Bifurcation Analysis of Planar Piecewise Linear System with Different Dynamics

In this paper, we investigate the bifurcation phenomena of a planar piecewise linear system. This piecewise linear system comprises two linear subsystems. The two linear subsystems have different types of dynamics. One subsystem has node or saddle dynamic and the other has focus dynamic. Some sufficient and necessary conditions for the existence of periodic orbit are given by studying the properties of Poincaré maps. Our results show that two crossing periodic orbits can bifurcate from this piecewise linear system. Moreover, we establish some sufficient and necessary conditions for the existence of sliding periodic orbit, crossing–sliding periodic orbit and sliding homoclinic orbit passing through a pseudo saddle and so on. We find that this piecewise system can appear multiply as two limit cycle bifurcation, buckling bifurcation, critical crossing cycle bifurcation, sliding homoclinic bifurcation, pseudo homoclinic bifurcation and so on. To our knowledge, sliding bifurcation phenomena are usually ignored when people study piecewise linear systems.