Caustics and catastrophes

Abstract

On p. 63 of his book Stabilit6 structurelle et morphog~n~se, R. Thorn explains why the elementary catastrophes do occur as singularities of propagating wave fronts. The purpose of the present paper is to provide some of the mathematical details. Although at least two similarly motivated papers do already exist, namely those of Porteous [4] and of Guckenheimer [2], I believe that there is still much to be said about this problem. I am here only concerned with a rather limited aspect of the question, which was not given much attention in the above mentioned papers. I consider a wave front propagation mechanism on a manifold, completely described by a positive and homogeneous Hamiltonian on the cotangent bundle and look at the local gradient models given by ray length, as suggested by Thorn. What, precisely, is the relationship between the caustics of the propagation and the bifurcation sets of these caustic gradient models? And what conditions on the Hamiltonian guarantee the occurrence of a given universal unfolding as such a caustic model? After recalling some basic facts in § 1, I consider the first, preliminary question in § 2 (Theorem 1) and then use Mathers theory to give an answer to the second in § 3, (Theorem 2). The proper question of stability of caustics is not dealt with in this paper; I just consider very stable caustics and ask if all the usual catastrophes do occur as such. I think that weaker stability conditions deserve attention and can probably also be handled by standard Mather theory. Although I do not make explicit use of the papers by Guckenheimer and Porteous, they have of course helped me to understand caustics from the catastrophe point of view. I also profited from Christopher Zeeman, listening and reading, and, needless to say, from Ren~ Thorn.