Catastrophe conditions for vector fields in ${\mathbb{R}}^n$

Abstract

Practical conditions are given here for finding and classifying high codimension intersection points of $n$ hypersurfaces in $n$ dimensions. By interpreting those hypersurfaces as the nullclines of a vector field in ${\mathbb{R}}^n$, we broaden the concept of Thom's catastrophes to find bifurcation points of (non-gradient) vector fields of any dimension. We introduce a family of determinants $B_j$, such that a codimension $r$ bifurcation point is found by solving the system $B_1=...=B_r=0$, subject to certain non-degeneracy conditions. The determinants $B_j$ generalize the derivatives $\frac{{\partial }^j}{\partial x^j}F(x)$ that vanish at a catastrophe of a scalar function $F(x)$. We do not extend catastrophe theory or singularity theory themselves, but provide a means to apply them more readily to the multi-dimensional dynamical models that appear, for example, in the study of various engineered or living systems. For illustration we apply our conditions to locate butterfly and star catastrophes in a second order PDE.