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# Chaos on the Circle

by

Susan Niefield
Union College

February 12, 2007
4:45 pm
Bailey Hall 201

Refreshments will be served

## Abstract:

Take a point $P$ on the unit circle in the plane and double the angle $\theta$ to get a point $f (P)$.

Repeating this process gives a set $\{P, f(P), f(f(P)), \ldots\}$ of points, called the orbit of $P$ under $f$. What kind of orbits can we find? Of course, that depends on the starting point $P$. Some orbits are finite, while others are not. Among the infinite ones, there are even orbits that hit every point on the circle.

This map is an example of a chaotic dynamical system. After presenting a definition of chaos, we will show that points on the unit circle can be represented by binary sequences, and use this representation to prove that the angle-doubling map $f$ is chaotic.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.
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