# Two-Stroke Relaxation Oscillators

## TL;DR

This paper develops a general geometric singular perturbation framework for analyzing two-stroke relaxation oscillators, which feature a distinct slow and fast phase, extending the theory to non-standard singular perturbation problems.

## Contribution

It introduces a novel geometric singular perturbation approach tailored for non-standard two-stroke oscillators, providing existence and uniqueness results.

## Key findings

- Framework applicable to nonlinear transistor dynamics
- Proves existence and uniqueness of solutions
- Demonstrates applicability to mechanical oscillator models

## 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 compliment this by providing a general framework for the application of geometric singular perturbation theory in this non-standard setting and illustrate its applicability by proving existence and uniqueness results on a general class of two-stroke relaxation oscillators. 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.