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.

在本文中，我们研究具有Van der Pol阻尼的光滑振荡器的不连续极限情况，该振动器是具有两个不连续线的Filippov系统。对于这种不连续的分段光滑振荡器，获得了所有平衡的定性性质，包括在无穷远处的性质。通过对光滑系统和非光滑系统应用定性理论，我们给出了极限环和掠食环的存在的充要条件。特别地，证明了该振荡器在不同的参数区域中最多具有两个较大的极限周期，两个较小的极限周期，一个较大的双极限周期和三类放牧周期。我们完全展示了该振荡器模型的分叉图和所有全局相位图。