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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