Blowup analysis of a hysteresis model based upon singular perturbations

Abstract

In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity {vanishes}, hysteresis converges in an appropriate sense to Filippov's sliding vector-field. Recently {(2022)}, however, {Bonet and Seara} showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper -- which was developed by {Bonet et al in a paper from 2017} as an attempt to unify different regularizations of piecewise smooth systems -- involves two singular perturbation parameters and includes a combination of slow-fast and nonsmooth effects. The description of this model is therefore -- from the perspective of singular perturbation theory -- challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov's sliding {vector-field} to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).