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.

中文翻译：

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Filippov型蔡氏回路中具有边界同宿分岔的爆裂振荡

典型蔡氏电路的修改版本，具有周期性外部激励和分段非线性电阻器，被认为是为了研究 Filippov 系统中可能的突发振荡和动力学机制。当外部激励频率远小于自然频率时，观察到两种新的对称周期性爆发振荡。除了传统的Hopf分岔，当将整个激发项作为分岔参数时，讨论了两个非光滑分岔，即边界同宿分岔和非光滑折叠极限环分岔。从理论上分析了Filippov系统的滑动解和切换流形上滑动矢量场的伪平衡分岔。基于分岔和滑动解的分析，揭示了爆裂振荡的动力学机制。外部激励在产生爆裂振荡中起着重要作用。也就是说，只有当激发项通过边界同宿分岔时，才可能形成爆发振荡。否则，它们不会发生。此外，突发振荡的两个对称相邻尖峰之间的时间间隔和系统停留在稳定伪节点的持续时间取决于激励频率。