Chaotic Modeling and Simulation ─ Analysis of Chaotic Models, Attractors and Forms

Offers Both Standard and Novel Approaches for the Modeling of SystemsExamines the Interesting Behavior of Particular Classes of Models
Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms presents 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 very intriguing 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.

Introduction
Chaos in Differential Equations Systems
Chaos in Difference Equation Systems
More Complex Structures
Chaos and the Universe
Odds and Ends and Milestones
Models and Modeling
Introduction
Model Construction
Modeling Techniques
Chaotic Analysis and Simulation
Deterministic, Stochastic, and Chaotic Models
The Logistic Model
The Logistic Map
The Bifurcation Diagram
Other Models with Similar Behavior
Models with Different Chaotic Behavior
The GRM1 Chaotic Model
Further Discussion
The Delay Logistic Model
Introduction
Delay Difference Models
Time Delay Differential Equations
A More Complicated Delay Model
A Delay Differential Logistic Analogue
Other Delay Logistic Models
Model Behavior for Large Delays
Another Delay Logistic Model
The Hénon Model
Global Period Doubling Bifurcations in the Hénon Map
The Cosine-Hénon Model
An Example of Bifurcation and Period Doubling
A Differential Equation Analogue
Variants of the Hénon Delay Difference Equation
Variants of the Hénon System Equations
The Holmes and Sine Delay Models
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
The Lorenz Model
Nonchaotic Systems
Conservative Systems
Linear Systems
Egg-Shaped Forms
Symmetric 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
Isometries in Modeling
Reflection and Translation
Application in the Ikeda Attractor
Chaotic Attractors and Rotation-Reflection
Experimenting with Rotation and Reflection
Chaotic Circular Forms
Further Analysis
Chaotic Advection
The Sink Problem
Noncentral Sink
Two Symmetric Sinks
Chaotic Forms without Space Contraction
Other Chaotic Forms
Complex Sinusoidal Rotation Angle
A Special Rotation-Translation Model
Other Rotation-Translation Models
Chaos in Galaxies and Related Simulations
Introduction
Chaos in the Solar System
Galaxy Models and Modeling
Rotation-Reflection
Relativity in Rotation-Translation Systems
Other Relativistic Forms
Galactic Clusters
Relativistic Reflection-Translation
Rotating Disks of Particles
Rotating Particles under Distant Attracting Masses
Two Equal Attracting Masses in Opposite Directions
Galactic-Type Potentials and the Hénon–Heiles System
Introduction
The Hénon–Heiles System
Discrete Analogues to the Hénon–Heiles System
Paths of Particles in the Hénon–Heiles System
Other Forms for the Hamiltonian
The Simplest Form for the Hamiltonian
Gravitational Attraction
A Logarithmic Potential
Hamiltonians with a Galactic Type Potential: The Contopoulos System
Another Simple Hamiltonian System
Odds and Ends
Forced Nonlinear Oscillators
The Effect of Noise in Three-Dimensional Models
The Lotka–Volterra Theory for the Growth of Two Conflicting Populations
The Pendulum
A Special Second-Order Differential Equation
Other Patterns and Chaotic Forms
Milestones
References
List of Figures
Index