Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms

by Skiadas; Christos H.

• ISBN13: 9781420079005
• ISBN10: 142007900X
• Format: Hardcover
• Copyright: 2008-10-20
• Publisher: Chapman & Hall/

Summary

Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Formspresents the main models developed by pioneers of chaos theory, along with new extensions and variations of these models. Using more than 500 graphs and illustrations, the authors show how to design, estimate, and test an array of models.Requiring little prior knowledge of mathematics, the book focuses on classical forms and attractors as well as new simulation methods and techniques. Ideas clearly progress from the most elementary to the most advanced. The authors cover deterministic, stochastic, logistic, Gaussian, delay, Hénon, Holmes, Lorenz, Rössler, and rotation models. They also look at chaotic analysis as a tool to design forms that appear in physical systems; simulate complicated and chaotic orbits and paths in the solar system; explore the Hénon'Heiles, Contopoulos, and Hamiltonian systems; and provide a compilation of interesting systems and variations of systems, including the veryintriguing Lotka'Volterra system.Making a complex topic accessible through a visual and geometric style, this book should inspire new developments in the field of chaotic models and encourage more readers to become involved in this rapidly advancing area.

Table of Contents

Introduction	p. 1
Chaos in Differential Equations Systems	p. 1
Chaos in Difference Equation Systems	p. 4
The logistic map	p. 5
Delay models	p. 6
The Henon model	p. 6
More Complex Structures	p. 8
Three-dimensional and higher-dimensional models	p. 8
Conservative systems	p. 8
Rotations	p. 10
Shape and form	p. 12
Chaos and the Universe	p. 14
Chaos in the solar system	p. 14
Chaos in galaxies	p. 19
Galactic-type potentials and the Henon-Heiles system	p. 23
The Contopoulos system	p. 25
Odds and Ends, and Milestones	p. 27
Models and Modelling	p. 29
Introduction	p. 29
Model Construction	p. 30
Growth/decay models	p. 30
Modelling Techniques	p. 32
Series approximation	p. 33
Empirical model formulation	p. 36
The calculus of variations approach	p. 38
The probabilistic-stochastic approach	p. 39
Delay growth functions	p. 40
Chaotic Analysis and Simulation	p. 41
Deterministic, Stochastic and Chaotic Models	p. 42
The Logistic Model	p. 47
The Logistic Map	p. 47
Geometric analysis of the logistic	p. 47
Algebraic analysis of the logistic	p. 51
The Bifurcation Diagram	p. 58
Other Models with Similar Behaviour	p. 61
Models with Different Chaotic Behaviour	p. 62
The GRM1 Chaotic Model	p. 64
GRM1 and innovation diffusion modelling	p. 64
The generalized rational model	p. 66
Parameter estimation for the GRM1 model	p. 68
Illustrations	p. 69
Further Discussion	p. 71
The Delay Logistic Model	p. 79
Introduction	p. 79
Delay Difference Models	p. 79
Simple delay oscillation scheme	p. 79
The delay logistic model	p. 81
Time Delay Differential Equations	p. 82
A More Complicated Delay Model	p. 84
A Delay Differential Logistic Analogue	p. 86
Other Delay Logistic Models	p. 86
Model Behaviour for Large Delays	p. 89
Another Delay Logistic Model	p. 91
The Henon Model	p. 99
Global Period Doubling Bifurcations in the Henon Map	p. 99
Period doubling bifurcations when b = -1	p. 100
Period doubling bifurcations when b = 1	p. 101
The Cosine-Henon Model	p. 101
An Example of Bifurcation and Period Doubling	p. 103
A Differential Equation Analogue	p. 103
Variants of the Henon Delay Difference Equation	p. 104
The third-order delay model	p. 104
Second-order delay models	p. 105
First-order delay variants	p. 107
Exponential variants	p. 108
Variants of the Henon System Equations	p. 109
The Holmes and Sine Delay Models	p. 110
The Holmes model	p. 110
The sine delay model	p. 111
Three-Dimensional and Higher-Dimensional Models	p. 117
Equilibrium Points and Characteristic Matrices	p. 117
The Lotka-Volterra Model	p. 118
The Arneodo Model	p. 119
An Autocatalytic Attractor	p. 121
A Four-Dimensional Autocatalytic Attractor	p. 122
The Rossler Model	p. 123
A variant of the Rossler model	p. 124
Introducing rotation into the Rossler model	p. 127
The Lorenz Model	p. 129
The modified Lorenz model	p. 131
Non-Chaotic Systems	p. 135
Conservative Systems	p. 135
The simplest conservative system	p. 137
Equilibrium points in Hamiltonian systems	p. 138
Linear Systems	p. 139
Transformations on linear systems	p. 140
Qualitative behaviour at equilibrium points	p. 142
Egg-Shaped Forms	p. 144
A simple egg-shaped form	p. 144
A double egg-shaped form	p. 145
A double egg-shaped form with an envelope	p. 146
Symmetric Forms	p. 148
More Complex Forms	p. 150
Higher-Order Forms	p. 153
Rotations	p. 157
Introduction	p. 157
A Simple Rotation-Translation System of Differential Equations	p. 158
A Discrete Rotation-Translation Model	p. 163
A General Rotation-Translation Model	p. 169
Rotating Particles inside the Egg-Shaped Form	p. 171
Rotations Following an Inverse Square Law	p. 173
Shape and Form	p. 179
Introduction	p. 179
Symmetry and plane isometries	p. 180
Isometries in Modelling	p. 184
Two-dimensional rotation	p. 184
Reflection and Translation	p. 186
Space contraction	p. 186
Application in the Ikeda Attractor	p. 187
Chaotic Attractors and Rotation-Reflection	p. 188
Experimenting with Rotation and Reflection	p. 191
The effect of space contraction on rotation-translation	p. 191
The effect of space contraction and change of reflection angle on translation-reflection	p. 192
Complicated rotation angle forms	p. 193
Comparing rotation-reflection	p. 194
A simple rotation-translation model	p. 196
Chaotic Circular Forms	p. 196
Further Analysis	p. 200
The space contraction rotation-translation case	p. 202
Chaotic Advection	p. 205
The Sink Problem	p. 205
Central sink	p. 205
The contraction process	p. 207
Non-Central Sink	p. 207
Two Symmetric Sinks	p. 208
Aref's blinking vortex system	p. 208
Chaotic Forms without Space Contraction	p. 213
Other Chaotic Forms	p. 213
Complex Sinusoidal Rotation Angle	p. 216
A Special Rotation-Translation Model	p. 219
Other Rotation-Translation Models	p. 219
Elliptic rotation-translation	p. 219
Rotation-translation with special rotation angle	p. 220
Chaos in Galaxies and Related Simulations	p. 223
Introduction	p. 223
Chaos in the Solar System	p. 225
Galaxy Models and Modelling	p. 228
A special rotation-translation image	p. 234
Rotation-Reflection	p. 235
Relativity in Rotation-Translation Systems	p. 236
Other Relativistic Forms	p. 241
Galactic Clusters	p. 247
Relativistic Reflection-Translation	p. 248
Rotating Disks of Particles	p. 248
A circular rotating disk	p. 248
The rotating ellipsoid	p. 250
Rotating Particles under Distant Attracting Masses	p. 253
One attracting mass	p. 253
The area of the chaotic region in galaxy simulations	p. 255
The speed of particles	p. 257
Two Equal Attracting Masses in Opposite Directions	p. 258
Symmetric unequal attracting masses	p. 260
Two Attracting Equal Nonsymmetric Masses	p. 263
Galactic-Type Potentials and the Henon-Heiles System	p. 267
Introduction	p. 267
The Henon-Heiles System	p. 268
Discrete Analogues to the Henon-Heiles System	p. 270
Paths of Particles in the Henon-Heiles System	p. 272
Other Forms for the Hamiltonian	p. 273
The Simplest Form for the Hamiltonian	p. 275
Gravitational Attraction	p. 275
A Logarithmic Potential	p. 279
Hamiltonians with a Galactic Type Potential: The Contopoulos System	p. 279
Another Simple Hamiltonian System	p. 281
Odds and Ends	p. 285
Forced Nonlinear Oscillators	p. 285
The Effect of Noise in Three-Dimensional Models	p. 285
The Lotka-Volterra Theory for the Growth of Two Conflicting Populations	p. 288
The Pendulum	p. 290
A Special Second-Order Differential Equation	p. 292
Other Patterns and Chaotic Forms	p. 292
Milestones	p. 297
References	p. 303
Index	p. 345
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