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

TL;DR

This paper introduces a new relaxation oscillation mechanism in substrate-depletion models, analyzed through geometric singular perturbation theory, revealing complex interactions near the nonsmooth limit.

Contribution

It presents a novel relaxation oscillation mechanism involving stable nodes and discontinuities, expanding understanding of nonsmooth dynamical systems.

Findings

Oscillations caused by interplay between stable nodes and discontinuity set

Use of geometric singular perturbation theory with blow-up techniques

Mechanism applicable to broader nonsmooth models

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 $\varepsilon\rightarrow 0$. This interplay produces a new mechanism for producing relaxation-type oscillations, which we also describe in a more general setting.