Global Dynamics of Degenerate Linear Differential Systems with Symmetry and Two Parallel Switching Lines

In this paper we investigate the global dynamics for a degenerate linear differential system with symmetry and two paralleled switching lines. After analyzing the qualitative properties of all equilibria including infinity and the number of closed orbits, we obtain all global phase portraits on the Poincaré disc. From these main results, we find necessary and sufficient conditions for the existence of crossing limit cycles, crossing heteroclinic loops and sliding heteroclinic loops, respectively, and prove that the numbers of these three types of closed orbits are all at most 1. Moreover, switching lines maybe pseudo singular lines or boundary singular lines.