Abstract: By introducing suitable values of the parameters, a fast-slow coupled oscillator has been obtained. Under the analysis of the equilibrium points as well as the characteristics of the coupled system and the subsystems, combining the theory of Clarke differential inclusions, the singularities on the non-smooth boundaries are explored, which reveals that the non-conventional bifurcation composed of Hopf bifurcation and Fold bifurcation may occur when the trajectory passes across the boundary for suitable parameters. Different types of bursting phenomena associated with the corresponding parameters conditions have been obtained, the oscillating characteristics of which is discussed in details. It is pointed out that the frequency of the spiking in bursters may depend on the frequency related to Hopf bifurcation of the fast subsystem on the non-smooth boundary, while the natural frequency of the slow subsystem may influence the oscillating period of the bursters. The bifurcation mechanism of the different periodic bursters caused by the non-smooth bifurcation are presented in the end.