Catastrophe conditions for vector fields and varieties in n-dimensions

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 $\mathcal{B}_j$, such that a codimension r bifurcation point is found by solving the system $\mathcal{B}_1 = \ldots = \mathcal{B}_r = 0$, subject to certain non-degeneracy conditions. The determinants $\mathcal{B}_j$ 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 partial differential equation.