BURSTING OSCILLATIONS AS WELL AS THE CLASSIFICATION IN THE FIELD WITH CODIMENSION-3 FOLD-FOLD-HOPF BIFURCATION
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摘要: 不同尺度耦合系统存在着广泛的工程背景, 通常表现为大幅振荡与微幅振荡交替出现的簇发振荡, 其产生机理一直是当前国内外研究的前沿课题之一. 传统的几何奇异摄动分析方法仅对时域上的两尺度耦合有效, 无法揭示频域上不同尺度之间的相互作用, 同时, 当前相关研究仅针对余维一fold或Hopf分岔展开. 本文针对频域两尺度耦合向量场存在余维三fold-fold-Hopf分岔时的复杂动力特性, 基于包含三阶非线性项以内的该分岔向量场的标准型及其普适开折, 给出相应的分岔集, 从而将双开折参数平面划分为对应于不同行为的子区域. 引入慢变周期激励项取代其中一个开折参数, 随慢变激励项的变化, 会存在两类轨迹访问子区域途径, 产生周期Hopf/LPC, Hopf/LPC/Hopf/LPC, fold/LPC/Hopf/Homoclinic和fold/LPC 4种簇发振荡类型. 在分析过程中, 发现系统轨迹上的真实分岔, 往往与理论上的分岔点之间存在着滞后效应, 这种滞后效应的滞后时间也会随着激励幅值的增大而延长, 因为激励幅值的增大, 会导致轨迹沿相应平衡态运动的惯性增大, 特别是, 当激励幅值增大到一定值后, 会导致轨迹沿某平衡态运动并穿越该区域, 也即相关分岔效应来不及出现, 从而导致振荡形式的改变. 本工作表明, 对于局部分岔下的快慢效应, 通过向量场标准型开折参数的周期扰动, 在一定程度可以对该分岔所导致的所有可能的各种簇发进行归类, 并得到其相应的产生机制.
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关键词:
- 频域两尺度耦合 /
- fold-fold-Hopf分岔 /
- 标准型及普适开折 /
- 簇发振荡 /
- 分岔机制
Abstract: The dynamical systems with the coupling of different scales observed widly in engineering problems often behave in the bursting oscilltions, characterized by the alternations between large-amplitude oscilltations and small-amplitude oscillations, the generation mechanism of which has been one of the hot topics in nonlinear science at home and abroad. The traditional geometric pertubation method can be employed to explore the mechanism of the oscillations only in the systems with two scales in time domain, which can not be directly used to investigate the interaction between different scales in frequency domain. Meanwhile, most of the results are obtained in the vector fields with codimension-1 fold or Hopf bifurcations. Here we focus on the complicated behaviors in the vector field with codimension-3 fold-folfd-Hopf bifurcation when two scales in frequency domain exist. Based on the normal form as well as its universal unfolding with the nonlinear terms up to the third order, all the possible bifurcations are derived, which divide the two unfolding parameter plane into several regions with different dynamics. By introducing a slow-varying periodic excitation instead of one of the unfolding parameters, two types of routes for the tarjectory visiting those regions can be observed, which may result in four classes of bursting oscillations, i.e., periodic Hopf/LPC, Hopf/LPC/Hopf/LPC, fold/LPC/Hopf/Homoclinic and fold/LPC bursting attractors with the variation of the exciting term. It is found that there may exist delay between the locations of the theoretical bifurcation points and the real bifurcation points on the trajectory. The delay may increase with the exciting amplitude, since the inertia of the movement along the equilibrium states may increase. Especiallty, when the exciting amplitude increases to an extent, the trajectory may pass acorss the corresponding regions before the related bifurcation occurs, which leads to the qualitative change of the oscillations. It is shown that, the slow-fast effect with local bifurcations can be investigated by using the periodic perturbation on the unfolding parameters in the normal form of the vector field, which can therefore to present all possible types of bursting patterns as well as the mechanism. -
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