . The explicit coordinates for the image of all strategic points in Fig. 4.3 are tabulated in the following two lemmas: Lemma 4.10. (a) The explicit coordinates for the strategic points in the D0 unit are given by:  A0 = (1, p0 , 0)       γ 0 (γ 0 −σ0 −p0 ) γ 0 (1−p0 (σ 0 −γ 0 )) 1−γ 0 (γ 0 −σ 0 −p0 )  , (σ −γ )2 +1 , (σ −γ )2 +1 B =  0  (σ 0 −γ 0 )2 +1 0 0 0 0 (4.27) C0 = (0, 0, 1)    E0 = (1, σ 0 , 0)      σ20 +1  F = γ 0 (γ 0 −2σ0 ) , γ 0 (1−σ0 (σ0 −γ 0 )) , . 0

(σ 0 −γ 0 )2 +1

(σ 0 −γ 0 )2 +1

(σ 0 −γ 0 )2 +1

(b) The explicit coordinates for the strategic points in the D1 unit are given by:  A1 = (1, p1 , 0)     B  1 = (1, σ 1 , 0)   D1 = (0, 0, 1)   (4.28) γ 1 (γ 1 −σ1 −p1 ) γ 1 (1−p1 (σ 1 −γ 1 )) 1−γ 1 (γ 1 −σ 1 −p1 )   , , E = 2 2 2 1  (σ (σ1 −γ 1 ) +1 (σ1 −γ 1 ) +1   1 −γ 1 ) +1   σ 21 +1  F = γ 1 (γ 1 −2σ 1 ) , γ 1 (1−σ 1 (σ 1 −γ 1 )) , 1

(σ 1 −γ 1 )2 +1

(σ 1 −γ 1 )2 +1

(σ 1 −γ 1 )2 +1

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where


 ki 2   pi = σ i + γ i σ i + 1 , i = 0, 1 γ (p −σ ) ki = i σ 2i+1 i , i = 0, 1 i   2 qi = (σ i − γ i ) + 1, i = 0, 1.

Note that k0 and k1 can be easily calculated from the relationship 1 γ˜ k0 = = k = − 0. k1 γ˜1

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(4.29)

(4.30)

These explicit expressions for the coordinates of the strategic points in the D0 and the D1 units will be used for the derivation of the Poincar´e maps for the Chua’s system (4.1)–(4.2). 4.2.4

Canonical piecewise linear normal form

From the circuit-theoretic point of view, the set L represents the family of all third-order piecewise linear circuits whose vector fields satisfy the properties (P-0) to (P-6) of Lemma 4.1. For simplifying the study of these circuits, we will give some results about the equivalence classes in the family L using two forms of equivalence; namely, linear equivalence and linear conjugacy introduced in the following definitions: Definition 4.5. (a) Two circuits are said to be linearly equivalent if and only if, except possibly for a uniform change in the time scale, their respective orbits are qualitatively identical. (b) Two circuits are said to be linearly conjugate if and only if, for the same time scale, their respective orbits are qualitatively identical. Mathematically, the above definitions can be reformulated as follows: Definition 4.6. (a) (Linear equivalence): Two vector fields ξ and ξ ′ in L are said to be linearly equivalent if and only if there exists a nonsingular linear transformation G : R3 −→ R3 and a real number v > 0 such that
G ◦ ξ = v ξ′ ◦ G . (4.31)

(b) (Linear conjugacy): Two vector fields ξ and ξ ′ in L are said to be linearly conjugate if and only if v = 1 in (4.31).

Using the above definitions, we find the essential following theorems that characterize the equivalence classes in the family L: Theorem 4.1. (Linear conjugacy criteria):

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(a) For each set of eigenvalues defined by the six eigenvalue parameters (˜ σ0, ω ˜ 0 , γ˜0 , σ ˜1, ω ˜ 1 , γ˜1 )

(4.32)

there exists a vector field ξ ∈ L having these eigenvalue ⇔ ω ˜ 0 > 0, ω ˜ 1 > 0, γ˜0 γ˜ 1 < 0.

(4.33)

′

(b) Two vector fields ξ and ξ in L are linearly conjugate of each other ⇔ they have identical eigenvalues, i.e.,  σ ˜ 0 = σ ′0 , ω ˜ 0 = ω ′0 , γ˜ 0 = γ ′0 (4.34) ′ σ ˜ 1 = σ1 , ω ˜ 1 = ω′1 , γ˜1 = γ ′1 . For proving Theorem. 4.1, one need to state the following definition: Definition 4.7. (Normalized eigenvalue parameters) The five normalized eigenvalue parameters (σ 0 , γ 0 , σ 1 , γ 1 , k) for each set of eigenvalues defined by the six eigenvalue (˜ σ0 , ω ˜ 0 , γ˜0 , σ ˜1, ω ˜ 1 , γ˜ 1 ), are defined by  σ ˜i   σ i = ω˜ i , i = 0, 1 γ ˜i γ i = ω˜ i , i = 0, 1 (4.35)  γ ˜0  k = − γ˜ . 1

Note that one more parameter must be specified before the eigenvalues associated with (4.9) can be uniquely recovered. Theorem 4.2. (Linear equivalence criteria): (a) There exists a continuous vector field ξ ∈ L having (4.35) as normalized eigenvalue parameters⇔ γ 0 γ 1 < 0, k > 0.

(4.36)

′

(b) Two vector fields ξ and ξ in L are linearly equivalent ⇔ they have identical normalized eigenvalues parameters. Obviously, the eigenvalues of two distinct vector fields having identical normalized eigenvalue parameters are generally not identical, and two vector fields having identical normalized eigenvalue parameters are generally not linearly conjugate. Some tedious calculations lead to the following theorem: Lemma 4.11. Theorems 4.1 and 4.2 are equivalent. Proof. Use the effect that any linearly conjugate vector fields must have identical eigenvalue parameters and identical normalized eigenvalue parameter. 

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For the proof of Theorem 4.1, we need to state the following lemma: Lemma 4.12. Let L [µ] denote the family of all vector fields in L having the same normalized eigenvalue parameters, µ = (σ 0 , γ 0 , σ 1 , γ 1 , k) given by −− → −→ −−→ −−→ (4.35). Let OP , OA, OB, OE denote the four vectors from the origin O in Fig. 4.3 to the four fundamental points P, A, B, and E, respectively. Then the following properties hold: (a) All polyhedrons whose vertices consist of the origin and the four fundamental points of vector fields belonging to the family L [µ] are similar in the sense that −−→ −→ −−→ −−→ OP = lOA + mOB + nOE (4.37) where I = I(µ), m = m(µ), and n = n(µ) are real numbers which depend only on µ and hence are identical for all vector fields in L [µ] . (b) The numbers k0 , k1 , and k are related by 1 (4.38) k = k0 = . k1 (c) There exists a vector field ξ ∈ L [µ] ⇔ γ 0 γ 1 < 0, k > 0.

(4.39)

To prove Lemma 4.12, we need the following lemma: −→ −−→ −−→ Lemma 4.13. The vectors OA, OB, and OE are linearly independent. Proof. The proof can be done by reduction to absurdity using the analytical expressions for the coordinates of each point. For the proof of Lemma 4.13, use the effect that the continuity of ξ implies that its restrictions on both the D0 and D1 units are equal in U1 .  In the end, Theorem 4.1 allows us to partition all vector fields in L into linearly conjugate equivalence classes, each parameterized by the eigenvalues σ ˜0 + jω ˜ 0 , γ˜ 0 , σ ˜1 + jω ˜ 1 , and γ˜1 . Then it suffices to investigate only one member in each class. Hence the canonical piecewise linear equation involving twelve parameters, each of which is expressed explicitly in terms of only six eigenvalue parameters (˜ σ0 , ω ˜ 0 , γ˜ 0 , σ ˜1, ω ˜ 1 , γ˜ 1 ) is given in the following theorem: Theorem 4.3. (Normal form equation for L): Every equivalence class of linearly conjugate vector fields in L defined by (˜ σ0 , ω ˜ 0 , γ˜ 0 , σ ˜1, ω ˜ 1 , γ˜ 1 ) satisfying ω ˜ 0 > 0, ω ˜ 1 > 0, γ˜ 0 γ˜1 < 0

(4.40)

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can be described analytically by the following canonical piecewise linear equation:  ′      x x a11 a12 a13 b1  y ′  =  a21 a22 a23   y  + (|z − 1| − |z + 1|)  b2  (4.41) ′ z a31 a32 a33 z b3

where

and

 (σ˜ 2 +˜ω2 )[(˜σ 0 −˜γ )2 +˜ω2 ]   a11 = γ˜1 − σ˜ 21+˜ω21 γ˜ − σ˜ 21+˜ω2 γ˜0  (  1 1) 1 ( 0 0) 0  2 2 2 2  (˜ σ −˜ γ ) +˜ ω γ ˜ σ ˜ +˜ ω  ] [ ) ( 1 0 1 1 1   a12 = γ˜ σ˜ 2 +˜ω2 γ˜ − σ˜ 2 +˜ω2 γ˜1  (( ) ( ) )  0 1 0 1 1 0 0      (σ˜ 21 +˜ω21 )γ˜ 1 [2(˜σ 0 −˜γ 1 )+˜γ 1 −˜γ 0 ] (σ˜ 21 +˜ω21 )γ˜ 1   γ1 + a13 = σ˜ 2 +˜ω2 γ˜ . −˜   ( 0 0) 0 (σ˜ 21 +˜ω21 )γ˜ 1 −(σ˜ 20 +˜ω20 )γ˜ 0   2 2  γ ˜ σ ˜ +˜ ω  [(˜σ0 −˜γ )2 +˜ω2 ] ) ( 0 0 0   a21 = − γ˜ σ˜ 2 +˜ω2 γ˜ − σ˜ 21+˜ω2 γ˜0   1 (( 1 1) 1 ( 0 0) 0) (σ˜ 20 +˜ω20 )[(˜σ 1 −˜γ 0 )2 +˜ω21 ] a22 = γ˜0 + σ˜ 2 +˜ω2 γ˜ − σ˜ 2 +˜ω2 γ˜     ( 1 12) 1 2 ( 0 0 ) 0   2  ˜1 σ ˜ 1 +˜ ω 21 )γ (σ˜ 0 +˜ω0 )γ˜ 0 [2(˜σ 1 −˜γ 0 )+˜γ 1 −˜γ 0 ] (   . −˜ γ + a =  23 0  (σ˜ 20 +˜ω20 )γ˜ 0 (σ˜ 21 +˜ω21 )γ˜ 1 −(σ˜ 20 +˜ω20 )γ˜ 0     a31 = a21     σ 1 −˜ γ )2 +˜ ω2 ] σ ˜ 20 +˜ ω 2 )[(˜ (   a32 = σ˜ 2 +˜ω20 γ˜ − σ˜ 20+˜ω2 γ˜1   ( 1 1) 1 ( 0 0) 0    (σ˜ 21 +˜ω21 )γ˜ 1 [2(˜σ 1 −˜σ 0 )+˜γ 1 −˜γ 0 ]   a33 = (σ˜ 21 +˜ω21 )γ˜ 1 −(σ˜ 20 +˜ω20 )γ˜0

Proof.

 (σ˜ 2 +˜ω2 )γ˜ −(σ˜ 2 +˜ω2 )γ˜  b1 = 1 21 σ˜ 21+˜ω2 0γ˜ 0 0 a13   ( 1 1) 1   (σ˜ 2 +˜ω2 )γ˜ −(σ˜ 2 +˜ω2 )γ˜ b2 = 1 21 σ˜ 21+˜ω2 0γ˜ 0 0 a23 ( 1 1) 1   2 2 2 2    b3 = (σ˜ 1 +˜ω1 )γ˜ 1 −(σ˜ 0 +˜ω0 )γ˜ 0 a33 . 2 2 2(σ ˜ 1 +˜ ω 1 )γ ˜1

(4.42)

(4.43)

The proof can be obtained directly from the proof of Lemma 4.6. 

To show the effect of the three regions characterizing the dynamics of the family L, the following lemma shows another equivalent form of the normal form given in (4.41): Lemma 4.14. Eq.(4.41) is equivalent to the following equation:    x  M1 (x, y, z − s) , z ≥ 1 ξy = M0 (x, y, z) , |z| ≤ 1  z M1 (x, y, z + s) , z ≤ −1,

(4.44)

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where

           

 a11 a12 a13 M1 =  a21 a22 a23  a31 a32 a33   a11 a12 (1 − s) a13  M0 =  a21 a22 (1 − s) a23       a31 a32 (1 − s) a33    (σ˜ 2 +˜ω2 )γ˜   s = 1 − σ˜ 20 +˜ω20 γ˜ 0 ( 1 1) 1

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

(4.45)

and aij (1 ≤ i, j ≤ 3) is defined by (4.42) and (4.43). More detailed analysis about the normal form of Chua’s system (4.1)– (4.2) and its applications can be found in [Chua and Ying (1983), Chua and Lin (1990), Lin (1991), Altman (1993), Bohme and Schwarz (1993), Chua et al. (1993a-1993b), Kocarev et al. (1993), Pospisil et al. (1995-1999), Chua et al. (1995), Wu and Chua(1996), and Hobson and Lansbury (1996)]. 4.2.5

Poincar´ e and half-return maps

In this section, and for the rigorous proof of chaos in Chua’s system given by Eqs. (4.1)–(4.2) using the so-called Poincar´e and half-return maps introduced in Sec. 1.3.2, we will henceforth restrict our analysis to the following subset L0 ⊂ L of vector fields, called the double-scroll family ξ (σ 0 , γ 0 , σ 1 , γ 1 , k), defined by the following: Definition 4.8. (The double-scroll family): This family is described by: L0 = {ξ (σ 0 , γ 0 , σ 1 , γ 1 , k) , σ 0 < 0, γ 0 > 0, σ 1 > 0, γ 1 < 0, k > 0} . (4.46) Because Theorem 4.3 implies that the study of the global dynamics of the double-scroll family L0 (4.46) is equivalent to the study of the canonical piecewise linear equation given by (4.44), we have the following: Lemma 4.15. The eigenvalue pattern of any member of the double-scroll family at the equilibrium point P + (resp., P − ) must be a mirror image (except for scales) of that at the origin O. The eigenspaces of a typical vector field ξ ∈ L0 are shown in Fig. 4.4(a) along with two typical trajectories. Property (P.l) implies that all trajectories occur in odd-symmetric pairs so that Fig. 4.4(a) need only show half of the salient features. This property helps when looking for the Poincar´e map.

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Fig. 4.5 D0 unit: Thick arrows denote the direction of the vector field at various points along L20 = ψ0 (L2 ) where all vectors lie on the V0 -plane. (b) Graph of a possible inverse return-time function u = u+ (v, t). Here, I + (v) denotes the set of first return times which are not connected whenever u+ (v, t) is not a monotone function. Adapted from [Chua et al. (1986)].

4.3

The dynamics of an orbit in the double-scroll

In order to construct explicitly the corresponding Poincar´e map, we remark that Fig. 4.4(a) shows the following action for a typical orbit in the doublescroll equation: (1) The upper trajectory Γ1 in Fig. 4.4(a) originates from some point on U1 , moves downward, turns around before reaching U−1 , and returns to U1 after a finite time. It then continues to move upward before turning around and returns once more to U1 . (2) The lower trajectory Γ2 in Fig. 4.4(a) also originates from U1 , moves downward, penetrates U−1 , and after a finite time, turns around and returns to U−1 a second time.

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The typical trajectory Γ1 defines a return map called a Poincar´e map introduced in Sec. 1.3 from some subset S ⊂ U1 into S. We can decompose2 this Poincar´e map into two components: a “half-return map,” which maps the initial point on U1 to the first return point on U1 , and a “second halfreturn map,” which maps the first return point to the second return point on U1 . On the other hand, and by the odd-symmetry of the vector field ξ, we can identify each return point x in U−1 , by its reflected image −x in U1 . Similarly, the portion of Γ2 below U−1 can be identified with a corresponding version of Γ1 above U1 . Through this identification scheme, both typical types of trajectories Γ1 and Γ2 actually define the same Poincar´e map π, which in turn is simply the composition of two half-return maps π 0 and π 1 . Generally, the half-return maps in Fig. 4.4(a) cannot be calculated by an explicit formula or direct algorithm because the coordinates of the return points can only be found by solving a pair of transcendental equations. Thus we must find a new coordinate system so that these half-return maps π 0 and π 1 can be easily calculated and their errors rigorously estimated. The existence of such a coordinate system for any ξ ∈ L0 was proved in [Matsumoto et al. (1985)]. The approach used for deriving this new coordinate system is to work with the greatly simplified but equivalent real Jordan forms of the regions D0 and D1 in Fig. 4.4(a), namely, the D0 unit and the D1 unit in Fig. 4.4(b). 4.3.1

The half-return map π 0

Before the construction of the half-return map π 0 , we note that throughout this section, “downward component ” or “moving down” (resp., “upward component ” or “moving up”) means the vector field enters the boundary plane V0 from above (resp., leaves V0 from below). The symbol \ denotes the set difference operator. Odd-symmetry in R3 means symmetry with respect to the origin. Hence two points (x, y, z) and (x′ , y ′ , z ′ ) are odd symmetric if and only if (x′ , y ′ , z ′ ) = (−x, −y, −z). Consider first the D0 unit at the bottom of Fig. 4.4(b) representing the image of D0 in Fig. 4.4(a) under the affine transformation ψ 0 . The three fundamental points A, B, and E in D0 map into A0 , B0 , and E0 , respectively. Lemma 4.16. The vector field ξ 0 (x) has a downward component for all x 2 In

the sense of the usual composition operation.

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to the right of L20 and an upward component to its left. Proof. This lemma is a result of three things: First, the line L2 maps into the straight line L20 passing through B0 and E0 , and second, the last relation of (4.7), and third, the qualitative nature of trajectories in D0 .  Hence any trajectory originating inside the triangular region ∆A0 B0 E0 defined by ∆A0 B0 E0 = {x ∈ V0 : x is bounded within triangle A0 B0 E0 }

(4.47)

must move down initially. But because the z-axis in the D0 unit is the image of an unstable eigenvector, this trajectory must move toward V0 as depicted by the upper trajectory in the D0 unit. This trajectory defines a map π + 0 as follows: Definition 4.9. The map π + 0 is defined by π+ 0 : ∆A0 B0 C0 −→ V0

(4.48)

T π+ 0 (x) = ϕ0 (x)

(4.49)

by the obvious image

where ϕ0 (x)T denotes the flow in the D0 unit from x to the first return point where the trajectory first intersects V0 at some time T > 0, where  (4.50) T = T (x) = inf t > 0 : ϕt0 (x) ∈ V0 .

Here, we assume that ϕt0 (x) does not hit U−1 before time T . The more general case is fully treated in [Kahlert and Chua (1987)] where some techniques developed in the theory of Poincar´e half-maps are modified and applied to Chua’s circuit. Both transfer and return maps, induced by the trajectories inside the intermediate region D0 in state space, are formulated as implicit equations with explicit formulas for their domains boundaries. More investigations were done in [Kahlert (1988, 1989, 1990)] using the method for calculating the initial points of touching trajectories subject to appropriate switching dynamics. A charting of the canonical parameter space of the dynamics that acts on the intermediate region D0 is indicated, and it reveals the existence of another type of chaotic attractor different from the double-scroll and R¨ ossler attractors previously discovered from Chua’s circuit (4.1)–(4.2). Before continuing the proof, we must define the following elements of Euclidian geometry [Lopshits (1963)]:

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Definition 4.10. (a) A polygon is a plane figure that is bounded by a closed path composed of a finite sequence of straight line segments, i.e., by a closed polygonal chain. (b) A quadrilateral is a polygon with four sides or edges and four vertices or corners. (c) A trapezium is a quadrilateral, which is defined as a shape with four sides that has one set of parallel lines for sides. (d) A polyhedron is a geometric object with flat faces and straight edges. (e) A wedge is a polyhedral solid defined by two triangles and three trapezoidal faces. A wedge has five faces, nine edges, and six vertices. Now, let us consider a typical trajectory originating from a point in the infinite wedge (angular region) ∠A0 B0 E0 = {x ∈ V0 : x lies within the wedge-like extension of ∆A0 B0 E0 } (4.51) to the right of A0 E0 in the D0 unit as depicted in Fig. 4.4(b). This trajectory must move downward (because it originates to the right of L20 ) and eventually intersects V0− . This trajectory corresponds to the portion of Γ2 within D0 in Fig. 4.4(a) and defines the map π − 0 as follows: Definition 4.11. The map π − 0 is defined as follows: π− 0 : ∠A0 B0 E0 \∆A0 B0 E0 −→ V0

(4.52)

via the obvious image T π− 0 (x) = ϕ0 (x)

(4.53)

where  T = T (x) = inf t > 0 : ϕt0 (x) ∈ V0−

(4.54)

is the time this trajectory first penetrates V0− .

By identifying this return point in V0− with its reflected odd-symmetric image in V0 , we can define the following half-return map π 0 : Definition 4.12. The half-return map π 0 is defined by π 0 : ∠A0 B0 E0 −→ V0 by the image π 0 (x) =



π+ 0 (x) , x ∈ ∆A0 B0 E0 − −π 0 (x) , x ∈ ∠A0 B0 E0 \∆A0 B0 E0 .

(4.55)

(4.56)

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− To derive an algorithm for calculating π + 0 (x) and π 0 (x), magnify the triangular region ∆A0 B0 E0 on V0 and the angular region ∠A0 B0 E0 on V0 as shown in Fig. 4.5(a). Since the z-coordinate of each point (x, y, z) on V0 is simply z = 1 − x, it suffices to specify each point on V0 , by its (x, y) coordinate. Our next crucial step is to define a “local ” coordinate system (u, v) on V0 , so that each point x0 = (x, y) ∈ ∠A0 B0 E0 , is uniquely − specified in terms of (u, v) such that π + 0 (x) and π 0 (x) can be expressed in terms of u and v. We will define our local (u, v) coordinates as a weighted sum of the four corner points A0 , B0 , E0 and F0 given in (4.27) as follows:   x0 (u, v) = u [vA0 + (1 − v) E0 ] + (1 − u) [vB0 + (1 − v) F0 ] (4.57) 0 ≤ u < +∞  0≤v≤1

and

 x0 (1, 1) = A0    x0 (1, 0) = E0 x (0, 1) = B0    0 x0 (0, 0) = F0 .

(4.58)

From (4.57) and (4.58) one has:

Lemma 4.17. All points along the line segments E0 A0 and F0 B0 have a ucoordinate equal to 1 and 0, respectively. Similarly, all points along the line segments B0 A0 and F0 E0 have a v-coordinate equal to 1 and 0, respectively. A typical point H with a (u0 , v0 ) coordinate can be identified as the intersection between the u = u0 coordinate line and the v = v0 coordinate line. Thus one has the following result: Lemma 4.18. All points inside the triangular region ∆A0 B0 E0 have 0 < u < 1, and all points inside the angular region ∠A0 B0 E0 outside of ∆A0 B0 E0 have 1 < u < +∞. Hence in terms of the (u, v) coordinate systems (4.47) and (4.51), assume the following equivalent form: ∆A0 B0 E0 = {x0 (u, v) : (u, v) ∈ [0, 1] × [0, 1]}

(4.59)

∠A0 B0 E0 = {x0 (u, v) : (u, v) ∈ (0, +∞) × [0, 1]} .

(4.60)

Then one can calculate the π + 0 return map as follows:

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Theorem 4.4. (Calculating the π + 0 return map): Given x0 = (x0 , y0 ) ∈ ∆A0 B0 E0 , the return map π + (x ) 0 is given by: 0   cos t − sin t + σ0 t π 0 (x0 (u, v)) = e x0 (u, v) (4.61) sin t cos t

where (u, v) is the local coordinate of (x0 , y0 ) = (x0 (u, v) , y0 (u, v)) where (u, v) ∈ [0, 1] × [0, 1] and t is the first return time. First, we have the following result:

Lemma 4.19. Given any return time t0 , 0 ≤ t0 < +∞, and any coordinate line v = v0 , then there exists a unique u = u0 = u+ (v, t) such that the trajectory ϕt00 (x0 (u, v)) starting from x0 (u, v) at t = 0 would hit V0 at t = t0 . Proof.

This is a result of the conditions (4.62).



Second, the first return time t is calculated explicitly as follows: (a) Use the second local coordinate “v” to calculate the inverse return-time function defined by hϕt (B0v ) , hi − 1 (4.62) u+ (v, t) = t 0 hϕ0 (B0v − A0v ) , hi where ϕt0 (x) denotes the location of the trajectory in R3 which originates from x. A0v = x0 (1, v) denotes the location in R3 of a point along the line segment E0 A0 “v” units from E0 . B0v = x0 (0, v) denotes the location in R3 of a point along the line segment F0 B0 “v” units from F0 . T h = (1, 0, 1) denotes the normal vector from the origin to V0 and h., .i denotes the usual vector dot product in R3 . (b) Use the first local coordinate “u” such that 0 < u < 1 to calculate  t = inf t ≥ 0, u+ (v, t) = u . (4.63) Following the same notation and proof as Theorem 4.5, we obtain the following theorem: T

Theorem 4.5. (Calculating the π − 0 return map): Given x0 = (x0 , yo ) ∈ ∠A0 B0 E0 \∆A0 B0 E0 , the return map π − 0 (x0 ) is given by   cos t − sin t − σ0 t π 0 (x0 (u, v)) = e x0 (u, v) (4.64) sin t cos t

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where (u, v) is the local coordinate of (x0 , y0 ) = (x0 (u, v) , y0 (u, v)) where (u, v) ∈ (1, +∞) × [0, 1] and t is the first return time. First, we have the following result: Lemma 4.20. Given any return time t0 , 0 ≤ to < +∞, and any coordinate line v = v0 , then there exists a unique u = u0 = u− (v, t) such that the trajectory ϕt00 (x0 (u, v)) starting from x0 (u, v) at t = 0 would hit V0 at t = t0 . Proof.

This is a result of the conditions (4.65).



Second, the first return time t is calculated explicitly as follows: (a) Use the second local coordinate “v” to calculate the inverse return-time function defined by u− (v, t) =

hϕt0 (B0v ) , hi + 1 . hϕt0 (B0v − A0v ) , hi

(4.65)

(b) Use the first local coordinate “u” such that 1 < u < +∞ to calculate  t = inf t ≥ 0, u− (v, t) = u . (4.66) It follows from Theorems 4.4 and 4.5 that the half-return map π 0 defined in (4.56) can be explicitly calculated, i.e., without solving any system of nonlinear equations. The plot of the graphs of the inverse return time functions u+ (v, t) and u− (v, t) permit us to find the first return times t. This method is simple compared with the direct method of solving a system of transcendental equations. But we will see that for the rigorous proof and analysis in the following sections, it is never necessary to calculate the first return time t. Instead, the image under π 0 of various constant-v lines, which is given explicitly by (4.57), (4.61), (4.62), and (4.65), is used directly. In the following, we give two examples, the first of which is concerned with a π 0 -return map for a monotone inverse return-time function, and the second of which is concerned with a π 0 -return map for a nonmonotone inverse return-time function. This phenomenon can be explained by looking at the associated inverse return-time function and examining its maxima and minima. Example 4.1. (π 0 with monotone inverse return-time function) Consider the vector field ξ with (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3, 1.5, 0.2, −2.0, 0.75).

(4.67)

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Fig. 4.6 π 0 associated with a monotone inverse return-time function in the V0 -plane. (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3, 1.5, 0.2, −2.0, 0.75) . Adapted from [Chua et al. (1986)].

The images of the line segments B0 A0 and F0 E0 in the V0 -plane under the half-return map π 0 = π + 0 are shown in Fig. 4.6 as two spirals from B0 to \ \ C0 denoted by [B0 C0 ], and from F0 to C0 , denoted by [F 0 C0 ], respectively, where [] denotes that both end points are included. The images of the line segment A0 A0∞ and E0 E0∞ (where A0∞ and E0∞ denote the extension of the respective straight lines to +∞) in the V0 plane under the half-return map π 0 = −π − 0 are also shown in Fig. 4.6 by the ′ ′ ] and [C\ ′ ], where A′ spirals [C\ A E 0 0∞ 0 0∞ 0∞ and E0∞ denote the extension of the respective curves to +∞. The graphs of the inverse return-time functions u = u+ (1, t) along B0 A0 and u = u− (1, t) along A0 A0∞ are shown in Fig. 4.7(a). A magnification of these curves in Fig. 4.7(b) shows that both functions are monotone functions. Example 4.2. (π 0 with nonmonotone inverse return-time function) Consider the vector field ξ with (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.2, 0.75, 0.2, −1.0, 0.75) .

π+ 0

(4.68)

The image in the V0 -plane under the half-return map π 0 = of the line \ segment B0 A0 is shown by the spiral [B0 C0 ] in Fig. 4.8. Its corresponding inverse return-time function u+ (1, t) as shown in Fig. 4.9(a) and magnified in Fig. 4.9(b) is a monotone function as in Example 4.1. However, the

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Fig. 4.7 π 0 associated with a monotone inverse return-time function with (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3, 1.5, 0.2, −2.0, 0.75) . (a) Graph of the inverse return-trme functions u = u− (1, t) and u = u+ (1, t). (b) Magnification of (a) over the region 0.90 < u < 1.10. Adapted from [Chua et al. (1986)].

image in the V0 -plane under the half-return map π 0 = −π − 0 of the line \ ′ C ] and segment A0 A0∞ consists of the union of two disconnected curves [b 0 ′ \ [bA0∞ ]. This phenomenon can be explained by looking at the associated inverse return-time function u− (1, t) in Fig. 4.9(a) whose magnification in Fig. 4.9(b) shows a nonmonotonic curve with a local minimum at t1 and

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Fig. 4.8 The V0 -plane of the π 0 associated with a nonmonotone inverse return-time function with (σ 0 , γ 0 , σ1 , γ 1 , k) = (−0.2, 0.75, 0.2, −1.0, 0.75) . Adapted from [Chua et al. (1986)].

a local maximum at t2 . The image of the line segment x0 (u1 , 1) x0 (u2 , 1) d in Fig. 4.8. If we plot the second and the under π 0 = −π0 is the spiral [ab] d third return maps of x0 (u1 , 1) x0 (u2 , 1), we would obtain the curves [b´ a] [ ′ ′ during the time interval t1 < t < t2 , and [a b ] during the time interval t2 < t < t3 , where t3 = inf {t ≥ t1 , u− (1, t) = u1 } .

4.3.2

Half-return map π1

In this section, we consider the D1 unit at the top of Fig. 4.4(b) representing the image of D1 in Fig. 4.4(a) under the affine transformation ψ 1 . The three fundamental points A, B, and E in D1 map into A1 , B1 , and E1 , respectively. Here, D1 denotes both the top region in Fig. 4.4(a) and a point on the z-axis in the D1 unit in Fig. 4.4(b). The same notation as in the preceding section is used with the exception that each subscript ‘0’ corresponding to the D0 unit should be changed to ‘1’ for the D1 unit. Hence we again define a local coordinate system (u, v) such that the line segments E1 F1 and A1 B1 in V1 in Fig. 4.4 correspond to the v = 0 and v = 1 coordinate line, respectively. Similarly, the line segments F1 B1 and E1 A1 correspond to the u = 0 and the u = 1 coordinate liness, respectively. Any point x1 inside the wedge (angular region) bounded by B1 A1∞ and

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Fig. 4.9 π 0 associated with a nonmonotone inverse return-time function with (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.2, 0.75, 0.2, −1.0, 0.75) . The positions a, ´b, x0 (u1 , 1) and x0 (u2 , 1) are not exact but are exaggerated to give more space. (a) Graph of the inverse return-time functions u = u− (1, t) and u = u+ (1, t). (b) Magnification of (a) over the region 0.90 < u < 1.10. Adapted from [Chua et al. (1986)].

B1 E1∞ is uniquely identified by   x1 (u, v) = u [vA1 + (1 − v) E1 ] + (1 − u) [vB1 + (1 − v) F1 ] 0 ≤ u < +∞  0 ≤ v ≤ 1.

(4.69)

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Under this local coordinate system, we can define the triangular region ∆A1 B1 E1 and the angular region ∠A1 B1 E1 as follows: ∆A1 B1 E1 = {x1 (u, v) : (u, v) ∈ [0, 1] × [0, 1]}

(4.70)

∠A1 B1 E1 = {x1 (u, v) : (u, v) ∈ (0, +∞) × [0, 1]} .

(4.71)

Finally, we define the second half-return map π 1 as follows: Definition 4.13. The second half-return map π 1 is defined as follows: π 1 : ∠A1 B1 E1 −→ V1

(4.72)

with the obvious inverse image π 1 (x) = ϕ−T 1 (x)

(4.73)

ϕ−T 1

where (x) denotes the flow in the D1 unit from x to the first return point where the trajectory first intersects V1 at some reverse time −T < 0, where  T = T (x) = inf t > 0 : ϕ1−t (x) ∈ V1 . (4.74)

The next theorem shows that π 1 can be calculated by an explicit algorithm similar to that for π 0 . T

Theorem 4.6. (Calculating the π 1 return map): Given x1 = (x1 , y1 ) ∈ ∆A1 B1 E1 , the return map π 1 (x1 ) is given by:   cos t − sin t π 1 (x1 (u, v)) = eσ 1 t x1 (u, v) (4.75) sin t cos t where (u, v) is the local coordinate of (x1 , y1 ) = (x1 (u, v) , y1 (u, v)) where (u, v) ∈ [0, 1] × [0, 1] and t is the first return time. Similar to the case of π 0 , the first return time t for π 1 can be calculated explicitly as follows: (a) Use the first local coordinate “u” to calculate the inverse return-time function defined by

−t  ϕ1 (E1u ) , h − 1  (4.76) v (u, t) = −t ϕ1 (E1u − A1u ) , h where A1u = x1 (u, 1) denotes the location in R3 of a point along the line segment B1 A1 “u” units from B1 . B1u = x1 (u, 0) denotes the location in R3 of a point along the line segment F1 E1 “u” units from F1 .

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V1 -plane. (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.4, 0.3, 0.2, −1.0, 0.3), where F1\ W 1 D1 =
\ ´ π 1 E1 F1 , e1 F\ 2 B1 = π 1 (e2 a2 ) , E1 A1 = π1 E1 A1 , and f1 = 1 W2 D1 = π 1 (e1 a1 ), e[ π −1 1 (F1 ) . The position of f1 is exaggerated in this figure for clarity. The actual position of f1 is very close to a1 . Adapted from [Chua, et al. (1986)]. Fig. 4.10




(b) Use the second local coordinate “v” such that 0 ≤ v ≤ 1 to calculate t = inf {t ≥ 0, v (u, t) = v} .

(4.77)

The following is an example of the finding the map π 1 with a nonmonotonic inverse return-time function: Example 4.3. (π 1 with nonmonotonic inverse return-time function): Consider the vector field ξ with (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.4, 0.3, 0.2, −1.0, 0.3).

(4.78)

Then one has the following lemma: Lemma 4.21. The map π 1 (x) given in Example 4.2 is discontinuous along the line segment E1 F1 in Fig. 4.10, whereas it is continuous along the line segment F1 B1 . Proof. Because π 1 (x) is defined to be the reverse flow, the vector field ξ 1 (x) on V1 becomes −ξ 1 (x) in following the image of x under ψ 1 (x). Hence the direction of ξ (x) along the line ψ 0 (L2 ) = L20 in Fig. 4.4(a) must be reversed in the corresponding line ψ 1 (L2 ) = L21 in Fig. 4.10, with  the effect that E1 F1 corresponds to the v = 0 coordinate line.

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This is the opposite of the map π 0 (x) and leads to the following lemma: Lemma 4.22. (a) The map π 0 (x) is discontinuous along F0 B0 but continuous along E0 F0 .
(b) The image π 1 F1 B1 = the spiral [F1\ W1 D1 ] in Fig. 4.10 is tangent to the line E1 B1 at F1 . Proof. Use the explicit expression in term of the coordinates of F1 and B1 and the rigorous expression of the map π 1 .  The image of the line segment E1 A1 is shown in Fig. 4.10 as part of a large spiral E1 A′1 . The continuation of this spiral to the right of A′1 is the image of the extension of E1 A1 beyond A1 . From Fig. 4.11, one can see the following properties: (1) The inverse return-time function v = v (0, t) in Fig. 4.11(a) and its magnification in Fig. 4.11(b) show that it is a monotonic increasing function of t. (2) The inverse return-time function v = v (1, t) in Fig. 4.12(c) and its magnification in Fig. 4.12(d) show that it is not monotonic and has a value larger than 1 for t3 ≤ t ≤ t4 where t3 = inf {t > 0, v (1, t) = 1} is the time it takes A1 to go to A′1 . Hence the time interval (t3 , t4 ) corresponds to the time where the extension of the outer spiral E1 A′1 lies to the right of the line segment B1 A1 , i.e., the v = 1 coordinate line, because F1 B1 and E1 A1 correspond to our u = 0 and u = 1 coordinate lines, respectively. (3) The graph of the inverse return-time function v = v (u1 , t) is shown in Fig. 4.11(c), and its magnification in Fig. 4.11(d) shows that it is d2 v monotonic with an inflection point, i.e., dv dt = 0 and dt2 = 0 at some time t1 . (4) The graph of the inverse function v = v (u2 , t) is shown in Fig. 4.12(a), and its magnification in Fig. 4.12(b) shows that it is nonmonotonic with a maximum value v = l at t = t2 , where t2 is the time it takes to go from a2 to B1 . (5) Now let f1 be the inverse image of F1 in Fig. 4.10, i.e., π 1 (f1 ) = \ F1 . Similarly, let the inverse image of F1 B1 be denoted by [f 1 a2 ], d namely, the curve f1 a2 in Fig. 4.10. Since the region bounded by the closed curve e1 e\ 2 a2 f1 e1 is found to map into the region bounded by the F arc e[ F , line 1 B1 , arc B1 e2 , and line e2 e1 , whereas the neighboring 1 1 region bounded by the closed curve f1\ a1 a2 f1 is mapped into the region

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bounded by the closed curve F1 W\ 1 D1 W2 F1 in Fig. 4.10, it follows that π 1 (x) is discontinuous along the curve fd 1 a2 , in addition to already being discontinuous along E1 F1 . (6) There exist 0 < u1 < u2 < 1 such that the corresponding coordinate lines a1 e1 (u = u1 line) and a2 e2 (u = u2 line) are mapped under π 1 into the two curves e1 F1 W1 D1 and e2 B1 , respectively, and one has the following result: Lemma 4.23. (a) π 1 (a1 e1 ) is a spiral e1 F1 W1 D1 which is tangent to E1 B1 at F1 . (b) π 1 (a2 e2 ) is a curve e2 B1 which is tangent to A´1 A1 at B1 . Proof. Use the explicit expression in terms of the coordinates of a1 and e1 in (4.73) and a2 and e2 in (4.74) and the rigorous expression of the map π 1 given by (4.75).  Note that the additional discontinuity points of π 1 (x) in Fig. 4.10 occur when one chooses parameters close to those which gave the double-scroll. They may not occur inside ∆A1 B1 E1 for other choices of parameters. We summarize the behavior of π 1 in Fig. 4.10 as follows: Lemma 4.24. (1) π 1 (∆A1 B1 E1 ) = a fan-like closed region A′1 B1 E1 shown in Fig. 4.10. (4.79) (2)
(4.80) π 1 B1 a2 = D1 .

(3) We will define (as in π 0 )

π 1 (x) = x, for all x ∈ E1 F1 .

(4.81)

π 1 (f1 ) = π 1 (F1 ) = F1 .

(4.82)

In particular,

With this definition, π 1 becomes continuous at E1 F1 . (4) The map π 1 is one-to-one at all points inside the triangular region ∆A1 B1 E1 and its boundary except the points along [B1 a2 ) and the isolated n the line segment o point f1 , i.e., on ∆A1 B1 E1 \ [B1 a2 ) ∪ {f1 } .

(5) π −1 is well-defined at all points in the fan-like region A′1 B1 E1 , 1 except for the two isolated points F1 and D1 .

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v

v

3.0

1.005

2.0 1.000 1.0 v = v(0, t) 0.0

0 1 2 3 4 5 6 7 8 9 10 (a)

v = v(0, t) t

0.995

v

0 1 2 3 4 5 6 7 8 9 10 (b)

t

v

3.0

1.005

2.0 1.000 1.0 Slope = 0

v = v(u1, t) 0.0

0 1 2 3 4 5 6 7 8 9 10 (c)

v = v(u1, t) t

0.995

0 1 2 3 4 5 t 6 7 8 9 10 1 (d)

t

Fig. 4.11 Graphs of the inverse return-time function v = v (u, t). The parameter values are the same as those of Fig. 4.10. (a) v = v (0, t) . (b) Magnification of (a) over the region 0.995 < u < 1.005. (c) v = v (u1 , t), where u1 = 0.570. (d) Magnification of (c) over the region 0995 < u < 1.005. Adapted from [Chua et al. (1986)].

(6) The spiral F1\ W1 D1 is the set of discontinuous points of π −1 1 . The −1 d function π 1 is discontinuous at these points because π −1 1 (x) −→ f1 a2 from the right as x −→ W1 from the right, whereas π −1 1 (x) −→ F1 B1 from the right as x −→ W1 from the left. Proof. (1) Use the explicit formula for π 1 .
(2) We remark that π 1 B1 a2 actually maps into the origin in the unstable eigenspace ψ 1 (E c (P + )), which becomes a stable equilibrium under the reverse flow ψ −t 1 . It is logical and convenient to identify the origin with D1 = ψ 1 (P + ) in V1 . (3) We remark that π 1 is discontinuous along E1 F1 .

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(6) This follows becausethe return map π 1 is discontinuous   along the −1 d d curve f1 a1 , and because π 1 f1 a1 = F1 B1 = π 1 F1\ W1 D1 . 

Using the above properties, we can now define the inverse half-return map π −1 1 as follows:

Definition 4.14. The inverse half-return map π −1 1 is defined as follows: −1 ′ π 1 : A1∞ B1 E1∞ −→ ∠A1 B1 E1 (4.83) where A′1∞ B1 E1∞ = {(x, y, z) ∈ V1 : y ≥ σ 1 x + γ 1 (1 − x) , x ≤ 1} (4.84) ′ is the region above the line B1 E1∞ and to the left of A1 A1∞ in Fig. 4.10, and where  −1 π 1 (D1 ) = B1 (4.85) π −1 1 (F1 ) = f1 . \ Lemma 4.25. The map π −1 1 is discontinuous along [F1 W1 D1 ].

4.3.3

Connection map Φ

Since the D0 unit and the D1 unit in Fig. 4.4 have different reference frames, let us match the two units by defining the affine connection map: Definition 4.15. The connection map Φ is defined by: −1 (4.86) Φ = (ψ 1 |U1 ) ◦ (ψ 0 |U1 ) where ψ 1 |U1 and ψ 0 |U1 denote the restriction of ψ 0 and ψ 1 on U1 . Hence one has the following result: Lemma 4.26. The expression of the connection map Φ is given by:       x0 − 1 1 x0 =L + Φ (4.87) y0 y 0 − p0 p1 where    w1 w2 (σ 21 +1)k1   L =   (σ20 +1)(k0 +1)Q1 γ 1 w3 + w4 w5    2   Qi = (σ i − γ i ) + 1, i = 0, 1    w1 = −γ 1 (k0 + 1) [Q0 + γ 0 (σ 0 − γ 0 ) (k1 + 1)] (4.88)  w2 = γ 1 γ 0 (k0 + 1) (k1 + 1)     w3 = −γ 0 (k1 + 1) (σ 0 − γ 0 ) [σ 1 (σ 1 − γ 1 ) + 1]     w  4 = −γ 1 (k0 + 1) (σ 1 − γ 1 ) [σ 0 (σ 0 − γ 0 ) + 1]   w5 = γ 0 (k1 + 1) [Q1 + γ 1 (σ 1 − γ 1 ) (k0 + 1)] .

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v

v

3.0

1.005

2.0 1.000 1.0 v = v(u2, t)

v = v(u2, t) 0.0

t

0 1 2 3 4 5 6 7 8 9 10

0.995

0 1 2 3 4 t2 5 6 7 8 9 10

(a)

t

(b) v

v 1.005

3.0

2.0 1.000 1.0 v = v(1, t) 0.0

v = v(1, t) 0.995

t

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 t 5 t 6 7 8 9 10

t

4

3

(d)

(c)

Fig. 4.12 Graphs of the inverse return-time function v = v (u, t). The parameter values are the same as those of Fig. 4.10. (a) v = v (u2 , t), where u2 = 0.786. (b) Magnification of (a) over the region 0.995 < u < 1.005. (c) v = v (1, t). (d) Magnification of (c) over the region 0.995 < u < 1.005. Adapted from [Chua et al. (1986)].

Proof. Because zi = l − xi and Φ is affine, then it suffices to find the explicit formula relating (x0 , y0 ) ∈ D0 to (x1 , y1 ) ∈ D1 . Since A0 = (1, p0 , 0) −→ A0 = (1, p1 , 0), then 

x1 y1



=Φ



x0 y0



=L



x0 − 1 y 0 − p0



+



1 p1



.

(4.89)

Hence, for any (x0 , y0 ) ∈ D0 , one has 

x1 − 1 y 1 − p1



=L



x0 − 1 y 0 − p0



.

(4.90)

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Now, since B0 = (B0x , B0y ) −→ B1 = (B1x , B1y ) and E0 = (E0x , E0y ) −→ E1 = (E1x , E1y ), it follows from the action of L in (4.90) that     B0x − A0x B1x − A1x   =L  B − A0y B − A1y    0y  1y (4.91)  E0x − A0x E1x − A1x   . =L E0y − A0y E1y − A1y It follows from (4.91) that L=



B1x − A1x E1x − A1x B1y − A1y E1y − A1y



B0x − A0x E0x − A0x B0y − A0y E0y − A0y

−1

.

(4.92)

Substituting (4.27) and (4.28) for the respective components of Ai , Bi , Ei into (4.91), we obtain the formula for L given in (4.92) expressed directly in terms of the normalized eigenvalue parameters (σ 0 , γ 0 , σ 1 , γ 1 , k).  4.4

Poincar´ e map π

We will now use the half-return maps π 0 and π 1 and the connection map Φ given in Definition 4.15 to define a Poincar´e map π: Definition 4.16. The Poincar´e map π of the double-scroll Eqs.(4.1)–(4.2) is defined by π : V1′ −→ V1′

(4.93)

V1′ = {(x, y) ∈ V1 : x ≤ 1}

(4.94)

−1 π −1 (x) , x ∈ ∠A1 B1 E1 1 Φπ 0 Φ −1 −1 Φπ 0 Φ π 1 (x) , x ∈ V1′ \∠A1 B1 E1 .

(4.95)

where

via the formula π (x) =



Here, V1′ denotes the V1 -plane to the left of x = 1. Therefore, it is not hard to prove the following result: Lemma 4.27. We have π (∠A1 B1 E1 ) ⊂ ∠A1 B1 E1 , and π 1−1 is well defined for all x ∈ V1′ \∠A1 B1 E1 in view of (4.34).

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More detailed analysis of the Poincar´e map for Chua’s system can be found in [Matsumoto et al. (1985), Chua et al. (1986), Chua (1992), Kahlert and Chua (1987), Kahlert (1988), Kahlert (1989), Kahlert (1990), Genot (1993), Silva and Chua (1988), Chua and Tichonicky (1991), Kuznetsov et al. (1993), Sharkovsky and Chua (1993), Wu and Rul’kov (1993), Chua et al. (1993), and Kuznetsov and Satayev (1994)], where the most significant results obtained from these works were discussed in several places of this book. 4.4.1

V1 portrait of V0

In our study of the global dynamics of the double-scroll family in the following sections, we will often need to look at the image by Φ given by (4.36) of the half-return map of several line segments defined as follows: Lemma 4.28. We have: (a) 

−1 \  A1 B1 = Φπ 0 A0 B0 B 1 C1 = Φπ 0 Φ  

 −1   F1 E1 = Φπ 0 F0 E0 F[ 1 C1 = Φπ 0 Φ 

−1 ′ A1 A1∞ = Φπ 0 A0 A0∞ C\ 1 A1∞ = Φπ 0 Φ 

 −1 ′  E1 E1∞ = Φπ 0 E0 E0∞ C\  1 E1∞ = Φπ 0 Φ  
 ′ \ E 1 A1 = π 1 E1 A1 .

(4.96)

(b) Cl = Φ (C0 ) = ψ 1 (C).

Definition 4.17. The V1 portrait of V0 refers to the images ′ ′ \ \′ \ [ \ B 1 C1 , F1 C1 , C1 A1∞ , C1 E1∞ , and E1 A1 for a typical set of normalized eigenvalue parameters (σ 0 , γ 0 , σ 1 , γ 1 , k) of a vector field ξ ∈ L0 . The V1 portrait of V0 is shown in Fig. 4.13, and it consists of four distinct sets of points: (1) Set 1: two boundary lines B1 A1∞ and B1 E1∞ representing the V1 coordinates of points along the boundary lines B0 A0∞ and B0 E0∞ of the infinite wedge ∠A0 B0 E0 . (2) Set 2: the boundary line E1 A1 of the triangular region ∆A1 B1 E1 . (3) Set 3: four spirals representing the image of points in Set 1 under the π 0 -map in the D0 unit, but translated into the coordinates on V1 . (4) Set 4: a partial spiral representing the image of the points in Set 2 under the π 1 -map. In Sec. 4.5, we will consider the important case where Set 4 includes the point C1 , i.e., C1 ∈ E1 A′1 .

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Fig. 4.13 V1 portrait of V1 for (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.4, 0.5, 0.05, −2.0, 0.25). Adapted from [Chua et al. (1986)].

4.4.2

Spiral image property

The various spirals in Figs. 4.6, 4.8, 410, and 4.13 were calculated by computer for various specific sets of parameters. In general, the image under − π+ 0 , π 0 , or π 1 of any bounded straight line segment along a u = u0 or v = v0 coordinate line is always a spiral. To prove this important property, it is convenient to rewrite (4.61) and (4.75) in a more compact form by identifying a point x = (xi , yi ) in the Vi -plane (i = 0, l) as a complex number (phasor) X = xi + jyi . For example, (4.75) can be rewritten in the equivalent form π 1 (X1 (u, v)) = X1 (u, v) exp (− (σ 1 + j.1) t) where

(

X1 (u, v) = x1a (u, v) + jy1b (u, v) T x1 (u, v) = (x1a (u, v) , y1b (u, v)) .

(4.97)

(4.98)

Now for t ∈ (0, +∞), X1 (u0 , v (u0 , t)) represents one point along the u = u0 coordinate line. The following geometrical properties can be seen from Figs. 4.11 and 4.12: (1) If v (u0 , t) increases monotonically from v = 0 to v = 1 as in Fig. 4.11(a) when u0 = 0, then X1 (u, v) moves monotonically from v = 0 to v = 1 as t increases from 0 to +∞.

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(2) If v (u0 , t) is not monotonic but is bounded between va and vb as in Fig. 4.12(d), X1 (u0 , v (u0 , t)) will move back and forth along portions of the u = u0 coordinate line while moving from va to vb . 2 (3) In either case, since x21a (u, v) + y1b (u, v) < +∞, π 1 (u0 , v (u0 , t)) −→ 0 as t −→ +∞. The loci of points under π 1 along u = u0 is therefore a shrinking spiral whose amplitude is modulated in accordance with x1 (u0 , v (u0 , t)). (4) If x1 (u0 , v (u0 , t)) varies only slightly for all t ∈ (0, +∞), as in the cases, shown in Figs. 4.10 and 4.13, the shrinking spiral would look almost like a logarithmic spiral. The same interpretations (1) to (4) − given above apply to π + 0 and π 0 . (5) In view of the odd symmetry of the vector field ξ, spiral images under − π+ 0 , π 0 , and π 1 always occur in odd-symmetric pairs. This proves formally that the cross section along the U1 and U−1 boundary planes of the double-scroll attractor consists of two tightly wound odd-symmetric spirals, thereby justifying the choice of the name double-scroll. − (6) Since the image of π + 0 , π 0 , and π 1 of an arbitrary curve or line segment in U1 is, in general, a curve with no special properties, it is indeed remarkable that the images along the u = u0 and v = v0 coordinate lines are always spirals. It is precisely this observation that justifies the choice of the unconventional local coordinate system given in (4.57). 4.5 4.5.1

Method 1: Sil’nikov criteria Homoclinic orbits

In this section, we will prove that the double-scroll family (4.1)–(4.2) is chaotic by showing that the conditions of Sil’nikov’s Theorem given in Sec. 1.6 are satisfied. In particular, we will prove that there exist parameters such that the trajectory along the unstable real eigenvector E r (O) from the origin will enter the stable eigenspace E c (O) in Fig. 4.4(a), and hence will return to the origin. By symmetry, the trajectory along the other unstable real eigenvector will behave in the same way. These two special trajectories are shown in Fig. 4.14(b) and are therefore both homoclinic orbits. Theorem 4.7. (Homoclinic orbits in the double-scroll family) Let ξ be any vector field in the double-scroll family L0 = {ξ (σ 0 , γ 0 , σ 1 , γ 1 , k) , σ 0 < 0, γ 0 > 0, σ 1 > 0, γ 1 < 0, k > 0} . (4.99)

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Assume ξ satisfies the following conditions: (1) Let C1 = ψ 1 (C) map under π −1 into a point on the line segment A1 E1 in the D1 unit as shown in Fig. 4.14(a). (2) In the D1 unit (Fig. 4.4(b)), no trajectory starting from points on the line segment A0 E0 in the eigenspace z = 0 intersects the boundary line x = −1. Then ξ has a homoclinic orbit through the origin. If, in addition, (3) |σ 0 | < γ 0 ,

(4.100)

then ξ is chaotic in the sense of Sil’nikov’s Theorem. Proof. First, Theorem 4.2 guarantees that the vector field ξ ∈ L0 is continuous and the half-return map π 1 is well defined. Second, consider the trajectory Γ0 , through the origin and moving upward along the unstable real eigenvector E r (O) in Fig. 4.4(a) until it hits U
1 at point C. Since A E = ψ AE C1 = ψ 1 (C) and C1′ = π −1 (C ) ∈ (see Fig. 4.14(a)) in 1 1 1 1 1 view of condition (1), it follows that the trajectory ΓC through C must land at a point C2 on segment AE in Fig. 4.4(a). But AE lies on the stable eigenspace E c (O) at the origin, and since condition (2) guarantees that the trajectory ΓC2 through C2 will not intersect the lower boundary U−1 , it follows that ΓC2 must remain on the eigenspace E c (O) and converge to the origin as t −→ +∞. Since Γ = Γ0 ∪ ΓC ∪ ΓC2 tends to the origin as t −→ +∞ and as t −→ −∞, it is a homoclinic orbit. If, in addition, |σ 0 | < γ 0 , then the hypotheses of Sil’nikov’s Theorem 1.13 are satisfied, and hence ξ is chaotic.  Theorem 4.8. (Chaos in the double-scroll) The double-scroll system (4.1)– (4.2) is chaotic in the sense of Sil’nikov’s Theorem for some parameters m0 , m1 , α and β. In particular, if m0 = − 71 , m1 = 27 , and α = 7, then there exists some β in the range 6.5 ≤ β ≤ 10.5 such that the hypotheses of Sil’nikov’s Theorem are satisfied. Theorem 4.7 implies the existence of a homoclinic orbit at the origin, and a horseshoe is embedded in a neighborhood of this homoclinic orbit. Homoclinic orbits through the other two equilibrium points P + and P − can also occur for appropriate parameter values as shown in Theorem 4.8. As a result of the existence of these orbits, there are positively and negatively invariant Cantor sets containing the following:

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Fig. 4.14 Homoclinic orbits. (a) V1 portrait of V0 . (b) Two odd-symmetric homoclinic orbits through the origin. Adapted from [Chua et al. (1986)].

(1) infinitely many saddle-type (unstable) periodic orbits of arbitrarily long periods, (2) uncountably many bounded nonperiodic orbits, and (3) a dense orbit. Moreover, the horseshoe persists under perturbations. This theorem was confirmed in [Mees and Chapman (1987)] where they also carefully analyzed the dynamics of the double-scroll system (4.1)–(4.2) and confirmed the existence of heteroclinic orbits as shown in Sec. 4.5.3 below. Additional insights and conditions for the appearance of the doublescroll attractor are given in [Kahlert and Chua (1987)]. In [Matsumoto et al. (1988)], three key inequalities (4.100) above and (4.130) and (4.131) below are studied in Sec. 4.8 below by giving verifiable error bounds to the quantities involved in the inequalities with the assistance of a computer. This provides another rigorous proof that the double-scroll is chaotic in the sense of Sil’nikov. Based on the choice of the type of computations (logical), the computer performs interval analysis, which is a method of computing intervals containing the true values introduced in Sec. 1.3.3. The application of this method to the double-scroll is described in Sec. 4.8 below. Before we can prove Theorem 4.8, we need state and prove four lemmas. To avoid repetition, we make the following assumption:

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Standing assumption: The parameters for all lemmas are 2 1 m0 = − , m1 = , α = 7, β ∈ J = [6.5, 10.5] . 7 7 Also, we will use the abbreviated notation λ ↑ in a ≤ λ ≤ b, (resp.λ ↓ in b ≥ λ ≥ a)

(4.101)

(4.102)

to mean that the variable λ = λ (β) increases (resp., decreases) monotonically and satisfies a ≤ min (λ) ≤ max (λ) ≤ b

(4.103)

as β increases monotonically in the range J. Lemma 4.29. As β increases monotonically from β 1 = 6.5 to β 2 = 10.5, the following parameters also vary monotonically as indicated: (1)  ˜ 0 ↑ in − 1.066296 ≤ σ ˜ 0 ≤ −0.906832 σ (4.104) ω ˜ 0 ↑ in 1.382371 ≤ ω ˜ 0 ≤ 2.228686  γ˜ 0 ↓ in 2.132590 ≥ γ˜ 0 ≥ 1.813664

(2)

 

(3)

(4)

σ ˜ 1 ↓ in 0.295297 ≥ σ ˜ 1 ≥ 0.138551 ω ˜ 1 ↑ in 1.879726 ≤ ω ˜ 1 ≤ 2.527628  γ˜ 1 ↑ in − 3.590593 ≤ γ˜1 ≤ −3.277103

 σ0 ↑    σ1  γ0   γ1 ↑

(

k1 γ1

k0 γ0

in − 0.771352 ≤ σ 0 ≤ −0.406890 ↓ in 0.157096 ≥ σ 1 ≥ 0.054814 ↓ in 1.542704 ≥ γ 0 ≥ 0.813782 in − 1.910168 ≤ γ 1 ≤ −1.296513

↑ in 0.384997 ≤ ↓ in − 0.881427 ≥

k0 γ0 k1 γ1

≤ 0.680079 ≥ −1.393659.

(4.105)

(4.106)

(4.107)

Moreover, the above bounds can be calculated to any desired accuracy. Proof. It follows from (4.1)–(4.2) that the real eigenvalue γ˜ i corresponding to m = mi (i = 0, l) is a real root of the characteristic polynomial equation x3 + (αm + 1) x2 + (αm − α + β) x + αβm = 0.

(4.108)

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Solving (4.108) for β, we obtain β = β (x) = α − x (x + 1) −

α2 m . x + αm

(4.109)

It follows from (4.109) that if α > 0 and αm > l, then β : (−∞, αm) −→ R is an increasing bijection (i.e., one-to-one and onto), and if α > 0 and αm < 0, then β : (−αm, +∞) −→ R is a decreasing bijection. Hence for α = 7, m0 = − 17 , (resp., m1 = 72 ), γ˜ 0 (resp., γ˜1 ) decreases (resp., increases) and satisfies 1.813664 ≤ min (˜ γ 0 ) ≤ max (˜ γ 0 ) ≤ 2.13259

(4.110)

resp., −3.590593 ≤ min (˜ γ 1 ) ≤ max (˜ γ 1 ) ≤ −3.277103

(4.111)

as β increases from 6.5 to 10.5. Now the solutions of (4.108) are related to its coefficients as follows:  2˜ σ i + γ˜i = − (αmi + 1)  2 (4.112) σ ˜ +ω ˜ 2i + 2˜ σ i γ˜ i
= αmi − α + β  i γ˜ i σ ˜ 2i + ω ˜ 2i = −αβmi .

Solving for σ ˜ i and ω ˜ 2i from (4.112), we obtain for i = 0, l ( σ ˜ i = − 12 (αmi + 1 + γ˜ i ) 2 2 mi . ω ˜ 2i = − 41 (αmi − 1 − γ˜ i ) − γ˜ α+αm i

(4.113)

i

Combining (4.111) and (4.113), we obtain properties (1) and (2). Property (3) follows directly from properties (1) and (2) and the assumptions σ 0 < 0, γ 0 > 0, and ω ˜ 0 > 0. Property (4) follows from properties (1) and (2) and the relationships    

k0 γ0



γ ˜0 γ ˜



= −  γ˜ 10  = − ωγ˜˜ 0 

ω ˜0



1

γ ˜i    γk1 = −  γγ˜˜10  = − ωγ˜˜ 1 . 1

ω ˜1

(4.114)

0

Finally, note that the bounds in properties (1)–(4) can be calculated exactly to any number of digits because (4.109) and (4.113) are rational expressions. 

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Examination of the loci of points

First, we must define what we mean by a locus: Definition 4.18. A locus is a condition that defines a continuous figure or figures, that is, a curve. For example, a line is the locus of points equidistant from two fixed points or from two parallel lines. Second, we examine the loci of points obtained by applying the half-return map π 1 given by (4.75) to the segment E1 A1 (i.e., u = 1,0 ≤ v ≤ 1) on V1 : they are obtained by substituting u = 1 and v = v(1, t) for t ∈ I (1) into (4.75), where I (1) denotes the set of firstreturn times for v ∈ [0, 1] ,   cos t sin t x (t) = π 1 (x1 (1, v (1, t))) = exp (−σ 1 t) x1 (1, v (1, t)) − sin t cos t (4.115) for t ∈ I (1). Using the phasor notation (4.97), Eq. (4.115) assumes the following compact form: X (t) = X1 (1, v (1, t)) e−(σ 1 +j1)t , t ∈ I (1) .

(4.116)

Similarly, it follows from (4.61) that the loci of points obtained by applying the half-return map π + 0 to the segment B0 A0 (i.e., v = 1,0 ≤ u ≤ 1) on V0 assumes the following compact form:
(4.117) X (t) = X0 u+ (1, t) , 1 e−(σ 0 +j1)t , t ∈ I + (1)

where X0 is the phasor associated with x0 and I + (1) is the set of “firstreturn times ” for u ∈ [0, 1]. We have already identified the set of points in (4.116) and (4.117) as portions of a shrinking spiral whose amplitude is modulated in time. We will now show that these two spirals are sandwiched between two logarithmic spirals. To simplify the notation, all vectors in the following three lemmas (Lemmas 4.30, 4.31, and 4.32) are projected onto the xy-plane, and hence they −−→ −−→ represent two dimensional vectors. For example, OE1 and OE0 should be −−−→ −−−→ interpreted as D1 E1 and C0 E0 . Lemma 4.30. (1) For each β ∈ J and for any time t ∈ I (1), the magnitude of x(t) of the spiral (4.116) in V1 is bounded by two exponentials |A1 | e−σ1 t ≥ |x (t)| ≥ |E1 | e−σ1 t .

(4.118)

|A0 | e−σ0 t ≥ |x (t)| ≥ |B0 | e−σ0 t .

(4.119)

+

(2) For each β ∈ J and for any time t ∈ I (1), the magnitude of x(t) of the spiral (4.117) in V0 is bounded by two exponentials

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(3) For each β ∈ J,

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|E1 | ≤ |E1 | e−σ1 θ1

(4.120) −−→ −−→ where θ1 denotes the angle subtended by the two vectors OE1 and OA1 on the xy-plane. (4) For each β ∈ J, |A0 | ≤ |E0 | e−σ0 θ0

(4.121) −−→ −−→ where θ0 denotes the angle subtended by the two vectors OE0 and OA0 , on the xy-plane. Proof. The proof of this lemma is very long and uses the previous results of this chapter. Here are some suggestions for beginning the proof: To prove (1) and (2), it suffices to show that |A1 | ≥ |x1 (1, v (1, t))| ≥ |E1 | −−→ −−−→ 2 because x1D(1, v) = OEE1 + v E1 A1 , v ∈ [0, 1] . Hence calculate |x1 (1, v)| , −−→ −−−→ and show OE1 , E1 A1 > 0 using the first two coordinates of E1 and A1 .   To prove (3), note that θ1 < tan (θ1 )
for all θ1 ∈
0, π2 . To prove (4), note that (4.121) is equivalent to 1 + p20 − 1 +
σ 20 e2σ0 θ0 > 0, and define the 2σ0 θ 0 2 2 , where t ∈ [0, θ0 ] and function g (t) = 1 + tan
(ϕ + t) − 1 + σ 0 e π −1 ϕ = tan σ 0 ∈ − 2 , 0 . Some functional analysis shows that g (θ0 ) > 0.

Lemma 4.31. For each β ∈ J, the double-scroll system (4.1)–(4.2) is a member of the double-scroll family (4.99) and satisfies hypotheses (2) and (3) of Theorem 4.7. Proof. It suffices to prove only that hypothesis (2) of Theorem 4.7 holds for all β ∈ J using the local coordinate of A0 .  Lemma 4.32. Let C1 = ψ 1 (C) = (xC , yC ) and F1 = ψ 1 (F ) = (xF , yF ) on the xy-plane in Fig. 4.4(b). Then for every β ∈ J, we have xC < xF < 1.

(4.122)

Moreover, C1 is a continuous function of β for all β ∈ J. Proof. Calculate exactly the values of xC , yC , xF , yF in terms of the vector (σ 0 , γ 0 , σ 1 , γ 1 , k1 ), and note that C1 = ψ 1 (C0 ) = Φ (0, 0) when projected onto the xy-plane, where Φ is the connection map defined in (4.86).  To better understand the main structure of the double-scroll chaotic attractor, let us introduce the following topological definitions [Malykhin (2001)]:

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Fig. 4.15 The circles bounding Sa and Sb on the V1 -plane and related arcs. Adapted from [Chua et al. (1986)].

Definition 4.19. (a) The space A is said to be connected if it cannot be represented as the disjoint union of two or more nonempty open subsets. (b) The space A is said to be path-connected if for any two points x1 and x2 in A, there exists a continuous function h : [0, 1] −→ A with h (0) = x1 and h (1) = x2 . (c) The space A is said to be simply-connected if it is path-connected and every path between two points can be continuously transformed into every other. This means that an object is simply-connected if it consists of one piece and does not have any holes that pass all the way through it. These definitions help in the characterization of the structure of the double-scroll chaotic attractor. Now we ready to gives the proof for Theorem 4.8. Proof. Proof of Theorem 4.8: This proof is purely constructive and uses some of the above results. It follows from Lemma 4.31 that it suffices to prove that hypothesis (1) of Theorem 4.7 holds for some β ∈ J,
i.e., we must prove that there exists some β ∈ J such that C1 ∈ π 1 A1 E1 as depicted in the V1 portrait of V0

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Fig. 4.16 V1 portrait of V0 and the two bounding circles Sa and Sb (which appear as ellipses due to unequal
horizontal and vertical scales). The parameters are (α, β, m0 , m1 ) = 10.5, 7, − 17 , 27 . Adapted from [Chua et al. (1986)].

Fig. 4.17 V1 portrait of V0 and the two bounding circles Sa and Sb (which appear as ellipses due to unequal
horizontal and vertical scales). The parameters are (α, β, m0 , m1 ) = 8.6, 7, − 17 , 27 . Adapted from [Chua et al. (1986)].

in Fig. 4.14(a) when this happens. To do this, draw two concentric circles Sa and Sb with their centers at D1 = (0, 0) in the V1 -plane and with radii equal to |A1 | and |E1 | e−2πσ1 , respectively, as shown in Fig. 4.15. Let l be the horizontal line through D1 (i.e., the x-axis) and l′ be the vertical line through F1 . Clearly, l′ is to the

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Fig. 4.18 V1 portrait of V0 and the two bounding circles Sa and Sb (which appear as ellipses due to
unequal horizontal and vertical scales). The parameters (α, β, m0 , m1 ) = 6.5, 7, − 71 , 27 . Adapted from [Chua et al. (1986)].

left of the x = 1 line in view of Lemma 4.32. Let Sa intersect l and l′ at points a and a′ , respectively. Let Sb intersect l at a point b to the left of D1 . Depending on the values of |E1 | and σ 1 , Sb either intersects l′ at two points, in which case the upper point is labelled b′ , or otherwise, let b′ be the point where Sb intersects l to the right of D1 as shown in Fig. 4.15. Let g be the upper point where Sb intersects the y-axis. Let R denote the ′ b′ gba (if b′ lies region enclosed by the closed contour formed by either aa\ ′ ′ ′ ′ \ on l ) or aa f b gba (if b lies on l). In other words, R denotes the portion of the ring (area between Sa and Sb ) above the x-axis and to the left of l′ . Hence R is a simply-connected region. Consider next the two logarithmic spirals XE (t) = E1 exp (− (σ 1 + j1) t) , t ≥ 0

(4.123)

XA (t) = A1 exp (− (σ 1 + j1) t) , t ≥ 0.

(4.124)

and

Note that XA (t) and XE (t) correspond to the two shrinking spirals ′′ ′ ´ [′ A\ 1 d d d (starting from A1 at t = 0) and E1 cc (starting from E1 at t = 0), respectively, as shown in Fig. 4.15. It follows from Lemma 4.32 and 4.33 that d′′ lies on the extension of the line D1 E1 . Since both |XA (t)| and |XE (t)| shrink exponentially with the same rate σ 1 , the time tEl c it takes XE (t) to go from E1 to c (where it first intersects

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l) is equal to the time td′′ d it takes XA (t) to go from d′′ to d (where it first intersects l). Note that tEl c = td′′ d = ∠E1 D1 d (in radians), where ∠E1 D1 d is the angle between D1 E1 and D1 d. Since ∠E1 D1 d < 2π, it follows that d must lie to the left of c, which in turn must lie to the left of b. Depending on σ 1 , the continuation of the shrinking spiral from points d and c may either intersect l′ or l. Let these points of intersection be d′ and c′ , respectively. Let td′′ d′ denote the time it takes to go from d′′ to d′ , and let tEl c denote the time it takes to go from E1 to c′ . Since td′′ d′ < 2π and tEl c < 2π, both d′ and c′ must lie outside of Sb in Fig. 4.15, and c′ must be below d′ in view of Lemma 4.32. Hence d must lie between a and c, whereas d′ must lie between a′ and c′ in Fig. 4.15. Recall next that the image under π 1 of the line segment E1 A1 = {(x (u, v) , y (u, v)) : u = 1, 0 ≤ v ≤ 1}

(4.125)

and its extension beyond A1 (v > 1) is given by X (t) = X1 (1, v (1, t)) exp (− (σ 1 + j1) t) , t ≥ 0.

(4.126)

\ A part of this image is shown by the bold spiral E 1 ee′ in Fig. 4.15 (it ′ \ corresponds to a part of E1 A1 in Fig. 4.10 and the last equation of (4.96)). Here, e = X (t1 ) is the point at which X (t) first intersects l at some time t1 and e′ = X (t2 ) is the point at which X (t) first intersects either l′ or l to the right of D1 (if it does not intersect l′ ) at some time t2 . Since both e and e′ lie to the left of x = l, its associated starting point X1 (1, v (1, t)) must lie to the left of the v = 1 line. Hence we must have 0 < v (1, ti ) < 1, i = 1, 2, and X1 (1, v (1, ti )) ∈ A1 E1 , i = 1, 2.

(4.127)

H = {(x, y) : x ≤ xF , y ≥ 0} .

(4.128)

′

It follows that e must lie between c and d, and e must lie between c′ and d′ in Fig. 4.15 for β ∈ J. If we can show that there exists some β ∈ J c′ , we will be done. Since such that C1 = ψ 1 (C) lies on the bold spiral ee C1 is a function of β (assuming α, m0 , and m1 are fixed), we will denote this function by C1 (β). Now suppose it is possible to find a β 1 ∈ J such that C1 (β 1 ) is located outside of Sa and a β 2 ∈ J such that C1 (β 2 ) is located inside of Sa . Lemma 4.32 guarantees that C1 (β) must lie in the simply-connected region Since C1 (β) is a continuous function (Lemma 4.32), the set (assuming without loss of generality β 1 < β 2 ) ΓC = {C1 (β) : β 1 ≤ β ≤ β 2 } ⊂ H

(4.129)

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is a plane curve (parameterized by β) starting from a point β = β 1 outside Sa and ending at a point β = β 2 inside Sb . Since this curve must lie within c′ spiral at some point β 0 , β 1 ≤ β 0 ≤ β 2 . Hence H, ΓC must cross the ee hypothesis (1) of Theorem 4.7 is satisfied when β = β 0 . It remains for us to show there exist β 1 and β 2 with the above stipulated properties. When β = 10.5, we calculate (xC , yC ) using (4.125) and (4.126) and obtain C1 (10.5) = 0.7064 < 0.8 < |E1 | exp (−2πσ 1 ) = 0.9151.

(4.130)

Similarly, when β = 6.5, we obtain C1 (6.5) = ˜1.4155 > 1.3 > |A1 | ≈ 1.2477. Hence β 1 = 6.5 and β 2 = 10.5 represent one of many valid choices.

(4.131) 

Two important remarks: First, by computer simulation, we have found the approximate value of β 0 = 8.6. The V1 portraits of V0 corresponding to β = 10.5, 8.6, and 6.5 are shown in Figs. 4.16, 4.17, and 4.18, respectively. It follows from Theorem 4.8 that the double-scroll system (4.1)–(4.2) has a homoclinic orbit when m0 = − 71 , m1 = 27 , α = 7,
and β = 8.6. Second, using the parameters (α, β, m0 , m1 ) = 7, 8.6, − 71 , 27 , we have confirmed by computer simulation the existence of a double-scroll attractor similar to those reported in [Matsumoto (1984)] and [Zhong and Ayrom (1985)]. However, there are difficulties when looking for the exact quantitative bounds and analytical expressions corresponding to homoclinic orbits in the bifurcation parameter space. These difficulties are addressed in [Gribov and Krishchenko (2002)] where necessary and sufficient conditions are determined for the existence of a homoclinic orbit in Chua’s Eqs.(4.1)–(4.2) that describes the operation of an electronic oscillator with the conditions m0 < 0, m1 > 0. \ \ Fig.4.19 shows that the separatrix loop OM 1 M2 O = OM1 ∪ M1 M2 ∪ 3 [ M 2 O is a curve in R , where OM1 is a straight-line segment passing through \ the point O(0, 0, 0) parallel to the vector e′1 , M 1 M2 is the part of the integral curve M1 M2 in region D1 such that the point M1 is in the plane x = 1, and the point M2 is in the trace on x = 1 of the plane ω that passes through the point O parallel to the vectors e′2 and e′3 (or to their real and [ imaginary components if p′2,3 is a complex number), and M 2 O is the part of the integral curve that lies in the plane  ω and does notleave region D0 . m0 0 = Here, O = (0, 0, 0) , O1 = 1− m m1 , 0, m1 − 1 , and O2   m0 m0 are the equilibrium points of the system (4.1)–(4.2). m1 − 1, 0, 1 − m1

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Fig. 4.19

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Geometrical explanation of Theorem 4.9.

Let pi = {˜ σ1 ± j ω ˜ 1 , γ˜1 } be the eigenvalues in regions D1 and D−1 and ′ pi = {˜ σ0 ± j ω ˜ 0 , γ˜0 } be the eigenvalue in region D0 , with the conditions Im p′1 = 0, Re p′1 > 0; Re p′k < 0, k = 2, 3. Let ei = (λi , η i , ξ i ), and ei = (λi , η i , ξ i ) be the eigenvectors corresponding to the eigenvalues pi and p′i . Let W (e1 , e2 , e3 ) be the matrix whose columns are the eigenvectors, and ω = detW (e1 , e2 , e3 ), ω ′ = detW (e′1 , e′2 , e′3 ). Finally, let wi (e1 , e2 , e3 |a) be the determinant obtained from detW by replacing column i with column vector a. Hence the following theorems were proved in [Gribov and Krishchenko (2002)]: Theorem 4.9. A separatrix loop exists in system (4.1)–(4.2) only if the system parameters α, β, m0 , m1 , and the time τ satisfy the system of equations p τ 
m0 X i λi ω i e1 , e2 , e3 O1 M1 exp = m1 ω i=1 i=3


ω1 e′1 , e′2 , e′3 OM2 = 0

(4.132)

(4.133)

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where

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           

O1 M1 =



OM2 =

T ξ1 ´1 ´ m0 η , , ´1 λ ´1 m1 λ (1, r1 , r2 )T

i=3 X



η i ωi e1 , e2 , e3 O1 M1 exp pωi τ r1 =    i=1   i=3  X 

 m 0  r = ξ i ω i e1 , e2 , e3 O1 M1 exp pωi τ . − 1 +  2 m1 

(4.134)

i=1

Theorem 4.10. A separatrix loop exists if the conditions of Theorem 4.10 are satisfied and for all t ∈ (0, τ )   i=3 X
m0 pi t > (4.135) λi ωi e1 , e2 , e3 O1 M1 exp ω m 1 i=1 and for all θ > 0

i=3 X i=1


´ i ω i e′ , e′ , e′ OM2 exp λ 1 2 3



p′i t ω ´



< 1.

(4.136)

Proof. For the proof, use the coordinates of each point and the definition of the three regions D−1 , D0 , and D1 , and the fact that the trajectory passing through the point M2 does not leave region D0 , and that the trajectory \ M  1 M2 hits the boundary of region D1 only at the instant τ . Theorem 4.10 confirms Chua’s results [Chua et al. (1986)] for α = 7, given above and gives the following additional result: Theorem 4.11. The double-scroll system (4.1)–(4.2) is chaotic in the sense of Sil’nikov’s Theorem for m0 = − 71 , m1 = 27 , and α ∈ {6, 7, 8, 9, 10} and minimum values of β ∈ {7.229, 8.591, 9.986, 11.36, 12.75} in this order, respectively, such that the hypotheses of Sil’nikov’s Theorem are satisfied. 4.5.3

Heteroclinic orbits

In searching for a heteroclinic orbit in Chua’s circuit (4.1)–(4.2) and applying the Sil’nikov’s Theorem, a method similar to the one used for homoclinic orbits was introduced in [Mees and Chapman (1987)]. A demonstration of the method was given in [Mees and Chapman (1985)], where a computer-calculated hole filling heteroclinic orbit is obtained as in Fig. 4.22 for (α, β, m0 , m1 ) = (9.439, 13.987, −0.614, −1.256). A transition diagram as shown in Fig. 4.20 simplifies the search for all possible homoclinic and

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P+

211

x

12 z 0

21 y

P− Fig. 4.20

Transition diagram for the classification of homoclinic and heteroclinic orbits.

heteroclinic orbits, where the piecewise linear structure of the system (4.1)– (4.2) allows one to get conditions for the existence of any such orbits. The equality L0 = L1 in Fig. 4.3 is the answer to this problem, and it requires varying two parameters. The conditions for this bifurcation can be derived without having to solve any transcendental equations. For example, the curve OP + O (which is the same as P + OP + ) represents a heteroclinic orbit; the curve OO represents a homoclinic orbit. The orbit OOO means an orbit leaves O, comes nearly back to it but escapes, and finally returns. Symbolically, if we denote 0 for O, 1 for P + , and 2 for P − , then one can see that every finite sequence of symbols from the set {0, 1, 2} represents a homoclinic or a heteroclinic orbit. For example the code 12102 represents a P + P − P + OP − P + orbit. The orbits 12 and 21 are shown in Fig. 4.20. These sequences have the following properties: (1) Sequences that are cyclic permutations of one another represent the same orbit. (2) Every homoclinic orbit is accompanied by another orbit corresponding to the complement module 3 of its symbol sequence, provided these are distinct. This is due to the odd-symmetry of the system (4.1)–(4.2). For example, 0121 is accompanied by 0212, but 12 is on its own because its complement 21 represents the same orbit, being a cyclic permutation of 12.

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Fig. 4.21 Behavior of motion in the plane E c (P − ). The straight line QQ′ is the intersection with the plane U−1 . Trajectories cannot enter in reverse time to the left of Q, and if they enter to the right of Q′ , they leave the linear region again.

(3) A given sequence may represent more than one possible orbit in the original system (4.1)–(4.2). For example, the simplest orbit O may occur at widely separated parameter values, perhaps with different topologies in the sense that it crosses U±1 a different number of times or in different ways. The following result has been proved in [Mees and Chapman (1987)]: Theorem 4.12. If L0 = L1 in Fig. 4.3, then one has a 01 heteroclinic orbit (and its 02 partner). Proof.

The linear systems in the three domains of R3 are dX = Mi (X − e), i = −1, 0, 1, dt

where e is an equilibrium point and    −α (m + 1) α 0     1 −1 1  , i = −1, 0, 1  Mi =  0 −β 0    m = m , 0 if e = O   m = m1 , if e = P ± .

(4.137)

(4.138)

For an eigenvalue γ of Mi , the right eigenvector may be taken as T

v = (β + γ (1 + γ) , γ, −β) .

(4.139)

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Solving for L0 and L1 , one has    γ0 −β  L0 = 1, ,   (β+γ 0 ((1+γ 0 ))) (β+γ 0 ((1+γ 0 )))  L1 = 1, (1−k)γ 1 , −k−(1−k)β β+γ ((1+γ )) β+γ ((1+γ ))

(4.140)

1

1

1

1

because the eigenvalue γ satisfies the characteristic equation  3 γ + (α (m + 1) + 1) γ 2 + (−α + β + α (m + 1)) γ + c1 = 0. c1 = −β + β (α (m + 1) + 1) Then the condition L0 = L1 requires   m0 − m1 γ0 = 1 − γ1. m0 + 1

(4.141)

(4.142)

The variables β, α, m0 , and m1 can be varied with the conditions that P ± exists and γ 0 > 0 and γ 1 < 0, and the complex conjugate pairs of eigenvalues behave correctly. For example, m0 < −1 < m1 is a good choice for satisfying condition (4.142).  Fig.4.22 indicates that the 01 orbit corresponds to a trajectory leaving O and tending to P + together with a trajectory leaving P + and tending to O which always happens because some trajectory on E c (P + ) always passes through point A, the point of intersection of E c (P + ), U1 , and E r (0). Lemma 4.33. If m=



1 1 m0 , if e = O = , + 1 = 0, or m1 , if e = P ± α β

(4.143)

then the 01 and the corresponding 02 orbits cannot exist. Proof. We remark that for m0 = −1 or m1 = −1, the structure of Fig. 4.3 is destroyed (either one has a line of equilibria through O, or P ± does not exist). Therefore, the 01 and the corresponding 02 orbits cannot exist.  For the general case of P + P − P + orbits, i.e., a 12 and 21 heteroclinic orbit as shown in Fig. 4.22, the method is as follows: Because Fig. 4.3 shows that the required trajectory must spiral out on E c (P − ), pass through the part of the vector field governed by O, and come into P + along E r (P + ), it must contain the line segment DP + . Integrating backwards from L1 implies that this orbit hits the intersection of U−1 and E c (P − ) since then it will spiral (in reverse time) into P − . Thus the necessary condition for the existence of the heteroclinic orbits 12 and 21 is that a point hit a line in the plane U−1 . For this purpose, let τ be the

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smallest positive number such that ϕ−τ (L1 ) ∈ U−1 where ϕτ is the flow of Eqs. (4.1)–(4.2). If x(t), y(t), and z(t) are the components of the flow ϕt , then the time τ is a solution of the equation −1 = x(−τ ) = ς 1 e−γ 0 τ + ς 2 e−γ 1 τ cos (−ω1 τ + ̟)

(4.144)

where ς 1 , ς 2 , and ̟ are found from the coordinates of L1 . Thus, for any fixed values of (α, β, m0 , m1 ), one can solve Eq.(4.144) numerically for a τ that gives the coordinates of ϕ−τ (L1 ). The condition ϕ−τ (L1 ) ∈ U−1 is simply a linear equation in the coordinates of ϕ−τ (L1 ) , namely   m0 − m1 m0 − m1 (4.145) , 0, w.ϕ−τ (D) = w. − m0 + 1 m0 + 1 where w is the normal to E c (P − ). Since the corresponding left eigenvector of γ is the normal in question, one has w = (β + γ 1 (1 + γ 1 ) , αγ 1 , α)T .

(4.146)

Finally, the 12 homoclinic orbit can be found if the values of τ and (α, β, m0 , m1 ) are known and (4.144), (4.145), and (4.146) are satisfied simultaneously. Finally, it is easy to prove the following result [Mees and Chapman (1987)]: Theorem 4.13. If (4.144) and (4.145) and (4.146) are satisfied simultaneously, then one has a 12 heteroclinic orbit (and its 21 partner). 4.5.4

Geometrical explanation

Geometrically, it is easy to see that the point ϕ−τ (L1 ) cannot lie anywhere on the line U−1 ∩ E c (P − ), but rather it must be in the interval (Q, Q′ ) as shown in Fig. 4.21. Here Q is the point where the flow ϕt in E c (P − ) is tangent to the line U−1 ∩ E c (P − ), and Q′ is the point where the trajectory through Q meets the line as it spirals out from P − . Assume that the reverse-time trajectory from D cannot enter to the left of Q in Fig. 4.21 but does enter to the right of Q′ because then it would otherwise reenter the O region to the left of Q. In the latter case, it could not return to E c (P − ). The above geometric and analytic statements can be treated with necessary side conditions in any search for bifurcation parameter values. A numerical optimization subroutine [Mees and Chapman (1987)] that optimizes a nonlinear function subject to nonlinear equality and inequality constraints was able to find parameter values satisfying the equations

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Fig. 4.22 Approximation of a 12 heteroclinic orbit of (4.1)–(4.2) integrated through the point L1 with (α, β, m0 , m1 ) = (9.530, 13.976, −0.646, −1.286) .

and the side conditions. The distance between (α, β, m0 , m1 ) and the values (9, 14 72 , − 75 , 78 ) of [Matsumoto et al. (1985)] was minimized subject to the stated constraints and a set of values for which ϕ−τ (L1 ) is close to Q′ is (α, β, m0 , m1 ) = (9.439, 13.987, −0.614, −1.256), while a set for which ϕ−τ (L1 ) lies well within the interval (Q, Q′ ) in Fig. 4.21 is (α, β, m0 , m1 ) = (9.530, 13.976, −0.646, −1.286). Fig. 4.22 shows a good approximation to a 12 heteroclinic orbit integrated through the point L1 with the latter set of values. 4.5.5

Dynamics near homoclinic and heteroclinic orbits

In [Blazquez and Tuma (1993d)], the chaotic behavior of the solutions of a 3-D dynamical system, in particular Chua’s circuit (4.1)–(4.2), was studied in the neighborhood of a homoclinic orbit and near a heteroclinic orbit connecting an equilibrium point of the saddle-focus type to a saddle-focus point that forms a closed contour, respectively. This study suggests that one can find a subfamily of solutions of the considered system such that, for each given sequence of natural numbers, there exists an orbit of this family such that the “number of turns ” around the equilibrium points is given by the sequence in each lap.3 In particular, there exists an infinite number of periodic solutions and homoclinic orbits for these solutions. 3 Under

some conditions this orbit touches one stable manifold, and it tends to the equilibrium point as shown in Theorem 4.14 below.

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Fig. 4.23 The dual double-scroll attractor obtained for α = 9, β = 13.8, m0 = m1 = − 71 .

2 , 7

and

Consider the equation x′ = f (x) , x ∈ R3 , f ∈ C k (k ≥ 3) .

(4.147)

Assume that there exist two isolated, hyperbolic equilibrium points Pi , i = 1, 2 of the saddle-node type with a spectrum σ (Li ) = {λi , ρi ± iωi } of the linear part Li = Df (Pi ) , i = 1, 2. Then one has the following results proved in [Blazquez and Tuma (1993d)]: Theorem 4.14. Assume that (1)  ρi < 0, λi > 0, ωi 6= 0, i = 1, 2 ρ1 ω 2 + λ2 ω 1 > 0, ρ2 ω 1 + λ1 ω 2 > 0.

(4.148)

(2) There exist two connections Γi = {pi (t) , t ∈ R} , i = 1, 2 such that  limt→−∞ p1 (t) = P1 , limt→∞ p1 (t) = P2 (4.149) limt→−∞ p2 (t) = P2 , limt→∞ p2 (t) = P1 . Then there exists a subsystem of solutions in one-to-one correspondence with the set Ωa = {(..., jn , ...) : jn ∈ N∗ , jn+1 < ajn }

(4.150)

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for some real a, such that

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 λ2 ω1 λ1 ω2 . (4.151) ,− 1 < a < min − ρ2 ω 1 ρ1 ω 2 In other words, Theorem 4.14 says that one can find a subfamily of solutions of (4.147) in general, and in Chua’s circuit (4.1)–(4.2) in particular, such that for each given sequence of natural numbers, there exists an orbit of the family such that the “number of turns” around the equilibrium points is given by the sequence in each lap. In particular, there exists an infinite number of periodic solutions and homoclinic orbits for those solutions. 

Theorem 4.15. Assume that (1)  ρi < 0, λi > 0, ωi > 0, i = 1, 2 ρ1 ω 2 + λ2 ω 1 > 0, ρ2 ω 1 + λ1 ω 2 > 0.

(4.152)

(2) There exist two connections Γi = {pi (t) , t ∈ R} , i = 1, 2 such that  limt→−∞ p1 (t) = P1 , limt→∞ p1 (t) = P2 (4.153) limt→−∞ p2 (t) = P2 , limt→∞ p2 (t) = P1 . Then for every m ∈ Z, there exists a subsystem of solutions in one-to-one correspondence with the set ∗ Ωm a = {(..., jm ) : jn ∈ N , jn+1 < ajn , n < m}

(4.154)

which represents orbits asymptotic to P1 if m = 1 mod (2) and to P2 if m = 0 mod (2) . In other words, Theorem 4.15 says that for each given infinite sequence, there exists a solution with the behavior described in Theorem 4.14, but it soon touches one stable manifold and then tends to the equilibrium point. Theorem 4.16. Assume that (1)  ρi < 0, λi > 0, ωi > 0, i = 1, 2 λ1 λ2 − ρ1 ρ2 > 0.

(4.155)

(2) There exist two connections Γi = {pi (t) , t ∈ R} , i = 1, 2 such that  limt→−∞ p1 (t) = P1 , limt→∞ p1 (t) = P2 (4.156) limt→−∞ p2 (t) = P2 , limt→∞ p2 (t) = P1 . Then there exists a subsystem of solutions in one-to-one correspondence with Ωa for some real a such that λ1 λ2 . (4.157) 1
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The two Theorems 4.14 and 4.15 can be generalized to a Banach space as follows: Definition 4.20. A linear operator A on a Banach space X is said to be sectorial
if A is a closed, densely defined operator such that for some φ ∈ 0, π2 and some µ ≥ 1 and a real number a, the sector Sa,φ = {λ : φ ≤ |arg (λ − a)| ≤ π, λ 6= a}

(4.158)

µ , for all λ ∈ Sa,φ . |λ − a|

(4.159)

is in the sesolvant set of A and

−1

(λ − A) ≤

Now, consider an equation of the form of the Chua’s system z ′ + Az = f (z)

(4.160)

where A is a sectorial operator in a Banach space X and f ∈ C k (X α , X) , 0 ≤ α < 1 such that there exist two equilibrium points P1 and P2 and if we linearize about Pi , i = 1, 2, we have the operator Li = A − Df (Pi ) , i = 1, 2. Then one has [Blazquez and Tuma (1993d)] the following: Theorem 4.17. Assume that (1) There exists µi1 , µi2 , ..., µin eigenvalues of Li with
negative real part,
i i i µ1 , µ2 complex conjugates and Re µj < ρi = Re µi1 < 0, i = 1, 2, j = 3, 4, ..., n. (2) The remainder of the spectrum has positive real part and there exists + a simple  i eigenvalue λi ∈ R such that Re (µ) > λi > 0 for µ ∈ σ (L) − µj , λi . (3)  ρ1 ω2 + λ2 ω1 > 0 (4.161) ρ2 ω 1 + λi ω 2 > 0.

(4) There exists two heteroclinic points Γi , i = 1, 2. Γ
1 connects P1 with P2 and Γ2 connects P2 with P1 such that dim Wc+i ∩ Wc−i = 1, where Wc+i and Wc−i are the tangent spaces at ci ∈ X to the stable and unstable manifolds. Then there exists a subsystem of solutions in one-to-one correspondence with the set Ωa = {(..., jn , ...) : jn ∈ N∗ , jn+1 < ajn }

(4.162)

  λ1 ω2 λ2 ω1 1 < a < min − . ,− ρ2 ω 1 ρ1 ω 2

(4.163)

for some real a such that

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Note that the proof of these theorems can be deduced from the proof of Theorem 4.14 based on the return maps defined for the system (4.147). A generalization of these theorems to Banach space with infinite dimension gives the same results as above in view of the existence of subsystems of solutions in one-to-one correspondence with the sets Ωa and Ω+ a [Blazquez and Tuma (1993a)].

4.6

Subfamilies of the double-scroll family

Another subfamily of L0 was studied exactly with the same method as above by extending in a more general fashion than is done in [Chua et al. (1986)] the Poincar´e half-map technique4 allowing one to detect homoclinic and heteroclinic orbits and to locate the region in parameter space where stable attracting sets exist. In addition, the boundaries between the return/transfer/escape regions and a period-one limit cycle were predicted using this map [Parker and Chua (1987)]. This family is called the dual double-scroll family,5 as in the following definition: Definition 4.21. The dual double-scroll family is the subset of L0 that satisfies the properties (P − i)i=0,6 , and that has γ˜0 < 0, and γ˜1 > 0. The dynamics of the double-scroll family [Chua et al. (1986)] and the dynamics of the dual double-scroll family [Parker and Chua (1987)] are quite different. For example, in the double-scroll family, entry points either transfer or return, while for the dual double-scroll family, the opposite stability type of the equilibrium point in region D1 leads to the third possibility of escape. An example of the dual double-scroll attractor is shown in Fig. 4.23. The double-hook family 6 Fs [Bartissol and Chua (1988), Silva (1991)], is a derivative of the well-known double-scroll family presented in Sec. 4.3 that exhibits chaotic behavior both numerically and experimentally [Bartissol and Chua (1988)] as well as analytically [Silva (1991)]. The analysis of the 4 These maps are defined with no assumptions on the system dynamics and with the largest possible domain. 5 The name “dual double-scroll family” comes from the fact that this family describes the same circuit as the double-scroll equation (4.1) except for the single nonlinear element which is the dual of the double-scroll nonlinearity. 6 So-called because its basic structure appeared as two “fishhooks” in tandem, connected at their pointed ends.

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double-hook is the same as was done as above using Poincar´e maps and searching for homoclinic and heteroclinic orbits thus proving horseshoes. The double-hook family Fs is obtained by replacing property (P.5) given in Lemma 4.1 by the following: (P.5)’: M0 has three real eigenvalues denoted η 0 , µ0 and ν 0 , where η 0 µ0 > 0 and η 0 ν 0 > 0, while M±1 has a pair of complex conjugate eigenvalues σ 1 ± jω1 , and one real eigenvalue γ 1 with ω 1 > 0 and γ 1 6= 0. In this case, P ± are still saddle-unstable foci, and the origin is a saddlenode instead of a saddle-focus. This implies a dramatic change in the behavior and analysis of the resulting system. Definition 4.22. The double-hook family Fs , is defined as the subset of L0 such that η 0 < 0, µ0 > 0, ν 0 > 0, σ ˜ 1 > 0, and γ˜1 < 0.

(4.164)

The geometry associated with the canonical dynamical vector field ξ for the double-hook family Fs is shown in Fig. 4.24, and an example of a typical orbit is shown in Fig. 4.25. Changing the conditions (4.164), one obtains the following definition: Definition 4.23. The dual double-hook family Fs∗ is the subfamily of L0 defined by the condition η 0 > 0, µ0 > 0, ν 0 < 0, σ ˜ 1 < 0, and γ˜1 > 0.

(4.165)

In this case, the stabilities of the various eigenlines and eigenplanes associated with the equilibria O and P ± is reversed. Therefore, by some minor sign and flow reversal changes, the results for Fs∗ are found from the analysis of the subfamily Fs . The remaining vector fields in L0 outside of Fs ∪ Fs∗ cannot exhibit horseshoe chaos through the Sil’nikov mechanism because the equilibrium points P ± are no longer saddle foci. 4.7

The geometric model

In Sec. 1.2, we discussed the three types of strange attractors, and in this section we continue in a more specific way the presentation of the so-called geometric model defined for Chua’s Eqs.(4.1)–(4.2). Indeed, in [Belykh and Chua (1992)], a type of strange attractor generated by an odd-symmetric three-dimensional vector field with a saddle-focus having two homoclinic orbits at the point O was reported, and it was shown that this type is

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Fig. 4.24 Geometry associated with the canonical dynamical vector field ξ for the double-hook family Fs . Adapted from [Bartissol and Chua (1988)].

related to the double-scroll, and it is different from the Lorenz-type or quasi-attractors introduced in Sec. 1.2. This type contains unstable points7 in addition to the Cantor set structure of hyperbolic points. In particular, the corresponding two-dimensional Poincar´e map has a strange attractor with no stable orbits. For the purpose of illustration, let us look at Fig. 4.26 and define the following subsets of R2 : 7 This

implies that the points from the stable manifolds of the hyperbolic points must necessarily attract the unstable points.

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Fig. 4.25 The double-hook attractor obtained for α = −4. 915 0, β = −3. 642 5, m1 = −1.3475, and m0 = −0.5013.

      

 G = |x| ≤ h, y 2 + z 2 ≤ r2 i  2∂G =2 ∪i D2i ∪ dh , i = 1, 2. D1(2) = y + z = r : 0 < x < h (0 > x > h) (4.166)   1(2)  2 2 2  d = y + z ≤ r : x = h (x = −h)  h    D = D1 ∪ D2 ∪ y 2 + z 2 = r 2 : x = 0 . Here ∂G is the boundary of the cylinder G. The set D1(2) denotes the upper (lower) cylinder bounding surfaces except the common boundary at 1(2) x = 0. The set dh denotes the top and bottom disks. The set D denotes the cylinder boundary surface. Now consider a class of three-dimensional piecewise linear system (PLsystem) satisfying the following conditions: (1) Inside the cylinder G, the PL-system is defined by the following linear system in the normal form:  x′ = γx  ′ y = −σy − ω0 z (4.167)  ′ z = ω 0 y − σz. (2) Outside of the cylinder G, the PL-system generates an odd-symmetric 1(2) linear Poincar´e map S such that S d1(2) = S1(2) : dh → D. The h two points (±h, 0, 0) lying on the one-dimensional unstable manifold Γ1 (Γ2 ) of (4.167) have the images P1(2) = S1(2) (±h, 0, 0) ⊂ D. These two points are shown in Fig. 4.26 for the case where the x-coordinate is zero. Hence the global unstable manifold Γ1 (Γ2 ) of the PL-system returns to ∂G, i.e., P1(2) = Γ1(2) ∩ D.

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Fig. 4.26 Global geometric model of the PL-system (4.167) showing two odd-symmetric homoclinic orbits through the origin and their tubular neighborhoods for flows under the linear map S. The return points P1 and P2 are drawn for the case µ = 0. For the general case µ > 0, the center P1 (P2 ) of the right (left) circle is translated upward (downward) by an amount equal to µ. Adapted from [Belykh and Chua (1992)].

Some typical trajectories of the PL-system which are homeomorphic to the corresponding trajectories from Chua’s circuit (4.1)–(4.2) are shown in Fig. 4.26. To derive a suitable Poincar´e map, define an application T [Sil’nikov (1965)] which maps the PL-system trajectories inside G, originating from 1(2) the cylinder surfaces D1(2) into the top and bottom disks dh , namely 1(2) T D1(2) = T1(2) : D1(2) → dh . Hence the global Poincar´e map is given by f = S ◦ T, where f D = f1(2) = S1(2) ◦ T1(2) : D1(2) → D. 1(2)

Using polar coordinates

y = ρ cos φ, z = ρ sin φ

(4.168)

and solving the Sil’nikov boundary problem, we obtain the following formu-

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las for the mapping T : (φ (0) , x (0)) → (y (τ ) , z (τ )) :  y (τ ) = ρ (τ ) cos φ (τ )      z (τ ) = ρ (τ ) sin φ (τ )   v  ρ (τ ) = a1 |x (0)| φ (τ ) = φ (0) + φ1 − ω ln |x (0)|      v = σγ , ω = ωγ0 , a1 = rh−v , φ1 = ω ln h    −1 r = γ −1 ln h |x (0)| .

(4.169)

The parameter r here is the elapse of motion from the cylinder surface D 1(2) to the top ( bottom) disk dh along the trajectories of (4.167). Consider 1(2) the following simplest linear mapping S : (y, z) ∈ dh → (φ, x) ∈ D which realizes the global picture of the PL-system shown in Fig. 4.26:   S U = S + S1i U, i = 1, 2    i  0i     U = u1 , S = (1 − i) π + ψ 0i u (4.170) (−1)i+1 µ  2 i   i+1   (−1) α sin θ (−1) α cos θ   .  S0i = α cos θ α cos θ

Here (u1 , u2 ) denotes the coordinates (y, z) of the initial point on the top or bottom disk, S0i is the coordinate vector of the return points Pi , i = 1, 2, where the x and φ coordinates of the return point are translated by two constant parameters µ and ψ for the sake of generality, θ is the torsion angle of the twisting of the disks D1(2) as it maps into D, α2 is the contraction (expansion) coefficient of the linear maps S, and the signs of the det S1 , are chosen to match the orientation of the coordinates (y, z) . Finally, the explicit formulas in term of (x, φ) for the discontinuous map f of the PLsystem is given by:
 v φ = − π2 + π2 + a |x| cos (φ + ϕ − ω ln |x|) sgn (x) , x 6= 0, x ∈ D v x = µ sgn (x) − a |x| sin (φ + ϕ − ω ln |x|) , x 6= 0, x ∈ D (4.171) where sgn (.) denotes the signum function and  π  ϕ = ψ + φ1 − θ − 2 , a = αa1 (4.172) φ = φ (0) − ψ, x = x (0)  φ = φ (τ ) − ψ, x = x (τ ) . We remark that the Poincar´e map f is discontinuous at x = 0 because any point located at an infinitesimal distance above x = 0 must map into a neighborhood of P1 , whereas any point located below x = 0 must map into a neighborhood of P2 , and f is undefined at x = 0. However, we can define f (x) at x = 0 as follows:

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lim− φ, x = (0, µ) , lim+ φ, x = (−π, −µ) .

x→0

x→0

(4.173)

Note that whereas v and ω are local parameters of (4.167 ), µ, a, and ϕ are global parameters: µ controls the return points P1 (0, µ) and P2 (−π, −µ), a is called the separatrix value, and ϕ is the phase shift . Let us define the sets:   1(2)  Cη = x = η (−η) , φ ∈ S 1  1(2) (4.174) Lξ = {φ = ξ, 0 ≤ x ≤ η (−η ≤ w ≤ 0)}    1 Dη = |x| ≤ η ≤ h, φ ∈ S . Consider the one-to-one 1-D map g : R → R

x = µ + axv , v < 1, a > 0, µ > 0.

(4.175) 1

The map g has a unique stable fixed point xs > x1 = (va) 1−v > µ. Then one has the following results proved in [Belykh and Chua (1992)]: 1(2)

Lemma 4.34. (1) The images f1(2) Cη

are circles.

1(2)

are shrinking spirals connecting the cir(2) The images of f1(2) Lη 1 cle f1 Cη to the point P1 , where it rotates in a clockwise direction as |x| decreases (f2 Cη1 to the point P2 , where it rotates in a counter-clockwise directions as |x| decreases, respectively). (3) If 1

v < 1, a 1−v < h, 0 ≤ µ ≤ h − ahv , then (3-1) f (D) ⊂ D (f1(2) D1(2) = d1(2) ⊂ D ). (3-2) There exist domains di = fi Dxs defined by   2 i 2 2 2v di = x + (−1) µ + (φ − (1 − i) π) ≤ a xs , i = 1, 2

(4.176)

(4.177)

such that d1 ∪ d2 ⊂ f (Dη ) for η > xs and f (Dη ) ⊂ d1 ∪ d2 for η < xs . (4) The mapping f has an attractor in the minimum attracting domain d1 ∪ d2 , i.e., A = lim f p D ⊂ d1 ∪ d2 .

(4.178)

µ < axvs ,

(4.179)

p→∞

For

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Fig. 4.27 Each shaded region denotes the intersection between the image and the preimage of one Sil’nikov snake. Adapted from [Belykh and Chua (1992)].

Fig. 4.28 Schematic diagram showing the double-horseshoes resulting from the map defined by the geometric model for small µ and large l. Adapted from [Belykh and Chua (1992)].

let us consider the images  dij = fi dj , i, j = 1, 2    dijk = dij ∩ Dk , i, j, k = 1, 2 (4.180) −1 d = dijkl , d−1  ijkl = fi dijkl , i, j, k = 1, 2   ijk VL = dijkl ∩ di−1 ′ j ′ k ′ l′ where l is the number of domain intersections dijk ∩ {φ = 0, φ = −π} as 1(2) 1(2) |x| decreases along l0 and l−π . L is the index vector m = 0, 1, 2 with the coordinate values i, j, k, i′ , j ′ , k ′ = 1, 2, l, l′ = 1, 2, ... The case m = 0 corresponds to one component tangent intersection, whereas m = 1 and m = 2 correspond to left and right transversal intersections as shown in Fig. 4.27. Note that any subset dij has a snake-like spiral shape, and it is called an S-snake or Sil’nikov snake. Definition 4.24. Let X be a subset of D which is homeomorphic to a union of a disk in R2 . We define a topological B-operator as the following map:

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B (X) = (f (X) ∩ X) ∪ f −1 (f (X) ∩ X) .

(4.181)

Some properties of the B-operator are given in Exercise 4.13. In this case the B-operator was used to determine whether any limiting set component is stable, unstable, or hyperbolic. Observe that the limiting set Ω = lim B p (∪L VL ) ,

(4.182)

p→∞

i.e., the attractor A, is rather complicated because of the inevitable existence of the tangent components VL (m = 0) for which hyperbolicity does not hold [Gavrilov and Sil’nikov (1973), Newhouse (1979)]. Let us divide the union ∪L VL into two parts ∪L VL = V h ∪ V 0 , such that V h (V 0 ) is the union of all transversal tangent intersections VL for m = 1 (m = 0, respectively). Hence one has [Belykh and Chua (1992)]: Theorem 4.18. The limiting set of (f |V h ) (Ωh ) = lim B p V h p→∞




(4.183)

is a hyperbolic set conjugate to the topological Markov chain with an infinite (finite) number of symbols for µ = 0 (µ > 0, respectively). In this case, the set Ωh encloses the “twin” Smale’s horseshoes,8 coupled to each other by their preimages. This geometric object is considered as an attribute of the double-scroll attractor, namely the double-horseshoe shown in Fig. 4.28. To study the general case of Ω and the special case Ω (f |V 0 ) , note that for µ = 0 each one-side map fi : Di → Di when restricted to the small halfneighborhood Ui ⊂ Di of the homoclinic points Pi , i = 1, 2, is the subject of various theorems from [Ovsyannikov and Sil’nikov (1986-1991)]: Corollary 4.1. (1) For 12 < v < 1, the set of parameters which implies the existence of structurally unstable periodic orbits and a countable set of stable periodic orbits of each map fi |Ui , i = 1, 2 is dense. (2) For v < 21 , the map fi |Ui , i = 1, 2 has no stable points. To examine the strangeness of the attractor A, Corollary 4.1 isolates only a small subset of A in a small nonattracting vicinity of Ui . For the general case, we need to study the feedback mapping D1 → D2 → D1 , not 8 An

infinite number for µ = 0.

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for Ui only, but also for the whole attracting regions d1 and d2 . Indeed, it is possible to prove that the set A is a strange attractor. For this purpose, consider the map Fk : R2 → R2 defined for each k = 0, 1, 2, ... as follows:  v x = µ sgn (x) − a |x| sin (φ + ϕ − ω ln
|x|) v π π φ = − 2 − πk + 2 + πk + a |x| cos (φ + ϕ − ω ln |x|) sgn (x) , k = 0, 1, 2... (4.184) Observe that F0 = f. For k ≥ 1, Fk is a generalization of f. However, all previous results are still valid for each k ≥ 1, with only difference being that the distance between the centers of d1 and d2 is equal to −π (2k + 1) . The following result was proved in [Belykh and Chua (1992)]: Theorem 4.19. (1) Assume that 1 . 2 Then the map Fk for any k = 0, 1, 2... has a strange attractor if v (xs − µ)2 >1 xs where xs is the unique stable fixed point of the map g. (2) For v<

(4.185)

(4.186)

1

(4.187) µ > a (2a) 1−v , we have d1 ⊂ {x > 0} and d1 ⊂ {x < 0} . Hence A = A1 ∪ A2 , Ai ⊂ di , i = 1, 2, such that A1 and A2 are two separate noninteracting spiral-type strange attractors. (3) For small values of δ 1 > 0 and parameters of f from the “resonance zones ” defined by π (2n + 1) (4.188) ∆1 = ϕ − ω ln xs − < δ 1 , n ∈ Z, 2

the attractor A has stable periodic orbits and is therefore not strange under the condition xs − µ < 1, v < 1 (4.189) for any µ ≥ 0 in the case of even n and for some small µ ≥ 0 in the case of odd n. (4) Assume that for small δ 2 > 0 the condition ∆1 = |ϕ − ω ln xs − πn| < δ 2 , n ∈ Z (4.190) holds. If 2 1 v (xs − µ) 1−2v > qm ,v < (4.191) xs 2 where qm = xxms < 1, xm is the maximum value of |x| for dij , i, j = 1, 2, then the attractor A is strange.

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Fig. 4.29 (a) A typical strange attractor of the spiral (Rˆ ossler) type generated by the geometric model (4.167) inside the region d1 ∪ d2 for the indicated parameters. (b) A typical strange attractor of the double-scroll type generated by the geometric model (4.167) inside the region d1 ∪ d2 for the indicated parameters. Adapted from [Belykh and Chua (1992)].

Examples of the strange attractors generated by the geometric model are shown in Fig. 4.29. Finally, and as mentioned before, the geometric model (4.167) is much more complicated than the Lorenz-type attractors [Afraimovich et al. (1983)], although the above model represents the simplest idealization of the double-scroll attractor. 4.8

Method 2: The computer-assisted proof

In [Matsumoto et al. (1988)], the inequalities (4.100), (4.130), and (4.131) that involve the eigenvalues of the system (4.1)–(4.2) and many arithmetic operations are analyzed using a computer assisted proof, and it thus provides yet another rigorous proof that the double-scroll circuit is chaotic in the sense of Sil’nikov. Eq.(4.108) can be solved exactly using the Cardan method, but exact numerical values are impossible to obtain because the

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formula involves cube roots and square roots. Thus estimation of the exact error is impossible. In [Chua et al. (1986)], computations of Eqs. (4.100), (4.130), and (4.131) are done in double precision floating-point format, so the resulting error is inevitable. However verifiable error bounds for all the quantities involved in these inequalities are given in [Matsumoto et al. (1988)] based on the choice of the type of computations (logical). It is clear that the verification of the inequalities (4.100), (4.130), and (4.131) requires computations of the eigenvalues σ ˜i + jω ˜ i , and γ˜i , i = 0, 1 and then the quantities kC1 (β)k2 , kA1 (β)k2 , kE1 (β)k2 , and e−4πσ 1 (β) . Now, and specifically, any possible errors are induced by the eigenvalue computations, the arithmetic operations, i.e., +, −, ×, .. , and the conversion of a real number to and from a machine-represented number (sign, radix, exponent, and mantissa). To control all possible errors, all the computations were reduced to the following four logical operations: AND, OR, NOT, XOR, and performed using the interval analysis [Moore (1979)] introduced in Sec. 1.3.3.2, which is a method of computing intervals containing the true values of the parameter β for which the three inequalities (4.100), (4.130), and (4.131) hold. 4.8.1

Estimating topological entropy

For the evaluation of the Poincar´e map H and its periodic orbits9 of a piecewise linear system, the so-called mean value form introduced in Sec. 1.3.4 is the best method10 because it is capable of eliminating some types of overestimates of the resulting solution sets called the wrapping effect observed in the Lohner method [Galias (2002b)]. This method was applied to Chua’s circuit (4.1)–(4.2) [Galias (2002b)] with a full determination of the corresponding periodic orbits with length less than 16.11 Chua’s equation taken here is given by (4.4)–(4.5). Here the hyperplanes Σ1 , Σ2 are given by   3  Σ1 = x ∈ R : x1 = −1 (4.192) Σ = x ∈ R3 : x1 = 1  2 Σ = Σ1 ∪ Σ2 . The generalized Poincar´e map H is defined by following steps:

9 This is possible in the case where the generalized Poincar´ e map H is continuous in the region containing the attractor. 10 Compared with the following methods: the direct evaluation of analytical formulas in interval arithmetic, bisection of the return time, generalized bisection in the Poincar´ e plane, the Lohner method, and a combination of these methods. 11 These short periodic orbits are embedded within the attractor.

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(1) Find a time t1 such that ϕ(s, x) ∈ / Σ for all x ∈ X and s ∈ (0, t1 ]. (2) Find a time t2 > t1 such that for all x ∈ X, the point ϕ(t2 , x) belongs to another linear region. (3) The interval t = [t1 , t2 ] is the enclosure of the return time for all points in X, i.e.,  t ⊃ {τ (x) : x ∈ X} (4.193) R = exp (Ak t) (x0 − pk ) + pk , k = −1, 0, or 1. (4) Apply the mean value given in Sec. 1.3.4 to the second equation of (4.193) to obtain a narrow enclosure y of {H(x) : x ∈ X}, i.e., y = {H(x) : x ∈ X} = R ∩ Σ.

(4.194)

(5) Compute the Jacobian of H at x ∈ Σ using the following formula   Ak (y − pk ) eT1 ′ H (X) = I − T exp (Ak τ (x)) , k = −1, 0, or 1. e1 Ak (y − pk ) (4.195) ′ (6) Compute the enclosure of {H (x) : x ∈ X} using the last equation with the interval quantities t and y. (7) Locate the trapping region Γ containing the numerically observed attractor. (8) Find the graph representation of the dynamics of the system in the trapping region Γ. Now, with parameters values   C1 = 1, Ga = −3.4429 , Gb = −2.1849, L = 0.06913, R = 0.33065, R0 = 0.00036

(4.196)

where for C2 = 7.65 Chua’s circuit (4.4)–(4.5) displays the R¨ ossler-type attractor shown in Fig. 4.30(a) and for C2 = 9.3515 the system gives a double-scroll attractor as shown in Fig. 4.30(b). In this case, let us define the following intervals [Galias (1996)]:   A1 = (−0.1950, −2.6942956550)          A2 = (−0.1761, −2.2243882059)          A = (−0.2376, −2.9659317744) 3       A4 = (−0.2410, −3.02489461290) . (4.197)  A5 = (−0.3181, −4.1785885539)         A6 = (−0.3315, −4.0981421985)          A = (−0.3597, −4.4381670543)   7     A8 = (−0.3472, −4.5294652668)

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Fig. 4.30 Chaotic attractors from (4.4)–(4.5) (a) R¨ ossler-type attractor for C2 = 7.65 (b) Double-scroll attractor for C2 = 9.3515.

Let N1 be the quadrangle A1 A2 A3 A4 and N0 be the quadrangle A5 A6 A7 A8 . Let N1U = A1 A2 , N1D = A3 A4 , N0U = A5 A6 , N0D = A7 A8 be the horizontal sides of N1 and N0 , respectively. Then the following result was proved in [Galias (1996)]: Theorem 4.20. For all parameter values in a sufficiently small neighborhood of (4.196) with C2 = 9.3515, (1) There exists a continuous Poincar´e map P defined on N0 ∪ N1 .

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Fig. 4.31 Computer generated trajectory of the Poincar´ e map H for C2 = 7.65 (b) The corresponding trapping region Γ composed of two polygons Γ1 and Γ2 . Adapted from [Galias (2003)].

(2) The Poincar´e map P has an infinite number of periodic points. (3) The Poincar´e map P satisfies for all parameter values in a sufficiently small neighborhood of (4.196) with C2 = 9.3515 the following inequality: √ 1+ 5 . (4.198) H (P ) ≥ log 2 (4) For all parameter values in a sufficiently small neighborhood of (4.196) with C2 = 9.3515, the Poincar´e map P is chaotic in the sense that it has

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Fig. 4.32 Computer generated trajectory of the Poincar´ e map H for C2 = 9.3515. Adapted from [Galias (2003)].

Fig. 4.33

The sets N0 and N1 .

positive topological entropy. An example of calculating the topological entropy H (P ) for C2 = 7.65 and C2 = 9.3515 of the Poincar´e map based on the number of short cycles is shown in Fig. 4.35.

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ht 0.1

0.05

0

t 0

20

40

60

80 (a)

100

120

140

20

40

60

80 (a)

100

120

140

ht 0.1

0.05

0

0

t

Fig. 4.34 Topological entropy of the Poincar´ e map based on the continuous-time system for (a) the R¨ ossler-type attractor when C2 = 7.65, and (b) the double-scroll attractor when C2 = 9.3515. Adapted from [Galias (1996)].

The topological entropy of the flow is defined by [Fomin et al. (1980)] the following: Definition 4.25. Topological entropy of a flow f : X → X is a topological entropy of the map g defined by g : X → X, g (x) = f (x, t) ., i.e., ht (f ) = H (g) .

(4.199)

The existence of an infinite number of periodic orbits for Chua’s circuit (4.4)–(4.5) can be used to prove that the topological entropy of the flow is positive, namely the following theorem [Galias (1996)]: Theorem 4.21. For all parameter values in a sufficiently small neighborhood of (4.196) with C2 = 9.3515, the topological entropy of the flow generated by Chua’s circuit (4.4)–(4.5) is positive. On the other hand, using the formula (4.199), one has that the topological entropy for the R¨ ossler-type attractor obtained for C2 = 7.65 is H (P ) ≈ 0.22, and for the case of the double-scroll attractor obtained for C2 = 9.3515, one has H (P ) ≈ 0.11 as shown in Fig. 4.34.

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1

H 0.5

0 0

n 10

20

10

20

(a)

30

40

30

40

1

H 0.5

0 0

n (b)

Fig. 4.35 Topological entropy of the Poincar´ e map based on the number of short cycles for (a) the R¨ ossler-type attractor when C2 = 7.65, and (b) the double-scroll attractor when C2 = 9.3515. Adapted from [Galias (1996)].

4.8.2

Formula for the topological entropy in terms of the Poincar´ e map

In [Yang and Li (2004)], a formula for the topological entropy of a Chua’s circuit (4.1)–(4.2) in terms of the Poincar´e map was presented for α = 10, β = 14.87, m0 = 1.27, and m1 = −1.68 using the results about symbolic dynamics introduced in Sec. 1.5.3 and the following result [Robinson (1995)]: Lemma 4.35. Let X be a compact metric space. If a map f : X → X is semi-conjugate to the m-shift σ, then H (f ) ≥ H (σ) = log m.

(4.200)

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Fig. 4.36 The cross sections v1 , v2 , and their images v1′ and v2′ under the half-return Poincar´ e map. Adapted from [Yang and Li (2004)].

For the definition of the Poincar´e map P , define the following subsets of R3 :   v1 = (x, y, z) ∈ R3 : x = 1, y ∈ [−1.9, −1.75] , z ∈ [0.175, 0.185] v2 = (x, y, z) ∈ R3 : x = −1, y ∈ [1.75, 1.9] , z ∈ [−0.185, −0.175] . (4.201) Now consider the two half-return Poincar´e maps Pv1 : v1 → v2 and Pv2 : v2 → v1 as shown in Fig. 4.36. Finally, the entropy of Chua’s circuit (4.1)– (4.2) was studied by virtue of the following composite map P = Pv2 ◦ Pv1 : v1 → v1

(4.202)

that gives the following result [Yang and Li (2004)]: Theorem 4.22. In terms of the Poincar´e map P , Chua’s system (4.1)– (4.2) has an entropy as follows: H (P ) ≥ 2 log 2.

(4.203)

Proof. The verification of this assertion is given by a computer-assisted proof, but it is not mathematically rigorous. 

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Exercises

(1) (a) Prove properties (P.i), i = 0, 4 of Lemma 4.1. (b) Find sufficient conditions such that system (4.1)–(4.2) satisfy properties (P.5) and (P.6) of Lemma 4.1. (c) Show that E c (0) is a 2-D eigenspace, E r (0) is a 1-D eigenspace, E c (P + ) is a 2-D eigenspace, and E r (P + ) is a 1-D eigenspace. (2) Prove Lemma 4.2. (3) Prove Lemmas 4.3 and 4.4. (4) Prove Lemmas 4.5 and 4.6. (5) Prove Lemmas 4.7, 4.8, and 4.9. (6) Prove Lemma 4.10. (7) Prove Theorems 4.1 and 4.2. (8) Prove Lemma 4.12. (9) Prove Lemmas 4.14 and 4.15. (10) Prove Lemmas 4.17 and 4.18. (11) Prove Lemmas 4.25 and 4.27. (12) Prove Lemma 4.28. (13) Show that the B-operator defined in (4.181) has the following properties: (a) B (X) ⊂ X (b) If f (X) ⊂ X, then B = f and A = limp→∞ B p (D) . (c) If Ωx is the limiting set of f |x such that f (Ωx ) = f −1 (Ωx ) = Ωx , then B (Ωx ) = Ωx . (d) ∪L VL = B (d1 ∪ d2 ) .

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Chapter 5

Rigorous analysis of bifurcation phenomena

5.1

Introduction

In this chapter, we discuss different bifurcation phenomena proved analytically for Chua’s system (4.1)–(4.2). The main sources for this chapter are the following papers: [Hahn (1967), Sil’nikov (1965, 1970), Marsden and McCracken (1976), Hurley (1982), Sparrow (1981), Holmes and Whitley (1984), Matsumoto (1984), Cvitanovic (1984), Mees and Chapman (1985), Zhong and Ayrom (1985), Matsumoto et al. (1985), Chua et al. (1986), Pei et al. (1986), Broucke (1987), Matsumoto, (1987), Ogorzalek (1987), Yang and Liao (1987), MacKay and van Zeijts (1988), Lozi and Ushiki (1988, 1989, 1991, 1993), Chua and Tichonicky (1991), Kahlert (1991), Chua and Huynh (1992), Chua et al. (1993a, 1993b), Duchesne (1993), Genot (1993), Madan (1993), Wu and Rul’kov (1993), Brown (1993a, 1993b), Kuznetsov et al., (1993a, 1993b, 1994), Misiurewicz (1992, 1993), Krishchenko (1995, 1997), Kahan and Sicardi-Schifino (1998), Krishchenko and Shalneva (1999), Galias (1996, 2001), Bougaba and Lozi (2000), Silva (2003), S´ anchez (2004), and Carmona et al. (2005)]. In Sec. 5.2 we study the asymptotic stability of equilibria and the deduction of chaotic regions. In Sec. 5.3 we discuss some types of chaotic attractors of Chua’s system (4.1)–(4.2) with a rigorous mathematical proof of their existence and bifurcation phenomena. Sec.5.4 provides some analytical methods for finding trapping regions for both the double-scroll and the R¨ ossler type attractors based on the so-called pull-up map and confinors theory used to predict bifurcations and chaos in dynamical systems, in addition to finding bifurcation diagrams for the observed collision process. In Sec. 5.5 we study bifurcation phenomena using a 1-D approximation of the 2-D Poincar´e map defined in Sec. 4.4. The final Section 5.6 discusses

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the methods based on computer assisted proofs for studying the properties of the 2-D Poincar´e map defined in Sec. 4.4.

5.2

Asymptotic stability of equilibria

The asymptotic stability of the equilibrium points of Chua’s system (4.1)– (4.2) was studied in [S´anchez, 2004] as follows: Lemma 5.1. If m0 < 0, m1 > 0, then the equilibria of Chua’s system (4.1)–(4.2) are given by     O = (0, 0, 0)  m0 + 0 , 0, − 1 P = 1− m m1 m1      P − = m0 − 1, 0, m0 − 1 .

(5.2)

The Jacobian matrices are given by   −αmi α 0 Mi (α) =  1 −1 1  , i = 0, 1, 0 −β 0 and use Hurwitz’s criterion [Hahn (1967)].

(5.4)

m1

(5.1)

m1

(a) The point O is always unstable. (b) The points P + and P − have the same stability type. (c) The points P + and P − are asymptotically stable if and only if the inequalities  1 + αm1 > 0  (5.3) (1 + αm1 ) (β − α + αm1 ) − αβm1 > 0  αβm1 ((1 + αm1 ) (β − α + αm1 ) − αβm1 ) > 0 hold. Proof.



Hence here is a series of lemmas that give sufficient conditions for the convergence to an equilibrium of every positive semi-orbit of Chua’s system (4.1)–(4.2) [S´anchez (2004)]: Lemma 5.2. (a) The equilibria P + and P − are asymptotically stable if and only if  m1 ≥ q 1, orm1 < 1  (5.5) 1−m1 − (1−m1 )2 −4(m21 −m1 )β α < G = . 2(m21 −m1 ) (b) Chua’s system (4.1)–(4.2) is dissipative provided that (5.5) holds.

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β

G = G (m1) = 1 m1 m1 = 1

Fig. 5.1

Region of stability of P + and P − .

Proof. (a) Note that m1 is positive. ´ = (b) Note that Chua’s system (4.1)–(4.2) can be rewritten as X 1 Mi X + g (x) , i = 0, 1 where g (x) = 2 (m0 − m1 ) (|x + 1| − |x − 1|), and use a Lyapunov function V (X) = X T QX for some positive definite symmetric matrix Q, and the fact that the function g is bounded.  The schematic representation of the conditions is given in Fig. 5.1. To prove the theorem about the convergence to an equilibrium of every positive semi-orbit of Chua’s system (4.1)–(4.2), we need the following lemma: Lemma 5.3. Let X ′ = F (X) be a system having a finite number of equilibria q1 , ..., qm , where F : Rn → Rn is locally Lipschitz-continuous and smooth in a neighborhood of each qi . Suppose that every equilibrium is hyperbolic. If every positive semi-orbit converges to some qi , then there is at least one of them that is asymptotically stable. Proof. Let us suppose that none of the points qi is asymptotically stable. s Denote by Wloc (qi ) the local stable manifold of qi , which then has a dimension strictly lesser than n. From [Marsden and McCracken (1976)], we know that this local stable manifold is the graph of a Lipschitz-continuous function and that it has Lebesgue measure equal to zero. The global stable manifold defined as s W s (qi ) = ∪k∈N ϕ−k (Wloc (qi )) (5.6) has also zero measure since the induced flow ϕt is locally Lipschitz. But s this contradicts the equality Rn = ∪m i=1 W (qi ) implied by the convergence to an equilibrium of every positive semi-orbit. 

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Theorem 5.1. Let P be a symmetric matrix of order 3 having one positive eigenvalue and two negative eigenvalues, and suppose that there exist real numbers λ0 and λ1 such that the matrices MiT P + P Mi + λi P, i = 0, 1

(5.7)

are positive definite. Assume also that  m1 ≥ 1, orm1 < 1   q  α
1−m1 −

(1−m1 )2 −4(m21 −m1 )β

(5.8)

2(m21 −m1 ) m0 > − α1 .

Then every positive semi-orbit of Chua’s system (4.1)–(4.2) converges to an equilibrium. Proof.

See [S´anchez (2004)] for a detailed proof and explanation.



In [Ogorzalek (1987)], by transforming Chua’s system (4.4)–(4.5) into the Lur’e form
X ′ (t) = AX (t) + Bf C T X (t) (5.9)

where

     G G − 0 − −1  C C  1 1      A =  CG − CG2 C12  , B =  0   2R   1  0 0 0 −    L   1 V1 (t)    0  , X (t) =  V2 (t)   C =      0 i3 (t)   µ = Ga ,

(5.10)

the displacement of the equilibria and their stability sectors were studied using the so-called transfer function introduced in [Matsumoto et al. (1985)]. In this case, the transfer function of the linear part of system (5.7) is given by G (s) =

1 sG C2 + LC2 1 2 )G + s LC s2 (C1C+C 1 C2 2

s2 +

s3 +

+

G LC1 C2

.

(5.11)

Hence one has the following result [Ogorzalek (1987)]: Lemma 5.4. (a) If R varies and C1 =

1 1 , C2 = 1, L = , 9 7

(5.12)

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then one has that the linearized system with the function f (x1 ) = µx1 of conductance G given by (5.11) is Routh–Hurwitz stable if    

10G + µ > 0 µG + 7 > 0  9G + µ > 0   (µ + 10Gµ + 7) G > 0.

(b) If C1 varies and A =

1 C1

(5.13)

and

C2 = 1, L =

1 1 ,R = , 7 0.7

(5.14)

then the linearized system with the function f (x1 ) = µx1 of conductance G given by (5.11) is Routh–Hurwitz stable if    

0.7A + µ + 0.7 > 0 0.7A + 7 > 0  4.9A + 7µ > 0   0.7µ + 0.49 (A + 1) µ + 4.9 > 0.

(5.15)

However, for different values of G in (5.13) and A in (5.15), we obtain different stability sectors as shown in Figs. 5.2 and 5.3. Note that the chaotic regions lie within the regions of existence of two stability sectors. For example, when ( √ √ 7 7 < G < 5 3 (5.16) 6.556 < A < 14.286, the chaotic ranges 0.606 < G < 0.817

(5.17)

in [Matsumoto et al., (1985)] and 6.8 < A < 10.75

(5.18)

lie within the resulting stability sectors. Furthermore, chaotic attractors occur only in a small stability sector, and for the onset of chaotic oscillations, the nonlinear characteristic f intersect this sector. The transformation of the inequalities (5.13) and (5.15) to the usual αβ-plane is given as a question in Exercise 5.3.

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Fig. 5.2 Stability regions of the double-scroll circuit with f (x1 ) = µx1 as a function of conductance G (conductance bifurcation). The√equations of the curves in the figure are √ 7 µ1 (G) = −5G + 25G2 − 7, µ2 (G) = −5G − 25G2 − 7, µ3 (G) = −9G, µ4 (G) = − G . Shaded areas indicate regions where the system (5.9) is stable. Adapted from [Ogorzalek (1987)].

5.3

Types of chaotic attractors in the double-scroll

For the double-scroll Eqs.(4.1)–(4.2), there are several types of chaotic attractors, including those cited previously in [Matsumoto (1984), Matsumoto et al. (1986), Parker and Chua (1987), Bartissol and Chua (1988), Silva (1991)], period-doubling types, and periodic window types: (1) The first type of chaotic attractor is a R¨ ossler screw-type attractor that is sandwiched between the eigenspace through P + and the eigenspace through 0 as shown in Fig. 4.4(a) because it has a screw-like structure first reported by R¨ ossler [R¨ossler (1979)]. An odd-symmetric image of this attractor has also been observed between the eigenspaces through P − and O as expected. These two R¨ ossler screw-type attractors are separated by the eigenspace through O. (2) The second type of chaotic attractor is the double-scroll, which has already been extensively reported [Matsumoto (1984), Matsumoto et al. (1986)] and which spans all three regions D−1 , D0 , and D1 in Fig. 4.4(a). (3) The other types of chaotic attractors of Chua’s system (4.1)–(4.2) are

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Fig. 5.3 Stability (Hurwitz) sectors of the double-scroll circuit as a function of capacitance (A = C1 ). The equations of the curves in the figure are µ1 (A) = −0.35 (A + 1) + 1 q q 0.5 0.49 (A + 1)2 − 28, µ2 (A) = −0.35 (A + 1) + 0.5 0.49 (A + 1)2 − 28, µ3 (A) = −0.7A, µ4 (A) = −10. Shaded areas indicate regions where the system (5.9) is stable. Adapted from [Ogorzalek (1987)].

discussed in Sec. 4.6 along with rigorous proofs of chaos. As we increase the value of α in Chua’s system (4.1)–(4.2) for fixed β, m0 , and m1 , one can observe that the two disjoint R¨ ossler screw-type attractors grow in size until they eventually collide and give birth to the double-scroll [Matsumoto et al. (1986)]. As α increases further, the doublescroll grows while the coexisting unstable saddle-type periodic orbit shrinks in size until eventually they too collide with each other and the double-scroll disappears [Matsumoto et al. (1986)]. This evolution scenario is called the birth and death of the double-scroll and has been found to be quite typical over wide ranges of β, m0 , and m1 . 5.4

Method 1: Rigorous mathematical analysis

In this section, the analytical tools developed in Chapter 1 are used to rigorously analyze the above bifurcation phenomena, with a rigorous derivation of the locations of the R¨ ossler screw-type attractor and the double-scroll

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attractor. In addition, an exact determination is made of the so-called birth and death boundaries in the αβ-plane which separates the double-scroll attractors and their periodic windows from other attractors (both chaotic and periodic). See Fig. 5.29. To realize that, note that from Fig. 4.4(a), one can observe the following actions for the typical trajectories Γ1 and Γ2 : (1) The orbits Γ1 and Γ2 originate from a point on U1 to the right and the left, respectively, of the boundary line L0 passing through A and E. (2) The line L0 bifurcates the set of all trajectories which return to D1 from those which continue downward to D−1 . (3) All trajectories originating from U1 to the left of L2 (passing through E and B) must move down, while those to the right of L2 must move up. (4) If |γ 1 | is large, as is the case with the R¨ ossler screw-type attractor and the double-scroll, all trajectories originating on either side of the top eigenspace E c (P − ) get sucked in rapidly toward E c (P + ) and eventually cross U1 along an infinitesimally thin slit centered at the line L1 passing through A and B.

5.4.1

The pull-up map

In this section, we will define the so-called pull-up map introduced in [Chua et al. (1986)] and show that the triangle ∆ABE bounded by the three lines L0 , L1 , and L2 is essential in predicting the asymptotic behavior of the trajectories. For that, the new reference frames corresponding to the D1 unit and D0 unit in Fig. 4.4(b) were reconsidered because the Poincar´e map π in (4.95) and its associated half-return maps π 0 in (4.56) and π 1 in (4.75) have analytic equations. Moreover, since it is essential to follow the dynamics originating from ∆A0 B0 E0 = ψ 0 (∆ABE) and taking place in the D0 unit but viewed from the reference frame in the D1 unit, the V1 portrait of V0 defined in Sec. 4.4.1 will play a crucial role in our analysis. In particular, the dynamics taking place within the D0 unit can be translated into the D1 unit by the “pull-up map” defined by π 2 = Φπ −1 0 Φ : ∠A1 B1 E1 → V1 where Φ is the connection map (4.86).

(5.19)

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Fig. 5.4 V1 portrait of V0 with trapping region ∆ = ∆A1u B1 E1u . Adapted from [Chua et al. (1986)].

5.4.2

Construction of the trapping region for the doublescroll

In this section, we will discuss the trapping region of the Poincar´e map π and its role in locating the attractors of the double-scroll system (4.1)–(4.2). The V1 portrait
of V0 corresponding to the parameters (α, β, m0 , m1 ) = 4.0, 4.53, − 71 , 27 , which corresponds to (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.721, 1.075, 0.074, −1.600, 0.530)

(5.20)

is shown in Fig. 5.4. See Exercise 5.4. Hence one has the following result: Lemma 5.5. The pull-up map π 2 satisfies the following properties: (a) 

\ [ B  1 C1 = π 2 A1 B1 , F1 C1 = π 2 F1 E1  

 ′′ ′′ \ C\ 1 A1u = π 2 A1 A1u , C1 E1u = π 2 E1 E1u (5.21)

′′ E ′′ = π  \ A\  2 A1u E1u , F1 W1 D1 = π 2 F1 B1 1u 1u   Cl = π 2 (A1 ) = π 2 (E1 ) . (b) Any point on the line B1 F1 is a fixed point of π 2 .

Now, let us define the following subsets of the Fig. 5.4: Definition 5.1. (a) The subset Sa denotes the “snake-like” area bounded \ [ by B 1 C1 , F1 C1 and B1 F1 .

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A general trapping region corresponding to (α, β, m0 , m1 ) = 4, 4.85, − 71 , 27 \ ′′ \ and u = 2: V1 portrait of V0 . The snake A′′ 1u C1 E1u intersects the spiral F1 W1 D1 =
π 1 F1 B1 , which coincides with the set of discontinuous.points of π −1 . Adapted from 1 [Chua et al. (1986)]. Fig. 5.5

′′ (b) The subset Sb denotes the “snake-like” area bounded by C\ 1 A1u , ′′ ′′ ′′ \ \ C1 E1u , and A1u E1u . (c) The subset A´1u B1 E1u denotes the “fan-like” region bounded by ´ A1 B1 and B1 E1u and


′ A\ 1u A1u = π 1 E1u A1u .

(5.22)

Definition 5.2. (d) The subset

S1 = Sa ∪ Sb = π 2 (∆A1u B1 E1u )

A′1u B1 E1u .

(5.23)

denotes the “double-snake” area bounded within (e) The subset S = ψ −1 1 (S1 ) is a snake-like area and is the set of all points ψ −1 (S ) where returning trajectories of the type Γ1 originating from a 1 ∆ABE intersect the U1 -plane, and the set ψ −1 1 (Sa ) represents the oddsymmetric image of the set of all points where returning trajectories of the type Γ1 originating from ∠ABE\∆ABE intersect the U−1 -plane. Now, we have the following lemma: Lemma 5.6. π −1 1 (S1 ) ⊂ ∆A1u B1 E1u .

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Fig. 5.6 A general trapping region corresponding to (α, β, m0 , m1 ) = 4, 4.85, − 71 , 27 and u = 2: Illustration of π (∆A1u B1 E1u ). The snake-like area π (∆A1u B1 E1u ) is actually an infinitesimally thin set located very near B1 A1u . Adapted from [Chua et al. (1986)]. ′ Proof. Note that π −1 1 (A1u B1 E1u ) = ∆A1u B1 E1u , and use the assumption that S1 ⊂ A′1u B1 E1u . 

Consequently, if we restrict the Poincar´e map π: V1′ → V1′ to the region ∆ = ∆A1u B1 E1u ,

(5.24)

π (∆) ⊂ ∆,

(5.25)

then one has i.e., there are an isolated small area on the subset V1′ where the Poincar´e map π maps into itself. Henceforth, the following definition is obtained: Definition 5.3. The subset ∆ is called the trapping region of the Poincar´e map π. Here, Fig. 5.4 is obtained by the following assumptions: (1) The curve line E1u A1u is near E1 A1 (where u is closer to 1), and hence the corresponding fan-like region A′1u B1 E1u could actually cross the double-snake area S1 .

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(2) The subset S1 ⊂ A′1u B1 E1u , then u must be chosen to be sufficiently large. For the parameters associated with Fig. 5.4, i.e., (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.721, 1.075, 0.074, −1.600, 0.530), the value u = 1.53 is a satisfactory choice. Thus the trapping region ∆ of the Poincar´e map π has the following properties: Theorem 5.2. (1) π: ∆ → ∆ is a continuous map. (2) The subset π (∆) is a compact (i.e., bounded and closed) subset of ∆. Proof. (1) In Fig. 4.10, if x tends to F1 from the inside of the curvilinear −1 \ wedge region bounded by W 1 F1 and F1 e1 , then π 1 (x) tends to F1 , and −1 −1 so lim π 1 (x) = F1 6= f1 = π 1 (F1 ). However, if x tends to F1 from the outside of this curvilinear wedge region, then π −1 1 (x) tends to f1 , and so −1 −1 lim π 1 (x) = f1 = π 1 (F1 ). Since the double-snake area S1 = Sa ∪ Sb , in Fig. 5.4 lies outside of this curvilinear wedge region near F1 , it follows that π −1 1 |S1 : S1 → ∆A1u B1 E1u

(5.26)

is a homeomorphism from the compact domain S1 into ∆A1u B1 E1u . Since π 2 |∆A1u B1 E1u : ∆A1u B1 E1u → S1 is continuous, we have that π |∆ : ∆ = ∆A1u B1 E1u → ∆

(5.27)

is continuous. (2) Since the image of a compact set under a continuous map is compact, π (∆) is a compact subset of ∆.  Now, if we set Λ = ∩n≥0 π n (∆) ,

(5.28)

then one has the following lemma: Lemma 5.7. Λ ⊂ π (Λ). Proof. Take x ∈ Λ = ∩n≥0 π n (∆). Since x ∈ π n+1 (∆) and since π n (∆) is compact, the set Yn = π −1 (x) ∩ π n (∆)

(5.29)

is nonempty and compact. Since Yn+1 ⊂ Yn , we have Y = ∩n≥0 Yn ⊂ ∩n≥0 π n (∆) is not empty, and π (Y ) = x. Therefore, x = π (Y ) ∈ π (∩n≥0 π n (∆)) = π (Λ) . 

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Theorem 5.3. The subset Λ = ∩n≥0 π n (∆) is π-invariant in the sense that π (Λ) = Λ.

(5.30)

Proof. Note that the
double-snake area S1 does not intersect the spiral F1\ W1 D1 = π 1 F1 B1 except at F1 , and use the equalities
π (∩n≥0 π n (∆)) = π ∆ ∩ π 1 (∆) ∩ π 2 (∆) ∩ .... ⊂ ∩n≥0 π n (∆) (5.31)

and the fact that Λ ⊂ π (Λ) , given in Lemma 5.7, where π 0 (∆) = ∆.



If we define Λ1 = Λ ∪ π 2 (Λ)

and

(5.32)

 
 ˜ = Closure of ∪t≥0 ϕt ψ −1 (Λ1 ) ∪ −ψ −1 (Λ1 ) Λ (5.33) 1 1 t ˜ where ϕ is the flow associated with (4.1)–(4.2), then Λ can be interpreted as the closure of the R¨ ossler screw-type attractors or the double-scroll, depending on the parameters. ˜ is an attractor of the double-scroll system Definition 5.4. The subset Λ (4.1)–(4.2). Then one has the following results: ˜ of Λ ˜ which satisfies Lemma 5.8. There exists an open neighborhood N (Λ)   ˜ . ˜ = ∩t≥0 ϕt N (Λ) (5.34) Λ

Proof. In Fig. 5.4, we observe that π −1 1 maps S1 \{B1 } into the interior of ∆A1u B1 E1u . However, the point B1 maps into the point a2 on B1 A1 in Fig. 4.10. From this we have π (∆) ⊂ {a2 } ∪ interior (∆) .

(5.35)

π (a2 ) = π −1 1 π 2 (a2 ) ∈ interior (∆) .

(5.36)

π n (∆) ⊂ interior (∆) , n ≥ 2

(5.37)

π 1−1 (B(C1 )) ⊂ interior (∆) .

(5.38)

Since B1 6= a2 , we have Thus one has

using the facts that π (interior (∆)) ⊂ interior (∆) and π (∆) ⊂ ∆. To prove ˜ which satisfies (5.34), take a the existence of an open neighborhood N (Λ) small open ball B(C1 ) at C1 such that

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Then the set N1 = B(C1 ) ∪ interior (S1 ) ∪ interior (∆)

(5.39)

is an open neighborhood of Λ1 , in the V1 -plane which satisfies Λ1 = ∩t≥0 π n (N1 ) .

(5.40)

˜ of Λ ˜ in the double-scroll system such Choose a small neighborhood N (Λ) ˜ that any trajectory originating in N (Λ) intersects U1 ∪ U−1 only at points
−1 belonging to the set ψ −1 (N ) ∪ −ψ (N ) , where N1 is defined in (5.39). 1 1 1 1 ˜ satisfies (5.34). Then N (Λ)  From these lemmas and theorems, one concludes on the one hand, that ˜ possesses the properties of an attractor defined by several the subset Λ researchers including Hurley [Hurley (1982)]. On the other hand, and be˜ in the literature which contains no cause there exists some attractor Λ dense orbits, computer simulations strongly suggest that both the R¨ ossler screw-type and the double-scroll attractors contain at least one dense orbit. Lemma 5.9. The region ∆ in (5.24) is a trapping region of Λ. Proof. Note that ∆ is a neighborhood of Λ and every trajectory originating from ∆ tends to Λ under the Poincar´e map π.  5.4.3

Finding trapping regions using confinors theory

The concept of confinors and anticonfinors was introduced and initially defined in [Lozi and Ushiki (1988, 1989)] for constrained ordinary differential equations. The purpose of this notion is to understand the dynamics and bifurcations of strange attractors of dissipative dynamical systems with chaotic behavior, especially the intermediate structures between attractors and their basin of attraction linked by schemes which are induced by symbolic dynamics given the following properties: (1) Ability to model both transient and asymptotic regimes. (2) Traceability even in the presence of chaos or an infinite number of periodic ω-limit sets. (3) Robustness versus the approximate computation of the solutions of ODE’s. (4) Allowance for the mathematician to choose the “level of analysis” of the phase portrait. (5) Accounting for the shape of the signal.

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Fig. 5.7 The three attractors (red, blue, and cyan in the original source) plotted together for α = 15.60, β = 28.58, m0 = − 71 , m1 = 72 , and initial conditions (x0 , y0 , z0 ) = (−0.005, 0.017, 0.019) for the (red in the original source) attractor, (x0 , y0 , z0 ) = (−0.002, 0.014, 0.010) for the (blue in the original source) attractor, and (x0 , y0 , z0 ) = (−0.008, 0.020, 0.028) for the (cyan in the original source) attractor. Adapted from [Lozi and Ushiki (1993)].

(6) Finding coexisting attractors [Lozi and Ushiki (1993)]. The result about the coexistence of three chaotic attractors for Chua’s system (4.1)–(4.2) was obtained without mathematical proof for the parameters α = 15.60, β = 28.58, m0 = − 71 , and m1 = 72 using 60000 iteration of π as shown in Fig. 5.7. For the constrained differential equations, the frontiers of confinors are points and are then easily traceable, because the corresponding Poincar´e map is 1-D. While for the double-scroll Chua’s attractor, the frontiers of confinors of this system are curves because the Poincar´e half-maps π 0 and π 1 defined in Sec. 4.3.1 and Sec. 4.32 are two-dimensional. Proof of the existence of such 2-D confinors is based on some inequalities and computerassisted proofs. In the case of 3-D confinors such as those involved in the double-scroll family [Lozi and Ushiki (1991)], the cross section is 2-D, and it requires curves as boundaries. The existence proof of such 3-D confinors can be found in [Lozi and Ushiki (1991)]. Here are some definitions of confinors theory, the details of which can be found in [Lozi and Ushiki (1988, 1989, 1991, 1993)]: Definition 5.5. A subset C˜ is a main confinor if it is compact and positively invariant under the flow defined by the vector field   (5.41) ∀t > 0, ϕt C˜ ⊂ ∆

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Fig. 5.8 Projection in the xy-plane of the R¨ ossler-type attractors of the double-scroll family obtained for (x0 , y0 , z0 ) = (−0.001, 0.001, 0.001) and (a) (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3080, −0.8576, 0.1239), (b) (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3826, −0.9173, 0.1054), (c) (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3921, −0.9236, 0.1034), (d) (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.5146, −0.9795, 0.0824) . Adapted from [Lozi and Ushiki (1991)].

if it is trapping, and such that all the trajectories starting from points in C˜ have a sequence of non-null length for a given symbolic dynamics D of the flow. A main confinor can be decomposed in the following (nonunique) way: Definition 5.6. Let Xn+p be a vector with dim X = n + p, Xn+p = (X1 , ..., Xn , Xn+1 , ..., Xn+p ) , n ∈ N∗ , p ∈ N

(5.42)

for which Xi , 1 ≤ i ≤ n, is either a “letter” or a “word” of the symbolic ˜ used to define the main confinor C˜ and Xn+j (1 ≤ j ≤ p) is a dynamics D ˜ Let M ˜ n+p be a transition “letter” of an additional symbolic dynamics H. matrix given by:   ˜n,p ˜n,n N N ˜ (5.43) Mn+p = Ip,n 0

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with mij ∈ N, 1 ≤ i, j ≤ n + p, such that for every 1 ≤ i ≤ n, there exists at least one j ∗ with mij ∗ > 0. o n ˜ n+p is called a scheme of symbolic Definition 5.7. (a) Sn+p = Xn+p , M ˜ dynamics D. (b) A partition of C˜ in confinors and anticonfinors is called a “level of analysis. ” (c) A subset A˜0 of C˜ is a confinor if it is compact, positively invariant, if there exists a scheme Sn+p of the symbolic dynamics D such that all the itineraries of the trajectories in A˜0 can be generated by this scheme, and if the submatrix In,p is identically a null-matrix. (d) A subset A˜0 of C˜ is an anticonfinor if it is open, negatively invariant ˜ if there exists a scheme Sn+p of the symbolic under the flow reduced to C, dynamics D such that all itineraries of the trajectories belonging to A˜0 during a limited or infinite interval of time can be generated by this scheme, and if the submatrix In,p is identically a null-matrix. For the double-scroll Chua’s attractor (4.1)–(4.2), Lozi and Ushiki [Lozi and Ushiki (1991)] chose a particular symbolic dynamics of intervals based on the patterns in the waveforms which are bounded in time because they are very easily obtained for a large range of parameter values in Chua’s circuit (4.1)–(4.2) [Matsumoto (1987)]. The symbolic dynamics may be described as Iτ ⇆ Iθ where Iτ = [τ min , τ max ] and Iθ = [θ min , θmax ] , where τ min , τ max , θmin and θ max were defined in Theorem 5.4 below and are related to the existence of a main confinor for Chua’s system (4.1)–(4.2). In [Pei et al. (1986)], these patterns were used to describe a sequence of bifurcation phenomena for a negative resistance circuit. Definition 5.8. A pattern consists of m successive peaks belonging to the upper sheet of the attractor, followed by n successive peaks belonging to the lower sheet per period when the waveform is periodic. For chaotic states, some portion of the waveform contains oscillations from nearby periodic states as shown in Fig. 4.5(a). For some parameter values of Chua’s circuit (4.1)–(4.2), there are patterns in the waveform which are bounded from above and below [Lozi and

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Patterns in the measured time waveform. Adapted from [Matsumoto, (1987)].

Ushiki (1991)] such as  (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3080, −0.8576, 0.1239)    (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3826, −0.9173, 0.1054) (5.44)  (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3921, −0.9236, 0.1034)   (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.5146, −0.9795, 0.0824) where Chua’s Eqs.(4.1)–(4.2) has the R¨ ossler-type strange attractors as shown in Fig. 5.8. The number of peaks shown in Fig. 5.9 of similar appearance can be assumed to be the number of “turns” of the corresponding solution around some axis. In the case of the piecewise linear systems, especially Chua’s system (4.1)–(4.2), the number of “turns” has a clear meaning1 because it is possible to compute the time spent in going from a point to some other point, divided by the time needed to make an exact turn around the axis of rotational symmetry of the linear part of the vector field, which amounts to a real number of turns instead of an integer [Lozi and Ushiki (1991)]. If one has an estimate for this real number of turns,2 then the simplest case of such a description of patterns is given in terms of “pattern intervals” in Theorem 5.4 [Lozi and Ushiki (1991)]: tmin ≤ τ (M ) ≤ tmax , and θ min ≤ θ (M ′′ ) ≤ θmax defined in Theorem 5.4(b) below. Theorem 5.4. (a) There is a confinor for the double-scroll Eqs. (4.1)– (4.2) for the following parameter values:3 (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.3250, 0.1350, 0.3921, −0.9236, 0.1034). (5.45) 1 In

some cases, the number of “turns” might not be well defined. example, one can have tmin and tmax such that tmin ≤ t ≤ tmax holds for all t needed in going from some set, especially a separating plane, into another set. 3 These values are called theoretical Chua’s parameters. 2 For

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(b) There exists τ min , τ max , θmin , θmax , such that for each point M ∈ ∆ ⊂ U1∗ , if M ′ = π n (M ) , n ≥ 2 then tmin ≤ τ (M ) ≤ tmax , and if M ′′ = π 0 (M ′ ) then θmin ≤ θ (M ′′ ) ≤ θmax . Proof. The proof is purely technical and requires several mathematical tools such as dynamical and anti-dynamical manifolds, selecting and antiselecting curves, spirals and anti-spirals, and the theorem of the turnstile for the snake and for the anti-snake.  5.4.4

Construction of the trapping region for the R¨ osslertype attractor

The trapping region for the R¨ ossler-type attractor 4 is not discussed in [Chua et al. (1986)], but it was introduced in [Bougaba and Lozi (2000)] for (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.465716, 0.932544, 0.4152731, −0.3446764, 0.3279262) (5.46) corresponding to the parameters 2 1 α = −6.800, β = −1.520, m0 = − , m1 = . 7 7 Consider a fixed triangular region of the plane U1∗ U1∗ = {(ξ, η, ζ) : ζ = 0}

(5.47)

(5.48)

in a new system of coordinates (ξ, η, ζ) ∈ R3 [Bougaba and Lozi (2000)]. In this case, the strange R¨ ossler-type attractor shown in Fig. 5.10 was obtained, and the cross-section5 corresponding to it in the plane U1∗ has four components (Ai )1≤i≤4 as shown in Fig. 5.11. It was found that this attractor is located in two trapping regions P1 and P3 constructed using a method based uniquely on the isochronic lines. The main results are the following:  A1 ⊂ P1    A3 ⊂ P3 (5.49)  π (P1 ) ⊂ P3   π (P3 ) ⊂ P1 . 4 This trap of the solutions is useful and appropriate for the application of the theory of confinors given in Sec. 5.6 to the double-scroll system (4.1)–(4.2) [Lozi and Ushiki (1991, 1993)]. 5 The Poincar´ e section map.

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Fig. 5.10 Projection onto the xy-plane of the R¨ ossler-type attractor obtained for α = −6.800, β = −1.520, m0 = − 71 , m1 = 27 , and initial conditions (−0.001, 0.001, 0.001) . Adapted from [Bougaba and Lozi (2000)].

To describe the method used in this case, we need the following tools: If S = π 0 (∆) , then assume that S ∩ ∆ = F B. Given that t > 0, let  ai(t) = exp ((σ 0 − γ 0 ) t) − cos t i = 0, 1. bi(t) = exp (σ 0 t) (1 − exp (−γ 0 t)) .

(5.50)

(5.51)

Let N0 ⊂ V0 be the intersection point of E0 B0 with the straight line defined by sin (−t) y ′ = a0(−t) x′ + b0(−t)

(5.52)

where the triangle △AEB corresponds to two regions: △A0 B0 E0 in the (x′ , y ′ , z ′ ) coordinate system and △A1 B1 E1 in the (x, y, z) coordinate system. Let G0 ∈ V0 , the intersection point of the line (5.52) with either A0 B0 or A0 E0 which belongs to the triangular region ∆A0 B0 E0 given by (4.59). Then one has the following result:
Lemma 5.10. (a) π 0 N0 G0 is the segment π 0 (N0 ) π 0 (G0 ) included in the straight line sin (t) y ′ = a0(t) x′ + b0(t).

(5.53)

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Fig. 5.11 Cross-section view of the R¨ ossler-type attractor. Adapted from [Bougaba and Lozi (2000)].

(b) π 0 (N0 ) and π 0 (G0 ) are determined by  ′ ′ ′  x (t) = exp (σ 0 t) (x0 cos t − y0 sin t) ′ ′ ′ y (t) = exp (σ 0 t) (x0 sin t + y0 cos t)  z ′ (t) = z0′ exp (γ 0 t) .

Let us also consider  the straight lines sin (t) y = a1(t) x + b1(−t) sin (t) y = a1(t) x + b1(t). Then one has the following definition:

(5.54)

(5.55)

Definition 5.9. (a) For π 0 , the straight lines defined by Eqs. (5.52) and (5.53) are called “isochronic lines .” (b) For π 1 , the straight lines defined by Eqs. (5.55) are called “isochronic lines.” In [Lozi and Ushiki (1991)], the explicit formula of the affine transformation ψ 0 : D0 → U1∗ defined in Sec. 4.2.3 is given by  ′  ξ = (1+k q)0 zσ2 +1   ( )# 0 0  # " "   ′ ′ γ (1−σ (σ −γ ))+k (1+σ2 )(σ −γ )−p q  η=     

y′ −

0

0

0

0

0 0 (1+k0 )(σ 2 0 +1)

ζ=

0

0

0 0

(σ0 −p0 ) γ 0 (x′ +z ′ −1) (1+k0 )(σ0 −p0 )

z ′ −p0 −

(1+σ0 p0 )γ 0 (x +z −1) (σ 0 −p0 )

(5.56)

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Fig. 5.12 Representation in U1∗ of the Poincar´ e section map for the attractor of the corresponding snake S, and the anti-snake As. Adapted from [Bougaba and Lozi (2000)].

and the explicit formula of the affine transformation ψ 1 : D1 → U1∗ defined in Sec. 4.2.3 is given by      ξ =     

"

y−

"

(

)

γ 1 (1−σ 1 (σ1 −γ 1 ))+k1 1+σ2 1 (σ1 −γ 1 )−p1 q1 (1+k1 )(σ2 +1) 1

η= ζ=

#

z−p1 −

(σ1 −p1 ) q1 z (1+k1 )(σ21 +1) γ 1 (x+z−1) (1+k1 )(σ1 −p1 )

(1+σ1 p1 )γ 1 (x+z−1) (σ1 −p1 )

#

(5.57)

Lemma 5.11. Assume that (5.50) holds.  Then (a) π 0 is a continuous map on ∆ − AE ∪ F B . (b) π 0 is one-to-one. ossler-type systems is when S ∩  But the situation that arises for R¨ ∆ − F B 6= ∅, as shown in Fig. 5.13(a), i.e., where there exist interior points of ∆ belonging to more than one isochronic lines and having virtual orbits which intersect the snake S more than once. A formal representation of this situation is shown in Fig. 5.13(b), which shows the following things: (1) Because π 0 is continuous, then the points of Ω1 limited by Ld belong to isochronic lines which do not intersect themselves in ∆.

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Fig. 5.13 (a) Isochronic lines for π 0 (in T ). (b) A formal representation explaining the discontinuity of π 0 . Adapted from [Bougaba and Lozi (2000)].

(2) The points belonging to Σ or to the left of Ω2 (relative to Ld ) have virtual orbits which coincide with the real orbits for all t′ , t′ < t0 , t0 defined for π 0 . (3) The points of Ld are points of discontinuity of π 0 . Fig.5.13 shows graphical evidence of π(P1 ) included in P3 and π(P3 ) included in P1 . Concerning the images by the Poincar´e map π of the isochronic lines, consider the following quantities:  F0 = B1 (L′ sin θ + P ′ cos θ) , (θvariable)     G0 = (C1 L′ + A1 P ′ ) cos θ     ′  H = (p1 L + P ′ ) cos θ + (p1 P ′ − L′ ) sin θ + Q′ exp (σ 0 θ)   0   L′ = LA1  ′ P = B1 P − LC1   ′  Q = QA1 B1 − P ′ − L′ p1     L = C0 sin (τ ∗ ) − a0(τ ∗ ) A0     P = D0 sin (τ ∗ )    Q = p0 sin (τ ∗ ) − a0(τ ∗ ) − b0(τ ∗ )

(5.58)

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         C0 =            C1 =   

(1+k0 )(1+σ20 ) q0 γ 0 (1−σ0 (σ0 −γ 0 ))+k0 (1+σ 20 )(σ 0 −γ 0 )−p0 q0 q0

A0 = −

D0 = σ 0 − p0 (1+k1 )(1+σ21 ) A1 = − q0 γ 1 (1−σ1 (σ1 −γ 1 ))+k1 (1+σ 21 )(σ 1 −γ 1 )−p1 q1

(5.59)

q1

B1 = σ 1 − p1

The explicit formulas for p0 , p1 , q0 , q1 , k0 , and k are given, respectively, by Eqs. (4.29) and (4.30).

and

 ˜ ˜ θ, (θvariable)     F1 = K cos θ+ M sin     ˜ cos θ ˜ + A1 M ˜ ˜ G1 = C1 M − A1 K sin θ + C1 K           ˜ +M ˜ cos θ + N ˜ exp (σ 1 θ) ˜ −K ˜ sin θ + p1 K H1 = p 1 M     ′′′  ˜ = K A1 K ˜ M = M ′′′ B1 − K ′′′ C1     ˜ ˜ − Kp ˜ 1 N = N ′′′ A1 B1 − M     ′′′ ′′  K = A0 K + C0 M ′′     M ′′′ = D0 M ′′    N ′′′ = K ′′ + p0 M ′′ + N ′′  ′′ K = M ′ cos τ − K ′ sin τ , (τ variable)     M ′′ = K ′ cos τ + M ′ sin τ      N ” = N ′ exp (σ 1 τ )     K ′ = M A0  ′ M = KD0 − M C0  ′   N = N − M ′ − K ′ p0     K = B1 sin θ∗     M = C1 sin θ ∗ + a1(θ∗ ) A1    N = p1 sin θ ∗ − a1(θ∗ ) − b1(θ∗ )

(5.60)

(5.61)

Then we have the following lemma:

Lemma 5.12. (a) Given Tk , an isochronic line for π 0 in the D0 unit and the corresponding fixed time τ ∗ , the image by π of Tk in the reference frame (ξ, η, ζ) is the curve described by the following equation: F0 .ξ + G0 .η + H0 = 0.

(5.62)

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Fig. 5.14 Schematic representation of the traps in the plane U1∗ . Adapted from [Bougaba and Lozi (2000)].

(b) Given Dh , an isochronic line for π 1 in the D1 unit and the corresponding fixed time θ∗ , the image by π of Dh in the reference frame (ξ, η, ζ) is the curve described by the following equation: F1 .ξ + G1 .η + H1 = 0.

(5.63)

The method is based on the construction in the basin of attraction of each “component” Ai of a trap Pi (i = 1, .., 4). In this case, each Pi contains the component Ai and attracts the trajectories of the flow of the system. Hence, using the four traps, one can describe the circulation of the trajectories according to the iterations of the Poincar´e map depicted as follows:  π 0 (P3 ) ⊂ P4 →π1 π 1 (P4 ) ⊂ P1 →π0 π 0 (P1 ) ⊂ P2 →π1 π 1 (P2 ) (5.64) ⊂ P3 →π0 π 0 (P3 ) ⊂ P4

and shown in Fig. 5.11. The construction of the boundaries of the traps Pi can be done using the distinguishing properties of the isochronic lines. Now, the method can be summarized as follows: Let Tki represent the isochronic lines for π 0 for P1 , and Tmi for P3 , as parallel lines in the Euclidean sense to Oη, and let Dhi represent the isochronic lines for π 1 as parallel lines in the Euclidean sense to Oξ as shown in Fig. 5.17. For a formal representation of the traps P1 and P3 , successive extremities of the “segment boundaries ” of Pi shown in Figs. 5.18 and 5.19 are labeled by successive natural numbers. Construction of P1 and P3 :

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Fig. 5.15 (a) Illustrative scheme of the algorithm for the method. (b) Formal representation of the isochronic lines. Adapted from [Bougaba and Lozi (2000)].

(1) Each Tki (respectively Tmi ) be defined by the abscissa of the intersection point with the axis Oξ:
  Tki ∩ Oξ = ξ ki , 0 (5.65) ξ ki+1 = ξ ki − λξ , λξ = step = 0.00156625. (2) Every isochronic line Dhi for π 1 , whose equation in D1 is given by Eq. (5.63), corresponds to θhi given by  θhi = θ hi (M ) = inf t > 0 : ϕ−1 (5.66) 1 (M ) ∈ V1 ∩ Dhi . ϕ−1 1 (M ) denotes the flow from M to the first return point where the trajectory first intersects V1 at the reverse time −θ < 0 with θhi+1 = θhi − λθ , λθ = step = 0.006250.

(5.67)

(3) For a segment [N, N + 1] belonging to both the boundary of Pi and an isochronic line for π 1 , we have Tki and Tkj (i < j), two isochronic lines for π 0 passing respectively by N and N + 1, and thus defining on Oξ a segment [kj ; ki ] as shown in Fig. 5.15. We consider [ξkj , ξki ] = ∪j−1 l=i [ξkl+1 , ξkl ] and a subdivision of twenty-one points of each interval [ξkl+1 , ξkl ] . (4) For all intersection points of [N, N + 1] with the isochronic lines for π 0 which pass by the points of the subdivision, we compute the successive images by π 0 and then by π 1 . For M ∈ [N, N + 1] ⊂ Dh , π 1 ◦ π 0 (M ) is a point on the curve shown in Fig. 5.15 and defined by the Eq.(5.63). (5) For a segment [N + 1, N + 2] belonging to both the boundary of Pi and an isochronic line for π 0 , we have Dhp and Dhq (p < q), the isochronic line for π 1 passing by N + 1 and N + 2 and  respectively  , θ defining on Oη a segment θ shown in Fig. 5.15. We conhp hq   q−1  sider θhp , θhq = ∪l=p θ hl , θhl+1 and a subdivision of six points

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Fig. 5.16 Isochronic lines for π 1 and their images by π 0 and by π in U1∗ . Adapted from [Bougaba and Lozi (2000)].

  of each interval θhl , θhl+1 . For all intersection points on this subdivision, we also compute the images by π 0 and then by π 1 . For M ∈ [N + 1, N + 2] ⊂ Tk , π 1 ◦ π 0 (M ) is on a curve shown in Fig. 5.16 defined by (5.62). The results are shown in Figs. 5.18 and 5.19. 5.4.5

Macroscopic structure of an attractor for the doublescroll system

˜ associated with (4.1)–(4.2) has been careThe macroscopic structure of Λ fully analyzed by computer simulations in [Matsumoto et al. (1985)], where ˜ consists of two they discovered that each x =constant cross section of Λ tightly-wound spirals , hence the name double-scroll for some parameter values. For example, the double-snake area Sa ∪ Sb defined in (5.23) and shown in Fig. 5.4 (see also the upper snakes Sa and Sb in Fig. 5.21 corresponds to the x = 1 cross section. The microscopic (local) structure of ˜ however, is much more complicated. Indeed, since it contains infinitely Λ, many horseshoes at least for some parameters (recall Theorem 4.8), we ˜ consists of a product between a can expect that the local structure of Λ manifold and a Cantor set similar to that described in [Williams (1974)].

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Fig. 5.17 Isochronic lines for π 0 and their images by π 0 and by π in U1∗ . Adapted from [Bougaba and Lozi (2000)].

Lemma 5.13. If the magnitude of the real eigenvalue γ˜1 at P + (˜ γ 1 < 0) is very large compared to the real part of the other eigenvalues, then the set Λ1 must be tightly squeezed near the curve 
 ′ \ \ (5.68) ΛE = A1 B1 ∪ A1 A1∞ ∪ B 1 C1 ∪ C1 A1∞ .

Proof. This is a result of the strong rate of contraction of the trajectory component along the real eigenvector E r (P + ) in Fig. 4.4(a) and the fact that trajectories passing through points on Λ1 represent the asymptotic behavior, i.e., long after the trajectory component along E r (P + ) has shrunk to an infinitesimal value, thereby ensuring that the trajectories through Λ1 are literally coasting on the surface of E r (P + ) in Fig. 4.4(a). This mechanism explains why the double-scroll in [Matsumoto et al. (1985)] must cross the U1 and U−1 -plane along a very thin contour.6  Computer simulations show the following behavior: ˜ can enter D0 from D1 (1) All trajectories originating from the attractor Λ
A1 B1 ⊂ only through the infinitesimally-thin gate centered at ψ −1 1
L1 , or at ψ 1−1 A1 A1∞ ⊂ L1 . 6 Note

that for the parameter assumed in Fig. 5.4, one can replace A1∞ by A1u .

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η ο

ξ

P1 U1* (a)

7

1 5

Dh65 Dh81 Dh97

4 3 2 1

Dh113 314D 308

P1

h129

Dh145 Dh161 Dh177 Dh193 Dh209

Tk8 Tk0 (b)

Fig. 5.18 Magnification of the structure of the trap P1 . (a) Original structure of P1 . (b) A formal representation of P1 . Adapted from [Bougaba and Lozi (2000)].

(2) Returning trajectories exiting from D0 to D1 or those returning trajectories exiting from D0 to D−1 can the    do so only through −1 −1 \ ′ \ infinitesimally-thin gates centered at ψ 1 B1 C1 and −ψ 1 C1 A1∞ (by symmetry of the vector field ξ), respectively.

The following gates will play a crucial role in the bifurcation analysis of Chua’s system (4.1)–(4.2): Definition 5.10. (a) The infinitesimally-thin at    gates centered 

−1 −1 −1 −1 \ ′ \ ψ 1 A1 B1 ⊂ L1 , ψ 1 A1 A1∞ ⊂ L1 , ψ 1 B1 C1 and −ψ 1 C1 A1∞ are called upper entrance gate, lower entrance gate, upper exit gate, and lower exit gate, respectively. ′ \ \ (b) The curves A1 B1 , A1 A1∞ , B 1 C1 , and C1 A1∞ are called upper entrance gate, lower entrance gate, upper exit gate, and lower exist gate, respectively.

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η

ο

ξ

P3

U1*

(a) 64 63 89

P3 Dh305

94 95 105

Dh313 Dh321

104

22 21

20

16

12 9 134

Tm64

Tm60

Tm52

Tm44

Tm36

Tm28

Dh337 Dh341

8

6

7 138

2

Dh353

1

Tm20 Tm16 Tm12

(b)

Fig. 5.19 Magnification of the structure of the trap P3 . (a) Original structure of P3 . (b) A formal representation of P3 . Adapted from [Bougaba and Lozi (2000)].


 ′ \ \ (c) The subset ΛE = A1 B1 ∪ A1 A1∞ ∪ B is called 1 C1 ∪ C1 A1∞

˜ Λ-gates.

5.4.6 5.4.6.1

Collision process Birth of the double-scroll

Computer simulations given in [Matsumoto et al. (1986)] show the following collision process: BDS: As α increases (for fixed β, m0 , and m1 ), the two R¨ ossler screwtype attractors eventually collide with each other, and the double-scroll suddenly emerges after any further infinitesimal increase in α. Then one has the following definition: Definition 5.11. The collision process BDS is called the birth of the double-scroll .

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First, Fig. 5.20 shows the qualitative picture of the structure of a R¨ ossler screw-type attractor corresponding to the value of α at the collision point as follows: (1) The attractor “funnels through” the upper entrance gate AB where its extreme left point on U1 coincides with A. (2) Any further increase in α would cause this attractor to expand with its extreme left point on U1 appearing to the left of A, thereby causing this trajectory to move downward and eventually link up with its twin from the D−1 region. Second, the parameter value α corresponding to the birth of the doublescroll was determined as a geometrical condition using the qualitative picture of the structure of a R¨ ossler screw-type attractor shown in Fig. 5.20. Hence the following theorem gives the bifurcation value α which heralds this event: Theorem 5.5. The birth of the double-scroll occurs
at a parameter value α when the upper snake Sa is tangent to π 1 E1 A1 .


′ \ Proof. We have assumed that E 1 A1 = π 1 A1 E1 intersects the line ψ 1 (L1 ) = {(x, y) : x = 1} at A′1 as shown in Fig. 5.21. The snake area
\′ \ [ Sa bounded by B 1 C1 , F1 C1 , and B1 F1 is tangent to E1 Q1 A1 = π 1 E1 A1 at Q1 . Since the R¨ ossler screw-type attractor above the eigenspace E c (O) is not connected to its twin below E c (O), only one snake Sa is shown in Fig. 5.21. The π 1−1 image of the upper snake Sa gives rise to another snakelike region S˜a = π 1−1 (Sa ) in Fig. 5.21. Since S˜a = π −1 1 π 2 (∆A1 B1 E1 ) = π (∆A1 B1 E1 ), the lower snake S˜a is the image of the triangular region ∆A1 B1 E1 under the Poincar´e map π. Consequently, S˜a must be tangent  to E1 A1 at Q′1 = π −1 1 (Q1 ). Example 5.1. A computer-calculated example of this situation is shown in Fig. 5.22, which
corresponds to the parameter values (α, β, m0 , m1 ) = 8, 8, 14.3, − 71 , 72 . 5.4.6.2

Death of the double-scroll

Computer simulations given in [Matsumoto et al. (1985)] show the following collision process:

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Fig. 5.20 Geometrical structure at the birth of the double-scroll: Macroscopic picture of the original system. Adapted from [Chua et al. (1986)].

DDS: An unstable (saddle-type) periodic orbit actually coexists with the double-scroll, and as α increases for fixed β, m0 , and m1 , the periodic orbit shrinks while the double-scroll grows in size. At the parameter α0 (or just below to be precise) where they collide with each other, the doublescroll suddenly disappears, while the unstable periodic orbit continues to exist. Then one has the following definition: Definition 5.12. The collision process DDS is called the death of the double-scroll . If we consider the Fig. 5.23, especially the subset Λ1 and the two points and H1− , then one has the following theorem:

H1+

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Fig. 5.21 Geometrical structure at the birth of the double-scroll: Enlargement of the

′ \ ˜ \ V1 portrait of V0 . B 1 C1 = π 2 B1 A1 is tangent to E1 A1 = π 1 A1 E1 at Q1 . Sa = −1 π 1 (Sa ) is an “infinitesimally” thin set (infinitely many layers compressed into a sheet) whose actual location is very close to A1 B1 . Adapted from [Chua et al. (1986)].

Theorem 5.6. The death of the double-scroll occurs when Λ1 intersects the points H1+ and H1− . Note that Theorem 5.5 is equivalent to the condition that Λ1 touches the stable manifolds W s (H1+ ) because x ∈ W s (H1+ ) ∩ Λ1 implies H1+ = limn→∞ π n (x) belongs to W s (H1+ ) ∩ Λ1 ⊂ Λ1 . The proof of Theorem 5.5 is based on Figs. 5.23, 5.24, and 5.25, and it requires the following lemmas: If the stable manifold is given by
 (5.69) W s H1− = x ∈ ∠A1 B1 E1 : π n (x) −→ H1− asn −→ +∞ ,

then one has


Lemma 5.14. W s H1− ≈ E1u0 A1u0 , i.e., the curve
E1u0 A1u0 is an excellent approximation of the stable manifold W s H1− . Proof. Suppose that the magnitude of the real eigenvalue γ˜ 1 at P + (˜ γ1 < 0) is very large compared to the real part of the other eigenvalue. This is

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Fig. 5.22 Geometrical structure at the birth of the double-scroll: V1 portrait of V0 for
′ \ \ (α, β, m0 , m1 ) = 8, 8, 14.3, − 71 , 27 . B 1 C1 is tangent to E1 A1 at Q1 . Adapted from [Chua et al. (1986)].

equivalent to considering the limit as γ˜1 → −∞. Hence, upon substituting γ 1 = ωγ˜˜ 11 and σ 1 = ωσ˜˜ 11 into the coordinates for F1 and E1 given by      γ 1 (γ 1 −2σ 1 ) γ 1 (1−σ 1 (σ1 −γ 1 )) ˜ 1 (ω ˜ 1 −σ 1 (˜ σ 1 −˜ γ 1 )) γ ˜ 1 (˜ γ 1 −2˜ σ1 ) γ  F = , , = 2 2 2 2 1 2 2  (σ 1 −γ (σ 1 −γ 1 ) +1 (˜ σ 1 −˜ γ ω1 (˜ σ1 −˜ γ 1 ) ++˜  1 ) +˜ ω1   1 ) +1 γ ˜ 1 (˜ γ 1 −˜ σ1 −p1 ω ˜ 1) 1 (γ 1 −σ 1 −p1 ) γ 1 (1−p1 (σ 1 −γ 1 )) = , , q E1 = γ(σ 2 1 (σ 1 −γ 1 )2 +1 (˜ σ 1 −˜ γ 1 )2 +˜ ω 21 1 −γ 1 ) +1   γ ˜ 1 (˜ ω 1 −p1 (˜ σ 1 −˜ γ 1 ))  q = 1

(˜ σ 1 −˜ γ 1 )2 ++˜ ω 21

(5.70) and then taking the limit as γ˜1 → −∞, one has F1 → B1 = (1, σ1 ) and E1 → A1 = (1, p1 ) . It follows from (5.70) that E1u0 = u0 E1 + (1 − u0 ) F1 → u0 A1 + (1 − u0 ) B1 = A1u0 .

(5.71)

H1+ A′1u0 shrinks to one point E1u0 = Under this condition, the arc E1u0\ −1 − H1 = A1u0 under π 1 , and therefore also under π. Therefore, the
s ′ \ H1+ as arc E1u 0 A1u0 , may be considered as the stable manifold W
′ \ ≈ W s H1+ . This implies that E1u0 A1u0 γ˜ 1 → −∞, i.e., E1u 0 A1u0  


′ \ = π 1−1 W s H1+ = W s H1− . E1u = π −1  0 A1u0 1 One can see, on the one hand, that the unstable manifold W u (H1− ) in this case must be a subset of S˜b because W u (H1− ) is an invariant set and the only invariant set in Fig. 5.24 other than W s (H1− ) which contains H1−

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Fig. 5.23 Geometrical structure at the death of the double-scroll: Macroscopic picture of the original system. Adapted from [Chua et al. (1986)].

is S˜b . On the other hand, and in the case of the period-doubling scenario to chaos, the two equilibrium points P − and P + in the outer regions D±1 have a 1-D stable manifold and a 2-D unstable manifold. Then if the stable eigenvalue γ˜ 1 is relatively large as in the case of Lemma 5.14, the resulting trajectories are contracted onto the 2-D eigenplane of the unstable eigenvalues, and the attractors tend to be flat in the outer regions D±1 as was observed in experiments [Madan (1993), Wu and Rul’kov (1993)]. Proof. (Theorem 5.5) First, Fig. 5.23 shows the double-scroll at the verge of colliding with the periodic orbit Γ∗ (shown dotted). Second, let Γ∗ intersect U1 at point H − in its downward swing and at point H + in its return upward swing. Note that H − must lie to the right of the line L1 because as Γ∗ moves down through H − in Fig. 5.23, it will first hit U−1 and turn around without hitting E c (P − ) and eventually hit U−1 in its upward

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Fig. 5.24 Geometrical structure at the death of the double-scroll: Enlargement of the V1 portrait of V0 . H1+ and H1− denote the position of the saddle-type periodic orbit.

′ \ ˜ ˜ \ B 1 C1 = π2 B1 A1 is tangent to E1u0 A1u0 = π 1 A1u0 E1u0 at Q1 . Sa ∪ Sb is an “infinitesimally” thin set (infinitely many layers compressed into a sheet) whose actual location is very close to A1u0 B1 . Adapted from [Chua et al. (1986)].

˜ − = −H − to the left of L ˜ 1 (odd symmetric image of swing at a point H − − − L1 ). Hence H ∈ ∠ABE. Let H1 = ψ 1 (H ) and H1+ = ψ 1 (H + ). Since H1− and H1+ are fixed points of π, one has

(5.72) H1+ = π 1 H1− = π 2 H1−

as shown in the V1 portrait of V0 in Fig. 5.24. Note that a double-snake area S1 = Sa ∪ Sb now appears in Fig. 5.24 because the double-scroll in Fig. 5.23 intersects U1 on both sides of the line L0 . The π −1 image of Sa and Sb is shown in Fig. 5.24 by another double-snake area S˜a = π −1 1 (Sa ) − and S˜b = π −1 (S ) . Now, given the coordinates of H as obtained by b 1 the shooting method, one can identify the corresponding local coordinates (u0 , v0 ) of H1− , namely H1− = x1 (u0 , v0 ) .

(5.73)

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Fig. 5.25 Geometrical structure at
the death of the double-scroll: V1 portrait of V0 for (α, β, m0 , m1 ) = 10.73, 14.3, − 71 , 27 . Adapted from [Chua et al. (1986)].

From this, one can define the local coordinates of A1u0 and E1u0 as follows:  A1u0 = u0 A1 + (1 − u0 ) B1 (5.74) E1u0 = u0 E1 + (1 − u0 ) F1 .
Since E1u0 A1u0 passes through the point H1− , π 1 E1u0 A
1u0 passes through the point H1+ as shown in Fig. 5.24. We have W s H1− ≈ E1u0 A1u0 . Now ˜ U1 denote the intersection of the double-scroll attractor with U1 and let Λ   ˜ U1 . Using the definition of the death of the doubledefine Λ1 = ψ 1 Λ \ scroll and the fact that the upper exit gate B 1 C1 approximates a portion \ of Λ1 as stated in Sec. 5.5.2, the parameter value where B 1 C1 touches
+ s ′ \ E1u0 A1u0 = π 1 A1u0 E1u0 = W (H1 ) gives an excellent approximation of the value at which the double-scroll disappears. 

Note that Fig. 5.24 shows the V1 portrait of V0 corresponding to the death of the double-scroll, and hence one can observe that the upper snake Sa must be tangent to Q1 and, correspondingly, the lower snake S˜a must be tangent to Q′1 . To show that the double-scroll would disappear if the parameter is further tuned so that Q′1 crosses the stable manifold W s (H1+ ) ≃ E1u0 A1u0 and moves below E1u0 A1u0 , we note that in this case the iterates of Q′1 under π would eventually leave the trapping region ∆ and fail to converge to an attractor within ∆.

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Fig. 5.26 A hole-filling double-scroll appears when (α, β, m0 , m1 ) = (10.73, 14.3, − 71 , 2 ): V1 portrait of V0 . Adapted from [Chua et al. (1986)]. 7

Example 5.2. A computer-calculated V1 portrait of V0 corresponding to the death of the double-scroll is shown in Fig. 5.25 where (α, β, m0 , m1 ) =
10.73, 14.3, − 71 , 27 . The point H1− is not identified in Fig. 5.25 because it is located very close to the point A1u0 . 5.4.6.3

The hole-filling double-scroll

All the double-scrolls given in [Matsumoto (1984), Zhong and Ayrom (1985)], have holes centered at P + and P − because the parameters were ˜ passes through the point D in Fig. 4.4(a) where such that no trajectory in Λ the real eigenvector E r (P + ) hits U1 . It is possible, however, to choose ˜ For example, when (α, β, m0 , m1 ) = parameters such
that D lies on Λ. 1 2 9.85, 14.3, − 7 , 7 , the corresponding V1 portrait of V0 is as shown in Fig. 5.26. Theorem 5.7. The double-scroll Eqs.(4.1)–(4.2) has a heteroclinic trajectory in one of the following cases: (a) The set Λ has a dense orbit under the discrete Poincar´e map π with a heteroclinic trajectory Υ originating from D0 in Fig. 4.4(a) which exits U1 at exactly the point D.

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Fig. 5.27 A hole-filling double-scroll appears when (α, β, m0 , m1 ) = (10.73, 14.3, − 71 , 27 ): The double-scroll with hole-filling orbits. Adapted from [Chua et al. (1986)].

Fig. 5.28 The V1 portrait of V0 which
gives rise to two odd-symmetric homoclinic orbits when (α, β, m0 , m1 ) = 4.1, 4.7, − 17 , 27 . Adapted from [Chua et al. (1986)].


\ (b) The point D1 lies on the upper exit gate C 1 B1 = π 2 A1 B1 .

′ Proof. Note that D1 = ψ 1 (D) lies on the lower exit gate C\ 1 A1∞ =
′ π 2 A1 A1∞ , and C\ 1 A1∞ converges (under π) rapidly to a point in Λ1 = Λ∪

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π 2 (Λ). It follows that one can make an infinitesimally small perturbation on β so that D1 lies on Λ1 = Λ ∪ π 2 (Λ). Suppose, in addition to D1 ∈ −1 ′ ′ C\ 1 A1∞ in Fig. 5.26, the point E1 = π 1 (B1 ) (that corresponds to the point a2 in Fig. 4.10, except that, for the parameter used in Fig. 4.10, a2 lies on the upper entrance gate) lies on the lower entrance gate A1 A1∞ in Fig. 5.26. ′ ′ \ This implies that B1′′ = π −1 1 (B1 ) lies on the lower exit gate C1 A1∞ . Now ′ assuming D1 lies between B1′′ and C1 on C\ 1 A1∞ , then the hole-filling orbit + starting from P would, after entering D0 from above, continue to move downward and eventually hit U−1 at D− = − ψ −1 1 (D1 ), where the lower r − eigenvector E (P ) intersects U−1 . By the odd symmetry of ξ, the return orbit would be a symmetric image and hence must exit U1 at D.  Such a trajectory would then follow the real eigenvector E r (P + ) and would converge rapidly toward P + . Since P + is an unstable focus when restricted to the eigenspace E c (P + ), it follows that the resulting doublescroll will not have a hole. Thus one has the following definition: Definition 5.13. The trajectory Υ is called a hole-filling orbit or heteroclinic orbit, and the chaotic attractor originating from it is called a hole-filling double-scroll. Theorem 5.8. (Chaos in the double-scroll ) The double-scroll system (4.l)– (4.2) is chaotic in the sense of the heteroclinic Sil’nikov’s Theorem for some parameters m0 , m1 , α, and β. In particular, if m0 = − 17 , m1 = 72 , and α = 9.85, then there exists a β = 14.3 such that the hypotheses of Sil’nikov’s heteroclinic theorem are satisfied. Example 5.3. The hole-filling double-scroll is shown in Fig. 5.27 for
(α, β, m0 , m1 ) = 9.85, 14.3, − 71 , 27 . The existence of such an attractor is guaranteed by Sil’nikov’s heteroclinic theorem [Sil’nikov (1965, 1970), Silva (2003)] because Sil’nikov’s Theorem also applies when the homoclinic orbit in the hypotheses is replaced by a heteroclinic orbit [Mees and Chapman, (1985), Sparrow (1981)]. Any rigorous demonstration of the existence of a heteroclinic orbit would also prove the existence of chaos in the doublescroll system (4.l)–(4.2) in the sense of Sil’nikov. Such a demonstration has been given in [Mees and Chapman (1985)], where a computer-calculated hole-filling heteroclinic orbit is shown.

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Homoclinic orbits

Theorem 4.7 gives sufficient conditions for the existence of at least one homoclinic orbit through the equilibrium point O. Example 5.4. Fig. 5.28 shows the V1 portrait on V0
associated with a homoclinic orbit, where (α, β, m0 , m1 ) = 4.1, 4.7, − 71 , 27 .

′ \ Note that the point C1 lies on E 1 A1 as required by hypothesis (i) of Theorem 4.7. Homoclinic orbits through the other two equilibrium points P + and P − can also occur under appropriate parameter values. In particular, one has the following theorem:

Theorem 5.9. (Homoclinic orbits in the double-scroll family ) Let ξ be any vector field in the double-scroll family L0 = {ξ (σ 0 , γ 0 , σ 1 , γ 1 , k) , σ 0 < 0, γ 0 > 0, σ 1 > 0, γ 1 < 0, k > 0} . (5.75) Assume ξ satisfies one of the following two conditions: ˜1′ = π −1 (B1 ) lies on the upper entrance gate A1 B1 as shown in (a) B 1 Fig. 5.26. ˜1” = π −1 (B ˜1′ ) and B1 on the upper exit gate B \ (b) D1 lies between B 1 C1 . 1 If ˜ ′ = π −1 (B1 ) lies on the lower entrance gate A1 A1∞ , (c) B 1 and \ (e) D1 lies on the upper exit gate B 1 C1 (between B1 and C1 ), then ξ has a homoclinic orbit through the two equilibrium points P + and P −. If, in addition, (e) |σ 1 | < γ 1 ,

(5.76)

then ξ is chaotic in the sense of Sil’nikov’s homoclinic theorem. Theorem 5.9 adds another way to prove rigorously the existence of chaos in the double-scroll (4.1)–(4.2) using a homoclinic orbit through the two equilibrium points P + and P − . See Sec. 4.5. 5.4.7

Bifurcation diagram

Fig. 5.29 shows the bifurcation diagram of the double-scroll Eqs.(4.1)–(4.2) obtained using the conditions derived in Sec. 5.5 and 6.3 and in Sec. 5.5 and

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6.4 for the birth and death of the double-scroll, and a detailed computer bifurcation analysis in double-precision of the αβ-plane with m0 = − 71 and m1 = 72 . Lemma 5.15. The eigenvalue at P + is pure imaginary if and only if β = (1 − m1 ) α (αm1 + 1) .

(5.77)

Eq.(5.77) determines the set of all (α, β) for which σ ˜ 1 = 0. Substituting m1 = 27 into (5.77), one can obtain curve (1) in Fig. 5.29, which is used with the Hopf bifurcation theorem7 to distinguish analytically the different dynamical behaviors of the double-scroll Eqs.(4.1)–(4.2). Now, Fig. 5.29 shows the following curves and regions: (1) The Hopf bifurcation curve (1), where it follows from the Hopf bifurcation theorem that any parameter (α, β) where P + and P − are sinks (i.e., σ ˜ 1 < 0 and γ˜ 1 < 0) lie above curve (1), and that for (α, β) in a small band to the right of this Hopf bifurcation curve, there exist nearly sinusoidal oscillations. (2) The birth boundary (2) , that is the sets of (α, β) which give rise to the birth of the double-scroll attractor. (3) The death boundary (3) , that is the sets of (α, β) which give rise to the death of the double-scroll attractor. (4) The region of period-doubling bifurcation, situated between the Hopf bifurcation curve (1) and the birth boundary curve (2). (5) The region of the R¨ ossler screw-type attractor, situated between the Hopf bifurcation curve (1) and the birth boundary curve (2). (6) The region of the double-scroll attractor, situated between the birth boundary curve (2) and the death boundary curve (3). Of course, there exist numerous periodic windows within the region between these two boundaries. 5.4.7.1

Comparison of numerical and analytical bifurcation diagrams

On the other hand, Fig. 5.30 shows the αβ bifurcation diagram derived from an approximate 1-D Poincar´e map π ∗ to be derived in Sec. 5.6. The map π ∗ yields results indistinguishable from those derived using the preceding exact analysis. By zooming into various regions of Fig. 5.30, one can find 7 The

Hopf bifurcation theorem has not been proved for PWL systems; the term is used here because the bifurcation possesses the essential features of a Hopf bifurcation.

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five regions in the αβ-plane having identical qualitative behaviors, while the αβ bifurcation diagram given in Fig. 5.29 gives only boundaries separating regions where certain qualitative behaviors could (but need not) occur. Those in Fig. 5.30 give a much finer structure showing that the chaotic region (in purple) is not a contiguous area, but rather is interspersed with countless periodic windows as predicted in Sec. 5.6. This picture also reveals a remarkable self-similar structure at different scales in the αβ-plane.

5.5 5.5.1

Method 2: One-dimensional Poincar´ e map Introduction

Chua’s circuit (4.1)–(4.2) has rich bifurcation phenomena and a very complicated structure of chaotic attractors, and its dynamics plays a crucial role in demonstrating some results of modern bifurcation theory, especially bifurcations of codimension 2, where the set of bifurcation points forms a manifold in the parameter space having a codimension of 2, where the codimension is the dimension of the parameter space minus the dimension of the manifold. Analytical results and numerical simulations [Chua et al. (1986), Genot (1993)] and experimental study [Madan (1993), Wu and Rul’kov (1993)] show that the return map π can be approximated by a 1-D map π ∗ due to the strong dissipative compression of the phase space of the double-scroll Eqs.(4.1)–(4.2). 5.5.2

Construction of the 1-D Poincar´ e map

In [Wu and Rul’kov (1993)], is a brief summary of some work dealing with the construction and the analysis of 1-D maps, numerically and theoretically from Chua’s circuit (4.1)–(4.2). Here we also summarize these earlier methods: (1) The first method given in [Chua et al. (1986), Genot (1993)] is based on the fact that the stable eigenvalue is relatively large. Hence for a suitable intersection plane in the outer region D±1 , the Poincar´e map π lives on a thin strip in the intersection plane, and it can then be approximated by collapsing the map π onto a 1-D line, which gives a 1-D map π ∗ that is useful in finding both stable and unstable periodic limit cycles, homoclinic and heteroclinic orbits, and more generally, the study of bifurcation phenomena in Chua’s system (4.1)–(4.2). This

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Fig. 5.29
(a) The exact bifurcation diagram on the αβ-plane (drawn with (m0 , m1 ) = − 17 , 27 ). Adapted from [Chua et al. (1986)].

method is explained in detail in Sec. 5.6. (2) For the simplification of the dynamics of Chua’s circuit (4.1)–(4.2), a class of 1-D maps can also be found which is similar to the 1-D map π ∗ . In [Brown (1993a)], Brown starts with the vector field of Chua’s circuit (4.1)–(4.2) and shrinks the width of the middle region D0 . When the middle region D0 has zero width, a 2-region piecewise linear (discontinuous at the plane x = 0 ) vector field η˜ is obtained and can be approximated as closely as needed by a C ∞ vector field ξ. Since the vector field ξ is odd-symmetric, the phase space is folded in half by associating the points x and −x. Then the stable eigenvalue is increased to infinity, flattening all the dynamics to the unstable eigenplane. This 2-D system then gives rise to 1-D Poincar´e maps, called Chua maps in [Brown (1993a)], at x = 0. The map has the form T ◦ F , where F is the reflection map with respect to a point on the eigenplane and T is a map which takes a point x as the initial condition and follows the trajectory of the 2-D system until it hits the line x = 0 using the intersection point as T (x). In [Misiurewicz (1993)], a subset of these maps is studied in more detail for a modified Chua’s circuit, and it was found that for certain parameter ranges, the Chua maps are unimodal with negative Schwarzian derivatives as in the logistic map. This implies that the dynamics will be similar to the logistic map. This method is not discussed here because this chapter is concerned only

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Fig. 5.30 Detailed αβ bifurcation diagram with (m0 , m1 ) = − 71 , 27 derived from the 1-D Poimcar´ e map π ∗ showing a self-similar structure, where the same pattern appears repeatedly as increasingly smaller clones arbitrarily close to the origin. The isolated thin color streak represents an artifact of the graphics software and is not therefore a part of the figure. Adapted from [Chua et al. (1986)].

with the piecewise linear Chua’s systems (4.1) with the three linear regions D±1 and D0 and the piecewise linear function h given by (4.2). In the sequel, the method given by [Chua et al. (1986), Genot (1993)] is studied in detail. The above analysis in Sec. 4.4 shows that the qualitative behavior of the double-scroll system (4.1)–(4.2) is determined essentially by the 2-D Poincar´e map π of points on an infinitesimally-narrow corridor centered along the two entrance gates A1 B1 and A1 A1∞ which correspond to the semi-infinite line L′1 ⊂ L1 , defined in Fig. 4.3 to be that part of L1

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V1 z

C1A1′∞ = π 2 (A1 A1∞) y B1C1 = π 2 (B1 A1)

D1 W1

y (u)

A1′∞

Y′ (u) C1

to N1∞

d2 N1 Y (u)

X (u)

d1

Y (0) X (0)

u=0 F1

0

θ

v = 0 E1 A1 x (u) A1∞

B1 M1 u=1

x

v=1

Fig. 5.31 Geometrical interpretation of the definition of the 1-D Poincar´ e map π ∗ : W1 plane in the D1 unit for (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.42, 0.5, 0.15, −1.5, 0.2). Adapted from [Chua et al. (1986)].

to the left of point B. Since this corridor is numerically indistinguishable from L′1 when |˜ γ 1 | is relatively large compared to the other eigenvalues, it is natural therefore to define a 1-D approximation π ∗ of the Poincar´e map π by restricting its domain to L′1 , and to compare its qualitative behavior with those of π. By brute-force computer integration of the system (4.1)–(4.2), a 1-D Poincar´e map π ∗ was constructed for many parameter values, and it predicts all of the qualitative behavior observed by computer simulation such as period-doubling, periodic windows, and by rigorous analysis such as the R¨ ossler screw-type attractors and the double-scroll. Hence a rigorous study of this map is given in this section. First, a much simpler analytical expression for π ∗ is possible if one chooses the domain of the function π ∗ to be another semi-infinite line segment P + N and its extension beyond N to N∞ at infinity as shown in Fig. 4.3. This line is constructed by connecting the point M = ψ 1−1 (1, 0, 0) and point P + by a straight line and extending it beyond N to infinity and deleting the portion P + M in Fig. 4.3. In other words, the 1-D Poincar´e map π ∗ is defined by π ∗ : P + N∞ → P + N∞ . Lemma 5.16. The conditions hold: ((a)) The spiral intersect the line L2

(5.78)

1-D Poincar´e map π ∗ is well defined if the following   −1 ′ ′ \ \ on U1 of Fig. 4.3 does not C A CA = ψ 1 ∞ 1 1∞ through points E, F , and B.

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Fig. 5.32 Geometrical interpretation of the definition of the 1-D Poincar´ e map π∗ : Graph of π ∗ for (σ 0 , γ 0 , σ1 , γ 1 , k) = (−0.42, 0.5, 0.15, −1.5, 0.2). Adapted from [Chua et al. (1986)].

((b)) The point D (where the real eigenvector hits U1 ) on U1 in Fig. 4.3 ′ . \ is located on the left-hand side of CA ∞ Proof. To prove that π ∗ in (5.78) is well defined under the above assumptions, it is more convenient to translate our analysis into the D1 unit in Fig. 4.4(b) by the coordinate transformation ψ 1 , which we redraw in Fig. 5.31. Consider the rectangular region  W1 = (x, y, z) ∈ R3 : x ≤ 0, y = 0 (5.79)

passing through the line segments ON1 and OD1 . Since O = ψ 1 (P + ), D1 = ψ 1 (D), and N1 = ψ 1 (N ), it follows that W1 corresponds to the plane W in Fig. 4.3 passing through the two line segments P + D and N D. Now, in terms of the local coordinates (u, v), points along the line B1 A1∞ are uniquely identified by a single coordinate u since v = 1 on this line. In particular, any point x(u) on this line is described by  x1 (u, 1) , 0 ≤ u ≤ 1, ifx (u) ∈ B1 A1 (5.80) x (u) = x1 (u, 1) , 1 < u < ∞, ifx (u) ∈ A1 A1∞ .

Since B1 A1∞ lies on the eigenspace ψ 1 (E c (P + )), all trajectories originating from B1 A1∞ must remain on the xy-plane in Fig. 5.31 while spiraling inwards (in backward time), and must eventually hit ON1 (on the negative x-axis) at some point a distance X(u) from O (We define X(u) as the

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Fig. 5.33 1-D Poincar´ e maps corresponding to the birth of the double-scroll when
(α, β, m0 , m1 ) = 8.8, 14.3, − 17 , 27 . Adapted from [Chua et al. (1986)].

distance from 0 since we want the domain of π ∗ to be part of the positive real axis) after a time interval π − θ, where θ = − arg x(u) = − tan−1 [xy (u)/xx (u)]. Here, xx (u) and xy (u) denote the x and y components of x(u), respectively. Clearly X (u) = |x (u)| exp (−σ 1 (π + arg (x(u)))) ≥ 0.

(5.81)

Now, if |˜ γ 1 | is relatively large, which is the case in the double-scroll, then the double spiral on W1 in reality is squeezed into a thin line sitting infinitesimally close to N1∞ O. Consequently, for all computational purposes, one can approximate Y ′ (u) as the point Y (u) on N1∞ O. Note that Y (u) is a positive real number given by Y (u) = |y (u)| exp (σ 1 (π − arg (y(u)))) ≥ 0, 0 ≤ u ≤ ∞. +

(5.82)

Since u = u (1, t) for 0 ≤ u ≤ 1 is given explicitly by (4.62) and since u = u− (1, t) for 1 < u < ∞ is given explicitly by (4.65), one can specify the graph of the Poincar´e map π ∗ for X(u) > X(0) by the following explicit parametric equations:  (X (u+ (1, t)) , Y (u+ (1, t))) , 0 ≤ t < ∞, 0 ≤ u < 1 (X (u) , Y (u)) = (X (u− (1, t)) , Y (u− (1, t))) , 0 < t < ∞, 1 < u < ∞. (5.83) Assumption (a) given by Lemma 5.16 is equivalent to the condition that ′ the lower exit gate C\ 1 A1∞ does not touch or intersect the line through

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Fig. 5.34 1-D Poincar´ e maps corresponding to the death of the double-scroll when
(α, β, m0 , m1 ) = 10.73.14.3, − 71 , 27 . Adapted from [Chua et al. (1986)].

Fig. 5.35 1-D Poincar´ e maps corresponding to a hole-filling double-scroll when
(α, β, m0 , m1 ) = 9.85, 14.3, − 17 , 27 . Adapted from [Chua et al. (1986)].

B1 , F1 , E1 in Fig. 5.31. It follows from the analysis of Figs. 4.6, 4.7, and 4.8 that both inverse-return functions u+ (1, t) in (4.62) and u− (1, t) in (4.65) are strictly monotone functions and hence have a unique inverse. Hence any point X(u) ≥ 0 on N1 X (0) maps uniquely into a point x(u) on B1 A1∞ by

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Fig. 5.36 1-D Poincar´ e maps corresponding to the existence of two odd-symmetric
homoclinic orbits when (α, β, m0 , m1 ) = 4.1, 4.7, − 71 , 27 . Adapted from [Chua et al. (1986)].

the flow ϕt1 , where X(0) is the limiting point which maps (under ϕt1 ) into B1 . Note that any point d1 between X(0) and O in Fig. 5.31 must map (under ϕt1 ) into a point d2 where d2 = e2πσ 1 d1

(5.84)

because the expanding logarithmic spiral from
d1 cannot touch B1 A1∞ . \ The upper exit gate B and the lower exit gate 1 C1 = π 2 B1 A1
\ C1 A1∞ = π 2 A1 A1∞ are shown in Fig. 5.31. Note that each point x(u) on B1 A1∞ maps under π 2 uniquely into a point y(u) with coordinates (yx (u), yy (u), yz (u)). Assumption (b) given by Lemma 5.16 is equivalent to the condition that the point D1 in Fig. 5.31 is located below (relative to ′ V1 -plane) the lower exit gate C\ 1 A1∞ . It follows from this condition that the t flow ϕ1 from y(u) must intersect the W1 rectangle at Y ′ (u). This translates d and into Fig. 4.3 to mean that trajectories starting from the exit gates BC −1 ′ \ CA∞ will always intersect the plane W = ψ 1 (W1 ). Hence the exit gates ′ \ \ B 1 C1 and C1 A1∞ in Fig. 5.31 must map into another double spiral on W1 as shown in Fig. 5.31, where each point y(u) maps into   (5.85) Y ′ (u) = − |y (u)| e[σ1 (π−arg(y(u)))] , 0, yz (u) e[γ 1 (π−arg(y(u)))] Eq.(5.83) defines the 1-D Poincar´e map π ∗ for all X(u) between X(0) and N1∞ . For points X(u) between X(0) and O, where u < 0 (for convenience

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we extend our local coordinate u > 0 to include negative u in order to parameterize the points between X(0) and O), we simply make use of (5.84), namely, Y (u) = e2πσ 1 X (u) , u < 0. Hence the proof is completed.

(5.86) 

Definition 5.14. The formulas (5.78), (5.83), and (5.84) are called the one-dimensional double-scroll Poincar´e map . Example 5.5. A typical graph of π ∗ corresponding to the normalized eigenvalue parameters (α, β, m0 , m1 ) = (−0.42, 0.50, 0.15, −1.5, 0.20) is shown in Fig. 5.33. Some things to note are the following: (a) Because σ 1 is a constant, the graph from X = O to X = X(0) is always a straight line with a slope equal to e2πσ1 . (b) The one-dimensional Poincar´e map π ∗ given in Definition 5.14 is valid not only for system (4.1)–(4.2), but also for the entire double-scroll family £0 . (c) In the V1 portrait of V0 in Fig. 5.31, the 1-D double-scroll Poincar´e map π ∗ is defined by (5.87) π ˜ ∗ : P + N∞ → P + N∞ ,   ´ onto P + π ´ , and it is a continuous ˜∗ B and it is a linear map from P + B

´ ∞ onto P + N∞ , where the point B ′ on P + N is nonlinear map from BN identified with the point X(0).

Several different graphs of the 1-D double-scroll Poincar´e map π ∗ , which illustrate the various qualitative behaviors analyzed in Sec. 5.5.6 are shown in Figs. 5.33, 5.34, 5.35, and 5.36. Note also that some of these figures were confirmed experimentally in [Madan (1993), Wu an Rul’kov (1993)]. 5.5.3

Properties of the 1-D Poincar´ e map π∗

In this section, some universal properties of the 1-D Poincar´e map π ∗ is presented and discussed, where the notation π ∗ means any variant of the 1-D Poincar´e map that has been studied in the literature.

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Feigenbaum’s number

Two period-doubling bifurcation trees are obtained: One is derived by a brute force method (integrating (4.1)–(4.2) numerically), and the other is derived using the 1-D Poincar´e map π ∗ . Note that they agree qualitatively. Hence, for computational efficiency, the 1-D map π ∗ was used to compute the associated Feigenbaum number [Cvitanovic (1984)]. The result was 4.6933.

5.5.3.2

Bimodality and self-similarity (vector scaling)

In [Cvitanovic (1984)], a renormalization group analysis was used to explain the universality of the period-doubling bifurcation in the logistic map with one parameter. To study the bifurcation phenomena in a map with more than one extremum, it is necessary to consider at least two bifurcation parameters. In [Kuznetsov et al. (1993b)], the 1-D map π ∗ with two bifurcation parameters is bimodal (having two local maxima) in the α′ βplane, where α′ = α − 0.68β. The critical points of doubly superstable period-2n cycles that form bifurcation points of codimension 2 are found as follows: Period 2 4 8 16 32

α 2.428631394 3.181220121 3.385570546 3.420523839 3.425635489

β 2.668405542 3.745645702 4.056225192 4.110073977 4.117966453

(5.88)

and so on.... The accumulation point of the period-doubling cascades for Chua’s map π ∗ is α = αc = 6.5408510. On the other hand, a twoparameter renormalization group analysis is performed, which results in universal properties describing the self-similarity of the bifurcation curves in the two-parameter plane. Experimental studies show also self-similarity in the 1-D map π ∗ in the two-parameter plane [Wu and Rul’kov (1993)]. See [MacKay and van Zeijts (1988), Kuznetsov et al. (1993a, 1993b)] for more details. Similar work following this approach for a modified Chua’s circuit can be found in [Misiurewicz (1992)]. The resulting Poincar´e map has the same properties as in the case of the 1-D Chua map π ∗ .

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Numerical examples for the 1-D Poincar´ e map π ∗ 1 -D Poincar´e map π ∗ for birth of the double-scroll


The graph of π ∗ for the parameters (α, β, m0 , m1 ) = 8.8, 14.3, − 71 , 27 is shown in Fig. 5.32, and it represents the birth of the double-scroll, because the maximum value of Y on the interval [0, X(1)] is equal to X(1), i.e., the point Y (u0 ) = max0≤u≤1 Y (u) coincides with the point X(1). Hence π ∗ (X(u0 )) = X(1) maps precisely through point A1 where u = 1. All other trajectories have Y (u) < X(1) and hence can only enter D0 through the upper gate B1 A1 . For this case, there are three families of secondary homoclinic orbits merging with the double-scroll and connecting unstable and stable manifolds of O [Kahan and Sicardi-Schifino (1998)]. Each branch of the n-H8 homoclinic family is associated with two wiggles of period-n orbits, one for the asymmetric and one for the symmetric cycles. Windows in this case explain the sequence of alternating chaotic and periodic behaviors of the system (4.1)–(4.2). Also, there are other secondary homoclinic orbits of the Sil’nikov type of higher order accumulated at some point on the BDS line. The method of analysis is the construction of a 1-D Poincar´e map just like π ∗ and defining some reference point H for both the R¨ ossler-type and the double-scroll attractors. 5.5.4.2

1-D Poincar´e map π ∗ for death of the double-scroll


The graph of π ∗ for the parameters (α, β, m0 , m1 ) = 10.73, 14.3, − 71 , 27 is shown in Fig. 5.34, and it represents the death of the double-scroll because XH (since XH > X(1), XH corresponds to u > 1) is an unstable fixed point of π ∗ and the maximum value max Y (u) on the interval [0, X(1)] is equal to XH , and this situation corresponds to the case where the unstable (saddle-type) periodic orbit through XH collides with the double-scroll. 5.5.4.3

1-D Poincar´e map π ∗ for a hole-filling orbit


The graph of π ∗ for the parameters (α, β, m0 , m1 ) = 9.85, 14.3, − 71 , 27 is shown in Fig. 5.35, and it represents the occurrence of the hole-filling orbit because on the interval [X(1), ∞], the minimum value of Y (u) is zero, ˜ is a double-scroll. Now and max0≤u≤1 Y (u) > X(1), and the attractor Λ max0≤u≤1 Y (u) = 0 implies that the spiral through Y ′ (u) associated with 8 An n-H orbit is an orbit spawned by the point H, for which the symmetrical flow winds n times around one of the nontrivial equilibrium points P ± and injects exactly at the trivial equilibrium point O on the stable manifold [Freire et al. (1993)].

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this point is tangent to the z-axis. This situation corresponds to the case ′ in Fig. 4.3 passes through D. \ where CA ∞ 5.5.4.4

1-D Poincar´e map π ∗ for homoclinic orbit


The graph of π ∗ for the parameters (α, β, m0 , m1 ) = 4.1, 4.7, − 71 , 27 is shown in Fig. 5.36, and it predicts the existence of the homoclinic orbit because X(1) is a fixed point and hence Y (1) = π ∗ (X(1)) = X(1). Since u = 1 at the point A1 , this implies that the trajectory originating from X(1) would enter D0 through A1 on the stable eigenspace through O and hence converge to O. This trajectory continues along the unstable eigenvector through O until it hits U1 at C, which is identified with C1 in Fig. 5.31. Since Y (1) = X(1), the trajectory continuing from C1 must intersect W1 at a point Y ′ (1) whose projection Y (1) is precisely equal to X(1). Hence this trajectory is a homoclinic orbit of the origin. 5.5.5

Periodic points of the 1-D Poincar´ e map π ∗

In this section, the correspondence between the periodic points of the 1-D Poincar´e map π ∗ and the periodic orbits in the double-scroll system (4.1)– (4.2) is described. Lemma 5.17. If |γ 1 | is relatively large compared to the other eigenvalues and Λ is infinitesimally thin, then each periodic orbit of the double-scroll system has at least one stable direction (i.e., the magnitude of at least one characteristic exponent is less than one). Proof. Note that the 1-D Poincar´e map π ∗ gives an excellent approximation (Lemma 5.16) under the mentioned condition.  In particular, a stable periodic point of π ∗ corresponds to a stable periodic orbit, and an unstable periodic point of π ∗ corresponds to a saddle-type periodic orbit of the double-scroll (4.1)– (4.2). Since Ymax = max0≤u≤1 Y (u) corresponds to the out˜ ermost orbit of Λ, if the period-n points {X = (π ∗ )n (X), ∗ ∗ n−1 π (X), ..., (π ) (X)} satisfy (π ∗ )i (X) ≤ Ymax , 0 ≤ i ≤ n − 1,

(5.89)

then the periodic orbit of the double-scroll system (4.1)–(4.2) corresponding ˜ Define to X is located on the attractor Λ. a = X (1) .

(5.90)

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As shown later, the type of periodic orbit of the double-scroll system is determined by the position of the point a relative to the periodic points of π∗ . 5.5.5.1

Fixed points of the map π ∗

Definition 5.15. The point X1 is said to be a fixed point for the map π ∗ if X1 = π ∗ (X1 ). Case (i): 0 < X1 < a. Fig.5.37(a) shows a fixed point X1 of π ∗ with X1 = X(u) for some 0 < u < 1. The corresponding period-1 orbit in the double-scroll system is depicted in Fig. 5.37(b). The trajectory originating from X1 would enter D0 through a point on the upper entrance gate AB, return to D1 , and hit X1 . By symmetry, one has a pair of periodic orbits as shown in Fig. 5.37(b). The essential features of this situation are summarized in the abstract sketch ´ 1 = −X1 . shown in Fig. 5.37(c), where N − = −N , a ´ = −X(1), and X Case (ii): a < X1 < ∞. Fig.5.38(a) shows a fixed point of π ∗ with X1 = X(u) for some u > 1. The trajectory originating from X1 would enter D0 through a point on the lower entrance gate AA∞ and continue its downward motion until it hits X1′ = −X1 . Therefore, one has a period-1 orbit as shown in the abstract sketch in Fig. 5.38(b). 5.5.5.2

Period-2 points of π ∗

Definition 5.16. The points X1 and X2 are said to be period-2 points of π ∗ if X1 = π ∗ (X2 ) and X2 = π ∗ (X1 ). Case (i): 0 < X1 < X2 < a. Two period-2 points X1 and X2 satisfying (i) are shown in Fig. 5.39(a). The trajectory originating from X1 would enter D0 through the upper entrance gate, return to D1 , and hit X2 . The trajectory continuing from X2 would enter D0 again through the upper entrance gate and eventually return to X1 . Therefore, one has a pair of period-2 orbits as depicted in Fig. 5.39(b). Case (ii): a < X1 < X2 Two period-2 points satisfying (ii) are shown in Fig. 5.40(a). The trajectory originating from X1 would enter D0 through the lower entrance gate, continue its downward motion through D0 , and hit X2′ = −X2 .

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Y

a X1

0

X (1)

X

(a) N

N

a X1

a = X (1) X1

P− D1

U1 A

P+

E F B O

D0

B− F− E−

A− P−

U−1 P− D−1

a′

X1′

X1′ = −X1 a′ = −X (1)

N− N−

(b)

(c)

Fig. 5.37 Fixed point X1 of π ∗ with 0 < X1 < a. (a) Graph of the 1-D Poincar´ e map π ∗ . (b) Corresponding periodic orbits in the original double-scroll system (4.1)–(4.2). (c) Abstraction of the main features of (b). Adapted from [Chua et al. (1986)].

Some things to note are the following: (a) The points X1 and X1′ (resp. X2 and X2′ ) of the double-scroll system (4.1)–(4.2) are identified as one point X1 (resp. X2 ) in the graph of π ∗ . (b) The trajectory continuing from X2′ would enter and continue its upward motion through D0 before returning to X1 . Therefore, one has a pair of period-1 orbits as depicted in Fig. 5.40(b), even though π ∗ in Fig. 5.40(a) seems to suggest that one has a period-2 orbit. Hence the period-doubling of a fixed point X of π ∗ with a < X < ∞ (as in Fig. 5.38(a) and Fig. 5.40(a)) corresponds to the splitting of the single odd-symmetric period-1 orbit in Fig. 5.38(b) into two period-1 orbits in Fig. 5.40(b).

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N x1

Y

a P+ x1

a P− a′ 0

X X (1) (a)

x1′

N− (b)

Fig. 5.38 One period-1 fixed point X1 of π ∗ with a < X1 < ∞. (a) Graph of the 1-D Poincar´ e map π ∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)].

(c) Each of the orbits in Fig. 5.40(b) is not odd symmetric, but the two orbits are odd-symmetric images of each other in view of the symmetry of the vector field ξ. The orbit in Fig. 5.38(b) exists by itself because it already exhibits odd symmetry. Case (iii): X1 < a < X2 . Two period-2 points satisfying (iii) are shown in Fig. 5.41(a). This situation corresponds to a period-3 orbit in the double-scroll system as depicted in Fig. 5.41(b), because the trajectory originating from X1 would enter D0 through the upper entrance gate, return to D1 , and hit X2 . The trajectory continuing from X2 would then enter D0 through the lower entrance gate, pass D0 , and hit X1′ = −X1 . Finally, the portion of the trajectory from X1′ to X1 must be symmetric with the portion of the trajectory from X1 to X1′ with respect to the origin. 5.5.5.3

Period-n point of the map π ∗

Definition 5.17. The points X1 , X2 , X3 , ..., Xn are said to be a period-n points of π ∗ if X2 = π ∗ (Xn−1 ), ..., i.e., the set X =

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N

a x2

a x2

P−

x1

x1

x1′

P+

0

a ′ x2′

X

X (1)

N−

(a)

(b)

Fig. 5.39 Two period-2 points X1 and X2 with 0 < X1 < X2 < a. (a) Graph of the 1-D Poincar´ e map π ∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)]. N x2 x1 a Y

P+ x2

a

x1

P− a′ x1′ x2′ 0

X X (1)

N−

(a)

(b)

Fig. 5.40 Two period-2 points X1 and X2 with a < X1 < X2 < ∞. (a) Graph of the 1-D Poincar´ e map π∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)].

n o n−1 (π ∗ )n (X), π ∗ (X), ..., (π ∗ ) (X) .

Let the above period-n points be ordered according to 0 < X1 < X2 < X3 < ... < Xn < ∞ (5.91) where one can assume X = X1 without loss of generality. Then the type of period-n orbit of π ∗ is uniquely characterized by a permutation of the indices (2, 3, ..., n) following the index 1. For example, the permutation

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N X2 Y

a X1

X2

P+

a

P−

X1

a′

0

X X (1)

(a)

X1′

X2′

N−

(b)

Fig. 5.41 Two period-2 points X1 and X2 with X1 < a < X2 < ∞. (a) Graph of the 1-D Poincar´ e map π∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)].

(1, 4, 2, 3, 5) corresponds to the following periodic points: 0 < X1 → X4 → X2 → X3 → X5 → X1 < ∞. (5.92) The type of periodic orbit of the double-scroll system is therefore determined by the position of the symbols (a, 0, X1 < X2 < X3 < ... < Xn , ∞}, where along the half line P + N , the point P + may be O, and N may be ∞. Hence Theorem 5.10. The total number NT of distinct types of periodic orbits of the double-scroll system (4.1)–(4.2) is equal to (n + 1)! NT = (n − 1)! × (n + 1) = (5.93) n Example 5.6. In the case of n = 3, we have eight different types of periodic orbits in the double-scroll system. Figs. 5.42(b) and 5.43(b) show two periodic  orbits corresponding to the following two dynamic routes: 0 < X1 < X2 < a < X3 < ∞, X1 → X2 → X3 → X1 (5.94) 0 < X1 < a < X2 < X3 < ∞, X1 → X2 → X3 → X1 . Another method was given in [Chua and Huynh (1992)] where the explicit solution of Chua’s system (4.1)–(4.2) was used to predict possible regions for the period-doubling bifurcations as follows: If  m0   m1 − 1, x > 1 (5.95) d= 0, |x| ≤ 1   1 − m0 , x < −1 m1

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and λ1 , λ2 , λ3 are the eigenvalues of the Jacobian matrix given in (5.4), then we have the following lemma: Lemma 5.18. The explicit solution of Chua’s system (4.1)–(4.2) is given by 
1 P3  λ2i + λi + β eλi t + d  x (t) = − β i=1 ci P 3 y (t) = − β1 i=1 λi ci eλi t  P3  z (t) = i=1 ci eλi t − d

(5.96)

where    z (0) + d c1  c2  = M   −βy (0) c3 −β [x (0) − y (0) + z (0)] 

(5.97)

and 

 M =

λ2 λ3 (λ2 −λ1 )(λ3 −λ1 ) −λ1 λ3 (λ1 −λ2 )(λ2 −λ3 ) λ1 λ2 (λ1 −λ3 )(λ2 −λ3 )

−λ2 −λ3 − (λ2 −λ 1 )(λ3 −λ1 ) λ1 +λ3 (λ1 −λ2 )(λ2 −λ3 ) −λ1 −λ2 (λ1 −λ3 )(λ2 −λ3 )

1 (λ2 −λ1 )(λ3 −λ1 ) −1 (λ1 −λ2 )(λ2 −λ3 ) 1 (λ1 −λ3 )(λ2 −λ3 )



 .

(5.98)

A computer program was developed based on Lemma 5.18 to find the trajectory of Chua’s circuit (4.1)–(4.2). All the pattern combinations take the form (code for O, code for P ± ), where R stands for a real eigenvalue, C stands for the real part of the complex conjugate eigenvalues, and O stands for 0. The letters are arranged in increasing order, and a bar on top is used to denote the one with the larger magnitude when R and C are on opposite sides of O. The coded patterns and their analytical conditions and possible eigenvalue pattern combinations are given in (5.99) and in Fig. 5.44, respectively. The diagram of the eigenvalue patterns can be divided into two groups: m0 < −1 in region O, and −1 < m1 < 0 in region P ± , because the diagram for m0 = − 87 gives a qualitative picture for all m0 < −1 in the sense of the resemblance in connections and positions of the curves. The same observation applies

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for m1 = − 75 and −1 < m1 < 0. The diagram is shown in Fig. 5.45.   RCO ⇐⇒ γ < σ < 0     CRO ⇐⇒ σ < γ < 0       COR ⇐⇒ σ < 0 < γ, |σ| > |γ|     COR ⇐⇒ σ < 0 < γ, |σ| < |γ|    OCR ⇐⇒ 0 < σ < γ    ORC ⇐⇒ 0 < γ < σ (5.99)  ROC ⇐⇒ γ < 0 < σ, |σ| > |γ|     ROC ⇐⇒ γ < 0 < σ, |σ| < |γ|     RRRO ⇐⇒ γ 1 < γ 2 < γ 3 < 0      RROR ⇐⇒ γ 1 < γ 2 < 0 < γ 3     RORR ⇐⇒ γ 1 < 0 < γ 2 < γ 3    ORRR ⇐⇒ 0 < γ 1 < γ 2 < γ 3
For (β, m0 , m1 ) = 16, − 78 , − 57 , the values of α at which period-doubling occurs are given by the following: Onset of period-α at onset 2 4 8 16 32 64 128 256

Ratio of difference 8.855726163 9.1080893023 9.1591511652 9.16997924215 9.17229337816 9.17278877717 9.172894866343 9.172917586935

4.94230 4.71569 4.67910 4.67126 4.66965 4.66930

(5.100)

and hence are used to verify the Feigenbaum number as in Sec. 5.6.3. The characteristic multipliers introduced in Sec. 1.3.1 were used to predict some routes to chaos, namely, period-doubling and quasi-period bifurcations corresponding to the birth of a torus and its breakdown to chaos [Duchesne (1993)]. 5.5.5.4

Localization of limit cycles

In [Krishchenko and Shalneva (1999)], a general method for estimation of domains with limit cycles and finding surfaces with the traces of all cycles was proposed. Indeed, consider the dynamical system x′ = f (x)

(5.101)

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N x3 a x2

Y x1

P+ x3 a P−

x2

a′

x1

0

X

x1′ x2′ x3′

X (1)

N−

(a)

(b)

Fig. 5.42 Three period-3 points X1 , X2 , and X3 with 0 < X1 < X2 < a < X3 < ∞. (a) Graph of the 1-D Poincar´ e map π ∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)].

N x2 x1 a

Y

P+ x2

a

x1

P− a′ x1′

0

x2′

X X (1)

N−

(a)

(b)

Fig. 5.43 Three period-3 points X1 , X2 , and X3 with 0 < X1 < a < X2 < X3 < ∞. (a) Graph of the 1-D Poincar´ e map π ∗ . (b) Abstraction of the corresponding periodic orbits in the original double-scroll system. Adapted from [Chua et al. (1986)].

T

where f (x) = (f1 (x) , ..., fn (x)) and the function ϕ ∈ C 1 (Rn ). Then the derivative of ϕ with respect to system (5.101) is given by Lf ϕ (x) =

n X i=1

fi (x)

∂ϕ (x) , ∂xi

(5.102)

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Fig. 5.44 Possible eigenvalue pattern combinations. Y: yes, blank: no. Adapted from [Chua and Huynh (1992)].

and let us define the following sets:   Sϕ = {x : Lf ϕ (x) = 0}      ϕsup = sup {ϕ (x) : x ∈ Sϕ } ϕinf= inf {ϕ (x) : x ∈ Sϕ }   ˜  Ωϕ = x : ϕinf ≤ ϕ (x) ≤ ϕsup    ˜ = ∩ϕ∈C 1 (Rn ) Ω ˜ ϕ. Ω

(5.103)

Then the following theorem was proved in [Krishchenko (1995, 1997)]: Theorem 5.11. (a) For any function ϕ ∈ C 1 (Rn ) , any cycles of system (5.101) contain at least two points of the set Sϕ . (b) For any function ϕ ∈ C 1 (Rn ) , all limit cycles of system (5.101) ˜ ϕ. belong to the set Ω ˜ (c) All limit cycles of system (5.101) belong to the set Ω. Now, if x, b, c ∈ Rn and A, A0 are two n×n matrices, then let us consider the piecewise linear system given by:   Ax + b, x1 ≤ −1 x′ = (5.104) A0 x, |x1 | ≤ 1  Ax + c, x1 ≥ 1.

By a linear transformation, system (5.104) can be put in the form z ′ = A′ z + T −1 c, then if Re λk 6= 0 for any eigenvalue λk of the matrix A, there exists a function ϕ (z) =

n X j=1

dj zj2

(5.105)

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Fig. 5.45 Eigenvalue pattern diagrams. Left column: m0 < −1 (O region). Right column: −1 < m1 < 0 (P ± region). 1: ORRR, 2:RROR, 3: RRRO. Adapted from [Chua and Huynh (1992)].

where dj 6= 0, |dj | > qj , such that Sϕ is an ellipsoid Elc for the system x′ = Ax + c and an ellipsoid Elb for the system x′ = Ax + b. For the system x′ = A0 x, the surface Cϕ = z : z T W z = 0 6= ∅ is a cone. Hence the following sets are defined by  Sϕ = (Elb ∩ {z : x1 (z) ≤ −1}) ∪ (Cϕ ∩ {z : |x1 (z)| < 1}) ∪ S (5.106) S = Elb ∩ {z : x1 (z) ≥ 1} Then the following result was proved in [Krishchenko and Shalneva (1999)]: Theorem 5.12. For the piecewise linear system (5.104), if (1) the system (5.104) is a continuous one,

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(2) Re λk 6= 0 for any eigenvalue λk of the matrix A, and (3) the intersection of the n-dimensional ellipsoid Elc and the hyperplane x1 (z) = 1 is a (n − 1)-dimensional ellipsoid, then the following hold: (a) The surface Sϕ given by (5.106) is a compact set. ˜ = Ωϕ ∩ Ωϕ′ containing all limit cycles (b) There exists a compact set Ω of the system (5.104) where ϕ and ϕ′ are different functions of the form (5.105). The application of Theorem 5.12 to Chua’s system (4.1)–(4.2) with α = 1 2 9, β = 100 7 , m0 = − 7 , and m1 = 7 , gives the following result [Krishchenko and Shalneva (1999)]: Theorem 5.13. For Chua’s system (4.1)–(4.2), there exists a compact set ˜ such that all limit cycles of the system belong to Ω, ˜ where Ω ˜ = Ωϕ ∪ Ωϕ′ Ω and  ϕ (x, y, z) = 3x2 + 133.71y 2 + 13.7z 2 + 34xy + 14xz + 2yz    ′ ϕ (x, y, z) = 5x2 + 115.71y 2 + 12.44z 2 + 34xy + 14xz + 2yz (5.107) Ωϕ = {(x, y, z) : −6.15 ≤ ϕ (x, y, z) ≤ 34338.32}    Ωϕ′ = {(x, y, z) : −7.30 ≤ ϕ′ (x, y, z) ≤ 30279.66} .

The study of the onset of a symmetric periodic oscillations in Chua’s oscillator (4.1)–(4.2) was given in [Carmona et al. (2005)], with a rigorous mathematical result that justifies the appearance of limit cycles in 3-D continuous piecewise linear systems. The following result about the bifurcation of limit cycles in Chua’s circuit (4.1)–(4.2) was proved and resolved an undesirable situation discussed in [Chua et al. (1993)], i.e., that the Poincar´e Andronov–Hopf bifurcation cannot be applied strictly to piecewise linear systems like Chua’s circuit (4.1)–(4.2), since smoothness of the state equations is required: Theorem 5.14. Assume that 

α>1 α 1 − 2α < m1 < 21 − 2α . Then an orbitally asymptotically stable limit cycle bifurcates for 1 2

β < −α2 m21 + α (α − 1) m1 + α. 5.5.5.5

(5.108)

(5.109)

Structure and order of the appearance of periodic orbits

The structure and the order of appearance of periodic orbits in the 1-D map π ∗ was studied in [Chua and Tichonicky (1991)] using an algorithm

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based on kneading theory.9 With this information, the structure and the period of the corresponding orbits of Chua’s system (4.1)–(4.2) was found. First, the observed bifurcations in system (4.1)–(4.2) are a codimension 1 bifurcation. Hence the 1-D map π ∗ is regular in the sense that the only bifurcations of periodic points are of the period-doubling and saddle-focus types. For α ∈ [7, 13] and β = 15, m0 = −0.14, m1 = 0.28, and if π ∗α is the corresponding 1-D map π ∗ for each value of α, then Fig. 5.46(a) shows a typical 1-D map π ∗α for α = 9.1073. In theory, the 1-D map π ∗α has an infinite number of extrema in a neighborhood of the point a of Fig. 5.46(a) appearing alternatively as follows: x0 is the origin and x1 = xmax1 , x2 = xmin1 , x3 = xmax2 , etc. In Fig. 5.46(b), there are ten extrema, five maxima and five minima. The interval of x between xj−1 and xj is called the j th interval. For example, the point a = 1.5089 is located in the fifth interval. For small values of α, the image of the first maximum π ∗α (xmax1 ) is less than the abscissa of the first minimum xmin1 . Therefore, if we choose the interval (0, xmin1 ) as the domain of the map π ∗α , then the image of each point in this interval will remain in it. In this case, the map π ∗α is unimodal. While if α increases, then π ∗α (xmax1 ) > xmin1 and more extrema are included in the domain of π ∗α , and thus it is not unimodal. Indeed, for α1 = 9.1073, one has π ∗α (xmax1 ) = xmin1 . Therefore, when α ≤ α1 , the map π ∗α is unimodal. In this case, the kneading invariant v (π ∗α ) is defined by the semi-infinite ∞ sequence v (π ∗α ) = (vn )n=0 = limx→x+ θ (x) , where max 1

 n+1 is increasing (orientation preserving) near x  +1, if (π ∗α ) m θ (x) = 0, if (π ∗α ) = 0 for some m with 0 ≤ m ≤ n  n+1 is decreasing (orientation reversing) near x. −1, if (π ∗α ) (5.110) On the other hand, any unimodal map has an attracting periodic orbit d 3 (π ∗ α)

if and only if v (π ∗α ) is periodic under the condition10 that (πdx∗3)′ − 32 α  ∗ 2 (π α )” < 0 [Holmes and Whitley (1984)]. For almost all cases, the ′ (π ∗ α) Schwarzian derivative is negative. When it is positive, then it is possible to apply kneading theory to order the set of kneading invariants. Note that each element of (vn )∞ n=0 is either +1 or −1 or 0 denoted by +, −, 0, respectively. For simplicity, we use the following notation: If (δ) is a finite ′ block of “−”’s and “+”’s, then (δ) is the sequence obtained by iterating (δ) 9 Kneading theory is an elaboration of symbolic dynamics [Guckenheimer and Holmes (1983)]. It gives information about the orbit of critical points of 1-D maps. 10 This is the map’s Schwarzian derivative.

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indefinitely, and δ is the sequence obtained by changing the sign of all el′ ements in (δ) . For example, (δ) = (+ + − + + − + + −...) = (+ + −) and
′ ∗ δ = (− − +) . The function α → v (π α ) is decreasing. When α ≤ α1 and if v (π ∗α ) has a n-periodic orbit, the map π ∗α has a stable n-periodic point n ′ x∗ and 0 ≤ ((π ∗α ) ) (x∗ ) ≤ 1, while if v (π ∗α ) has an n-antiperiodic orbit, n ′ then π ∗α has a stable n-periodic point and −1 ≤ ((π ∗α ) ) (x∗ ) < 0, or a ′  2n 2n-periodic point and 0 ≤ (π ∗α ) (x∗ ) ≤ 1. In order to search the bifurcation points starting from small value of α, i.e., α = 7, note that the birth ′ of a period-1 point at α = 7.4, the kneading invariant v = (+) does not change until the fixed point crosses the first maximum, and then v becomes ′ (+−) . As α increases, the stable periodic point becomes unstable and gives birth to two period-2 (one stable and one unstable) periodic points in a period-doubling bifurcation. In this case v = (+−)′ until one of the period2 points crosses a critical point, and v becomes (+ − −+)′ . The first four elements of the set of the kneading invariant are: (+)′ , (+−)′ , (+ − −+)′ , and (+ − − + − + +−)′ . If v (π ∗α ) is an admissible kneading invariant, then ′ one has the same type of behavior for all periodic invariants. Then v∗(+−) is the next lower admissible kneading invariant. The operation ∗ is de∞ fined as follows: for any sequence v = (vn )n=0 of period-m and a sequence ∞ ∞ γ = (γ n )n=0 , v ∗ γ = (τ n )n=0 where τ k×m+i = γ k vi , 0 ≤ i ≤ m. For ex′ ample, suppose v = (v0 v1 v2 ) with m = 3 and γ = (γ 0 γ 1 γ 2 γ 3 ...). Then we have τ = (τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 ...) = (γ 0 v0 γ 0 v1 γ 0 v2 γ 1 v0 γ 1 v1 γ 1 v2 ...) . Now if α > α1 , then the map π ∗α is multimodal, and hence the invariant interval has several minima and maxima. In this case, several coexisting stable periodic points of different periods are observed. For example when α = 9.337, there are two sets of stable periodic points, one of period-2 and the other of period-3. In this case, the dynamical representation of the first and fifth maximum are (+ − −)′ and (++)′ , respectively. To determine the period of the kneading invariant, the generalized itinerary µ (x) was used for each extremum. The results obtained are similar to those for the unimodal case, i.e., when a stable periodic point x∗ is found, then one of the maxima has  ′

2n

′

a periodic kneading invariant (δ) and 0 < (π ∗α ) (x∗ ) ≤ 1. When α ′  increases, (π ∗α )2n (x∗ ) will decreases towards 0. Hence (δ)′ is constant ′ 
′ 2n until (π ∗α ) (x∗ ) = 0, where it changes to δδ . As α is increased ′  further, (π ∗α )2n (x∗ ) reaches −1, and the periodic point become unsta-

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Fig. 5.46 (a) A typical 1-D map π ∗α for α = 9.1073. x1 is the first maximum xmax1 , x8 is the fourth maximum xmax4 , and xu is the most distant unstable fixed point of π ∗9.1073 . The point a is a critical point whose neighborhood contains infinitely many extrema in theory. (b) Magnification of the part around the point a located between x4 and x5 in (a). Adapted from [Chua and Tichonicky (1991)].

ble and gives birth to a stable 2n-periodic orbit. In this case, the same
′ maximum has a kneading invariant δδ , and the above behavior repeats itself. Therefore, for any periodic kneading invariant of any maximum, the next periodic kneading invariant of the same maximum is vi ∗ (+−)′ , and it is more difficult to classify the kneading invariants because the function α → v (π ∗α ) is no longer decreasing. For example, for α = 9.0765, ′ ′ v1 (π ∗α ) = (+ − − − −) , and for α = 9.1213, v1 (π ∗α ) = (+ + + + +) . As α increases, one can observe new stable periodic orbits always covering more and more intervals. Hence the kneading invariant changes since it is based on the itinerary of the stable periodic point.

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Using the property that the function α → µ (i) is increasing, one has that if αb is a bifurcation point and that there is a new stable periodic′ k point, then the kneading invariant (δ) is periodic with period-k, and the itinerary A (i) is periodic with the same period, and the generalized itinerary µ (i) is periodic with period-k or 2k. The sequence changes when k the sign of (π ∗α ) (xmax1 ) passes from + to −. The next sequence µ′ (i) has the same first terms as µ (i) and µ′k+1 (i) = µk+1 (i) + 1. For period2, we have α = 9.333, µ (7) = 7171..., α = 9.335, µ (7) = 718 − 1 − 818.... Continuing with this logic, the bifurcation points of the map π ∗α for α ∈ [7, 13] are shown in Table 3 of [Chua and Tichonicky (1991)]. For each stable periodic point x of the map π ∗α , the corresponding periodic trajectory in Chua’s system (4.1)–(4.2) was obtained using the inverse linear transformation x′′−1 x, where Q = (ea , eb , ec ) and ea , eb are the real and negative imaginary parts of the complex eigenvector corresponding to σ ˜± jω ˜ , eb and ec is the eigenvector corresponding to γ˜ . This operation permit one to transform the point x back into the original coordinate system and to observe the corresponding trajectory of (4.1)–(4.2) starting from this point. 5.5.6

Bifurcation diagrams using confinors theory

For drawing the bifurcation diagram of Chua’s system (4.1)–(4.2) for a wide range of the parameter values, several results were obtained using the 1-D Poincar´e map π ∗ [Chua et al. (1986), Yang and Liao (1987), Kahlert (1991), Broucke (1987), Lozi and Ushiki (1993)]. For example, in [Lozi and Ushiki (1993)], the so-called confinors theory introduced in Sec. 5.5.3 was used with the four finite numbers τ min , τ max , θmin , and θmax given in Theorem 5.4 to calculate both the images π 0 (M ) and π 1 (M ) of the point M ∈ △AEB. This method is used to understand the very precise structure of the attractors belonging to the main confinor and to obtain very accurate bifurcation diagrams for the double-scroll family. To obtain π 0 (M ) , we have to find the smallest t > 0 solution of the transcendental equation: eσ 0 t (x′0 cos t − y0′ sin t) + z0′ eγ 0 t = 1.

(5.111)

Generally, it is not possible to solve (5.111) explicitly for τ (x′0 , y0′ , z0′ ) or for τ 0 (x′0 , y0′ , z0′ ) . The method proposed in [Lozi and Ushiki (1993)] is based on the bisection method , knowing appropriate bounds for the initial values

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of τ , with these bounds coming from the knowledge of τ min , τ max , θmin , and θmax . When the image π 1 (M ) belongs to a small part of the snake bounded by the continuation of the isochronic lines corresponding to τ min and τ max , the bisection method fails in finding the solution of (5.111). Let M = (x′0 , y0′ , z0′ ) ∈ △AEB, and let H0 (t) = eσ0 t (x′0 cos t − y0′ sin t) + z0′ eγ 0 t − 1

(5.112)

be the transcendental equation to be solved with respect to τ . We want to find τ (M ) = min {t : H0 (t) = 0} . t>0

(5.113)

Then one has the following results proved in [Lozi and Ushiki (1993)]: Theorem 5.15. Suppose that we know 0 < ts < te such that H0 (ts ) < 0andH0 (te ) > 0

(5.114)

and that H0 (t) = 0 has only one solution satisfying 0 < t < te . In this case, the bisection method can be applied to find τ (M ) with ts < τ (M ) < te . Lemma 5.19. For each initial point M ∈ Int (△AEB) , the equation H0 (t) = 0 has only one solution in 0 < t < ∞. Proof. Use the fact that each orbit starting from M intersects with S0 = {(ξ, η) : ξ + η ≥ 1, ξ > 0, η ≥ 0} exactly once.  As a result, if one takes ts > 0 sufficiently small and te > 0 sufficiently large, then the bisection method works without a problem. The values τ min and τ max given in Theorem 5.4 serve as ts and te , respectively. The bisection method might be quite inefficient if we were to verify the a priori estimate ts and te for each iteration step of the half-Poincar´e mappings. Rather, we would like to have these values appropriate for every initial point M to be treated. Or at least we would like to find the a priori bounds ts and te with a small cost of computation. To solve this problem, consider the “inverse” problem, i.e., for given values ts and te with 0 < ts < te , find the region, ˜ of initial points M ∈ △AEB, for which the bisection method can be say ∆ applied to find π 0 (M ) . Let ∆t denote the component of △AEB − N0 G0 ˘ = ∆te − ∆ts . Then we have containing B0 as shown in Fig. 5.47(a), and ∆ the following results [Lozi and Ushiki (1993)]: Proposition 5.1. (a) If M ∈ ∆t , then τ (M ) < t. (b) For M ∈ △AEB, if M ∈ ∆t , then H0 (t) > 0, if M ∈ N0 G0 , then H0 (t) = 0, if M ∈

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η φ 0 (E0) = φ1 (E1)

te ℑ

ts F

∆

φ 0 (A0) = φ1 (A1)

π 0 (∆)

N0

ξ φ 0 (B0) = φ1 (B1)

G0 (a)

S

ℑ

(b)

˘ for (σ 0 , γ 0 , σ 1 , γ 1 , k) = Fig. 5.47 (a) The isochronic lines of π 0 and the region ∆ (0.1, −0.2, −0.86, 0.3921, 0.1034) . (b) The isochronic lines for π 0 when π 0 is not continuous for (σ 0 , γ 0 , σ 1 , γ 1 , k) = (0.1, −0.2, −0.86, 0.3921, 0.1034) . Adapted from [Lozi and Ushiki (1993)].


˘ then the bisection △AEB − ∆t ∪ N0 G0 , then H0 (t) < 0. (c) If M ∈ ∆, method can be applied to solve the equation H0 (t) = 0 with an initial guess t s < t < te . The same steps can be reconsidered for the image π 1 (M ) with the equation H1 (t) = eσ1 t (x0 cos t − y0 sin t) + z0 eγ 1 t − 1. (5.115) Let ℑ (t) denote the open half-plane of V1 which contains D1 for each t > 0 as shown in Fig. 5.48. Also, let ℑ = ℑ (ts ) − ℑ (te ) ∩ As. Then one obtains the following result [Lozi and Ushiki (1993)]: Proposition 5.2. (a) M is in ℑ (t) if and only if H1 (t) < 0. (b) If M ∈ ℑ, then the bisection method can be applied to find a solution

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ts

η φ 0 (E0) = φ1 (E1)

ℑ N1

ℑ

π 1 (ℑ)

φ 0 (A0) = φ1 (A1) Fig. 5.48 (1993)].

te G1 As

ξ

φ 0 (B0) = φ1 (B1)

The isochronic lines for π 1 and the region ℑ. Adapted from [Lozi and Ushiki

−0.01

1.01

−0.95

−0.55 γ 1

ξ

Fig. 5.49 Bifurcation diagram of Chua’s equation (4.1)-(4.2) for σ 0 = −0.325, γ 0 = 0.135, σ 1 = 0.3921, k = 0.1034. Adapted from [Lozi and Ushiki (1993)].

of H1 (t) the a priori guess ts < t < te . (c) If  = 0 starting from  M ∈ ℑ ∪ts
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Fig. 5.50 Magnification of Fig. 5.49 in the region −0.87 ≤ γ 1 ≤ −0.80, −0.01 ≤ ξ ≤ 1.01. Adapted from [Lozi and Ushiki (1993)].

Fig. 5.51 Magnification of Fig. 5.50 in the region −0.86 ≤ γ 1 ≤ −0.83, 0.65 ≤ ξ ≤ 0.80 showing period-doubling bifurcation and inverse cascade of bands of chaotic attractors with some anomalies. Adapted from [Lozi and Ushiki (1993)].

system (4.1)–(4.2) for the parameter values: (σ 0 , γ 0 , σ 1 , k) = (−0.325, 0.135, 0.3921, 0.1034)

(5.116)

with varying γ 1 . For each value of γ 1 , we take an initial point in △AEB and iterate the Poincar´e map π for this point to plot the ξ coordinate of

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Fig. 5.52 Magnification of Fig. 5.51 in the region −0.84 ≤ γ 1 ≤ −0.83, 0.70 ≤ ξ ≤ 0.80. Adapted from [Lozi and Ushiki (1993)].

Fig. 5.53 Magnification of Fig. 5.52 in the region −0.8375 ≤ γ 1 ≤ −0.8345, 0.70 ≤ ξ ≤ 0.75. Adapted from [Lozi and Ushiki (1993)].

the points of the orbit. 5.6

Exercises

(1) Prove Lemmas 5.1 and 5.2.

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Fig. 5.54 Magnification of Fig. 5.53 in the region −0.8364 ≤ γ 1 ≤ −0.83615, 0.72 ≤ ξ ≤ 0.74. Adapted from [Lozi and Ushiki (1993)].

Fig. 5.55 Magnification of Fig. 5.54 in the region −0.83633 ≤ γ 1 ≤ −0.83628, 0.725 ≤ ξ ≤ 0.735. Adapted from [Lozi and Ushiki (1993)].

(2) Prove Theorem 5.1. (3) (a) Transform the inequalities (5.13) and (5.15) to the usual αβ-plane. (b) Redraw Figs. 5.2 and 5.3 in the usual αβ-plane. (4) If (σ 0 , γ 0 , σ 1 , γ 1 , k) = (−0.721, 1.075, 0.074, −1.600, 0.530), then show that the local coordinates (u, v) are u = l along A1 E1 and u = 1.53

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(5) (6) (7) (8) (9) (10)

(11) (12) (13)

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along A1u E1u , respectively. Prove Lemma 5.5. Prove Lemma 5.10. Prove relations (5.56) and (5.57). Prove Lemmas 5.11 and 5.12. Prove relation (5.77). Use the Feigenbaum number properties to calculate the coordinates of points in the αβ-plane of the double superstable cycles of periods 64, 128, 256 and 512 according to the values given in (5.88). Prove Theorem 5.10. Prove Lemmas 5.18 and 5.19. (a) Using the notations λ1 = γ, λ2 = σ + iω, λ3 = σ − iω for the case of one real eigenvalues and two conjugate complex eigenvalues of Chua’s system (4.1)–(4.2), find the boundary in the αβ-plane for the following cases: σ = 0, γ = 0, γ = σ, γ = −σ, and ω = 0. (b) Using the notations λ1 = γ 1 , λ2 = γ 2 , λ3 = γ 3 for the case of three real eigenvalues, find the boundary in the αβ-plane for the following cases: γ 1 = 0, γ 2 = 0, and γ 3 = 0. Prove the relations given in (5.99). Compare the two approaches used in this chapter to calculate the Feigenbaum number. Prove Theorems 5.14 and 5.15. Prove Propositions 5.1 and 5.2.

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Pivka, L, Wu, C-W. and Huang, A. (1994). Chua’s oscillator: a compendium of chaotic phenomena. J. Franklin Instit., 331B, 6, pp. 705-741. Pivka, L., Wu, C-W. and Anshan, H. (1996). Lorenz equation and Chua’s equation. Inter. J. Bifur.Chaos., 6, 12B, pp. 2443-2489. Pastore, S. (1999). Detection of all the equilibrium points of dynamic 1-D autonomous rings. Proc. of the European Conference on Circuit Theory and Design. ECCTD’99. Politecnico di Torino. Part., 2, pp. 699-702. Phillipson, P.E. and Schuster, P. (2000). Bifurcation dynamics of threedimensional systems. Inter. J. Bifur.Chaos., 10, 8, pp. 1787-1804. Prokopenko, V.G. (2000). Expanding the Chua’s-circuit family. Radiotekhnika i Elektronika., 45, 10, pp. 1241-1244. Philominathan, P. and Neelamegam, P. (2001). Characterization of chaotic attractors at bifurcations in Murali-Lakshmanan-Chua’s circuit and one-way coupled map lattice system. Chaos Solitons & Fractals., 12, 6, pp. 10051017. Ramdani, S., Chua, L.O., Lozi, R. and Rossetto, B. (1999). A qualitative study comparing Chua and Lorenz systems. Proc.s of the 7th Inter. Specialist Workshop on Nonlinear Dynamics of Electronic Systems. Tech. Univ. Denmark. pp. 205-208. Rand, D. (1978). The topological classification of Lorenz attractors. Math. Proc.Camb. Phil. Soc., 83, pp. 451-460. Robinson, C. (1989). Homoclinic bifurcation to a transitive attractor of Lorenz type. Nonlinearity., 2, pp. 495-518. Robinson, C. (2004). Dynamical systems: stability, symbolic dynamics, and chaos. CRC Press. Rˆ ossler, O. E. (1979). Continuous chaos-Four prototype equations. Ann. N.Y. Acad. Sci., 31, pp. 376-392. Rossetto, B. (1993). Chua’s circuit as a slow-fast autonomous dynamical system. J. Circuits Systems & Computers., 3, 2, pp. 483-496. Russell, D. A., Hanson, J. D. and Ott, E. (1980). Dimension of Strange Attractors. Phys. Rev. Let., 45, pp. 1175-1178. Rul’kov, N. and Volkovskii. A. R.(1993). Experimental analysis of 1-D map from Chua’s circuit. In Chua’s Circuit: A Paradigm for Chaos (R. N. Madan, Ed.). Singapore: World Scientific, pp. 580-590. Rychlik, M. (1989). Lorenz attractors through a Sil’nikov-type bifurcation. Part 1. Ergod. Th. & Dynam. Sys.,10, pp. 793-821. S´ anchez, L. A. Convergence in a Chua’s system with three equilibria. Zeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP)., 55, 2, 2004, 183-200. Sancho, S., Suarez, A. and Roman, P.S. (2001). Floquet analysis of the intermittence route to chaos through a pitchfork bifurcation. IEEE Trans. Circuits Syst.-I: Fund. Th. Appl., 48, 3, pp. 374-377. Sarafian, G. and Kaplan, B.Z. (1995). Is the Colpitts oscillator a relative of Chua’s circuit?. IEEE Trans. Circuits Syst.-I: Fund. Th. Appl., 42, 6, pp. 373-376. Silva, C. P. and Chua, L. O. (1988). The overdamped double-scroll family, Part I: Piecewise-linear geometry and normal form. Inl. J. on Ckl. Th. and Appl.,16, pp. 233-302.

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Silva. C. P. (1991). Analytical study of the double-hook attracto. Circuits and Syst, Proc. 34th Midwest Symposium on, 2, 764-771. Silva, C. P. (2003). Shi’lnikov theorem -a tutorial. IEEE Trans. Circuits Syst.-I .,40, pp. 675-682. Sharkovsky, A.N. and Chua, L.O. (1993). Chaos in some 1-D discontinuous maps that appear in the analysis of electrical circuits. IEEE Trans. Circuits Syst.I: Fund. Th. Appl., 40,10, pp. 722-731. Shi’lnikov, L. P. (1965). A case of the existence of a countable number of periodic motions. Sov. Math. Docklady., 6, pp. 163-166 (translated by S. Puckette). Shi’lnikov, L. P . (1970). A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type. Math. U.S.S.R.Shornik., 10, pp. 91-102 (translated by F. A. Cezus). Shil’nikov, L. P. (1993a). Strange attractors and dynamical models. J. Circuits Syst. Comput., 3, 1, pp. 1-10. Shil’nikov, L.P. (1993b). Chua’s circuit: rigorous results and future problems. IEEE Trans. Circuits Syst.-I: Fund. Th. Appl., 40, 10, pp. 784-786. Shil’nikov, L.P. (1994). Chua’s circuit: rigorous results and future problems. Inter. J. Bifur.Chaos., 4, 3, pp. 489-519. Smillie, J. (1997). Complex dynamics in several variables. In Flavors of geometry., 31 of Math. Sci. Res. Inst. Publ., pp. 117-150. Cambridge University Press, Cambridge, 1997. With notes by Gregery, T. Buzzard. Smale, S. (1965). Diffeomorphisms with many periodic points,In Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, 63-70, S.S. Cairns (ed).. Princeton University Press, Princeton, NJ. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math.l Soc., 73, pp. 747-817. Smale, S. (1998). Mathematical Problems for the Next Century. Math. Intelligencer., 20, 2, pp. 7-15. Smale, S. (2000). Mathematical Problems for the Next Century. Mathematics: Frontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000. Spany, V. and Pivka, L. (1990). Boundary surfaces in sequential circuits. Inl. J. on Ckl. Th. and Appl., 18, 4, pp. 349-360. Spany, V. and Pivka, L. (1991). Invariant manifolds of sequential circuits. Elektrotechnicky Casopis., 42, 6, pp. 281-293. Spany, V. and Pivka, L. (1993). Two-segment bistability and basin structure in three-segment PWL circuits. IEE Proc.s-G Circuits Devices & Systems., 140, 1, pp. 61-67. Sparrow, C. (1982). The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer-Verlag, New York. Sprott, J. C. (1993a). Automatic generation of strange attractors. Comput. & Graphics., 17,3 , pp. 325-332. Sprott, J. C. (1993b). Strange Attractors: Creating Patterns in Chaos. M&T Books, New York. Sprott, J. C. (1993c). How common is chaos?. Phys. Lett. A., 173, pp. 21-24. Sprott, J. C. (1994a). Some simple chaotic flows. Phys. Rev. E., 50,2 , pp. R647-

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Index

2-D quadratic maps, 34

Borel transform, 81 bounded nonperiodic orbits, 199 bounded orbits, 105 bounds, 38 box, 12 breakdown of an invariant torus, 161 bubbles, 98

accessible periodic orbits, 74 accessible saddles, 74 adaptive branch-and-bound subdivision, 30 admissible transitions, 12 affinely conjugate, 52 Anosov diffeomorphism, 20 anti-integrable limit, 94 anticonfinors, 252 aperiodic points, 33 appearance of periodic orbits, 303 asymptotically stable, 108 attractive cycle, 101 attribute of the double-scroll attractor, 227 axiomatic definition, 14

canonical piecewise linear equation, 159 Cantor set, 21 Cantor-like set, 22 Cardan method, 120 catastrophes, 155 chaos detector, 46 chaos in the double-scroll, 198, 278 chaotic in the sense of Smale, 23 chaotic orbit, 7 chaotic scattering, 75 chaotic torus magnetic field line caused by the perturbation coil, 9 characteristic multipliers, 8 children, 96 Chua systems, 34 Chua’s circuits, 6 Chua’s diode, 160 co-existence of attractors, 74 codimension, 281 coexistence of three chaotic attractors, 253 coexisting phenomena, 74 complex susceptibility, 103

basic reversor, 61 basin boundaries, 99 basins of attraction, 99 bifurcation, 73 bifurcation diagram of the double-scroll, 279 bifurcation diagrams using confinors theory, 307 bimodal, 102 binary horseshoe, 94 birth boundary, 280 birth of the double-scroll, 268 bisection method, 307 337

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computer-assisted proof, 1 confinor (main), 253 confinors, 252 Conley index theory, 93 connecting orbit shadowing theorem, 36 connection, 24 connection map, 192 connection with respect to, 25 continuous homotopies, 15 continuous piecewise linear vector field, 8 correlation dimension, 49, 63 countable set of horseshoes, 27 Cremona map, 59 crises of chaotic attractors, 74 critical threshold, 74 cross number, 24 cubic polynomial, 160 D unit, 168 death boundary, 280 death of the double-scroll, 270 delayed logistic map, 112 density of periodic points, 34 deterministic, 45 diffeomorphism, 7 discontinuous function, 160 dissipative, 99 double-hook family, 219 double-horseshoe, 227 double-scroll attractor, 6 double-scroll family, 175 double-snake area, 274 downward component, 177 dual double-hook family, 220 dual double-scroll family, 219 eigenvalue, 163 enclosure for the Jacobian matrix, 12 entry and exit sets, 75 equivalence, 50 ergodic basin, 4 expanding fixed point, 28 explicit solution of Chua’s system, 298

exponentially small phenomena, 80 extrapolation method, 147 f-connected family with respect to, 25 fan-like region, 190, 248 Feigenbaum’s number, 290 first intersection, 80 first recurrence map, 8 fixed point, 7 fixed point index, 14 Floquet multiplier, 96 flow, 7 flow is expansive, 4 focus-focus equilibrium, 77 foliation, 101 forward orbit, 101 four-color theorem, 1 fractal, 99 fractal invariant set, 21 Friedland–Milnor classification theorem, 60 Frobenius norm, 45 fundamental points, 167 fundamental segment, 97 Gaussian algorithm, 10 generalization of Chua’s circuit, 160 generalized bisection technique, 12 generalized dimensions, 63 generalized fractal dimensions, 63 generalized H´enon mappings, 60 generalized Poincar´e map, 8 geometric Lorenz flows, 4 geometric models of Lorenz system, 4 geometric structure, 111 global graph transformation, 80 global Newton’s method, 81 graph representation, 12 H´enon map, 26, 49 half-return maps, 175 Hamiltonian system, 75 Hammerstein model, 42 Hausdorff dimension, 5 Heaviside function, 46 heteroclinic orbit, 2, 26

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339

Index

heteroclinic saddle-node bifurcation, 98 hole filling heteroclinic orbit, 210 hole-filling double-scroll, 276 homeomorphism, 18 homoclinic bifurcations, 21 homoclinic classes, 6 homoclinic orbit, 2, 26 homoclinic orbit shadowing theorem, 37 homoclinic orbits in the double-scroll family, 197, 279 homoclinic tangency, 32, 93 homology theory, 93 Hopf bifurcation curve, 280 Hopf bifurcation theorem, 280 Hopf bifurcations, 96 horseshoe chaos, 27 hyperbolic, 2 hyperbolic chaotic attractor, 3 hyperbolic saddle focus, 26 hyperbolicity-type condition, 37 hyperchaos, 38 inclusion mapping, 30 infinite wedge, 179 infinitesimally perturbed, 27 infinitesimally-narrow corridor, 283 infinitesimally-thin gate, 267 initial conditions, 99 intermittency, 161 interval analysis, 199 interval arithmetic, 11 interval matrix, 10 interval Newton operator, 10 interval Newton’s method, 10 interval vector, 11 invariant foliation, 5 invariant splitting, 34 invariant under, 17 inverse half-return map, 192 inverse return time functions, 182 invertible, 54 isochronic lines, 259 isolated islands, 155 isolating neighborhood, 18

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Jacobian conjecture, 59 Jacobian matrix, 8 Jordan canonical form, 166 Kaplan–Yorke conjecture, 146 Kaplan–Yorke dimension, 63 kneading invariant, 304 landing phenomenon, 74 Lebesgue (volume) measure, 4 level of analysis, 255 Li–York lemma, 26 limit cycle, 7 limiting set, 227 linear conjugacy criteria, 171 linear equivalence, 171 linear equivalence criteria, 172 linearly conjugate equivalence classes, 173 local product structure, 34 localization of limit cycles, 299 locus, 202 logical operations, 230 Lohner method, 11 Lorenz system, 4 Lorenz-manuscript, 5 Lorenz-type systems, 2 low-order polynomial maps, 49 lower entrance gate, 267 lower exist gate, 267 lower exit gate, 267 Lyapunov exponent, 5, 38 manifold, 33 Mann–Whitney rank-sum, 46 Markov chain, 227 Marotto theorem, 28 maximal invariant part, 15 maximal transitive sets, 5 MEan SquAred error histogram, 47 mean value form, 13 MESAH, 47 minimal system, 144 multistability, 99 multistructural, 162

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new stable periodic orbits, 306 Newhouse regions, 162 noisy parametric sweep through a period-doubling bifurcation, 75 nondeterministic, 45 nonparametric test, 45 nonprime singular values, 45 nonwandering point, 20 nonwandering set, 20 nonzero quadratic terms, 50 normal form equation, 173 normal form theory, 4 normal vector, 181 normalized eigenvalue parameters, 172 number of peaks, 256 number of turns, 215 numeric operators, 11 one-dimensional double-scroll Poincar´e map, 289 one-dimensional Poincar´e map, 281 onset of a symmetric periodic oscillations in Chua’s oscillator, 303 parallel computation, 9 parallel lines, 263 parent, 96 partially hyperbolic, 5 pattern, 255 pattern combinations, 298 period-doubling bifurcation, 96 periodic orbit, 7 periodic orbit of the double-scroll system, 292 periodic point, 7 periodic windows, 284 phase shift, 225 physical (or Sinai-Ruelle-Bowen) measure, 4 piecewise linear vector field family, 164 pitchfork bifurcation, 96 PL-system, 222 Poincar´e Andronov–Hopf bifurcation cannot be applied, 303

Poincar´e map, 7 Poincar´e map for a hole-filling orbit, 291 Poincar´e map for birth of the double-scroll, 291 Poincar´e map for death of the double-scroll, 291 Poincar´e map for homoclinic orbit, 292 Poincar´e section, 7 polynomial inverse, 60 polynomial map, 59 positive semi-orbit of Chua’s system, 241 preconnection, 24 priori bounds, 308 proof, 1 pseudo-orbit, 34 pull-up map, 246 quadratic area-preserving mapping, 60 quadratic nonlinearities, 50 quadratic planar map with a constant determinant, 58 quadratic symplectic mappings, 60 quasi-attractors, 2 quasi-period bifurcations, 299 quasi-periodic orbits, 102 quasi-periodic route to chaos, 145 quasiperiodic orbits, 9 R¨ ossler screw-type attractor, 244 rate of area contraction, 63 real Jacobian conjecture, 60 region of period-doubling bifurcation, 280 region of the double-scroll attractor, 280 region of the R¨ ossler screw-type attractor, 280 resonance zones, 228 return/transfer/escape regions, 219 rigorous numerics, 4 rigorous proof, 1 robust attractor, 4

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341

Index

robust transitive sets with singularities, 5 rotary homoclinic tangencies, 74 rotation numbers, 74 rotational bifurcation, 96 rounding errors, 33 Routh–Hurwitz stable, 243 saddle fixed point, 111 saddle-focus homoclinic loops, 6 saddle-node bifurcation, 95 scaled distributions, 45 scheme of symbolic dynamics, 255 scrambled set, 28 second half-return map, 177 second-derivative test, 38 segment boundaries, 263 self-similar structure, 281 semi-conjugacy, 14 separatrix loop, 209 separatrix value, 225 sequence of global period-doubling bifurcations, 74 shadowing lemma, 33 shadowing orbit, 34 shadowing techniques, 77 shift automorphism, 33 shift map, 17 shrimp-shaped robust isoperiodic domains, 102 Sil’nikov boundary problem, 223 Sil’nikov chaos, 26 Sil’nikov criterion, 26 Sil’nikov map, 27 Sil’nikov snake, 226 singular value decomposition, 44 singular-hyperbolic attractor, 4 Smale horseshoe, 4 Smale’s axiom A, 2 Smale–Moser theorem, 76 Smale–Williams attractor, 3 small nonautonomous perturbations, 5 small perturbations, 4 smooth Chua’s systems, 159 snake-like, 226

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snake-like area, 247 snap-back repellor, 28 spiral image property, 196 spiral-type strange attractors, 228 splitting constant, 81 stability sectors, 243 stable manifold, 9 standard fixed-size box-counting algorithms, 63 standing assumption, 200 stochastic counterpart, 43 stopping procedure, 9 straight line tangency property, 166 strange attractor, 2, 4 strategic points, 170 structurally stable, 2 structurally unstable homoclinic orbits, 6 subdivision algorithm, 93 subharmonic solution, 7 surrogate series, 45 symbolic dynamics, 21 symbolic space, 23 Takens’ theorem, 146 tightly-wound spirals, 265 topological basin of attraction, 4 topological degree, 9 topological entropy, 65 topological horseshoe, 77 topological invariant, 5 topologically equivalent, 50 total energy, 45 transformation, 53 transition matrix, 17 transition time, 97 transversal homoclinic orbit, 29 transverse homoclinic point, 21, 76 trapping region, 12 true hyperbolic periodic orbits, 37 true transversal connecting orbit, 37 TS-map, 16 unbounded orbits, 105 uniform hyperbolicity, 37 unimodal, 282

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2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach

unstable fixed point, 109 unstable periodic orbits, 5 unusual bifurcation phenomena, 74 upper entrance gate, 267 upper exit gate, 267 upward component, 178 V1 portrait of V0, 195 variational equation, 12

vector scaling, 290 verified optimization technique, 30 vertices, 12 weighted sum coordinate, 166 Wiener model, 39 winding number, 96 windows, 161 wrapping effect, 11

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