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.