Bifurcations of Sliding Heteroclinic Cycles in Three-Dimensional Filippov Systems

Abstract

Global bifurcations with sliding have rarely been studied in three-dimensional piecewise smooth systems. In this paper, codimension-2 bifurcations of nondegenerate sliding heteroclinic cycle $\mathrm{\Gamma }$ are investigated in three-dimensional Filippov systems. Two cases of sliding heteroclinic cycle are discussed: $(C_1)$ connecting two saddle-foci, $(C_2)$ connecting one saddle-focus and one saddle. It is proved that at most one sliding homoclinic or one sliding periodic orbit can bifurcate from $\mathrm{\Gamma }$ under certain conditions at the eigenvalues of the equilibria, but they cannot coexist. The asymptotic stability of the sliding periodic orbit and the structural feature of the bifurcation curves of homoclinic orbits are further studied. Finally, two numerical examples corresponding to cases $(C_1)$ and $(C_2)$, respectively, are simulated to verify the theoretical results.