一类混沌系统中的簇发振荡及其延迟叉形分岔行为

郑健康; 张晓芳; 毕勤胜

doi:10.6052/0459-1879-18-241

一类混沌系统中的簇发振荡及其延迟叉形分岔行为

doi: 10.6052/0459-1879-18-241

江苏大学土木工程与力学学院,江苏镇江 212013

基金项目: 国家自然科学基金资助项目(11632008);国家自然科学基金资助项目(11872188)

详细信息

作者简介:
2) 张晓芳,副教授,主要研究方向：动力学与控制.E-mail： xfzhang@ujs.edu.cn

中图分类号: O322
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出版历程
- 收稿日期: 2018-07-20
- 刊出日期: 2019-03-18

BURSTING OSCILLATIONS AS WELL AS THE DELAYED PITCHFORK BIFURCATION BEHAVIORS IN A CLASS OF CHAOTIC SYSTEM

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

摘要

摘要: 由于多时间尺度问题在实际工程系统中广泛存在,关于其复杂动力学行为及其产生机制的研究已成为当前国内外的热点课题之一.簇发振荡是多时间尺度系统复杂动力学行为的典型代表,而分岔延迟又是簇发振荡中的常见现象.本文为探讨非线性系统中分岔延迟所引发的簇发振荡的分岔机制,在一个三维混沌系统中引入参数激励,当激励频率远小于系统的固有频率时,系统产生了两时间尺度簇发振荡.将整个激励项看做慢变参数,激励系统转化为广义自治系统也即快子系统,分析快子系统平衡点的稳定性以及分岔条件,并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理.文中考察了4组参数条件下系统的动力学行为,研究发现当慢变激励项周期性地通过分岔点时,系统产生了明显的超临界叉形分岔延迟行为,随着参数激励振幅的增大,分岔延迟的时间也逐渐延长,当这种延迟的动态行为终止于不同的参数区域时,导致系统轨线围绕不同稳定吸引子(平衡点,极限环)运动,从而得到了不同的簇发振荡行为.
- 叉形分岔 /
- 慢变激励 /
- 簇发 /
- 吸引域 /
- 延迟
Abstract: Due to wide existence of multiple-time-scale problems in practical engineering, the complicated dynamic behaviors and their generation mechanism have become one of the hot topics at home and abroad. The systems with multiple time scales can often exhibit bursting oscillations with the bifurcation delay phenomenon. In order to investigate the bifurcation mechanism of bursting oscillations caused by bifurcation delay in a nonlinear system, a parametric excitation is introduced in a novel three-dimensional chaotic system. When the exciting frequency is far less than the natural frequency, the coupling of two time scales involves the vector field, which leads to the bursting oscillations. By considering the whole exciting term as a slow-varying parameter, the original system can be considered as a generalized autonomous system, which can be regarded as the fast subsystem. Upon the analysis of equilibrium points and bifurcation conditions of the fast subsystem, combining with the transformed phase portraits, the bifurcation mechanisms of bursting oscillations is presented. Four typical cases with different parameter conditions are discussed to reveal the evolution of the bursting oscillations. It is found that when the slow-varying exciting term passes across the bifurcation points, the delayed behaviors of super-critical pitchfork bifurcation can be observed. With the increase of the exciting amplitude, the occurring needed for the bifurcation delay is increased gradually. When the delayed behaviors end in different parameter regions, different types of bursting oscillations which may surround different attractors such as equilibrium points and limit cycles appear.
- pitchfork bifurcation /
- slow-varying excitation /
- bursting oscillation /
- domain of attraction /
- delay

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