Science in Society

Bryn Mawr College

Emergent Systems Working Group

November 1, 2002
Victor Donnay
Ideas on Regular and Chaotic Dynamical Systems: A Mathematician's Perspective on Issues in Complex Systems

Prepared by Panama Geer
Additions, revisions, extensions are encouraged in the Forum and/or at

Dynamical systems behave according to deterministic rules such as the force=mass*acceleration , spring-mass equations, or mass and gravity relationships. In general, people investigate how one state in the dynamical system develops into another state over time. El Nino is an example of a dynamical system.

As part of the discussion, Victor presented a theoretical mathematics approach to the billiard ball problem. The audience experimented with a computer simulation of the motion of a billiard ball on a table. The simulation showed one billiard ball moving at a constant rate bouncing off the edges of a billiard table. The user could control the shape of the billiard table, the initial location and direction of the ball, as well as how many times the billiard ball bounced.

At first the audience experimented with a circular table and described (regardless of the location, direction, and number of bounces of the billiard ball) the resulting path of the ball as "regular and predictable." With careful choice of initial location and direction, a vertical or horizontal line could be the only path traced by the ball. With different choices of the initial location and direction, shapes that were reminiscent of spirographs and stars were formed. The circular billiard is an example of an "integrable" dynamical system; these are the most regular possible.

The participants found that the path that the ball followed in an elliptical table was slightly less predictable, but still not very surprising. There were some theorems suggested about the trajectory of the ball depending on its initial location and direction. In some cases hyperbolic shapes were traced by the ball and in other cases elliptic shapes were traced by the ball. The elliptical billiard was also integrable.

Finally, the audience experimented with a stadium-shaped billiard ball table. It became clear right away that the behavior of the ball was dramatically different in this case. There did not appear to be any "regularity" in the path of the billiard ball. The path that the ball followed appeared chaotic. This system exhibited sensitive dependence on initial conditions: a small change in the initial location and direction of the ball would lead to a big change in the future behavior.

Based on relatively simple, deterministic rules the system exhibited behavior that was chaotic. This is in contrast to some of the other systems the group has discussed, where based on relatives simple, non-deterministic rules the systems have exhibited behavior that is very organized.

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