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.

A modified version of the typical Chua’s circuit, which possesses a periodic external excitation and a piecewise nonlinear resistor, is considered to investigate the possible bursting oscillations and the dynamical mechanism in the Filippov system. Two new symmetric periodic bursting oscillations are observed when the frequency of external excitation is far less than the natural one. Besides the conventional Hopf bifurcation, two non-smooth bifurcations, i.e., boundary homoclinic bifurcation and non-smooth fold limit cycle bifurcation, are discussed when the whole excitation term is regarded as a bifurcation parameter. The sliding solution of the Filippov system and pseudo-equilibrium bifurcation of the sliding vector field on the switching manifold are analysed theoretically. Based on the analysis of the bifurcations and the sliding solution, the dynamical mechanism of the bursting oscillations is revealed. The external excitation plays an important role in generating bursting oscillations. That is, bursting oscillations may be formed only if the excitation term passes through the boundary homoclinic bifurcation. Otherwise, they do not occur. In addition, the time intervals between two symmetric adjacent spikes of the bursting oscillations and the duration of the system staying at the stable pseudonode are dependent on the excitation frequency.

The main purpose of this paper is to investigate the complicated dynamical behaviours as well as the mechanism of a Filippov-type system. By introducing a non-smooth term and a periodic external excitation to a four-dimensional laser system, a new Filippov-type system with two scales can be obtained. When an order gap exists between the exciting frequency and the natural one, the whole excitation term can be considered as a slow-varying parameter. For smooth subsystems, the equilibrium branches along with bifurcations, such as fold bifurcation (FB) and Hopf bifurcation (HB) are derived. Boundary equilibrium bifurcation and non-smooth bifurcations, according to differential inclusion theory, are also explored by introducing an auxiliary parameter to the system. Several types of sliding movements, such as grazing-sliding and multisliding, can be observed with slow variation. In addition, due to the coexistence of stable attractors in the vector field, the symmetry breaking phenomenon of bursting attractors appears, the mechanism of which can be revealed by overlapping the transformed phase portrait and bifurcation curves.