Exact Stationary Response Solutions of Six Classes of Nonlinear Stochastic Systems under Stochastic Parametric and External Excitations

A systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external excitations of Gaussian white noises. The nonlinear system described here has the following form: $\ddot{x}$+g₀(x) +g₁(x)$\dot{x}$+g₂(x)$\dot{x}$²+g₃(x)$\dot{x}$³=k₁ξ₁(t) +k₂xξ₂(t) +k₃$\dot{x}$ξ₃(t), where ξ_i(t) = 1, 2, 3 are Gaussian white noise. The reduced Fokker-Planck equation is employed to get the governing equation of the probability density function. Based on this procedure, the primary focus of this paper is to find an undetermined function h(x, $\dot{x}$), which can satisfy the general FPK equation, so that the solution of the FPK equation can be found for each class of the nonlinear stochastic systems. By doing the parameter studies, we get the exact stationary response solutions of six classes of nonlinear stochastic systems. This paper will illustrate that previous certain classes of exact steady-state solutions available up until now have been special cases of exact stationary response solutions under specific conditions of stochastic parametric and external excitations.