Degenerate T-singularity bifurcation and crossing periodic orbits in a 3-dimensional piecewise smooth system

Abstract

In this paper we investigate the dynamics of a 3-dimensional piecewise smooth system, which is an unfolding of a degenerate T-singularity. In local aspect, under the same non-degenerate condition given in [6] and determined by the linear part of perturbation, we analyze the bifurcation of this degenerate T-singularity on the switching manifold and give the bifurcation diagram including the degenerate two-fold bifurcation curve and the transcritical bifurcation curve. When this condition is not satisfied, a further non-degenerate condition determined by terms no more than degree two of perturbation is constructed, under which we obtain the pseudo-equilibrium bifurcation curve besides previous two bifurcation curves. In non-local aspect, for a system satisfying the second non-degenerate condition we prove that there exist either one family or two families of non-isolated crossing periodic orbits or at most two crossing periodic orbits and it is reachable. Corresponding necessary and sufficient conditions are given as well as periods and locations of these crossing periodic orbits.