Two-stroke relaxation oscillators

Abstract

Two-stroke relaxation oscillations consist of two distinct phases per cycle—one slow and one fast—which distinguishes them from the well-known van der Pol-type ‘four-stroke’ relaxation oscillations. This type of oscillation can be found in singular perturbation problems in non-standard form, where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. The existing literature on such non-standard problems has developed primarily through applications—we complement this by illustrating the suitability of a more general framework for geometric singular perturbation theory to prove existence and uniqueness for a general class of two-stroke relaxation oscillators. While this result can be derived from a more general result in de Maesschalck et al (2011 Indagationes Math. 22 165–206), our methods emphasise the scope, simplicity and applicability of this non-standard approach. We apply this non-standard geometric singular perturbation toolbox to a collection of examples arising in the dynamics of nonlinear transistors and models for mechanical oscillators with friction.