On the cumulant-neglect closure method in stochastic dynamics

This paper examines the efficacy of cumulant-neglect closure methods for complex dynamical systems. Previous efforts to employ this method were limited by the complicated and lengthy nature of the governing moment equations. To address this difficulty, an algorithm was developed and is presented herein for the efficient and automated generation of the moment equations for dynamical systems with an arbitrary number of states and closed at an arbitrary level (limited only by available computational resources). Both stationary and non-stationary moment responses were obtained through solution of the closed set of moment equations. The Duffing oscillator subjected to additive Gaussian white noise and the linear oscillator subjected to the square of a Gaussian process were examined in order to demonstrate the effectiveness and accuracy of the algorithm. A number of additional systems were examined to determine the range of applicability of the cumulant neglect closure method. Some general observations regarding stability and performance of cumulant-neglect closure methods are given.