A First Course in Discrete Dynamical Systems

1. Front Matter

   Pages i-xi

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2. Introduction

   • Richard A. Holmgren

   Pages 1-7

3. A Quick Look at Functions

   • Richard A. Holmgren

   Pages 9-19

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 47-56

8. Parametrized Families of Functions and Bifurcations

   • Richard A. Holmgren

   Pages 57-66

9. The Logistic Function, Part I

   • Richard A. Holmgren

   Pages 67-77

10. Symbolic Dynamics and Chaos

    • Richard A. Holmgren

    Pages 79-93

11. The Logistic Function, Part II: Topological Conjugacy

    • Richard A. Holmgren

    Pages 95-103

12. The Logistic Function, Part III

    • Richard A. Holmgren

    Pages 105-114

13. Newton’s Method

    • Richard A. Holmgren

    Pages 115-139

14. Numerical Solutions of Differential Equations

    • Richard A. Holmgren

    Pages 141-153

15. The Dynamics of Complex Functions

    • Richard A. Holmgren

    Pages 155-180

16. The Quadratic Family and the Mandelbrot Set

    • Richard A. Holmgren

    Pages 181-189

17. Back Matter

    Pages 191-215

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An increasing number of colleges and universities are offering undergradu­ ate courses in discrete dynamical systems. This growth is due in part to the proliferation of inexpensive and powerful computers, which have provided access to the interesting and complex phenomena that are at the heart of dynamics. A second reason for introducing dynamics into the undergradu­ ate curriculum is that it serves as a bridge from concrete, often algorithmic calculus courses, to the more abstract concepts of analysis and topology. Discrete dynamical systems are essentially iterated functions, and if there is one thing computers do well, it is iteration. It is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Every effort has been made to exploit this opportunity to illustrate the beauty and power of mathematics in an interesting and engaging way. This work is first and foremost a mathematics book. Individuals who read it and do the exercises will gain not only an understanding of dynamical systems, but an increased understanding of the related areas in analysis as well.

• Calc
• Mathematica
• Microsoft Access
• bifurcation
• calculus
• chaos
• development
• dynamical systems
• function
• functions
• mathematical analysis
• mathematics
• proof
• sets
• system

• Department of Mathematics, Allegheny College, Meadville, USA

  Richard A. Holmgren

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