Levels of Structure in Catastrophe Theory Illustrated by Applications in the Social and Biological Sciences

Catastrophe theory is a method discovered by Thorn [14] of using singularities of smooth maps to model nature. In such models there are often several levels of structure, just as in a geometry problem there can be several levels of structure, for instance the topological, differential, algebraic, and affine, etc. And, just as in geometry the topological level is generally the deepest and may impose limitations upon the higher levels, so in applied mathematics, if there is a catastrophe level, then it is generally the deepest and likely to impose limitations upon any higher levels, such as the differential equations involved, the asymptotic behaviour, etc. Again, in geometry the complexity of the higher levels may render them inaccessible, so that they can only be handled implicitly rather than explicitly, while at the same time the underlying topological invariants may even be computable. Similarly in applied mathematics the complexity of the differential equations may sometimes render them inaccessible (even to computers), so that they can only be handled irnplicitly rather than explicitly, while the underlying catastrophe can be modeled, possibly even to the extent of providing quantitative prediction. Therefore catastrophe theory pifers two attractions: On the one hand it sometimes provides the deepest level of insight and lends a simplicity of understanding. On the other hand, in very complex systems such as occur in biology and the social sciences, it can sQmetimes provide a model where none was previously thought possible. In this paper we discuss various levels of structure that can be superimposed upon an underlying catastrophe and illustrate them with an assortment of examples. For convenience we shall mostly use the familiar cusp catastrophe (see [5], [13], [14], [24]).