共存吸引子对簇发振荡影响机理分析

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (6) : 224-231.

论文

共存吸引子对簇发振荡影响机理分析

张彬，张晓芳，马新东，毕勤胜

作者信息 +

Influence mechanism of coexisted attractors on bursting oscillations

ZHANG Bin,ZHANG Xiaofang,MA Xindong,BI Qinsheng

Author information +

文章历史 +

摘要

具有广泛工程和科学背景的多稳态系统存在着多个共存的吸引子及其相应的吸引域,其不同尺度下的行为会受到共存吸引子的影响而产生复杂的动力学特性，本文旨在探讨共存吸引子下簇发振荡的特性及其产生机制。基于经典的Hartley振子，引入周期外激，当激励频率远小于系统的固有频率时，随着参数的变化，系统呈现出不同簇发振荡模式。将周期激励项视为慢变参数，基于相应的广义自治快子系统的平衡点及其稳定性分析，得到其随慢变参数变化的平衡曲线及其相应的分岔图。发现在不同参数条件下，存在不同类型的多稳态吸引子共存情形。考虑三种典型情形下系统的快慢效应，分析各情形下共存稳态吸引子的特性及吸引域对分岔特性的影响，得到连接沉寂态和激发态的分岔模式，从而揭示相应的簇发振荡机理。指出当轨迹依次穿越不同共存吸引子的吸引域时，各稳定吸引子均会对簇发振荡的结构产生影响，使得簇发振荡变得复杂，而当轨迹仅穿越部分共存吸引子的吸引域时，根据初始条件的不同，会产生共存的簇发振荡。

Abstract

Since there exist coexisted stable attractors with different attracting basins, when the coupling of different scales involves the vector field, complicated dynamical behaviors can be observed in the multi-stable system with wide science and engineering background. The paper aims at the influence of the coexisted stable attractors on property of the bursting oscillations as well as the mechanism in the multi-stable system. Based on a classical Hatley oscillator, by introducing external periodic excitation, when the frequency is far less than the natural frequency, different forms of bursting oscillations can be obtained with the variation of parameters. Regarding the whole exciting term as a slow-varying parameter, we can derive the equilibrium branches and their bifurcations with the variation of slow-varying parameter by the stability analysis of the equilibrium point of the generalized autonomous fast subsystem. It is found that there exist several situations with different coexisted stable attractors. Here we consider three typical cases, where the characteristics of the coexisted attractors and the influence of their attracting basins on the bifurcation is presented. Accordingly, the bifurcations at the transitions between the quiescent states and spiking states are obtained, which can be used to explore the mechanism of the bursting oscillations. It is pointed out that when the trajectory passes across the attracting basins of the coexisted attractors in turn, the associated attractors may affect the structure of the bursting oscillations, which leads to complicated bursting oscillations. However, when the trajectory only passes across the attracting basins of some of the stable attractors, there may exist coexisted bursting attractors, which correspond to different initial conditions.

导出引用

张彬，张晓芳，马新东，毕勤胜. 共存吸引子对簇发振荡影响机理分析[J]. 振动与冲击, 2023, 42(6): 224-231

ZHANG Bin,ZHANG Xiaofang,MA Xindong,BI Qinsheng. Influence mechanism of coexisted attractors on bursting oscillations[J]. Journal of Vibration and Shock, 2023, 42(6): 224-231

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