三时间尺度下非光滑电路中的簇发振荡及机理

毛卫红; 张正娣; 张苏珍

doi:10.6052/0459-1879-20-331

三时间尺度下非光滑电路中的簇发振荡及机理

doi: 10.6052/0459-1879-20-331

江苏大学数学科学学院, 江苏镇江 212013

基金项目: 1) 国家自然科学基金资助项目(11872189)

详细信息

作者简介:
2) 张正娣, 教授, 主要研究方向: 非线性动力学. E-mail: dyzhang@ujs.edu.cn

通讯作者:
张正娣

中图分类号: O322
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出版历程
- 收稿日期: 2020-09-16
- 刊出日期: 2021-03-10

BURSTING OSCILLATIONS AND ITS MECHANISM IN A NONSMOOTH SYSTEM WITH THREE TIME SCALES

Faculty of Mathematical Sciences$,$ Jiangsu University$,$ Zhenjiang $212013$$,$ Jiangsu$,$ China

摘要

摘要: 实际工程应用中存在着诸如冲击、干摩擦、切换等非光滑因素,以此建立的动力学模型是包含非光滑项的系统. 目前针对非光滑动力系统的研究大多基于单一尺度或者两尺度, 而含有更多尺度的非光滑动力系统可能会存在更复杂的动力学现象. 本论文旨在探讨非光滑动力系统中的多尺度效应及其分岔机制.基于典型的非光滑蔡氏电路, 引入一个与系统固有频率存在量级差的周期变化的激励项, 同时通过选取适当的参数值,建立了一个三时间尺度耦合下的、含有两个分界面的四维分段线性电路系统模型, 研究了该系统存在的簇发振荡行为及其分岔机制. 首先,将对应快尺度与中间尺度的变量合并作为快变量, 将对应慢尺度的变量看作慢变量, 重新划分了快慢子系统,从而将三时间尺度耦合问题转化为两时间尺度耦合问题去分析. 然后根据双参数下的Hopf分岔情况, 对应于慢子流形的不同稳定性,给出了不同参数下系统存在的两种典型的簇发振荡行为. 最后, 基于快慢分析法, 结合转换相图以及慢子流形在非光滑分界面上的非光滑动力学行为的详细讨论, 分析了不同簇发振荡相互转化的分岔机制, 发现了一个新的簇发振荡的演化路径, 即由破坏性的擦边分岔诱导的簇发振荡.
- 非光滑系统 /
- 三时间尺度 /
- 非光滑分岔 /
- 簇发振荡
Abstract: Dynamic models established from practical engineering application are non-smooth systems owing to non-smooth factors, such as impact, dry friction and switching, etc. Up to now, most studies are in terms of the non-smooth dynamic systems with a single scale or two scales. While more complex dynamic phenomena may be observed in the non-smooth dynamic systems with more scales. The main purpose of this work is to explore multiscale effect in a non-smooth electric system and the related bifurcation mechanism. Upon the traditional Chua's circuit, by introducing a periodically excited oscillator with an order gap from the natural frequency of the system and taking suitable parameter values, a coupled 4-dimensional piecewise linear dynamic system with three time-scales and two boundaries is established to study the bursting oscillations as well as the corresponding bifurcation mechanism under three time-scales. Merging the variables corresponding to the fast scale and the variables related to the intermediate scale into the fast variables, while regarding the variable corresponding to the slow scale as the slow variable, the coupled problem with three time-scales is transformed into that with two time-scales. According to the relevant Hopf bifurcation curve under two independent parameters and the stability analyses of the slow submanifold of the fast subsystem, two different bursting oscillations of the coupled dynamic system are given in the case of two different parameter values. On the basis of the fast-slow analysis method, the transformed phase portrait and the non-smooth dynamics of the slow submanifold occurring on the non-smooth boundaries, the bifurcation mechanism of the mutual transformation of different bursting oscillations is analyzed in details, in which some helpful numerical simulations are given to illustrate the validity of our study simultaneously. At the same time, a new evolution of bursting oscillations is found, i.e., the bursting oscillation induced by destructive grazing bifurcation.
- non-smooth system /
- three time-scales /
- non-smooth bifurcation /
- bursting oscillations

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