Robustness of Grazing-sliding Bifurcations in Hybrid Systems

In the paper we consider how orbits characterised by the presence of so-called sliding motion, which orbits typically occur in hybrid systems of Filippov type, are affected by stable singular perturbations. To be able to pursuit our analysis we consider a planar minimal system and we tune a system parameter such that a stable periodic orbit of the system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. We then check the effect of stable singular perturbation on the grazing cycle. In the unperturbed system the periodic orbit remains stable, and its local return map becomes piecewise linear. The effect of an arbitrarily small stable singular perturbation is that the local return map changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.