ISBN-10:
9812838813
ISBN-13:
9789812838810
Pub. Date:
06/30/2010
Publisher:
World Scientific Publishing Company, Incorporated
Elegant Chaos: Algebraically Simple Chaotic Flows

Elegant Chaos: Algebraically Simple Chaotic Flows

by Julien Clinton Sprott

Hardcover

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Overview

This heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos.

No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study.

Product Details

ISBN-13: 9789812838810
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 06/30/2010
Pages: 304
Product dimensions: 6.10(w) x 9.10(h) x 0.70(d)

Table of Contents

Preface vii

List of Tables xv

1 Fundamentals 1

1.1 Dynamical Systems 1

1.2 State Space 2

1.3 Dissipation 7

1.4 Limit Cycles 8

1.5 Chaos and Strange Attractors 10

1.6 Poincaré Sections and Fractals 12

1.7 Conservative Chaos 16

1.8 Two-toruses and Quasiperiodicity 18

1.9 Largest Lyapunov Exponent 20

1.10 Lyapunov Exponent Spectrum 24

1.11 Attractor Dimension 29

1.12 Chaotic Transients 31

1.13 Intermittency 32

1.14 Basins of Attraction 32

1.15 Numerical Methods 36

1.16 Elegance 37

2 Periodically Forced Systems 41

2.1 Van der Pol Oscillator 41

2.2 Rayleigh Oscillator 43

2.3 Rayleigh Oscillator Variant 43

2.4 Duffing Oscillator 44

2.5 Quadratic Oscillators 47

2.6 Piecewise-linear Oscillators 48

2.7 Signum Oscillators 49

2.8 Exponential Oscillators 51

2.9 Other Undamped Oscillators 51

2.10 Velocity Forced Oscillators 53

2.11 Parametric Oscillators 55

2.12 Complex Oscillators 57

3 Autonomous Dissipative Systems 61

3.1 Lorenz System 61

3.2 Diffusionless Lorenz System 64

3.3 Rössler System 66

3.4 Other Quadratic Systems 68

3.4.1 Rössler prototype-4 system 68

3.4.2 Sprott systems 68

3.5 Jerk Systems 70

3.5.1 Simplest quadratic case 73

3.5.2 Rational jerks 76

3.5.3 Cubic cases 77

3.5.4 Cases with arbitrary power 79

3.5.5 Piecewise-linear case 80

3.5.6 Memory oscillators 82

3.6 Circulant Systems 83

3.6.1 Halvorsen's system 84

3.6.2 Thomas' systems 85

3.6.3 Piecewise-linear system 86

3.7 Other Systems 86

3.7.1 Multiscroll systems 87

3.7.2 Lotka-Volterra systems 88

3.7.3 Chua's systems 90

3.7.4 Rikitake dynamo 92

4 Autonomous Conservative Systems 95

4.1 Nosé-Hoover Oscillator 95

4.2 Nosé-Hoover Variants 97

4.3 Jerk Systems 98

4.3.1 Jerk form of the Nosé-Hoover oscillator 98

4.3.2 Simplest conservative chaotic flow 99

4.3.3 Other conservative jerk systems 99

4.4 Circulant Systems 101

4.4.1 Quadratic case 102

4.4.2 Cubic case 102

4.4.3 Labyrinth chaos 105

4.4.4 Piecewise-linear system 107

5 Low-dimensional Systems (D < 3) 109

5.1 Dixon System 109

5.2 Dixon Variants 110

5.3 Logarithmic Case 112

5.4 Other Cases 114

6 High-dimensional Systems (D > 3) 115

6.1 Periodically Forced Systems 115

6.1.1 Forced pendulum 116

6.1.2 Other forced nonlinear oscillators 118

6.2 Master-slave Oscillators 118

6.3 Mutually Coupled Nonlinear Oscillators 120

6.3.1 Coupled pendulums 121

6.3.2 Coupled van der Pol oscillators 123

6.3.3 Coupled FitzHugh-Nagumo oscillators 123

6.3.4 Coupled complex oscillators 124

6.3.5 Other coupled nonlinear oscillators 125

6.4 Hamiltonian Systems 126

6.4.1 Coupled nonlinear oscillators 128

6.4.2 Velocity coupled oscillators 129

6.4.3 Parametrically coupled oscillators 130

6.4.4 Simplest Hamiltonian 130

6.4.5 Hénon-Heiles system 132

6.4.6 Reduced Hénon-Heiles system 133

6.4.7 N-body gravitational systems 134

6.4.8 N-body Coulomb systems 138

6.5 Anti-Newtonian Systems 142

6.5.1 Two-body problem 142

6.5.2 Three-body problem 145

6.6 Hyperjerk Systems 147

6.6.1 Forced oscillators 147

6.6.2 Chlouverakis systems 148

6.7 Hyperchaotic Systems 152

6.7.1 Rössler hyperchaos 153

6.7.2 Snap hyperchaos 154

6.7.3 Coupled chaotic systems 154

6.7.4 Other hyperchaotic systems 156

6.8 Autonomous Complex Systems 156

6.9 Lotka-Volterra Systems 157

6.10 Artificial Neural Networks 159

6.10.1 Minimal dissipative artificial neural network 161

6.10.2 Minimal conservative artificial neural network 162

6.10.3 Minimal circulant artificial neural network 162

7 Circulant Systems 165

7.1 Lorenz-Emanuel System 165

7.2 Lotka-Volterra Systems 169

7.3 Antisymmetric Quadratic System 171

7.4 Quadratic Ring System 171

7.5 Cubic Ring System 171

7.6 Hyperlabyrinth System 173

7.7 Circulant Neural Networks 174

7.8 Hyperviscous Ring 176

7.9 Rings of Oscillators 176

7.9.1 Coupled pendulums 177

7.9.2 Coupled cubic oscillators 177

7.9.3 Coupled signum oscillators 178

7.9.4 Coupled van der Pol oscillators 179

7.9.5 Coupled FitzHugh-Nagumo oscillators 180

7.9.6 Coupled complex oscillators 182

7.9.7 Coupled Lorenz systems 182

7.9.8 Coupled jerk systems 185

7.10 Star Systems 185

7.10.1 Coupled pendulums 187

7.10.2 Coupled cubic oscillators 187

7.10.3 Coupled signum oscillators 188

7.10.4 Coupled van der Pol oscillators 190

7.10.5 Coupled FitzHugh-Nagumo oscillators 191

7.10.6 Coupled complex oscillators 191

7.10.7 Coupled diffusionless Lorenz systems 193

7.10.8 Coupled jerk systems 194

8 Spatiotemporal Systems 195

8.1 Numerical Methods 195

8.2 Kuramoto-Sivashinsky Equation 199

8.3 Kuramoto-Sivashinsky Variants 200

8.3.1 Cubic case 201

8.3.2 Quartic case 201

8.4 Chaotic Traveling Waves 201

8.4.1 Rotating Kuramoto-Sivashinsky system 203

8.4.2 Rotating Kuramoto-Sivashinsky variant 203

8.5 Continuum Ring Systems 204

8.5.1 Quadratic ring system 204

8.5.2 Antisymmetric quadratic system 205

8.5.3 Other simple PDEs 207

8.6 Traveling Wave Variants 212

9 Time-Delay Systems 221

9.1 Delay Differential Equations 221

9.2 Mackey-Glass Equation 223

9.3 Ikeda DDE 223

9.4 Sinusoidal DDE 225

9.5 Polynomial DDE 225

9.6 Sigmoidal DDE 227

9.7 Signum DDE 227

9.8 Piecewise-linear DDEs 229

9.8.1 Antisymmetric case 229

9.8.2 Asymmetric case 229

9.8.3 Asymmetric logistic DDE 230

9.9 Asymmetric Logistic DDE with Continuous Delay 232

10 Chaotic Electrical Circuits 233

10.1 Circuit Elegance 233

10.2 Forced Relaxation Oscillator 234

10.3 Autonomous Relaxation Oscillator 237

10.4 Coupled Relaxation Oscillators 239

10.4.1 Two oscillators 239

10.4.2 Many oscillators 241

10.5 Forced Diode Resonator 242

10.6 Saturating Inductor Circuit 243

10.7 Forced Piecewise-linear Circuit 246

10.8 Chua's Circuit 246

10.9 Nishio's Circuit 249

10.10 Wien-bridge Oscillator 251

10.11 Jerk Circuits 254

10.11.1 Absolute-value case 254

10.11.2 Single-knee case 255

10.11.3 Signum case 256

10.11.4 Signum variant 258

10.12 Master-slave Oscillator 259

10.13 Ring of Oscillators 261

10.14 Delay-line Oscillator 263

Bibliography 265

Index 281

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