Explicit expression of stationary response probability density for nonlinear stochastic systems

Identifying the exactly or approximately explicit expression of the stationary response probability density for general nonlinear stochastic dynamical systems is of great significance in the fields of stochastic dynamics and control. Almost all the existing methods are devoted to determine the exact or approximate solution for specific values of system and excitation parameters. Herein, aimed at stochastic systems with polynomial nonlinearity and excited by Gaussian white noises, a novel method is proposed to identify the stationary response probability density which explicitly includes system and excitation parameters. The stationary probability density is first written as an exponential function according to the maximum entropy principle, the power of the exponential function is then expressed as a linear combination of prescribed nondimensional parameter clusters constituted by system and excitation parameters, and state variables, with the coefficients to be determined. The undetermined coefficients are derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov equation. The application and efficacy of the proposed method are illustrated by a typical numerical example.