Underlying catastrophes: umbilics and pattern formation

Bifurcation theory provides a very general means to classify the local changes in numbers of zeros of vector fields, but not a general means to find where a given bifurcation occurs, at least at higher codimensions. Instead, it turns out, these bifurcations can be found by looking for their underlying catastrophes. Here I show that the concept of underlying catastrophes can be extended to the umbilics. The umbilics are important in opening up qualitatively different forms of bifurcations beyond the ‘corank 1’ catastrophes of folds, cusps, swallowtails, etc. An example is given showing how four zeros of a vector field bifurcating from a single point, may do so either via a 3-parameter swallowtail catastrophe involving equilibria of similar stabilities, or via a 4-parameter umbilic catastrophe involving equilibria of opposing stabilities. This opens an avenue to studying spatiotemporal pattern formation around high codimension bifurcation points, and I conclude with some illustrative examples.