Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms (Hardback) book cover

Chaotic Modelling and Simulation

Analysis of Chaotic Models, Attractors and Forms

By Christos H. Skiadas, Charilaos Skiadas

© 2008 – Chapman and Hall/CRC

364 pages | 500 B/W Illus.

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Offers Both Standard and Novel Approaches for the Modeling of Systems

Examines 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.


"This nicely written and generously illustrated book invites the reader with rather modest mathematical background to explore the exciting world of chaotic theory … The material is presented in a lucid and self-contained form, advancing from simpler to more complicated ideas and reflecting developments in the chaotic theory from its early steps to more recent discoveries. … This captivating book … provides an encyclopedic yet accessible … reading on chaotic theory and may well stimulate further interest … ."

Zentralblatt MATH, 1155

"…an extensive bibliography and a chronology of the most important papers in the field make the book an excellent reference source of general interest."

EMS Newsletter, June 2009

Table of Contents


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


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


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



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


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


Chaos in the Solar System

Galaxy Models and Modeling


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


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



List of Figures


Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
SCIENCE / Physics