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Chaos in Discrete Dynamical Systems

A Visual Introduction in 2 Dimensions

  • Ralph H. Abraham
  • Laura Gardini
  • Christian Mira

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Basic Concepts

    1. Front Matter
      Pages 1-1
    2. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 3-9
    3. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 11-27
    4. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 29-37
  3. Exemplary Bifurcation Sequences

    1. Front Matter
      Pages 39-39
    2. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 41-58
    3. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 59-83
    4. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 85-115
    5. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 117-150
    6. Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 151-151
  4. Back Matter
    Pages 153-248

About this book

Introduction

Chaos Theory is a synonym for dynamical systems theory, a branch of mathematics. Dynamical systems come in three flavors: flows (continuous dynamical systems), cascades (discrete, reversible, dynamical systems), and semi-cascades (discrete, irreversible, dynamical systems). Flows and semi-cascades are the classical systems iuntroduced by Poincare a centry ago, and are the subject of the extensively illustrated book: "Dynamics: The Geometry of Behavior," Addison-Wesley 1992 authored by Ralph Abraham and Shaw. Semi- cascades, also know as iterated function systems, are a recent innovation, and have been well-studied only in one dimension (the simplest case) since about 1950. The two-dimensional case is the current frontier of research. And from the computer graphcis of the leading researcher come astonishing views of the new landscape, such as the Julia and Mandelbrot sets in the beautiful books by Heinz-Otto Peigen and his co-workers. Now, the new theory of critical curves developed by Mira and his students and Toulouse provide a unique opportunity to explain the basic concepts of the theory of chaos and bifurcations for discete dynamical systems in two-dimensions. The materials in the book and on the accompanying disc are not solely developed only with the researcher and professional in mind, but also with consideration for the student. The book is replete with some 100 computer graphics to illustrate the material, and the CD-ROM contains full-color animations that are tied directly into the subject matter of the book, itself. In addition, much of this material has also been class-tested by the authors. The cross-platform CD also contains a software program called ENDO, which enables users to create their own 2-D imagery with X-Windows. Maple scripts are provided which give the reader the option of working directly with the code from which the graphcs in the book were

Keywords

Chaos Maple calculus chaos theory computer graphics dynamische Systeme geometry linear optimization numerical methods systems theory

Authors and affiliations

  • Ralph H. Abraham
    • 1
  • Laura Gardini
    • 2
  • Christian Mira
    • 3
  1. 1.University of California Santa CruzSanta CruzUSA
  2. 2.Instituto di Scienze EconomicheUniversitá di UrbinoUrbinoItaly
  3. 3.Dept. of Control EngineeringInstitut National des Sciences Appliquees de ToulouseToulouseFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-1936-1
  • Copyright Information Springer Science+Business Media New York 1997
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7347-9
  • Online ISBN 978-1-4612-1936-1
  • Buy this book on publisher's site