Two-Stroke Relaxation Oscillators
Samuel Jelbart, Martin Wechselberger
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.
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…
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