Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

Abstract

Abstract Stochastic dynamical systems enforced by Poisson white noise (PWN) are encountered widely in physics, chemistry, biology, and engineering fields, but it is hard to capture the probability density function (PDF) of the quantity of interest of these systems. Recently, the dimension-reduced probability density evolution equation (DR-PDEE) has shown significant advantages in probabilistic response determination of path-continuous processes, especially for systems of high dimensions and strong nonlinearity, but there are still challenges in path-discontinuous processes, such as PWN-driven systems, due to their random jumps. In the present paper, the DR-PDEE governing the PDF of any single component of state vector of interest for a high-dimensional system enforced by PWN is established. It is always a one-dimensional partial integro-differential equation regardless of the dimension of the system if merely one single quantity is of interest. The intrinsic drift function and intrinsic rate function (the latter is for parametric excitations) in the DR-PDEE can be identified numerically based on the data from representative deterministic dynamic analyses of the PWN-driven system. Then solving the DR-PDEE numerically yields the solution of transient PDF of the quantity of interest. Numerical examples are illustrated to verify the efficiency and accuracy of the proposed method.