An oscillator with two discontinuous lines and Van der Pol damping
An oscillator with two discontinuous lines and Van der Pol damping
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具有两条不连续线和范德波尔阻尼的振荡器
DOI：
10.1016/j.bulsci.2020.102867
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发表时间：
2020
影响因子：
1.3
通讯作者：
Tang Yilei
中科院分区：
文献类型：
--
作者：
Chen Hebai;Tang Yilei
在本文中，我们研究了具有范德波尔阻尼的平滑振荡器的不连续极限情况，这是一个具有两条不连续线的 Filippov 系统。对于这个不连续分段平滑振荡器，获得了包括无穷远平衡在内的所有平衡的定性特性。通过应用光滑系统和非光滑系统的定性理论，我们给出了极限环和放牧环存在的充分必要条件。特别地，证明了该振子在不同参数区域最多具有两个大极限环、两个小极限环、一个大双极限环和三类掠线环。我们完整地展示了该振荡器模型的分岔图和所有全局相图。
In this paper we investigate the discontinuous limit case of a smooth oscillator with a Van der Pol damping, which is a Filippov system with two discontinuous lines. The qualitative properties of all equilibria including that at infinity are obtained for this discontinuous piecewise smooth oscillator. By applying qualitative theory for smooth systems and for nonsmooth systems, we give necessary and sufficient conditions for the existence of limit cycles and grazing cycles. Particularly, it is demonstrated that this oscillator has at most two large limit cycles, two small limit cycles, one large double limit cycle and three classes of grazing cycles in different parameter regions. We present completely the bifurcation diagram and all global phase portraits of this oscillator model.