一类三维非线性系统的复杂簇发振荡行为及其机理

马新东; 姜文安; 张晓芳; 韩修静; 毕勤胜

doi:10.6052/0459-1879-20-231

一类三维非线性系统的复杂簇发振荡行为及其机理

doi: 10.6052/0459-1879-20-231

马新东^*,2), ,,
姜文安^*,†,
张晓芳^*,
韩修静^*,
毕勤胜^*

^*江苏大学土木工程与力学学院, 江苏镇江 212013
^†江苏科技大学船舶与海洋工程学院, 江苏镇江 212003

基金项目: 1) 国家自然科学基金资助项目(12002134);国家自然科学基金资助项目(12072132);国家自然科学基金资助项目(11872188);国家自然科学基金资助项目(11972173)

详细信息

作者简介:
2) 马新东, 讲师, 主要研究方向: 非线性动力学. E-mail: 1000004942@mail.ujs.edu.cn

通讯作者:
马新东

中图分类号: O322
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- 被引次数: 0
出版历程
- 收稿日期: 2020-06-29
- 刊出日期: 2020-12-10

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM

^*Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, Jiangsu, China
^†School of Naval Architecture & Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, Jiangsu, China);

摘要

摘要: 由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.
- 三维非线性系统 /
- 簇发振荡 /
- 快慢分析 /
- Hopf分岔滞后 /
- 叉形分岔滞后
Abstract: Bursting oscillation behavior induced by multiple time-scale coupling effect is one of the important topics in nonlinear dynamics research. In this paper, complicated bursting oscillation behaviors as well as their generation mechanism of a three-dimensional nonlinear dynamo system with slowly changing parametric excitation are revealed when the excitation frequency is much smaller than the natural frequency. The system can be used to describe the dynamic behaviors of two kinds of self-exciting homopolar dynamo systems, which are mathematically equivalent. By treating the parametric excitation as a slow-varying parameter, the generalized autonomous system corresponding to the nonautonomous system as well as the fast subsystem and the slow variable are got based on the fast-slow analysis method. Then, the stabilities and bifurcations of the fast subsystem are investigated theoretically, and the correctness of the theoretical analysis is verified by a one-parameter bifurcation diagram related to a typical parameter. With the help of the overlapping of the transformed phase diagram and bifurcation diagram, the mechanism of the symmetric delayed subHopf/fold cycle bursting oscillation as well as its dynamic transitions, i.e. delayed subHopf/fold cycle bursting oscillation, symmetric delayed pitchfork bursting oscillation of focus/focus type and delayed pitchfork bursting oscillation of focus/focus type are analyzed. The result shows that, two different forms of bifurcation delay phenomenon will appear under different parameter conditions, one is the subcritical Hopf bifurcation delay, and the other one is the pitchfork bifurcation delay. In addition, our research indicates that the stabilities of the equilibrium points and the width of the pitchfork bifurcation delay interval are both influenced by the control parameter. Meanwhile, we also find that the symmetry of the dynamic behaviors is affected by the choice of the initial values. The study of this paper further deepens the understanding and the comprehending of the different bursting patterns induced by bifurcation delay phenomenon.
- three-dimensional nonlinear system /
- bursting oscillations /
- fast-slow analysis /
- Hopf bifurcation delay /
- pitchfork bifurcation delay

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