The main purpose of this paper is to explore the patterns of the bursting oscillations and the non-smooth dynamical behaviours in a Filippov-type system which possesses parametric and external periodic excitations.We take a coupled system consisting of Duffing and Van der Pol oscillators as an example. Owing to the existence of an order gap between the exciting frequency and the natural one, we can regard a single periodic excitation as a slow-varying parameter, and the other periodic excitations can be transformed as functions of the slow-varying parameter when the exciting frequency is far less than the natural one. By analysing the subsystems, we derive equilibrium branches and related bifurcations with the variation of the slow-varying parameter. Even though the equilibrium branches with two different frequencies of the parametric excitation have a similar structure, the tortuousness of the equilibrium branches is diverse, and the number of extremepoints is changed from 6 to 10. Overlying the equilibrium branches with the transformed phase portrait and employing the evolutionary process of the limit cycle induced by the Hopf bifurcation, the critical conditionsof the homoclinic bifurcation and multisliding bifurcation are derived. Numerical simulation verifies the results well.

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua’s system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smoothbifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism.

To explore the bursting behaviours in a dynamic system with non-smooth factor, this paper takes Lü system as an example, introduces a non-smooth term and a periodic external excitation, ensures that there exists an order gap between the natural frequency and the exited frequency, then a non-smooth dynamic system with two time scales is established. Through the stability analysis of the equilibrium point, the conditions of fold bifurcation and Hopf bifurcation are given. The numerical simulations show the bursting oscillations of the system underdifferent parameter values and the dynamic behaviours of the trajectory at the non-smooth interface. In addition, combining with numerical calculation and related bifurcation theory, the bifurcation types of the system at the interface are determined. Finally, the mechanism of oscillations is revealed by the superposition of bifurcation curves and transformed phase portraits.