Synopses & Reviews
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.
To request a copy of the Solutions Manual, visit: http://global.oup.com/uk/academic/physics/admin/solutions
"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
"There is a great deal to like about this book, starting with the author's writing style, which I found particularly clear and enjoyable. ... All in all, this is a very valuable book. ... This is an excellent book and is highly recommended." --Mark Hunacek, MAA Reviews
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. He has recently begun a research project looking at trends in extreme precipitation events in Maine.
Table of Contents
I. Introducing Discrete Dynamical Systems
0. Opening Remarks
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
9. Chaos and the Logistic Equation
10. The Buttery Effect
11. The Bifurcation Diagram
13. Statistical Stability of Chaos
14. Determinism, Randomness, and Nonlinearity
15. Introducing Fractals
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
A. Review of Selected Topics from Algebra
B. Histograms and Distributions
C. Suggestions for Further Reading