Approximate stationary solution and stochastic stability for a class of differential equations with parametric colored noise

This paper aims to study a class of differential equations with parametric Gaussian colored noise. We present the general framework to get the solvability conditions of the approximate stationary probability density function, which is determined by the Fokker-Planck-Kolmogorov (FPK) equations. These equations are derived using the stochastic averaging method and the operator theory with the perturbation technique. An illustrative example is proposed to demonstrate the procedure of our proposed method. The analytical expression of approximate stationary probability density function is obtained. Numerical simulation is carried out to verify the analytical results and excellent agreement can be easily found. The FPK equation for the probability density function of order ε ⁰ is used to examine the almost-sure stability for the amplitude process. Finally, the stability in probability of the amplitude process is investigated by Lin and Cai's method.