Discrete Chaos : With Applications in Science and Engineering

ISBN-13: 9781584885924 / Angielski / Twarda / 2007 / 440 str.

Saber Elaydi; Elaydi Elaydi; Saber N. Elaydi

Opis

While maintaining the lucidity of the first edition, this update includes recent results on global stability, bifurcation, chaos, and fractals. It 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 CD-ROM contains PHASER software.

Spis treści

PREFACE
FOREWORD
The Stability of One-Dimensional Maps
Introduction
Maps vs. Difference Equations
Maps vs. Differential Equations
Linear Maps/Difference Equations
Fixed (Equilibrium) Points
Graphical Iteration and Stability
Criteria for Stability
Periodic Points and Their Stability
The Period-Doubling Route to Chaos
Applications
Attraction and Bifurcation
Introduction
Basin of Attraction of Fixed Points
Basin of Attraction of Periodic Orbits
Singer’s Theorem
Bifurcation
Sharkovsky’s Theorem
The Lorenz Map
Period-Doubling in the Real World
Poincaré Section/Map
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
Rössler’s Attractor
Saturn’s Rings
Stability of Two-Dimensional Maps
Linear Maps vs. Linear Systems
Computing Aⁿ
Fundamental Set of Solutions
Second-Order Difference Equations
Phase Space Diagrams
Stability Notions
Stability of Linear Systems
The Trace-Determinant Plane
Liapunov Functions for Nonlinear Maps
Linear Systems Revisited
Stability via Linearization
Applications
Appendix
Bifurcation and Chaos in Two Dimensions
Center Manifolds
Bifurcation
Hyperbolic Anosov Toral Automorphism
Symbolic Dynamics
The Horseshoe and Hénon Maps
A Case Study: Extinction and Sustainability in Ancient Civilizations
Appendix
Fractals
Examples of Fractals
L-System
The Dimension of a Fractal
Iterated Function System
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
Bibliography
Answers to Selected Problems
Index

O autorze

Saber N. Elaydi