# CHAOTIC DYNAMICAL SYSTEMS

### A presentation by Victor J. Donnay, Associate Professor of Mathematics, Bryn Mawr College, and students and colleagues at the College

In Dynamical Systems, an object moves according to a rule. Depending on the
rule motion, the object may move in a regular fashion or in a chaotic
fashion. We illustrate the ideas of chaos theory by letting the user play
with three different types of dynamical systems:

1. Billiards in which a ball moves around inside a billiard table.
The user can choose different shapes for the table (polygon, circle,
ellipse, stadium). For some shapes, the billiard motion is regular; for
others it is chaotic.

2. The Phase Space Game in which a point hops around inside a
rectangle. The moving point produces beautiful colored patterns. The user
can vary the rules of motion to produce either a regular pattern, a
chaotic pattern or a pattern that has a mixture of regular and chaotic
behaviour.

3. Iteration of a point on the real number line. A point moves on
the number line according to various rules that the user chooses. One can
display the motion either numerically or using the staircase method.

**Java Applets by:**
- Derya Davis, Mathematics and Physics, Bryn Mawr College
*Stadium Billiards, Circle Billiards*
- Carin Ewing, Philosophy, Bryn Mawr College
*Polygonal Billiards, Standard Map, Iteration Applet*
- Zhenjian He, Computer Science and Mathematics, Bryn Mawr College
*Elliptic Billiards*
- Tina Shen, Computer Science, Bryn Mawr College
*Polygonal Billiards, Staircase Game*

**Advisors:**
- Bogdan Butoi, Graduate Student in Mathematics, Bryn Mawr College
- Victor Donnay, Associate Professor of Mathematics, Bryn Mawr College
- Deepak Kumar, Assistant Professor of Computer Science, Bryn Mawr College

**Supported in part by a grant to the College from the Howard Hughes Medical Institute.**

Send any comments to: vdonnay@brynmawr.edu

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