Chaotic Dynamics Arising from Sliding Heteroclinic Cycles in 3D Filippov Systems

Abstract

Filippov systems are a representative class of piecewise smooth dynamical systems with sliding motion. It is known that such systems can exhibit complex dynamics, but how they generate chaos remains to be further studied. This paper establishes three Shilnikov-type heteroclinic theorems for 3-dimensional (3D) Filippov systems divided by a smooth surface, which admit heteroclinic cycles sliding on the switching surface. These theorems correspond to two typical scenarios of sliding heteroclinic cycles: (i) connecting two saddle-foci; (ii) connecting one saddle and one saddle-focus. In the presence of a sliding heteroclinic cycle, the corresponding Filippov system can be analytically proved to have a chaotic invariant set nearby the singular cycle under some assumed conditions. These results provide a reasonable explanation for the chaotic behaviors of 3D Filippov systems. Two numerical examples are presented to validate the theorems.