Chaotic Modelling and Simulation

Transcription

1 Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK

2 Contents 1 Introduction Chaos in Differential Equations Systems Chaos in Difference Equation Systems The logistic map Delay modeis The Henon model More Complex Structures Three-dimensional and higher-dimensional modeis Conservative Systems Rotations Shape and form Chaos and the Universe Chaos in the solar System Chaos in galaxies Galactic-type potentials and the Henon-Heiles System The Contopoulos System Odds and Ends, and Milestones 27 2 Models and Modelling Introduction Model Construction Growth/decay modeis Modelling Techniques Series approximation Empirical model formulation The calculus of variations approach The probabilistic-stochastic approach Delay growth functions Chaotic Analysis and Simulation Deterministic, Stochastic and Chaotic Models 42 3 The Logistic Model The Logistic Map Geometrie analysis of the logistic Algebraic analysis of the logistic The Bifurcation Diagram Other Models with Similar Behaviour 61

3 3.4 Models with Different Chaotic Behaviour The GRM1 Chaotic Model GRM1 and innovation diffusion modelling The generalized rational model Parameter estimation for the GRM1 model Illustrations Further Discussion 71 4 The Delay Logistic Model Introduction Delay Difference Models Simple delay oscillation scheme The delay logistic model Time Delay Differential Equations A More Complicated Delay Model A Delay Differential Logistic Analogue Other Delay Logistic Models Model Behaviour for Large Delays Another Delay Logistic Model 91 5 The Henon Model Global Period Doubling Bifurcations in the Henon Map Period doubling bifurcations when b = Period doubling bifurcations when b The Cosine-Henon Model An Example of Bifurcation and Period Doubling A Differential Equation Analogue Variants of the Henon Delay Difference Equation The third-order delay model Second-order delay modeis First-order delay variants Exponential variants Variants of the Henon System Equations The Holmes and Sine Delay Models The Holmes model The sine delay model Three-Dimensional and Higher-Dimensional Models Equilibrium Points and Characteristic Matrices The Lotka-Volterra Model The Arneodo Model An Autocatalytic Attractor A Four-Dimensional Autocatalytic Attractor The Rössler Model A variant of the Rössler model 124

4 6.6.2 Introducing rotation into thejlössler model The Lorenz Model The modified Lorenz model Non-Chaotic Systems Conservative Systems The simplest conservative System Equilibrium points in Hamiltonian Systems Linear Systems Transformations on linear Systems Qualitative behaviour at equilibrium points Egg-Shaped Forms A simple egg-shaped form A double egg-shaped form A double egg-shaped form with an envelope Symmetrie Forms More Complex Forms Higher-Order Forms, Rotations Introduction A Simple Rotation-Translation System of Differential Equations A Discrete Rotation-Translation Model A General Rotation-Translation Model Rotating Particles inside the Egg-Shaped Form Rotations Following an Inverse Square Law Shape and Form Introduction Symmetry and plane isometries Isometries in Modelling Two-dimensional rotation Reflection and Translation Space contraction Application in the Ikeda Attractor Chaotic Attractors and Rotation-Reflection Experimenting with Rotation and Reflection The effect of space contraction on rotation-translation The effect of space contraction and change of reflection angle on translation-reflection Complicated rotation angle forms Comparing rotation-reflection A simple rotation-translation model Chaotic Circular Forms Further Analysis 200

5 9.8.1 The space contraction rotation-translation case Chaotic Advection The Sink Problem Central sink The contraction process Non-Central Sink Two Symmetrie Sinks Aref's blinking vortex System Chaotic Forms without Space Contraction Other Chaotic Forms Complex Sinusoidal Rotation Angle A Special Rotation-Translation Model Other Rotation-Translation Models Elliptic rotation-translation Rotation-translation with special rotation angle Chaos in Galaxies and Related Simulations Introduction Chaos in the Solar System Galaxy Models and Modelling A special rotation-translation image Rotation-Reflection Relativity in Rotation-Translation Systems Other Relativistic Forms Galactic Clusters Relativistic Reflection-Translation Rotating Disks of Particles A circular rotating disk The rotating ellipsoid Rotating Particles under Distant Attracting Masses One attracting mass The area of the chaotic region in galaxy simulations The speed of particles Two Equal Attracting Masses in Opposite Directions Symmetrie unequal attracting masses Two Attracting Equal Nonsymmetric Masses Galactic-Type Potentials and the Henon-Heiles System Introduction The Henon-Heiles System Discrete Analogues to the Henon-Heiles System Paths of Particles in the Henon-Heiles System Other Forms for the Hamiltonian The Simplest Form for the Hamiltonian 275

6 12.7 Gravitational Attraction A Logarithmic Potential Hamiltonians with a Galactic Type Potential: The Contopoulos System Another Simple Hamiltonian System OddsandEnds Forced Nonlinear Oscillators The Effect of Noise in Three-Dimensional Models The Lotka-Volterra Theory for the Growth of Two Conflicting Populations ThePendulum A Special Second-Order Differential Equation Other Patterns and Chaotic Forms Milestones 297 References 303 Index, 345