Underlying Catastrophes

Underlying catastrophes provide a method for locating bifurcation points of vector fields by providing solvable conditions to find points where stationary points collide, disregarding other dynamical features. Here I summarise these conditions, showing how they develop on Thom’s catastrophe theory and the singularities of mappings. The method of finding underlying catastrophes is illustrated through some novel examples, including cases of coinciding saddle-node and saddle-focus bifurcations, and umbilic bifurcations. A gradient vector field that was given as an early refutation of Thom’s catastrophe theory is reconsidered in this framework. The geometry of the unfoldings is explored for the underlying catastrophes up to codimension 4.