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正负双向脉冲式爆炸及其诱导的簇发振荡

POSITIVE AND NEGATIVE PULSE-SHAPED EXPLOSION AS WELL AS BURSTING OSCILLATIONS INDUCED BY IT

  • 摘要: 簇发振荡普遍存在.探索通向簇发振荡的可能路径是簇发研究的热点问题之一."脉冲式爆炸(pulsed-shaped explosion,PSE)"是一种最近被报道的可以诱发簇发振荡的新机制,其特征为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.PSE会导致系统轨线急剧跃迁,从而诱发典型的簇发振荡.然而,目前报道的PSE中仅含有"单向的尖峰",未发现"双向的尖峰",且由其诱发的簇发振荡仅含单向的振荡簇.本文以多频激励Rayleigh系统为例,旨在揭示PSE的不同表现形式以及与此相关的簇发动力学.利用频率转换快慢分析法得到了Rayleigh系统的快子系统和慢变量.针对快子系统的分析表明,PSE表现出了较为复杂的动力学特性,其特征是PSE包含了正负双向两个不同的尖峰,此即所谓的正负双向PSE.其急剧量变行为,导致了系统轨线在单个振荡周期内出现正向和负向的多次跃迁,由此得到了由正负双向PSE所诱发的簇发振荡.根据吸引子类型分别揭示了点--点型和环--环型两类簇发振荡模式的产生机制.本文的研究给出了PSE的不同表现形式,丰富了多时间尺度下的簇发振荡的诱发机制.

     

    Abstract: Bursting oscillations, characterized by the alternation between large-amplitude oscillations and small-amplitude oscillations, are complex behaviors of dynamical systems with multiple time scales and have become one of the hot subjects in nonlinear science. Up to now, various underlying mechanisms of bursting oscillations as well as the classifications have been investigated intensively. Recently, a sharp transition behavior, called the "pulsed-shaped explosion (PSE)", was uncovered based on nonlinear oscillators of Rayleigh's type. PSE is characterized by pulse-shaped sharp quantitative changes appearing in the branches of equilibrium point and limit cycle. However, the previous work related to the PSE merely focused on the sharp transitions of unidirectional PSE, and more complex forms of PSE which may lead to more complicated bursting patterns need to be further investigated. Taking a parametrically and externally excited Rayleigh system as an example, we reveal different expression of PSE as well as bursting patterns induced by it. According to the frequency relationship between the two slow excitations, the fast subsystem and the slow variable are obtained by means of frequency-transformation fast-slow analysis. Bifurcation behaviors of the fast subsystem show that, there exist two originally disunited branches of equilibrium point and limit cycle, which extend steeply in different directions and inosculate as a integral structure according to the variation of system parameters. This inosculated integral structure inherits the "steep" properties in different directions from the originally disunited branches, and PSE is thus generated. Unlike the PSE phenomena studied in the previous works, the PSE reported here contains two different peaks in positive and negative directions, which can be named as "positive and negative PSE". Note that the positive and negative PSE essentially complicates bursting dynamics and plays a critical role in bursting. On the other hand, only with the properly chosen parameters could it be created. Based on this, two different types of bursting, i.e., point-point type and cycle-cycle type, are obtained. Subsequently, the transformed phase portraits are introduced to explore dynamical mechanisms of the bursting patterns. We show that, with the variation of the slow parameter, the trajectory may undergo sharp transitions between the rest and active states by positive and negative PSE, and therein lies the generation of bursting. Our results demonstrate the diversity of PSE and give a complement to the underlying mechanisms of bursting.

     

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