Van der Pol-Duffing系统共振双Hopf分岔

Resonant Double Hopf Bifurcation of Van der Pol-Duffing System

研究时滞反馈van der Pol-Duffing系统的共振双Hopf分岔,讨论时滞量和位移反馈增益变化对双Hopf分岔的影响。利用Hopf分岔定理得到系统出现1∶2共振双Hopf分岔的充要条件;借助中心流形定理和平均化方法约化了系统,从理论上分析共振双Hopf分岔点附近的动力学行为,得到共振双Hopf分岔引起的各种周期解的近似解析解和稳定性条件;通过数值实验,验证了理论分析的正确性。结果表明,时滞和位移反馈增益不仅导致共振双Hopf分岔,而且会使系统出现多稳态周期运动。

The dynamic behavior of a resonant double Hopf bifurcation is examined for a van der Pol-Duffing oscillator with delayed feedback, and the influence on double Hopf bifurcation is investigated with time delay and amplitude variation. Sufficient and necessary condition for a 1：2 double Hopf bifurcation interactions occurring in the system is obtained by using Hopf bifurcation theorem. With the aid of center manifold theorem and the averaging method, the system is reduced on the dimension. The dynamic behavior in the vicinity of the resonant double Hopf bifurcation is investigated by studying the possible solutions and their stability analytically, various periodic solutions and their stable condition are given in the resonant double Hopf bifurcation. The analytical predictions are found to be in good agreement with the results obtained by numerically integrating the original delayed system. The obtained results show that the time delay and feedback gain may not only lead to resonant double Hopf bifurcation, but also result in various periodic motions.