The geometry of the elementary catastrophes (1). the cuspoids

Abstract

The Elementary Catastrophes arise as stable singularities in a system of potentials parameterized by a manifold C (the Control Space) on a manifold X (the Behavior Space) and represented by a smooth map:

$$V:CxX \to R.$$

The present paper describes the geometry of these singularities for potentials of the type:

$$V = \frac{x}{{n + 2}}n + 2 + A\frac{x}{n}n + B\frac{x}{{n - 1}}n - 1 + ... + Rx.$$

and termed the Cuspoids.

This work was begun when the authors were both at the Mathematics Institute, University of Warwick, Coventry CV4 7AL, Warwickshire, England.