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. These type of oscillations 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. 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.

Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Nonsmooth Systems

“…In Section 2.1 we introduce and distinguish regular and singular perturbation theory. In Section 2.2 we expand on geometric singular perturbation theory (GSPT), providing a review in the context of planar systems based on [55,108] which is applicable for slow-fast systems in nonstandard form. In Section 2.3 we introduce a general class of smooth perturbation problems with sharp transitions, and cover key concepts in the theory of piecewise-smooth (PWS) systems.…”

“…In standard form problems such points are known as regular fold points [56,74], and have been studied in detail in [71]. The coordinateindependent characterisation above is studied in [55], and considered in further detail in Chapter 3; see also [19]. For our present purposes it suffices to consider the minimal extended system   Blow-up allows us to 'resolve' the degeneracy at Q by 'replacing' it with a higher dimensional manifold -in this case a sphere -on which a desingularised vector field with improved hyperbolicity properties is induced.…”

“…This is possible since all singularities of desingularised vector fieldX are at least partially hyperbolic, allowing for the application of well established methods in perturbation theory, Fenichel theory and centre manifold theory. For details, we refer again to [55,71]; see also Theorem 3.3.6 in Chapter 3.…”

“…Indeed, a combination of GSPT and blow-up has been applied to study a wide variety of slow-fast systems in nonstandard form 4 which do not satisfy (SF2); see e.g. [58,55,59,60,75,80,50] for applications and [43,108] for theoretical background. For problems in nonstandard form, the slow-fast structure cannot be reflected in terms of a separation of slow/fast variables.…”

“…Sometimes when studying singularly perturbed (but not slow-fast) systems of this kind, it is possible to derive corresponding qualitatively equivalent 'auxiliary problems' which are slow-fast via nontrivial state-dependent time transformations [55,60]. In general however, this is not always possible, and problems which do not satisfy (SF1) frequently require the use of blow-up methods to resolve degeneracies stemming from both loss of normal hyperbolicity and loss of smoothness along a switching manifold Σ.…”