Delay effect on the relaxation oscillations of a van der pol oscillator with delayed feedback
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摘要: 研究反馈控制环节时滞对van derPol振子张弛振荡的影响. 首先, 通过稳定性切换分析, 得到了系统的慢变流形的稳定性和分岔点分布图, 结果表明, 当时滞大于某临界值时, 系统慢变流形的结构发生本质的变化.其次, 基于几何奇异摄动理论, 分析了慢变流形附近解轨线的形状, 发现时滞反馈会引起张弛振荡中的慢速运动过程中存在微幅振荡, 其中微幅振荡来自于内部层引起的振荡和Hopf分岔产生的振荡两个方面; 同时, 时滞对张弛振荡的周期也具有显著的影响. 实例分析表明理论分析结果与数值结果相吻合.
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关键词:
- 张弛振荡
Abstract: This paper investigates the relaxation oscillations ofthe van der Pol oscillator with delayed feedback. Firstly, the stability ofslow manifold is determined and the bifurcation analysis on the fastsubsystem is carried out by means of stability switches. It is shown thatthe structure of the slow manifold changes essentially when the delayexceeds some critical value. Secondly, on the basis of geometric singularperturbation theory, it is shown that a small amplitude oscillation occursin the slow process of the relaxation oscillations due to the delay-inducedHopf bifurcation, and the period of the relaxation oscillations is shortenedas the delay increases. As demonstrated in the numerical simulations,oscillation with small amplitude in the slow process can be resulted frominner layer or Hopf bifurcation, or the both.-
Key words:
- relaxation oscillation.
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