There are two excellent web resources on Catastrophe Theory and on the Cusp Catastrophe in particular. One is a set of transparencies from a 1995, San Antonio lecture by E. C. Zeeman himself. Catastrophe Theory was discovered by René Thom in the 1960's. Along with his own contributions to the theory and ita pplications, Zeeman played St. Paul to Thom's Messiah and roamed the world as a tireless and eloquent expositor. Lucien Dujardin in Lille has a very rich and useful site Catastrophe Teacher complete with ingenious applets illustrating several "experiments" with the phenomena, including Zeenman's Catastrophe Machine.

The simplest example of a catastrophe, mathematically speaking,
occurs in the system consisting of a ball free to roll under gravity
in a double-well container that can be tilted from one side to
the other. Here the input is the tilt of the container, and the
output is the position of the ball. There is one catastrophe just beyond the configuration labelled 2. in the figure and on the graph. This is the point where the ball is just poised to fall to the left, but is still balanced on the right side of the well. If the ball is exactly at that point, the tiniest additional tilt will cause a large displacement of the ball (green arrow). A symmetrical catastrophe is just beyond the configuration labelled 6.
The catastrophe in this system is a one-parameter, or "co-dimension 1"
catastrophe: there is
The work of classifying mathematical catastrophes was done in the
1960's and 1970's. Together with its applications, it gave rise to
"Catastrophe Theory." For example, it is known that any one-parameter
catastrophe must be, qualitatively, exactly like the green arrow.
This is called the
In this column, we will investigate the only two-parameter catastrophe,
the |

SUNY at Stony Brook

- 1. What is a mathematical catastrophe?
- 2. An algebraic version of the double well
- 3. The Cusp Catastrophe
- 4. Doctor Zeeman's Original Catastrophe Machine
- 5. The Catastrophe Machine's unperturbed potential
function is
`y=x`^{4}

© copyright 2000, American Mathematical Society.