Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systemsgoverned by discontinuous but piecewise smooth vector fields) stable periodic motionwith sliding is not robust with respect to stable singular perturbations. We considera simple dynamical system that we assume to be a quasi-static approximationof a higher-dimensional system containing a fast stable subsystem. We tune asystem parameter such that a stable periodic orbit of the simple system touches thediscontinuity surface: this is the so-called grazing-sliding bifurcation. The periodicorbit remains stable, and its local return map becomes piecewise linear. However,when we take into account the fast dynamics the local return map of the periodicorbit changes qualitatively, giving rise to, for example, period-adding cascades orsmall-scale chaos.