ON THE SLIDING BIFURCATION OF A CLASS OF PLANAR FILIPPOV SYSTEMS

Abstract

In this paper, we study the sliding bifurcation phenomena of a class of planar piecewise smooth differential systems consisting of linear and quadratic subsystems. Using the differential inclusion and the qualitative theory of ordinary differential equations, we find some new interesting phenomena appearing in the piecewise smooth differential systems. In brief, we prove that the system may have sliding homoclinic bifurcation, sliding cycle bifurcation, semistable limit cycle bifurcation and heteroclinic cycle bifurcation. In addition, the mentioned systems can have at most two limit cycles, and the maximal number of limit cycles can be realized and central nested with one bifurcated from the sliding–crossing bifurcation of a sliding cycle and the other from the saddle homoclinic bifurcation. These two limit cycles collide and then both disappear. This novel scenario is verified by our systems.