An oscillator with two discontinuous lines and Van der Pol damping

In this paper we investigate the discontinuous limit case of a smooth oscillator with a Van der Pol damping, which is a Filippov system with two discontinuous lines. The qualitative properties of all equilibria including that at infinity are obtained for this discontinuous piecewise smooth oscillator. By applying qualitative theory for smooth systems and for nonsmooth systems, we give necessary and sufficient conditions for the existence of limit cycles and grazing cycles. Particularly, it is demonstrated that this oscillator has at most two large limit cycles, two small limit cycles, one large double limit cycle and three classes of grazing cycles in different parameter regions. We present completely the bifurcation diagram and all global phase portraits of this oscillator model.