The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

中文翻译：

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具有参数和外部激励的 Filippov 系统中的突发振荡及其机制

本文的主要目的是探索爆发振荡以及参数和外部激发 Filippov 型系统 (PEEFS) 的机制，其中不同类型的爆发振荡，如折叠/非光滑折叠 (NSF)/折叠/NSF ，可以观察到折叠/NSF/折叠和折叠/折叠爆裂振荡。通过使用变换相图和广义自治系统的平衡分支的重叠，研究了突发振荡的机制。我们的结果表明，折叠分岔和边界平衡分岔 (BEB) 可以导致静止状态和重复尖峰状态之间的转换。尖峰态的振荡频率可以通过它们的发生机制在理论上进行近似，这与数值模拟非常吻合。