Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

Abstract

The cusp singularity—a point at which two curves of fold points meet—is a prototypical example in Takens’ classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens’ classification occurring for the nerve impulse example—folds and folded saddles—is included to make this article self-contained.