Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noises

We develop the stochastic averaging method to investigate the asymptotic stationary solution and the stochastic bifurcations of a class of single degree of freedom system with combined Gaussian white and colored noises, and to derive the Fokker-Planck-Kolmogorov (FPK) equations. The general stationary solutions will be obtained analytically under some suitable conditions. Theoretically, a general algebraic expression of the stationary probability density function of amplitude for the dynamical system is presented to consider the influences of correlation time of the noise and the noise intensity on stochastic bifurcations. Then, an example is given to illustrate the averaging method, and the effectiveness of the averaging method is verified via comparing the analytical results with those from Monte Carlo simulation. Finally, stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution. It indicates that system parameters, noise intensity, and noise correlation time can be treated as bifurcation parameters, respectively.