Slow Unfoldings of Contact Singularities in Singularly Perturbed Systems Beyond the Standard Form

Abstract

We develop the contact singularity theory for singularly perturbed (or ‘slow–fast’) vector fields of the general form $$z' = H(z,\varepsilon )$$ , $$z\in {\mathbb {R}}^n$$ and $$0 < \varepsilon \ll 1$$ . 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 general slow–fast models of biochemical oscillators and the Yamada model for self-pulsating lasers.