Small-scale instabilities in dynamical systems with sliding

TL;DR

This paper shows that in Filippov systems, stable periodic motions with sliding are not robust under small perturbations, leading to complex behaviors like chaos when fast dynamics are considered.

Contribution

It demonstrates how stable sliding motions in Filippov systems become unstable when fast dynamics are included, revealing new bifurcation phenomena.

Findings

Stable periodic orbits with sliding are not robust under perturbations.

Fast dynamics induce qualitative changes in the return map.

Small-scale chaos can emerge from grazing-sliding bifurcations.

Abstract

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit 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 periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.