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

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  • Format: Hardcover
  • Copyright: 2008-10-20
  • Publisher: Chapman & Hall/

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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

Introductionp. 1
Chaos in Differential Equations Systemsp. 1
Chaos in Difference Equation Systemsp. 4
The logistic mapp. 5
Delay modelsp. 6
The Henon modelp. 6
More Complex Structuresp. 8
Three-dimensional and higher-dimensional modelsp. 8
Conservative systemsp. 8
Rotationsp. 10
Shape and formp. 12
Chaos and the Universep. 14
Chaos in the solar systemp. 14
Chaos in galaxiesp. 19
Galactic-type potentials and the Henon-Heiles systemp. 23
The Contopoulos systemp. 25
Odds and Ends, and Milestonesp. 27
Models and Modellingp. 29
Introductionp. 29
Model Constructionp. 30
Growth/decay modelsp. 30
Modelling Techniquesp. 32
Series approximationp. 33
Empirical model formulationp. 36
The calculus of variations approachp. 38
The probabilistic-stochastic approachp. 39
Delay growth functionsp. 40
Chaotic Analysis and Simulationp. 41
Deterministic, Stochastic and Chaotic Modelsp. 42
The Logistic Modelp. 47
The Logistic Mapp. 47
Geometric analysis of the logisticp. 47
Algebraic analysis of the logisticp. 51
The Bifurcation Diagramp. 58
Other Models with Similar Behaviourp. 61
Models with Different Chaotic Behaviourp. 62
The GRM1 Chaotic Modelp. 64
GRM1 and innovation diffusion modellingp. 64
The generalized rational modelp. 66
Parameter estimation for the GRM1 modelp. 68
Illustrationsp. 69
Further Discussionp. 71
The Delay Logistic Modelp. 79
Introductionp. 79
Delay Difference Modelsp. 79
Simple delay oscillation schemep. 79
The delay logistic modelp. 81
Time Delay Differential Equationsp. 82
A More Complicated Delay Modelp. 84
A Delay Differential Logistic Analoguep. 86
Other Delay Logistic Modelsp. 86
Model Behaviour for Large Delaysp. 89
Another Delay Logistic Modelp. 91
The Henon Modelp. 99
Global Period Doubling Bifurcations in the Henon Mapp. 99
Period doubling bifurcations when b = -1p. 100
Period doubling bifurcations when b = 1p. 101
The Cosine-Henon Modelp. 101
An Example of Bifurcation and Period Doublingp. 103
A Differential Equation Analoguep. 103
Variants of the Henon Delay Difference Equationp. 104
The third-order delay modelp. 104
Second-order delay modelsp. 105
First-order delay variantsp. 107
Exponential variantsp. 108
Variants of the Henon System Equationsp. 109
The Holmes and Sine Delay Modelsp. 110
The Holmes modelp. 110
The sine delay modelp. 111
Three-Dimensional and Higher-Dimensional Modelsp. 117
Equilibrium Points and Characteristic Matricesp. 117
The Lotka-Volterra Modelp. 118
The Arneodo Modelp. 119
An Autocatalytic Attractorp. 121
A Four-Dimensional Autocatalytic Attractorp. 122
The Rossler Modelp. 123
A variant of the Rossler modelp. 124
Introducing rotation into the Rossler modelp. 127
The Lorenz Modelp. 129
The modified Lorenz modelp. 131
Non-Chaotic Systemsp. 135
Conservative Systemsp. 135
The simplest conservative systemp. 137
Equilibrium points in Hamiltonian systemsp. 138
Linear Systemsp. 139
Transformations on linear systemsp. 140
Qualitative behaviour at equilibrium pointsp. 142
Egg-Shaped Formsp. 144
A simple egg-shaped formp. 144
A double egg-shaped formp. 145
A double egg-shaped form with an envelopep. 146
Symmetric Formsp. 148
More Complex Formsp. 150
Higher-Order Formsp. 153
Rotationsp. 157
Introductionp. 157
A Simple Rotation-Translation System of Differential Equationsp. 158
A Discrete Rotation-Translation Modelp. 163
A General Rotation-Translation Modelp. 169
Rotating Particles inside the Egg-Shaped Formp. 171
Rotations Following an Inverse Square Lawp. 173
Shape and Formp. 179
Introductionp. 179
Symmetry and plane isometriesp. 180
Isometries in Modellingp. 184
Two-dimensional rotationp. 184
Reflection and Translationp. 186
Space contractionp. 186
Application in the Ikeda Attractorp. 187
Chaotic Attractors and Rotation-Reflectionp. 188
Experimenting with Rotation and Reflectionp. 191
The effect of space contraction on rotation-translationp. 191
The effect of space contraction and change of reflection angle on translation-reflectionp. 192
Complicated rotation angle formsp. 193
Comparing rotation-reflectionp. 194
A simple rotation-translation modelp. 196
Chaotic Circular Formsp. 196
Further Analysisp. 200
The space contraction rotation-translation casep. 202
Chaotic Advectionp. 205
The Sink Problemp. 205
Central sinkp. 205
The contraction processp. 207
Non-Central Sinkp. 207
Two Symmetric Sinksp. 208
Aref's blinking vortex systemp. 208
Chaotic Forms without Space Contractionp. 213
Other Chaotic Formsp. 213
Complex Sinusoidal Rotation Anglep. 216
A Special Rotation-Translation Modelp. 219
Other Rotation-Translation Modelsp. 219
Elliptic rotation-translationp. 219
Rotation-translation with special rotation anglep. 220
Chaos in Galaxies and Related Simulationsp. 223
Introductionp. 223
Chaos in the Solar Systemp. 225
Galaxy Models and Modellingp. 228
A special rotation-translation imagep. 234
Rotation-Reflectionp. 235
Relativity in Rotation-Translation Systemsp. 236
Other Relativistic Formsp. 241
Galactic Clustersp. 247
Relativistic Reflection-Translationp. 248
Rotating Disks of Particlesp. 248
A circular rotating diskp. 248
The rotating ellipsoidp. 250
Rotating Particles under Distant Attracting Massesp. 253
One attracting massp. 253
The area of the chaotic region in galaxy simulationsp. 255
The speed of particlesp. 257
Two Equal Attracting Masses in Opposite Directionsp. 258
Symmetric unequal attracting massesp. 260
Two Attracting Equal Nonsymmetric Massesp. 263
Galactic-Type Potentials and the Henon-Heiles Systemp. 267
Introductionp. 267
The Henon-Heiles Systemp. 268
Discrete Analogues to the Henon-Heiles Systemp. 270
Paths of Particles in the Henon-Heiles Systemp. 272
Other Forms for the Hamiltonianp. 273
The Simplest Form for the Hamiltonianp. 275
Gravitational Attractionp. 275
A Logarithmic Potentialp. 279
Hamiltonians with a Galactic Type Potential: The Contopoulos Systemp. 279
Another Simple Hamiltonian Systemp. 281
Odds and Endsp. 285
Forced Nonlinear Oscillatorsp. 285
The Effect of Noise in Three-Dimensional Modelsp. 285
The Lotka-Volterra Theory for the Growth of Two Conflicting Populationsp. 288
The Pendulump. 290
A Special Second-Order Differential Equationp. 292
Other Patterns and Chaotic Formsp. 292
Milestonesp. 297
Referencesp. 303
Indexp. 345
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