A First Course in Discrete Dynamical Systems

1. Front Matter

   Pages i-xv

   PDF

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

    PDF

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.

• bifurcation
• calculus
• convergence
• differential equation
• dynamical systems
• iteration
• metric space
• real number

• Department of Mathematics, Allegheny College, Meadville, USA

  Richard A. Holmgren

Policies and ethics