Stochastic stability of variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise

Abstract

The stochastic asymptotic stability with probability one of the stochastic variable- mass Duffing oscillator under weakly parametric excitation is investigated by using the largest Lyapunov exponent, in which the small disturbance of mass is modeled as Gaussian white noise. First, the original system is replaced by an approximate system without mass disturbance based on an equivalent transformation. Then the averaged Ito equation of the approximate system is derived by applying the stochastic averaging method based on the generalized harmonic function. Finally, the Lyapunov exponent of the linearized averaged Ito equation is derived and a necessary and sufficient condition for the stability of the trivial solution is obtained approximately by letting this Lyapunov exponent be negative. The effects of mass disturbance on system stability are explored adequately. It turns out that mass disturbance deteriorates system stability remarkably. It is remarkable that the influences of linear stiffness on the stability of a variable-mass system are significantly different from that of an associated invariable-mass system.