NON-SMOOTH BURSTING OSCILLATION MECHANISMS IN A FILIPPOV-TYPE SYSTEM WITH MULTIPLE PERIODIC EXCITATIONS
摘要: 旨在揭示含双频周期激励的不同尺度Filippov系统的非光滑簇发振荡模式及分岔机制. 以Duffing和Van der Pol耦合振子作为动力系统模型,引入周期变化的双频激励项,当两激励频率与固有频率存在量级差时,将两周期激励项表示为可以作为一慢变参数的单一周期激励项的代数表达式,给出了当保持外部激励频率不变,改变参数激励频率的情况下,快子系统随慢变参数变化的平衡曲线及因系统出现的fold分岔或Hopf分岔导致的系统分岔行为的演化机制.结合转换相图和由Hopf分岔产生稳定极限环的演化过程,得到了由慢变参数确定的同宿分岔、多滑分岔的临界情形及因慢变参数改变而出现的混合振荡模式,并详细阐述了系统的簇发振荡机制和非光滑动力学行为特性.通过对比两种不同情形下的平衡曲线及分岔图,指出虽然系统有相似的平衡曲线结构, 却因参数激励频率取值的不同,致使平衡曲线发生了更多的曲折,对应的极值点的个数也有所改变,并通过数值模拟, 对结果进行了验证.
Abstract: The main purpose of this paper is to explore non-smooth bursting oscillations as well as the bifurcation mechanisms in a Filippov-type system with different scales and two periodic excitations. By using the coupling of Duffing and Van der Pol oscillators as the dynamical system model and introducing two periodically changed electrical sources, the two periodic excitations can be converted into a function of a single periodic exciting term which can be considered as a slow-varying parameter when there is an order gap between the exciting frequency and the natural one. The equilibrium branches as well as the bifurcation mechanisms which are caused by fold or Hopf bifurcations with the variation of the slow-varying parameter are obtained in the case of two different frequencies of parametric excitation when the amplitudes of two periodic excitations are constants. Based on the transformed phase portraits and the evolutions of stable limit cycles produced by Hopf bifurcations, the critical conditions of multisliding bifurcations and various oscillation modes determined by a slow-varying parameter are derived. The oscillation mechanisms and the analysis of the non-smooth dynamic behaviors are also described in detail. By contrasting the equilibrium branches with two different frequencies of the parametric excitation, we find the equilibrium branches become more tortuous although the equilibrium branches are similar in the structure. The number of the corresponding extreme points are also changed, and the results are verified by numerical simulations.
- 多频激励 /
- Filippov系统 /
- 簇发振荡 /
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