Multiple time scaling of the response of a Duffing oscillator to narrow-band random excitation

The response of a Duffing oscillator to narrow-band random excitation is considered. Two approaches, multiple time scaling and stochastic averaging, are used. Both lead to the same set of three state equations for the slowly varying local mean square response or envelope (and two other variables). The stationary solution of these equations is similar to results previously obtained by using statistical linearization. The stationary and nonstationary (from the three state equations) results compare, favourably with digital simulations. Random jumps are shown to occur if the excitation bandwidth is sufficiently small