Winter, 2003

*Chaos and Fractals: New Frontiers of Science*, H.-O. Peitgen, H. Jurgens, D. Saupe. (A copy of this book is also on reserve in the mathematics library, SS622)- The following three books are standard introductory texts for
courses such as this one. Refer to them if you want to see a more
mathematical and concise treatment than is given in
our text, as well as more examples
and a supply of exercises to help you think about the material.
*A First Course in Chaotic Dynamical Systems*, R. Devaney.*Exploring Chaos*, B. Davies.*Encounters with Chaos*, D. Gulick.

For a general discussion about the history behind 'chaos theory', the scientific ideas and the people involved, I would recommend the following books.

*Chaos: Making a New Science*, J. Gleick. (A journalist's lucid description of the main discoveries and contributors to 'chaos theory'. No mathematics here though.)*Does God Play Dice?*, I. Stewart. (A mathematician's (entertaining) survey of the historical development of the ideas that lead to 'chaos theory', along with accessible explanations of the mathematics involved.)

The Audio Visual library has several videos about chaos and fractals (just type the key words 'chaos' or 'fractal' on the on-line catalogue). We will watch one video,

There are many web sites devoted to chaos and fractals, so there is much to explore here (eg., 'text book' descriptions, pictures, applications in science, fractal music, etc). Note that many universities have web sites describing the research of 'dynamical systems' or 'nonlinear dynamics' groups working in mathematics, physics, chemistry, biology, computer science, and medicine. You can start with the links listed in the resources section of the course web page (the 'Internet Resources' appendix of the

Students should have second year calculus (can be taken concurrently) and a course in linear algebra. Differential equations and complex numbers will come up, but you are not expected to have studied these before.

- Final exam : 40% of final mark
- 2 term tests (50 min. each) : 20% of final mark. (Tentatively: February 26 and March 26)
- Homework : 40% of final mark

**Part I - Fractals**- Self-similarity, examples of classical fractals (Ch 2,3)
- Fractal dimension (Ch 4)
- Drawing fractals: Iterated Function Systems (Ch 5) and the Chaos Game (Ch 6)
- Fractals in nature: plants, landscapes, random fractals (Ch 7,9)

**Part II - Dynamics and Chaos**- Discrete dynamical systems (iteration of functions) in one and two dimensions: random vs deterministic systems, graphical analysis, invariant sets, attracting sets, fixed points, periodic points, analyzing deterministic chaos (Ch 10)
- Symbolic dynamics
- Charkovsky's Theorem. Bifurcations and the route to chaos. Fiegenbaum final state diagram. (Ch 11)
- Strange attractors, continuous dynamical systems (differential equations) (Ch 12)

**Part III - Complex Dynamics**- Complex numbers (section 13.2)
- Julia sets (Ch 13)
- The Mandelbrot set (Ch 14)