Journal of Nonlinear Science https://doi.org/10.1007/s00332-020-09647-4 Slow Unfoldings of Contact Singularities in Singularly Perturbed Systems Beyond the Standard Form Ian Lizarraga 1 · Robert Marangell 1 · Martin Wechselberger 1 Received: 3 April 2020 / Accepted: 22 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract We develop the contact singularity theory for singularly perturbed (or ‘slow–fast’) vector fields of the general form z  = H (z ,ε), z ∈ R n and 0 <ε  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. Keywords Multiple timescale dynamical systems · Singularity theory Mathematics Subject Classification 58K45 · 37C10 1 Introduction Mathematical models that are described by ordinary differential equations evolving on disparate timescales are considered singular perturbation problems. In this manuscript, we focus on the geometric approach to singular perturbation theory known as Geomet- ric Singular Perturbation Theory (GSPT) pioneered by Fenichel in his seminal work (Fenichel 1979). Under a particular choice of local coordinates, such a singularly Communicated by George Haller. B Ian Lizarraga ian.lizarraga@sydney.edu.au 1 School of Mathematics and Statistics, University of Sydney, Camperdown 2006, Australia 123