Efficient stochastic response analysis of high-dimensional nonlinear systems subject to multiplicative noise via the DR-PDEE

Significant challenges persist for the reliable probabilistic analyses of high-dimensional nonlinear dynamical systems subject to multiplicative white noise, particularly at the level of probability density. To address this issue, an efficient method is proposed by employing the dimension-reduced probability density evolution equation (DR-PDEE) to determine the probability density of the responses in such systems. The starting point of the method is that, in many cases, only a limited number of quantities in a system are of interest. Thus, the corresponding DR-PDEE is a one- or two-dimensional partial differential equation (PDE) that governs the instantaneous probability density function (PDF) of the quantity/response(s) of interest in high-dimensional stochastic dynamical systems. This is with the stipulation that the response meets the path continuity condition, and there is no restriction on the excitations being multiplicative or additive. The intrinsic drift and diffusion functions in the DR-PDEE are conditional expectation functions of these responses of the original high-dimensional systems that can be reliably estimated, where assessing the latter is specific to multiplicative noise problems. Interestingly, for a wide class of systems subject to multiplicative local noise, the intrinsic diffusion functions are analytically determinable. Subsequently, the instantaneous PDF of the quantity of interest can be efficiently obtained by numerically integrating the one- or two-dimensional DR-PDEE. The accuracy and efficiency of the DR-PDEE are verified by several typical nonlinear high-dimensional dynamical systems. Particularly, the DR-PDEE captures accurately the refined traits that are easily overlooked, and the tail range of response PDFs.