The main purpose of this paper is to investigate special forms of bursting oscillations with the trajectory sliding along the non-smooth surfaces as well as the associated mechanism in a Filippov-type model. Upon the traditional Chua’s circuit, a coupled three-dimensional non-smooth dynamical system with two time-scales was established. By considering the whole external excitation as the bifurcation parameter, the conditions of fold bifurcation and Hopf bifurcation were given on the basis of the stability analyses of the equilibrium point and the possible non-smooth bifurcations on the non-smooth surfaces was also discussed in details. With the help of the numerical simulations including the transformed phase portrait as well as the overlap of the transformed phase portrait and the bifurcation diagram of the slow sub-manifold of the system, the slow–fast analysis method was employed to reveal the bifurcation mechanism of the ‘point-cycle’-type periodic bursting oscillations, in whichdifferent non-smooth bifurcations of cycles can be observed under different values of parameters, leading to rich dynamic phenomena in non-smooth systems with multiple time-scales.