  In this paper a new method is presented for the formulation and solution of
two-time, response-excitation moment equations for a nonlinear dynamical system
excited by colored, Gaussian or non-Gaussian processes. Starting from equations
for the two-time moments (e.g. for Cxy(t,s), Cxx(t,s)), the method uses an
exact time-closure condition, in addition to a Gaussian moment closure, in
order to obtain a closed, non-local in time (causal) subsystem for the one-time
(t=s) moments. After solving this causal system, the two-time moments can be
calculated for all (t,s) pairs as well. The present method differs essentially
from the classical It\^o/FPK approach since it does not involve any specific
assumptions regarding the correlation structure of the excitation. In the case
where the input random process can be obtained as the solution of an It\^o
equation (as, e.g., happens with an Ornstein-Uhlenbeck process), the proposed
non-local system is localized, leading to moment equations identical with the
usual ones. The closed, non-local in time, moment system is numerically solved
by means of an appropriate, two-scale, iterative scheme, and numerical results
are presented for two families of colored stochastic excitations. The results
are confirmed by means of extensive Monte Carlo simulations. It is found that
both the correlation time and the details of the shape of the input random
function affect appreciable the response covariance. In the present paper we
focus on a monostable cubic half-oscillator, excited by a smoothly-correlated,
linear-plus-cubic-Gaussian (non-Gaussian) random input. The bistable case, as
well as more general nonlinear systems can be treated by the same method,
provided that a more elaborate moment closure will be used.
