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.

In this paper, the mechanism of system solutions approaching infinity is explored based on a modified Rayleigh–Duffing oscillator with two slow-varying periodic excitations. System solutions approaching infinity is a new novel route to bursting oscillation, and are not reported yet. The system can be separated into a fast subsystem and a slow subsystem according to the slow–fast analysis method. We find that there is a critical value for the fast subsystem, which limits the original region of the stable equilibrium point and the stable limit cycle, the right of which is the divergent region. When the control parameter slowly varies closely to the critical value $\delta_{\mathrm{CR}}$, both the stable equilibrium point and the stable limit cycle quickly leave the original region and approach positive infinity. The mechanism of two different bursting forms called bursting oscillation of point/point and bursting oscillation of cycle/cycle induced by system solutions approaching infinity are explored. This paper provides a new possible route to bursting oscillation unrelated to bifurcations and deepens the comprehension of bursting dynamics behaviours. Lastly, the accuracy of our study is verified by overlapping the transformed phase portraits onto the bifurcation diagrams.

By taking a low-frequency excited modified Chua’s circuit model with discontinuous vector field as an academic example, this paper introduces the periodic bursting patterns characterised by stick–slip motions as well as the underlying generation mechanism uniquely belonging to Filippov-type slow–fast dynamical systems (FSFDSs). Based on the tangencies and the visibilities, this paper is the first to clearly explain the unique transition routes to bursting phenomena involving stick–slip motions by introducing the subdivisions of the sliding region. The results indicate that such transition routes are heavily dependent on the local structures of pseudoequilibrium (PE), performing distinct non-conventional bifurcation schemes. Furthermore, two transition routes respectively corresponding to a stable node-typePEand a stable focus-typePEas well as the induced periodic bursting oscillations are further discussed in numerical simulations. Particularly, it should be pointed out that the second one is a novel route to bursting oscillation in FSFDSs.

The main purpose of this paper is to explore the mechanism of the non-smooth bursting oscillations in a Filippov neuronal model with the coupling of two scales and to try to explain some special phenomena appearing on the attractors. Based on a typical Hindmarsh–Rose neuronal model, when the recovery variable of the slow current is replaced by a slow-varying periodic excitation, which means the exciting frequency is far less than the natural frequency, the coupling of two scales in frequency exists, leading to the non-smooth bursting oscillations. By regarding the whole exciting term as a slow-varying parameter, we can define the full subsystem as Filippov type,which appears in generalised autonomous form. Equilibrium branches and their bifurcations of the fast subsystem can be derived by varying the slow-varying parameter.With the increase of the exciting amplitude, different types ofequilibrium branches and the bifurcations may involve the slow–fast vector field, whichmay cause qualitative changeof the bursting attractors, resulting in several types of periodic non-smooth bursting oscillations. By employing the modified slow–fast analysis method, the mechanism of the bursting oscillations is presented upon overlapping thetransformed phase and the equilibrium branches as well as their bifurcations of the generalised autonomous system. The sliding phenomenon in the bursting oscillations may occur since the governing system with different stableattractors may alternate between two subsystems located in two neighbouring regions divided by the boundary. Furthermore, the inertia of the movement along an equilibrium branch increases with the increase of the exciting amplitude, leading to the disappearance of the influence of the associated bifurcations on the attractors.