Discrete Chaos With Applications in Science and Engineering

ISBN-10: 1584885920

ISBN-13: 9781584885924

Edition: 2nd 2007 (Revised)

Authors: Saber N. Elaydi
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Description: While maintaining the lucidity of the first edition, Discrete Chaos, Second Edition: With Applications in Science and Engineering now includes many recent results on global stability, bifurcation, chaos, and fractals. The first five chapters provide the most comprehensive material on discrete dynamical systems, including trace-determinant stability, bifurcation analysis, and the detailed analysis of the center manifold theory. This edition also covers L-systems and the periodic structure of the bulbs in the Mandelbrot set as well as new applications in biology, chemistry, and physics. The principal improvements to this book are the additions of PHASER software on an accompanying CD-ROM and the Maple and Mathematica code available for download online.Incorporating numerous new topics and technology not found in similar texts, Discrete Chaos, Second Edition presents a thorough, up-to-date treatment of the theory and applications of discrete dynamical systems.

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Book details

List price: $99.99
Edition: 2nd
Copyright year: 2007
Publisher: CRC Press LLC
Publication date: 11/9/2007
Binding: Hardcover
Pages: 440
Size: 6.25" wide x 9.50" long x 1.00" tall
Weight: 1.628
Language: English

Preface
Author Biography
List of Symbols
Foreword
The Stability of One-Dimensional Maps
Introduction
Maps vs. Difference Equations
Maps vs. Differential Equations
Euler's Method
Poincare Map
Linear Maps/Difference Equations
Fixed (Equilibrium) Points
Graphical Iteration and Stability
Criteria for Stability
Hyperbolic Fixed Points
Nonhyperbolic Fixed Points
Periodic Points and Their Stability
The Period-Doubling Route to Chaos
Fixed Points
2-Periodic Cycles
2[superscript 2]-Periodic Cycles
Beyond [mu subscript infinity]
Applications
Fish Population Modeling
Attraction and Bifurcation
Introduction
Basin of Attraction of Fixed Points
Basin of Attraction of Periodic Orbits
Singer's Theorem
Bifurcation
Sharkovsky's Theorem
Li-Yorke Theorem
A Converse of Sharkovsky's Theorem
The Lorenz Map
Period-Doubling in the Real World
Poincare Section/Map
Belousov-Zhabotinskii Chemical Reaction
Appendix
Chaos in One Dimension
Introduction
Density of the Set of Periodic Points
Transitivity
Sensitive Dependence
Definition of Chaos
Cantor Sets
Symbolic Dynamics
Conjugacy
Other Notions of Chaos
Rossler's Attractor
Saturn's Rings
Stability of Two-Dimensional Maps
Linear Maps vs. Linear Systems
Computing A[superscript n]
Fundamental Set of Solutions
Second-Order Difference Equations
Phase Space Diagrams
Stability Notions
Stability of Linear Systems
The Trace-Determinant Plane
Stability Analysis
Navigating the Trace-Determinant Plane
Liapunov Functions for Nonlinear Maps
Linear Systems Revisited
Stability via Linearization
The Hartman-Grobman Theorem
The Stable Manifold Theorem
Applications
The Kicked Rotator and the Henon Map
The Henon Map
Discrete Epidemic Model for Gonorrhea
Perennial Grass
Appendix
Bifurcation and Chaos in Two Dimensions
Center Manifolds
Bifurcation
Eigenvalues of 1 or -1
A Pair of Eigenvalues of Modulus 1 - The Neimark-Sacker Bifurcation
Hyperbolic Anosov Toral Automorphism
Symbolic Dynamics
Subshifts of Finite Type
The Horseshoe and Henon Maps
The Henon Map
A Case Study: The Extinction and Sustainability in Ancient Civilizations
Appendix
Fractals
Examples of Fractals
L-system
The Dimension of a Fractal
Iterated Function System
Deterministic IFS
The Random Iterated Function System and the Chaos Game
Mathematical Foundation of Fractals
The Collage Theorem and Image Compression
The Julia and Mandelbrot Sets
Introduction
Mapping by Functions on the Complex Domain
The Riemann Sphere
The Julia Set
Topological Properties of the Julia Set
Newton's Method in the Complex Plane
The Mandelbrot Set
Topological Properties
Rays and Bulbs
Rotation Numbers and Farey Addition
Accuracy of Pictures
Bibliography
Answers to Selected Problems
Index
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