Table of Contents

I. Introducing Discrete Dynamical Systems
0: Opening Remarks
1: Functions
2: Iterating Functions
3: Qualitative Dynamics
4: Time Series Plots
5: Graphical Iteration
6: Iterating Linear Functions
7: Population Models
8: Newton, Laplace, and Determinism
II. Chaos
9: Chaos and the Logistic Equation
10: The Buttery Effect
11: The Bifurcation Diagram
12: Universality
13: Statistical Stability of Chaos
14: Determinism, Randomness, and Nonlinearity
III. Fractals
15: Introducing Fractals
16: Dimensions
17: Random Fractals
18: The Box-Counting Dimension
19: When do Averages exist?
20: Power Laws and Long Tails
20: Introducing Julia Sets
21: Infinities, Big and Small
IV. Julia Sets and The Mandelbrot Set
22: Introducing Julia Sets
23: Complex Numbers
24: Julia Sets for f(z) = z2 + c
25: The Mandelbrot Set
V. Higher-Dimensional Systems
26: Two-Dimensional Discrete Dynamical Systems
27: Cellular Automata
28: Introduction to Differential Equations
29: One-Dimensional Differential Equations
30: Two-Dimensional Differential Equations
31: Chaotic Differential Equations and Strange Attractors
VI. Conclusion
32: Conclusion
VII. Appendices
A: Review of Selected Topics from Algebra
B: Histograms and Distributions
C: Suggestions for Further Reading

About the Author

David Feldman joined the faculty at College of the Atlantic in 1998, having completed a PhD in Physics at the University of California. He served as Associate Dean for Academic Affairs from 2003 - 2007. At COA Feldman has taught over twenty different courses in physics, mathematics, and computer science.

Feldman's research interests lie in the fields of statistical mechanics and nonlinear dynamics. In his research, he uses both analytic and computational techniques. Feldman has authored research papers in journals including Physical Review E, Chaos, and Advances in Complex Systems. In 2011-12 he was a U.S. Fulbright Lecturer in Kigali, Rwanda.

Reviews

`Chaos and fractals are two intertwined concepts that have revolutionized many areas of science and renewed popular interest in mathematics over the past few decades. Feldman's book is a rich resource for anyone who wants a deeper understanding of these subjects without the need for advanced mathematics.'
Julien Clinton Sprott, University of Wisconsin-Madison
`David P. Feldman provides a delightful and thoughtful introduction to chaos and fractals requiring only a good background in algebra. The formal treatment of nonlinear dynamics, chaotic behavior, Lyapunov exponents, and fractal dimensions is leavened with creative analogies and many helpful and visually attractive figures and diagrams. Even more mathematically sophisticated readers will find this book a good starting point in exploring the complex and
beguiling realms of chaos and fractals.'
Robert C. Hilborn, Associate Executive Officer, American Association of Physics Teachers
`For the right audience and instructor, this is a wonderful book. With considerable effort on both sides it can take a wide audience with modest mathematics to a reasonable understanding of what is behind much of the complex phenomena seen in modern mathematical models of the physical universe.'
Thomas B. Ward, Durham University
`The book is very well produced, with excellent diagrams and very informative notes provided beside the main text. It also provides an extensive list of references for further reading.'
Scottish Mathematical Council Journal