Bifurcations and catastrophes of dynamical systems with centre dimension one

Elementary catastrophes occur in scalar or gradient systems, but the same catastrophes also underlie the more intricate bifurcations of vector fields, providing a more practical means to locate and identify them than standard bifurcation theory. Here we formalise the concept of these underlying catastrophes, proving that it identifies contact-equivalent families, and we extend the concept to difference equations (i.e. maps/diffeomorphisms). We deal only with bifurcations of corank one, and centre dimension one (meaning the system has one eigenvalue equal to zero in the case of a vector field, or equal to one in the case of a map). In this case we prove moreover that these underlying catastrophes identify topological bifurcation classes. It is hoped these results point the way to extending the concept of underlying catastrophes to higher coranks and centre dimensions. We illustrate with some simple examples, including a system of biological reaction diffusion equations whose homogenous steady states are shown to undergo butterfly and star catastrophes.