Catastrophe Theory

Abstract

In applied mathematics, engineering and physics, the subject of “shock waves” is plainly very important, since it appears in so many physical problems. Shock waves occur under a sudden change in some physical phenomenon (or physical system) from one state to another. Because of the discontinuous nature of a shock wave, it is a difficult phenomenon for physicists, engineers as well as mathematicians to comprehend. Recently, René Thom has classified the singularities for certain classes of functions, and this added enormously to our understanding of the qualitative aspect of discontinuity in natural processes. The main theoretical significance of Thom’s classification is that it allows one to determine the stable equilibria of a gradient system subject to a small number of constraints, and to describe how these equilibria change as the constraints vary. This classification theorem is at the heart of catastrophe theory.