David P. Feldman
This book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or physics. It introduces the key phenomena of chaos - aperiodicity, sensitive dependence on initial
conditions, bifurcations - via simple iterated functions. Fractals are introduced as self-similar geometric objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia Sets and the Mandelbrot Set. The last part
of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.
The book is richly illustrated and includes over 200 end-of-chapter exercises. A flexible format and a clear and succinct writing style make it a good choice
for introductory courses in chaos and fractals.
I. Introducing Discrete Dynamical Systems
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
There are no Instructor/Student Resources available at this time.
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.
He has recently begun a research project looking at trends in extreme precipitation events in Maine.
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