Discrete Chaos, Second Edition

Discrete Chaos, Second Edition: With Applications in Science and Engineering

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Features

  • Provides new chapters on controlling chaos and differential equations
  • Explores numerous applications in economics, ecology, neural networks, and physics
  • Offers both Maple and Mathematica examples in the text, along with the code available for download
  • Includes a CD-ROM with PHASER 3.0 and supplemental PHASER modules, enabling the formulation of graphs and the solution of problems
  • Presents various historical remarks and anecdotes throughout as well as an extensive set of exercises at the end of each section
  • Contains a complete solutions manual for qualifying instructors
  • Summary

    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

    Editorial 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

     

    Downloads Updates

    Resource OS Platform Updated Description Instructions
    Maple Programs.zip Cross Platform January 02, 2008
    Mathematica programs.zip Cross Platform January 02, 2008

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