• QINSHENG BI

Articles written in Pramana – Journal of Physics

• Non-smooth bursting analysis of a Filippov-type system with multiple-frequency excitations

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.

• Mixed-mode oscillations and the bifurcation mechanism for a Filippov-type dynamical system

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.

• Bursting oscillations with boundary homoclinic bifurcations in a Filippov-type Chua’s circuit

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.

• Two bursting patterns induced by system solutions approaching infinity in a modified Rayleigh–Duffing oscillator

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.

• Pramana – Journal of Physics

Volume 96, 2022
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Posted on July 25, 2019