Discontinuity mappings for stochastic nonsmooth systems

Abstract

For stability and bifurcation analysis involving recurrent behaviour such as periodic orbits, it is important to be able to quantify how nearby trajectories behave by means of a local mapping. In smooth systems these mappings can be computed using the system’s variational equations. For piecewise-smooth or hybrid systems the same technique cannot be used without some corrections. This is due to the fact that nearby trajectories can be topologically distinct because they can undergo different sequences of events associated with the system’s discontinuity boundaries. To account for this, one can derive zero-time discontinuity mappings associated with boundary interactions. In this paper we derive zero-time discontinuity mappings for piecewise-smooth vector fields and hybrid dynamical systems in which the position of the discontinuity boundary has a stochastic component. In particular, we consider systems with stochastically oscillating boundaries and systems with stochastic imperfections on the discontinuity boundary.