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Discrete Chaos, Second Edition

With Applications in Science and Engineering

By Saber N. Elaydi

Chapman and Hall/CRC – 2007 – 440 pages

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    978-1-58488-592-4
    November 9th 2007

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.

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

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

Name: Discrete Chaos, Second Edition: With Applications in Science and Engineering (Hardback)Chapman and Hall/CRC 
Description: By Saber N. Elaydi. 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...
Categories: Differential Equations, Combinatorics, Number Theory