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Chaos in Discrete Dynamical Systems

A Visual Introduction in 2 Dimensions

  • Book
  • © 1997

Overview

Authors:
  1. Ralph H. Abraham

    1. University of California Santa Cruz, Santa Cruz, USA

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  2. Laura Gardini

    1. Instituto di Scienze Economiche, Universitá di Urbino, Urbino, Italy

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  3. Christian Mira

    1. Dept. of Control Engineering, Institut National des Sciences Appliquees de Toulouse, Toulouse, France

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xxv
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  2. Basic Concepts

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 3-9

    3. Basic Concepts in 1D

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 11-27

    4. Basic Concepts in 2D

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 29-37

  3. Exemplary Bifurcation Sequences

    1. Front Matter

      Pages 39-39
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    2. Absorbing Areas

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 41-58

    3. Holes

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 59-83

    4. Fractal Boundaries

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 85-115

    5. Chaotic Contact Bifurcations

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 117-150

    6. Conclusion

      • Ralph H. Abraham, Laura Gardini, Christian Mira
      Pages 151-151

  4. Back Matter

    Pages 153-248
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Keywords

About this book

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

Authors and Affiliations

  • University of California Santa Cruz, Santa Cruz, USA

    Ralph H. Abraham

  • Instituto di Scienze Economiche, Universitá di Urbino, Urbino, Italy

    Laura Gardini

  • Dept. of Control Engineering, Institut National des Sciences Appliquees de Toulouse, Toulouse, France

    Christian Mira

Bibliographic Information

  • Book Title: Chaos in Discrete Dynamical Systems

  • Book Subtitle: A Visual Introduction in 2 Dimensions

  • Authors: Ralph H. Abraham, Laura Gardini, Christian Mira

  • DOI: https://doi.org/10.1007/978-1-4612-1936-1

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1997

  • Softcover ISBN: 978-1-4612-7347-9Published: 19 April 2013

  • eBook ISBN: 978-1-4612-1936-1Published: 29 June 2013

  • Edition Number: 1

  • Number of Pages: XXV, 246

  • Topics: Dynamical Systems and Ergodic Theory, Analysis, Theoretical, Mathematical and Computational Physics

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