Robust Chaos and Its Applications

by Elhadj, Zeraoulia; Sprott, Julien Clinton

• ISBN13: 9789814374071
• ISBN10: 9814374075
• Format: Hardcover
• Copyright: 2011-10-17
• Publisher: World Scientific Pub Co Inc

Summary

Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhoods in the parameter space of a dynamical system. This unique book explores the definition, sources, and roles of robust chaos. The book is written in a reasonably self-contained manner and aims to provide students and researchers with the necessary understanding of the subject. Most of the known results, experiments, and conjectures about chaos in general and about robust chaos in particular are collected here in a pedagogical form. Many examples of dynamical systems, ranging from purely mathematical to natural and social processes displaying robust chaos, are discussed in detail. At the end of each chapter is a set of exercises and open problems intended to reinforce the ideas and provide additional experiences for both readers and researchers in nonlinear science in general, and chaos theory in particular.

Table of Contents

Preface	p. vii
Poincaré Map Technique, Smale Horseshoe, and Symbolic Dynamics	p. 1
Poincaré and generalized Poincaré mappings	p. 1
Interval methods for calculating Poincaré mappings	p. 4
Existence of periodic orbits	p. 6
Interval arithmetic	p. 7
Smale horseshoe	p. 10
Dynamics of the horseshoe map	p. 12
Symbolic dynamics	p. 17
The method of fixed point index	p. 19
Exercises	p. 24
Robustness of Chaos	p. 27
Strange attractors	p. 27
Concepts and definitions	p. 27
Robust chaos	p. 29
Domains of attraction	p. 32
Density and robustness of chaos	p. 34
Persistence and robustness of chaos	p. 35
Exercises	p. 36
Statistical Properties of Chaotic Attractors	p. 39
Entropies	p. 39
Lebesgue (volume) measure	p. 39
Physical (or Sinai-Ruelle-Bowen) measure	p. 40
Hausdorff dimension	p. 42
The topological entropy	p. 43
Lyapunov exponent	p. 47
Ergodic theory	p. 50
Statistical properties of chaotic attractors	p. 51
Autocorrelation function (ACF)	p. 51
Correlations	p. 53
Exercises	p. 57
Structural Stability	p. 61
The concept of structural stability	p. 61
Conditions for structural stability	p. 62
A proof of Anosov's theorem on structural stability of diffeomorphisms	p. 64
Exercises	p. 77
Transversality, Invariant Foliation, and the Shadowing Lemma	p. 80
Transversality	p. 80
Invariant foliation	p. 82
Shadowing lemma	p. 84
Homoclinic orbits and shadowing	p. 88
Shilnikov criterion for the existence of chaos	p. 90
Exercises	p. 93
Chaotic Attractors with Hyperbolic Structure	p. 95
Hyperbolic dynamics	p. 96
Concepts and definitions	p. 96
Anosov diffeomorphisms and Anosov flows	p. 97
Anosov diffeomorphisms on the torus Tn	p. 102
Anosov automorphisms	p. 102
Structure of Anosov diffeomorphisms	p. 103
Anosov torus Tn with a hyperbolic structure	p. 104
Expanding maps	p. 105
The Blaschke product	p. 107
The Bernoulli map	p. 108
The Arnold cat map	p. 111
Classification of strange attractors of dynamical systems	p. 112
Properties of hyperbolic chaotic attractors	p. 114
Geodesic flows on compact smooth manifolds	p. 115
The solenoid attractor	p. 117
The Smale-Williams solenoid	p. 119
Plykin attractor	p. 120
Proof of the hyperbolicity of the logistic map for ? > 4	p. 121
Generalized hyperbolic attractors	p. 126
Generating hyperbolic attractors	p. 134
Density of hyperbolicity and homoclinic bifurcations in arbitrary dimension	p. 138
Hyperbolicity tests	p. 139
Numerical procedure	p. 141
Testing hyperbolicity of the Hénon map	p. 144
Testing hyperbolicity of the forced damped pendulum	p. 153
Uniform hyperbolicity test	p. 154
Exercises	p. 158
Robust Chaos in Hyperbolic Systems	p. 167
Modeling hyperbolic attractors	p. 167
Modeling the Smale-Williams attractor	p. 168
Testing hyperbolicity of system (7.1)	p. 176
Numerical verification of the hyperbolicity of system (7.1)	p. 183
Modeling the Arnold cat map	p. 187
Modeling the Bernoulli map	p. 194
Modeling Plykin's attractor	p. 202
Exercises	p. 204
Lorenz-Type Systems	p. 208
Lorenz-type attractors	p. 208
The Lorenz system	p. 210
Existence of Lorenz-type attractors	p. 211
Geometric models of the Lorenz equation	p. 220
Structure of the Lorenz attractor	p. 230
Expanding and contracting Lorenz attractors	p. 239
Wild strange attractors and pseudo-hyperbolicity	p. 241
Lorenz-type attractors realized in two-dimensional maps	p. 253
Exercises	p. 255
Robust Chaos in the Lorenz-Type Systems	p. 258
Robust chaos in the Lorenz-type attractors	p. 258
Robust chaos in Lorenz system	p. 260
Robust chaos in 2-D Lorenz-type attractors	p. 267
Exercises	p. 269
No Robust Chaos in Quasi-Attractors	p. 272
Quasi-attractors, concepts, and properties	p. 273
The Hénon map	p. 275
Uniform hyperbolicity of the Hénon map	p. 276
Hyperbolicity of Hénon-like maps	p. 281
Hénon attractor is a quasi-attractor	p. 282
The Strelkova-Anishchenko map	p. 286
The Anishchenko-Astakhov oscillator	p. 286
Chua's circuit	p. 288
Homoclinic and heteroclinic orbits	p. 294
The geometric model	p. 298
Exercises	p. 299
Robust Chaos in One-Dimensional Maps	p. 303
Unimodal maps	p. 303
S-unimodal maps	p. 305
Relation between unimodality and hyperbolicity	p. 307
Classification of unimodal maps of the interval	p. 309
Collet-Eckmann maps	p. 312
Statistical properties of unimodal maps	p. 314
The Barreto-Hunt-Grebogi-Yorke conjecture	p. 317
Counter-examples to the Barreto-Hunt-Grebogi-Yorke conjecture	p. 324
Robust chaos without the period-n-tupling scenario	p. 332
The B-exponential map	p. 335
Border-collision bifurcation and robust chaos	p. 342
Normal form for piecewise-smooth one-dimensional maps	p. 342
Border-collision bifurcation scenarios	p. 343
Robust chaos in one-dimensional singular maps	p. 349
Exercises	p. 352
Robust Chaos in 2-D Piecewise-Smooth Maps	p. 358
Robust chaos in 2-D piecewise-smooth maps	p. 358
Normal form for 2-D piecewise-smooth maps	p. 359
Border-collision bifurcations and robust chaos	p. 360
Regions for nonrobust chaos	p. 362
Regions for robust chaos: undesirable and dangerous bifurcations	p. 367
Proof of unicity of orbits	p. 370
Proof of robust chaos	p. 371
Robust chaos in noninvertible piecewise-linear maps	p. 378
Normal forms for two-dimensional noninvertible maps	p. 380
Onset of chaos: Proof of robust chaos	p. 382
Exercises	p. 395
Bibliography	p. 401
Index	p. 451
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