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A First Course in Discrete Dynamical Systems

  • Textbook
  • © 1996

Overview

Authors:
  1. Richard A. Holmgren

    1. Department of Mathematics, Allegheny College, Meadville, USA

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Part of the book series: Universitext (UTX)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    • Richard A. Holmgren
    Pages 1-8

  3. A Quick Look at Functions

    • Richard A. Holmgren
    Pages 9-20

  4. The Topology of the Real Numbers

    • Richard A. Holmgren
    Pages 21-29

  5. Periodic Points and Stable Sets

    • Richard A. Holmgren
    Pages 31-39

  6. Sarkovskii’s Theorem

    • Richard A. Holmgren
    Pages 41-46

  7. Differentiability and Its Implications

    • Richard A. Holmgren
    Pages 47-57

  8. Parametrized Families of Functions and Bifurcations

    • Richard A. Holmgren
    Pages 59-67

  9. The Logistic Function Part I: Cantor Sets and Chaos

    • Richard A. Holmgren
    Pages 69-86

  10. The Logistic Function Part II: Topological Conjugacy

    • Richard A. Holmgren
    Pages 87-93

  11. The Logistic Function Part III: A Period-Doubling Cascade

    • Richard A. Holmgren
    Pages 95-108

  12. The Logistic Function Part IV: Symbolic Dynamics

    • Richard A. Holmgren
    Pages 109-126

  13. Newton’s Method

    • Richard A. Holmgren
    Pages 127-151

  14. Numerical Solutions of Differential Equations

    • Richard A. Holmgren
    Pages 153-165

  15. The Dynamics of Complex Functions

    • Richard A. Holmgren
    Pages 167-192

  16. The Quadratic Family and the Mandelbrot Set

    • Richard A. Holmgren
    Pages 193-202

  17. Back Matter

    Pages 203-224
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Keywords

About this book

Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.

Authors and Affiliations

  • Department of Mathematics, Allegheny College, Meadville, USA

    Richard A. Holmgren

Bibliographic Information

  • Book Title: A First Course in Discrete Dynamical Systems

  • Authors: Richard A. Holmgren

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4419-8732-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York, Inc. 1996

  • Softcover ISBN: 978-0-387-94780-8Published: 15 August 1996

  • eBook ISBN: 978-1-4419-8732-7Published: 05 September 2012

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 2

  • Number of Pages: XV, 223

  • Number of Illustrations: 4 b/w illustrations

  • Topics: Analysis, Manifolds and Cell Complexes (incl. Diff.Topology)

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