Relaxation oscillations in substrate-depletion oscillators close to the nonsmooth limit

Abstract

In this paper, we describe a novel type of relaxation oscillations occurring in a model of substrate-depletion oscillators. Using geometric singular perturbation theory, with blow-up as a key technical tool, we show that the oscillations in this planar model are produced by a complicated interplay between two stable nodes and a discontinuity set in the singular limit $\epsilon \to 0$. This interplay produces a new mechanism for producing relaxation-type oscillations, which we also describe in a more general setting.