A modified Gaussian moment closure method for nonlinear stochastic differential equations

There are many physical phenomena in engineering applications that are mostly modeled as stochastic differential equations. Sensor noise and environmental disturbance are two main sources of randomness. Determination of a closed-form analytical solution for the dynamical systems under random excitation is significantly useful for further investigations of these systems. Linear systems with stochastic excitation obey simple classified rules, leading to straightforward procedures for derivation of their analytical solutions. However, it is not the case for nonlinear systems with stochastic excitation, where no comprehensive method has been developed to be applicable to all such systems. This article brings in a novel method for statistical analysis of stochastic nonlinear systems, especially the ones with multiple equilibria, for which the traditional methods such as statistical linearization give incorrect approximations. The proposed method is mainly based on the moment closure method and assumes the joint probability distribution function of the state vector, as a linear combination of several normal distribution functions. The method represents a solution for essentially nonlinear systems, which are far from being linearizable. Duffing oscillator with negative linear stiffness is discussed as a case study to illustrate the advantage of the proposed method compared to the traditional ones. Such nonlinear systems especially arise in energy harvesting applications, when linear harvester designs do not fulfill performance requirements.