双频1:2激励下修正蔡氏振子两尺度耦合行为

夏雨; 毕勤胜; 罗超; 张晓芳

doi:10.6052/0459-1879-18-017

双频1:2激励下修正蔡氏振子两尺度耦合行为

江苏大学土木工程与力学学院,江苏镇江 212013

基金项目: 国家自然科学基金重点项目(11632008)和国家自然科学基金项目(11472115, 11472116)资助.

详细信息

作者简介:
^*通讯作者：毕勤胜,教授,主要研究方向：动力学与控制. E-mail：qbi@ujs.edu.cn

通讯作者:
毕勤胜

中图分类号: O322;
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出版历程
- 收稿日期: 2018-01-12
- 刊出日期: 2018-03-17

BEHAVIORS OF MODIFIED CHUA’S OSCILLATOR TWO TIME SCALES UNDER TWO EXCITATOINS WITH FREQUENCY RATIO AT 1:2

Faculty of Civil Engineering and Mechanics,Jiangsu University, Zhenjiang 212013,Jiangsu, China

摘要

摘要: 不同尺度耦合系统存在的复杂振荡及其分岔机理一直是当前国内外研究的热点课题之一. 目前相关工作大都是针对单频周期激励频域两尺度系统,而对于含有两个或两个以上周期激励系统尺度效应的研究则相对较少. 为深入揭示多频激励系统的不同尺度效应,本文以修正的四维蔡氏电路为例,通过引入两个频率不同的周期电流源,建立了双频1:2周期激励两尺度动力学模型. 当两激励频率之间存在严格共振关系,且周期激励频率远小于系统的固有频率时,可以将两周期激励项转换为单一周期激励项的函数形式. 将该单一周期激励项视为慢变参数,给出了不同激励幅值下快子系统随慢变参数变化的平衡曲线及其分岔行为的演化过程,重点考察了3种较为典型的不同外激励幅值下系统的簇发振荡行为. 结合转换相图,揭示了各种簇发振荡的产生机理. 系统的轨线会随慢变参数的变化,沿相应的稳定平衡曲线运动,而fold分岔会导致轨迹在不同稳定平衡曲线上的跳跃,产生相应的激发态. 激发态可以用从分岔点向相应稳定平衡曲线的暂态过程来近似,其振荡幅值的变化和振荡频率也可用相应平衡点特征值的实部和虚部来描述,并进一步指出随着外激励幅值的改变,导致系统参与簇发振荡的平衡曲线分岔点越多,其相应簇发振荡吸引子的结构也越复杂.
- 多频激励 /
- 频域两尺度 /
- 簇发振荡 /
- 分岔机理
Abstract: The complicated behaviors as well as the bifurcation mechanism of the dynamical systems with different time scales have become one of the hot subjects at home and abroad, since they often behave in bursting attractors characterized by the combinations between large-amplitude oscillations and small-amplitude oscillations. Since the slow-fast analysis was employed to investigate the mechanism of the special forms of movements, a lot of results related to the bursting oscillations in autonomous systems with two scales in time domain have been obtained. Recently, based on the transformed phase portraits, different types of bursting oscillations as well as the mechanism in the vector fields with single periodic excitation have been presented. However, few works has been published related to the systems with multiple periodic excitations, the dynamics of which still remains an open problem. The main purpose of the manuscript is to explore effect of the multiple scales in such systems. As a example, based on a relatively simple four-dimensional Chua’s circuit, by introducing two periodically changed electric sources, when the two exciting frequencies are strictly resonant, both of which are far less than the natural frequency of the system, a dynamical model with scales under two periodic excitations is established. Note that the combination of the two exciting terms can be transformed as a function of a periodic term with single frequency, which can be regarded as a slow-varying parameter. The equilibrium branches as well as the associated bifurcations with the variation of the slow-varying parameter can be derived by employing the characteristics analysis of the equilibrium points. It is found that the distribution of the equilibrium branches as well as the bifurcation details may changed with the variation the amplitudes of the excitations, which may influence the attractors of the whole dynamical system. Three typical cases corresponding to the different situations of the equilibrium branches are considered, in which different forms of bursting oscillations are observed. Based on the transformed phase portraits, the bifurcation mechanism of the bursting oscillations has been presented. It is found that the trajectory may move almost strictly along one of the stable equilibrium branches, while jumping to another stable equilibrium branch may occur at the fold bifurcation points, the transient process of which leads to the large-amplitude oscillations corresponding to spiking states. Furthermore, it is pointed out that when more fold bifurcation points involve the behaviors of the system, more complicated bursting oscillations may appear.
- multi-frequency excitation /
- two scales in frequency domain /
- bursting oscillations /
- bifurcation mechanism

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参考文献(1)

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