Bifurcation of limit cycles in piecewise-smooth systems with intersecting discontinuity surfaces

Abstract

This paper deals with bifurcation of limit cycles for perturbed piecewise-smooth systems. Concentrating on the case in which the vector fields are defined in four domains and the discontinuity surfaces are codimension-2 manifolds in the phase space. We present a generalization of the Poincare map and establish some novel criteria to create a new version of the Melnikov-like function. Naturally, this function is designed corresponding to a system trajectory that interacts with two different discontinuity surfaces. This provides an approach to prove the existence of special type of invariant manifolds enabling the reduction of dynamics of the full system to the two-dimensional surfaces of the invariant cones. It is shown that there exists a novel bifurcation concerning the existence of multiple invariant cones for such system. Further, our results are then used to control the persistence of limit cycles for two- and three-dimensional perturbed systems. The theoretical results of these examples are illustrated by numerical simulations.