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.
Author(s): Julien Clinton Sprott
Publisher: World Scientific Pub Co (
Year: 2010
Language: English
Pages: 302
Contents......Page 10
Preface......Page 8
List of Tables......Page 16
1.1 Dynamical Systems......Page 18
1.2 State Space......Page 19
1.3 Dissipation......Page 24
1.4 Limit Cycles......Page 25
1.5 Chaos and Strange Attractors......Page 27
1.6 Poincare Sections and Fractals......Page 29
1.7 Conservative Chaos......Page 33
1.8 Two-toruses and Quasiperiodicity......Page 35
1.9 Largest Lyapunov Exponent......Page 37
1.10 Lyapunov Exponent Spectrum......Page 41
1.11 Attractor Dimension......Page 46
1.12 Chaotic Transients......Page 48
1.14 Basins of Attraction......Page 49
1.15 Numerical Methods......Page 53
1.16 Elegance......Page 54
2.1 Van der Pol Oscillator......Page 58
2.3 Rayleigh Oscillator Variant......Page 60
2.4 Du±ng Oscillator......Page 61
2.5 Quadratic Oscillators......Page 64
2.6 Piecewise-linear Oscillators......Page 65
2.7 Signum Oscillators......Page 66
2.9 Other Undamped Oscillators......Page 68
2.10 Velocity Forced Oscillators......Page 70
2.11 Parametric Oscillators......Page 72
2.12 Complex Oscillators......Page 74
3.1 Lorenz System......Page 78
3.2 Diffusionless Lorenz System......Page 81
3.3 RÄossler System......Page 83
3.4.2 Sprott systems......Page 85
3.5 Jerk Systems......Page 87
3.5.1 Simplest quadratic case......Page 90
3.5.2 Rational jerks......Page 93
3.5.3 Cubic cases......Page 94
3.5.4 Cases with arbitrary power......Page 96
3.5.5 Piecewise-linear case......Page 97
3.5.6 Memory oscillators......Page 99
3.6 Circulant Systems......Page 100
3.6.1 Halvorsen's system......Page 101
3.6.2 Thomas' systems......Page 102
3.7 Other Systems......Page 103
3.7.1 Multiscroll systems......Page 104
3.7.2 Lotka–Volterra systems......Page 105
3.7.3 Chua's systems......Page 107
3.7.4 Rikitake dynamo......Page 109
4.1 Nos–Hoover Oscillator......Page 112
4.2 Nos–Hoover Variants......Page 114
4.3.1 Jerk form of the Nos–Hoover oscillator......Page 115
4.3.3 Other conservative jerk systems......Page 116
4.4 Circulant Systems......Page 118
4.4.2 Cubic case......Page 119
4.4.3 Labyrinth chaos......Page 122
4.4.4 Piecewise-linear system......Page 124
5.1 Dixon System......Page 126
5.2 Dixon Variants......Page 127
5.3 Logarithmic Case......Page 129
5.4 Other Cases......Page 131
6.1 Periodically Forced Systems......Page 132
6.1.1 Forced pendulum......Page 133
6.2 Master–slave Oscillators......Page 135
6.3 Mutually Coupled Nonlinear Oscillator......Page 137
6.3.1 Coupled pendulums......Page 138
6.3.3 Coupled FitzHugh–Nagumo oscillators......Page 140
6.3.4 Coupled complex oscillators......Page 141
6.3.5 Other coupled nonlinear oscillators......Page 142
6.4 Hamiltonian Systems......Page 143
6.4.1 Coupled nonlinear oscillators......Page 145
6.4.2 Velocity coupled oscillators......Page 146
6.4.4 Simplest Hamiltonian......Page 147
6.4.5 Henon–Heiles system......Page 149
6.4.6 Reduced Henon–Heiles system......Page 150
6.4.7 N-body gravitational systems......Page 151
6.4.7.2 Restricted three-body problem......Page 153
6.4.8 N-body Coulomb systems......Page 155
6.4.8.1 Three spatial dimensions......Page 156
6.4.8.2 Two spatial dimensions......Page 157
6.5.1 Two-body problem......Page 159
6.5.2 Three-body problem......Page 162
6.6.1 Forced oscillators......Page 164
6.6.2.1 Snap systems......Page 165
6.6.2.2 Crackle systems......Page 167
6.6.2.3 Pop systems......Page 168
6.7 Hyperchaotic Systems......Page 169
6.7.1 Rossler hyperchaos......Page 170
6.7.3 Coupled chaotic systems......Page 171
6.8 Autonomous Complex Systems......Page 173
6.9 Lotka–Volterra Systems......Page 174
6.10 Artificial Neural Networks......Page 176
6.10.1 Minimal dissipative artificial neural network......Page 178
6.10.3 Minimal circulant artificial neural network......Page 179
7.1 Lorenz–Emanuel System......Page 182
7.2 Lotka–Volterra Systems......Page 186
7.5 Cubic Ring System......Page 188
7.6 Hyperlabyrinth System......Page 190
7.7 Circulant Neural Networks......Page 191
7.9 Rings of Oscillators......Page 193
7.9.2 Coupled cubic oscillators......Page 194
7.9.3 Coupled signum oscillators......Page 195
7.9.4 Coupled van der Pol oscillators......Page 196
7.9.5 Coupled FitzHugh–Nagumo oscillators......Page 197
7.9.7 Coupled Lorenz systems......Page 199
7.9.7.2 Diffusively coupled case......Page 200
7.10 Star Systems......Page 202
7.10.2 Coupled cubic oscillators......Page 204
7.10.3 Coupled signum oscillators......Page 205
7.10.4 Coupled van der Pol oscillators......Page 207
7.10.6 Coupled complex oscillators......Page 208
7.10.7 Coupled diffusionless Lorenz systems......Page 210
7.10.8 Coupled jerk systems......Page 211
8.1 Numerical Methods......Page 212
8.2 Kuramoto–Sivashinsky Equation......Page 216
8.3 Kuramoto–Sivashinsky Variants......Page 217
8.4 Chaotic Traveling Waves......Page 218
8.4.2 Rotating Kuramoto–Sivashinsky variant......Page 220
8.5.1 Quadratic ring system......Page 221
8.5.2 Antisymmetric quadratic system......Page 222
8.5.3 Other simple PDEs......Page 224
8.6 Traveling Wave Variants......Page 229
9.1 Delay Differential Equations......Page 238
9.3 Ikeda DDE......Page 240
9.5 Polynomial DDE......Page 242
9.7 Signum DDE......Page 244
9.8.2 Asymmetric case......Page 246
9.8.3 Asymmetric logistic DDE......Page 247
9.9 Asymmetric Logistic DDE with Continuous Delay......Page 249
10.1 Circuit Elegance......Page 250
10.2 Forced Relaxation Oscillator......Page 251
10.3 Autonomous Relaxation Oscillator......Page 254
10.4.1 Two oscillators......Page 256
10.4.2 Many oscillators......Page 258
10.5 Forced Diode Resonator......Page 259
10.6 Saturating Inductor Circuit......Page 260
10.8 Chua's Circuit......Page 263
10.9 Nishio's Circuit......Page 266
10.10 Wien-bridge Oscillator......Page 268
10.11.1 Absolute-value case......Page 271
10.11.2 Single-knee case......Page 272
10.11.3 Signum case......Page 273
10.11.4 Signum variant......Page 275
10.12 Master–slave Oscillator......Page 276
10.13 Ring of Oscillators......Page 278
10.14 Delay-line Oscillator......Page 280
Bibliography......Page 282
Index......Page 298
