Determining response probability density of nonlinear stochastic systems by modifying analytical result of degenerated systems

Abstract In this manuscript, an approximate technique is established to determine the stationary response probability density of nonlinear stochastic systems. The stationary probability density is assumed to be an exponential function with the power constituted by two specified parts: the first part is the power of the analytical solution of the associated degenerated system (e.g., the reduced linear/nonlinear system, equivalent linear/nonlinear system) while the second is an additional polynomial function of state variables with to-be-determined coefficients. Substituting the exponential-form expression into Fokker-Planck-Kolmogorov equation yields the residual error, and the coefficients can be determined by minimizing the mean-square value of this residual error. Furthermore, the accuracy of the proposed method can be improved by an iterative procedure. Several typical examples, including van der Pol-Duffing system, Column frictional system and a nonlinear system with complex damping and stiffness, are systematically investigated to demonstrate the validity and efficiency of the proposed method.