On a complex Duffing system with random excitation

In this paper, we consider a complex Duffing system subjected to nonstationary random excitation of the form, over(z, ̈) (t) + 2 ω ξ over(z, ̇) (t) + ω² z + ε{lunate} z (t) | z (t) |² = α F (t), where z(t) is a complex function, α = 1 + i, i denotes the imaginary unit, ω, ξ represent natural frequency and damping coefficient respectively, ε{lunate} is the small perturbation parameter and nonlinearity strength, and F(t) is a random function. This equation with F(t) = 0 has connection to the complex nonlinear Schrödinger equation which appears in many important fields of physics. The truncated Wiener-Hermite expansion is applied to derive the deterministic integro-differential equations. These equations have been solved by the small parameter perturbation approach to describe the root mean square response. The approximate solution moments for the original systems has been obtained analytically. Figures are presented to show the effect of the nonlinearity strength and the damping coefficients, respectively.