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2nd Edition

Discrete Chaos
With Applications in Science and Engineering




ISBN 9781584885924
Published November 9, 2007 by Chapman and Hall/CRC
440 Pages - 215 B/W Illustrations

 
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Book 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.

Table of Contents

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

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Reviews

“The present textbook gives an excellent introduction to this new, and potentially revolutionary, territory. It will take the reader, with clarity and precision, from simple beginnings with 1-dimensional difference equations (and their cascades of period doubling en route to chaos), on to 2- and 3-dimensional systems, and beyond this to fractals and relationships between geometry and dynamics. The final chapter deals with the Julia and Mandelbrot sets, where in my opinion mathematical elegance and pure aesthetic beauty begin to merge.”
—From the Foreword, Lord Robert M. May, Department of Zoology, University of Oxford, UK

"…the systematic and rigorous exposition on many concepts and results … is very effective for teaching purposes in many kinds of classrooms, both in the framework of theoretical courses and the ones devoted to applications. The main feature of Elaydi’s book is the huge spectrum of examples and exercises … the CD included that contains the program PHASER with several built-in examples as well as a user-friendly environment where the reader can experiment with his or her own examples is quite useful. … a nice guided tour that motivates the reader to a deeper journey into the rich spectrum of properties and applications of dynamical systems and deterministic chaos."
Mathematical Reviews, 2009b

“The long-awaited second edition of Discrete Chaos is finally here! Not only is this new edition, with its superb organization and exposition, a delight to read, but the accompanying electronic supplement, including a myriad of insightful computer experiments, is truly engaging. Discrete Chaos can serve as a textbook for undergraduate and beginning graduate courses, as well as a reference for researchers interested in discrete dynamical models. It provides rigorous coverage of stability, bifurcations, and chaos in one- and two-dimensional discrete dynamical systems. The power and the utility of theoretical considerations are successfully demonstrated in numerous problems and significant applications to models from ecology, epidemiology, physics, engineering, and social sciences. Rigorous yet eminently accessible, Discrete Chaos is the most up-to-date book in its class.”
—Huseyin Kocak, University of Miami, Florida, USA

“One of the features that makes this book unique is that, as a renowned and active researcher in discrete dynamical systems and difference equations, Elaydi integrates very skillfully these two fields, whenever possible, and provides in depth the stability theory for one- and two-dimensional dynamical systems. It is fascinating that, without sacrificing anything important, the author is able to simplify the treatment and compress the material on one-dimensional dynamics into chapter 1 that takes some texts many chapters.”
—Danrun Huang, St. Cloud State University, Minnesota, USA

"The book under review is an undergraduate-level text. It is a good starting point for scientists and students that would like to move into the field of studying the discrete chaos."

– Alexander O. Ignatyev, in Zentralblatt Math, 2009

 

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