2-d Quadratic Maps And 3-d Ode Systems: A Rigorous Approach

by Zeraoulia Elhadj, Julien Clinton Sprott

Overview

This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hénon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters.Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hénon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincaré map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincaré mapping in addition to other analytical methods.

Product Details

ISBN-13:	9789814307741
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/09/2010
Series:	World Scientific Series On Nonlinear Science Series A , #73
Pages:	356
Product dimensions:	6.10(w) x 9.10(h) x 1.00(d)

Table of Contents

Preface vii

Acknowledgements xiii

1 Tools for the rigorous proof of chaos and bifurcations 1

1.1 Introduction 1

1.2 A chain of rigorous proof of chaos 3

1.3 Poincaré map technique 7

1.3.1 Characteristic multiplier 7

1.3.2 The generalized Poincaré map 8

1.3.3 Interval methods 10

1.3.4 Mean value form 13

1.4 The method of fixed point index 14

1.4.1 Periodic points of the TS-map 16

1.4.2 Existence of semiconjugacy 17

1.5 Smale's horseshoe map 19

1.5.1 Some Basic properties of Smale's horseshoe map 20

1.5.2 Dynamics of the horseshoe map 22

1.5.3 Symbolic dynamics 23

1.6 The Sil'nikov criterion for the existence of chaos 26

1.6.1 Sil'nikov criterion for smooth systems 26

1.6.2 Sil'nikov criterion for continuous piecewise linear systems 27

1.7 The Marotto theorem 28

1.8 The verified optimization technique 30

1.8.1 The checking routine algorithm 30

1.8.2 Efficacy of the checking routine algorithm 31

1.9 Shadowing lemma 33

1.9.1 Shadowing lemmas for ODE systems and discrete mappings 35

1.9.2 Homoclinic orbit shadowing 36

1.10 Method based on the second-derivative test and bounds for Lyapunov exponents 38

1.11 The Wiener and Hammerstein cascade models 39

1.11.1 Algorithm based on the Wiener model 39

1.11.2 Algorithm based on the Hammerstein model 42

1.12 Methods based on time series analysis 43

1.13 A new chaos detector 46

1.14 Exercises 47

2 2-D quadratic maps: The invertible case 49

2.1 Introduction 49

2.2 Equivalences in the general 2-D quadratic maps 50

2.3 Invertibility of the map 59

2.4 The Hénon map 63

2.5 Methods for locating chaotic regions in the Hénon map 64

2.5.1 Finding Smale's horseshoe maps 64

2.5.2 Topological entropy 65

2.5.3 The verified optimization technique 68

2.5.4 The Wiener and Hammerstein cascade models 69

2.5.5 Methods based on time series analysis 70

2.5.6 The validated shadowing 71

2.5.7 The method of fixed point index 72

2.5.8 A new chaos detector 72

2.6 Bifurcation analysis 73

2.6.1 Existence and bifurcations of periodic orbits 73

2.6.2 Recent bifurcation phenomena 74

2.6.3 Existence of transversal homoclinic points 76

2.6.4 Classification of homoclinic bifurcations 94

2.6.5 Basins of attraction 99

2.6.6 Structure of the parameter space 100

2.7 Exercises 103

3 Classification of chaotic orbits of the general 2-D quadratic map 105

3.1 Analytical prediction of system orbits 105

3.1.1 Existence of unbounded orbits 105

3.1.2 Existence of bounded orbits 107

3.2 A zone of possible chaotic orbits 109

3.2.1 Zones of stable fixed points 111

3.3 Boundary between different attractors 112

3.4 Finding chaotic and nonchaotic attractors 123

3.5 Finding hyperchaotic attractors 131

3.6 Some criteria for finding chaotic orbits 139

3.7 2-D quadratic maps with one nonlinearity 140

3.8 2-D quadratic maps with two nonlinearities 148

3.9 2-D quadratic maps with three nonlinearities 149

3.10 2-D quadratic maps with four nonlinearities 151

3.11 2-D quadratic maps with five nonlinearities 153

3.12 2-D quadratic maps with six nonlinearities 153

3.13 Numerical analysis 154

3.13.1 Some observed catastrophic solutions in the dynamics of the map 155

4 Rigorous proof of chaos in the double-scroll system 159

4.1 Introduction 159

4.2 Piecewise linear geometry and its real Jordan form 164

4.2.1 Geometry of a piecewise linear vector field in R³ 164

4.2.2 Straight line tangency property 166

4.2.3 The real Jordan form 168

4.2.4 Canonical piecewise linear normal form 171

4.2.5 Poincaré and half-return maps 175

4.3 The dynamics of an orbit in the double-scroll 176

4.3.1 The half-return map π₀ 177

4.3.2 Half-return map π₁ 185

4.3.3 Connection map Π 192

4.4 Poincaré map π 194

4.4.1 V₁ portrait of V₀ 195

4.4.2 Spiral image property 196

4.5 Method 1: Sil'nikov criteria 197

4.5.1 Homoclinic orbits 197

4.5.2 Examination of the loci of points 202

4.5.3 Heteroclinic orbits 210

4.5.4 Geometrical explanation 214

4.5.5 Dynamics near homoclinic and heteroclinic orbits 215

4.6 Subfamilies of the double-scroll family 219

4.7 The geometric model 220

4.8 Method 2: The computer-assisted proof 229

4.8.1 Estimating topological entropy 230

4.8.2 Formula for the topological entropy in terms of the Poincaré map 236

4.9 Exercises 238

5 Rigorous analysis of bifurcation phenomena 239

5.1 Introduction 239

5.2 Asymptotic stability of equilibria 240

5.3 Types of chaotic attractors in the double-scroll 244

5.4 Method 1: Rigorous mathematical analysis 245

5.4.1 The pull-up map 246

5.4.2 Construction of the trapping region for the double-scroll 247

5.4.3 Finding trapping regions using confinors theory 252

5.4.4 Construction of the trapping region for the Rössler-type attractor 257

5.4.5 Macroscopic structure of an attractor for the double-scroll system 265

5.4.6 Collision process 268

5.4.7 Bifurcation diagram 279

5.5 Method 2: One-dimensional Poincaré map 281

5.5.1 Introduction 281

5.5.2 Construction of the 1-D Poincaré map 281

5.5.3 Properties of the 1-D Poincaré map π* 289

5.5.4 Numerical examples for the 1-D Poincaré map π* 291

5.5.5 Periodic points of the 1-D Poincaré map π* 292

5.5.6 Bifurcation diagrams using confinors theory 307

5.6 Exercises 312

Bibliography 315

Index 337