Mean solution of the noisy Duffing equation

Abstract

We present a method of obtaining the mean solution of the Duffing oscillator with an additive Gaussian white noise. An approximation using a known procedure for solving stochastic nonlinear equations was employed, yielding an oscillatory solution with a randomly varying phase. The mean value was obtained by taking the ensemble average of the Taylor expansion of the solution. Statistical considerations simplify the expansion into a mean solution in closed form. A decaying mean solution was obtained. This is compared to numerical simulations using the Euler-Maruyama scheme. The analytical and numerical mean trajectories are in close agreement only for initial amplitudes close to unity. This suggests that an intermediate scaling procedure might be devised to improve the numerical correspondence.