Some reflections on the efficacy of moment closure methods

In an earlier publication, the authors reported the successful solution of the Fokker-Planck equation by the finite element method for a class of second order systems subjected to both additive and multiplicative white noise excitations, giving a complete picture of the evolution of the transition probability density functions of those systems. More recently, these solutions have been used to determine the evolution of the moments of response, marginal distributions, and upcrossing rate of each system. In the course of that work, moment equations were developed for several systems employing both Gaussian and fourth order cumulant neglect closure (cumulants beyond fourth order neglected). These were integrated using a fourth order Runge-Kutta method, and the results were compared with those from the finite element analysis and with exact stationary solutions or, alternatively, from Monte Carlo simulations in the cases where exact solutions are not known (Bergman, et al. 1993, Spencer and Bergman 1993). Examination of the computed second moments of response revealed that the errors associated with the closed set of moment equations can be large, even at stationarity. Furthermore, a comparison of Gaussian and fourth order cumulant neglect closure and the finite element results showed that convergence to the exact stationary response value may be slow and, in fact, may not occur at all. Convergence of the closed moment equations was subsequently raised, and it was suggested that a closer examination would be useful. We have since attempted to examine the convergence properties of the cumulant neglect closure method. During that process, we have observed various phenomena that are, for us, unexpected, and we have attempted to understand them. We will, at present, confine our discussion to the parametrically and externally excited Duffing oscillator, the response moments of which have been studied in some detail and for which the stationary probability density function is known in the absence of parametric excitation (Caughey 1963). In the latter case, response moments of all orders are known a priori to exist.