Probabilistic solution of high-dimensional nonlinear systems excited by combined Gaussian and Poisson white noise

Determining the probability density function (PDF) solution for nonlinear systems excited to combined Gaussian and Poisson white noise is of great significance in stochastic dynamics. However, this task becomes computationally intractable for high-dimensional systems. To address this challenge, this study employs a dimension-reduction method previously developed for analyzing system responses under Gaussian white noise. Here, the method is extended to determine the stationary responses of nonlinear systems under combined Gaussian and Poisson white noise excitations. The proposed approach utilizes the generalized Fokker-Planck-Kolmogorov (GFPK) equation to describe the evolution of the system response PDF. By integrating the unconcerned state variables in the series-form GFPK equation, a one-dimensional integro-differential equation is obtained. Then, equivalent drift coefficients are introduced for this dimension-reduced GFPK equation, whose equivalent form is still a one-dimensional forward GFPK equation. Finally, the fourth-order finite difference method (FDM) is applied to solve this dimension-reduced GFPK equation, and its stationary velocity responses are obtained. A six degree-of-freedom (DOF) coupled Duffing-van der Pol system and an 8-DOF system with nonlinear terms in the velocities are demonstrated to validate the proposed approach. Numerical results show that the proposed approach is in good agreement with the Monte Carlo simulation (MCS) results. In addition, some parameters, such as nonzero mean impulse amplitudes of the Poisson noise, are discussed in the examples.