摘要: 张弛振荡现象普遍存在于自然科学以及工程技术的各个领域，探索张弛振荡的可能路径是张弛振荡研究的重要问题之一。最近，一种名为“脉冲式爆炸”（pulse-shaped explosion，简称PSE）的可以诱发张弛振荡的新机制被相继报道。PSE意味着平衡点和极限环表现出了与参数变化相关的脉冲式急剧量变，这导致系统出现急剧转迁现象，进而诱发张弛振荡。本文以多频激励Mathieu-van der Pol-Duffing系统为例，探讨了复合式的张弛振荡现象。当参数激励和外部激励存在相位差时，快子系统包含了两个不同的向量场部分，由此得到了系统的双稳定特性。特别地，在狭小的参数范围内，分岔会随着PSE的产生而产生，这使得PSE更具复杂性。基于此，揭示了两种复合式的张弛振荡，其特征是每一周期的演化过程包含了由PSE连接的两个张弛振荡簇。我们的研究深化了对PSE及张弛振荡复杂动力学行为的理解。
Compound relaxation oscillations connected by pulse-shaped explosion
- Received Date:
29 November 2019
Abstract: Relaxation oscillations are ubiquitous in various fields of natural science and engineering technology. Exploring possible routes to relaxation oscillations is one of the important issues in the study of relaxation oscillations. Recently, the pulse-shaped explosion (PSE), a novel mechanism which can lead to relaxation oscillations, has been reported. The PSE means pulse-shaped sharp quantitative changes related the variation of system parameters in branches of equilibrium points and limit cycles, which leads the system’s trajectory to undertake sharp transitions and further induces relaxation oscillations. Regarding externally and parametrically excited nonlinear systems with different frequency ratios, some work on PSE has been reported. The present paper focuses on the PSE and the related relaxation oscillations in a externally and parametrically excited Mathieu-van der Pol-Duffing system. We show that if there is an initial phase difference between the slow excitations with the same frequency ratio, the fast subsystem may compose of two parts with different s, each of which determines a different vector field. As a result, the bistable behaviors are observed in the system. In particular, one of the vector fields exhibits two groups of bifurcation behaviors, which are symmetric with respect to the positive and negative PSE, and each can induce a cluster in the relaxation oscillations. Based on this, we present several routes to compound relaxation oscillations, and obtain new types of relaxation oscillations connected by pulse-shaped explosion, which we call compound “subHopf/fold-cycle” relaxation oscillations and compound “supHopf/supHopf” relaxation oscillations, respectively. Our results show that the two clusters in the resultant relaxation oscillations are connected by the PSE, and the initial phase difference plays an important role in transitions to different attractors and the generation of relaxation oscillations. Although the research in this paper is based on a specific nonlinear system, we would like to point out that the bistable behaviors, the PSE and the resultant compound relaxation oscillations can also be observed in other dynamical systems. The reason is that the fast subsystem composes of two different vector fields induced by the initial phase difference, which dose not rely on a specific system. The results of this paper deepen the research about PSE as well as the complex dynamics of relaxation oscillations.