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

We develop the contact singularity theory for singularly perturbed (or 'slow-fast') vector fields of the general form z' = H(z, epsilon), z is an element of R-n and 0 < epsilon << 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.