时滞反馈下分数阶Rayleigh系统的稳定性分析

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (2) : 1-6.

论文

陈聚峰1,2，申永军1,3，张静4，李向红2，王晓娜5

作者信息 +

Stability analysis of a fractional-order Rayleigh system with time-delayed feedback

CHEN Jufeng1,2，SHEN Yongjun1,3，ZHANG Jing4，LI Xianghong2，WANG Xiaona5

Author information +

文章历史 +

摘要

本文主要研究了时滞反馈下分数阶Rayleigh系统的稳定性和Hopf分岔发生的条件。首先，得到了具有线性速度反馈的分数阶Rayleigh系统的平衡点渐近稳定的充要条件，发现它不仅与反馈增益有关，还与分数阶阶次有关。其次，以时滞作为分岔参数，对具有线性时滞速度反馈的分数阶Rayleigh系统进行了稳定性分析。在一定条件下，计算出了时滞的临界值，当时滞参数小于该值时，平衡点是稳定的，当时滞参数大于该值时，平衡点是不稳定的；进而，得到了Hopf分岔发生的条件。最后，选取了三组系统参数进行数值模拟，验证了所得理论结果的正确性。

Abstract

The stability and existence conditions of Hopf bifurcation of a commensurate Rayleigh system with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order Rayleigh system with linear velocity feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order Rayleigh system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing three typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results.

导出引用

陈聚峰1,2，申永军1,3，张静4，李向红2，王晓娜5. 时滞反馈下分数阶Rayleigh系统的稳定性分析[J]. 振动与冲击, 2023, 42(2): 1-6

CHEN Jufeng1,2，SHEN Yongjun1,3，ZHANG Jing4，LI Xianghong2，WANG Xiaona5. Stability analysis of a fractional-order Rayleigh system with time-delayed feedback[J]. Journal of Vibration and Shock, 2023, 42(2): 1-6

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