Contact singularities in nonstandard slow-fast dynamical systems

We develop the contact singularity theory for singularly-perturbed (or `slow-fast’) vector fields of the general form $z^{\mathrm{\prime }}=H(z,\epsilon )$, $z\in {\mathbb{R}}^n$ and $\epsilon \ll 1$. Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable factorization. This factorization can in turn be computed explicitly in a wide variety of applications. We demonstrate these computable criteria by locating contact folds and, for the first time, contact cusps in some nonstandard models of biochemical oscillators.