A New Method for the Probabilistic Solutions of Large-Scale Nonlinear Stochastic Dynamic Systems

This paper proposes a novel method for determining the probabilistic stationary solution of multi-dimensional nonlinear stochastic dynamic systems. In general, the PDF solution is governed by Fokker-Planck equations in multi-dimensions. By dividing the space of the state variables into two subspaces and integrating the Fokker-Planck equation over one of the subspaces, a reduced set of Fokker-Planck equations can be obtained in the state variables of the other subspace. This is achieved by manipulating the integrals and approximating the conditional PDFs resulted from integration. Hence, the reduced set of Fokker-Planck equation will have a smaller number of state variables at choices and can be solved by the exponential polynomial closure method. Examples of the nonlinear stochastic dynamic systems with polynomial nonlinearity are given to show the effectiveness of this novel subspace method. The paper attempts to provide a tool for analyzing the probabilistic solutions of some highly multi-dimensional nonlinear stochastic dynamics systems in various areas of science and engineering.