Global dynamics of a SD oscillator

In this paper, we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, homoclinic bifurcation, double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides, we show that the system has five limit cycles, including four small limit cycles and one large limit cycle. At last, we give all numerical phase portraits to illustrate our results.