The analysis of the stochastic evolutionary process of retarded Mathieu–Duffing oscillator

A Mathieu–Duffing oscillator with delayed feedback disturbed by Gaussian white noise is proposed and studied. Firstly, the areas where bifurcation may occur under the disturbance or not and the effect of noise on bifurcation threshold are analyzed, and we found that the noise has a stabilizing effect with the disregard of nonlinearities. Secondly, we obtain the stationary probability density through the average equation and Fokker–Planck–Kolmogorov equation, and we found that there is a D-Bifurcation in the Mathieu–Duffing stochastic delayed oscillator. Finally, the average equation is high approximate to an original oscillator with the help of numerical simulation work, which is proved by the comparison of the cumulative distribution function between the theory and the numerical.