Two-Stroke Relaxation Oscillators.

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. These type of oscillations can be found in singular perturbation problems in {\em non-standard form} where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. We provide a framework for the application of geometric singular perturbation theory to problems of this kind, and apply it to prove the existence, uniqueness and stability of the observed relaxation oscillations. The analysis of such two-stroke oscillations is motivated by applications which arise in the dynamics of nonlinear transistors, and models for mechanical oscillators with friction.