Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Itô equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.