具有时滞状态反馈的随机Van der Pol系统的动力学研究

研究了时滞及时滞反馈控制参数对Van der Pol系统极限环幅值的影响. 运用自适应的平均场近似方法给出了系统的线性化近似及系统参数Lyapunov稳定性的边界条件, 同时给出了Van der Pol系统的关联时间和功率谱密度的数值模拟结果. 通过与平均场近似下的解析结果比较后发现, 数值模拟结果和理论结果符合.进一步讨论了时滞反馈控制参数、噪声强度以及时滞对关联时间和功率谱密度的影响. 关键词： 平均场近似 关联时间 Lyapunov稳定性

The study of stochastic Van der Pol system with delayed state feedback

In this paper, we studied the amplitude control of the limit cycle in Van der Pol system with time-delay and time-delayed feedback strength. We show the critical stability boundary and the corresponding linearized equation of the Van der Pol system by mean-field approximation method. Mean-field approximate analytical expressions and numerical simulations for the power spectral density and the correlation time were given, and mean-field approximate analytical expressions in dependence upon noise intensity, delay time, and feedback strength are in good agreement with our numerical simulations. The time delayed feedback strength, noise intensity and time delay are taken as variable parameters to investigate their effects on the dynamics of the system, such as the power spectral density and the correlation time.