Elementary catastrophes underlying bifurcations of vector fields and PDEs

Abstract

A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their ‘underlying catastrophes’. Here we apply the idea to the homogeneous steady states of a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these ‘underlying’ catastrophes? We then show they essentially define a restricted class of ‘solvable’ rather than ‘all classifiable’ singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom’s elementary catastrophes in much broader contexts.