Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise

Abstract

The stochastic stability of a second-order linear parametric oscillator whose stiffness is perturbed by a stationary zero-mean colored Gaussian process is investigated in this paper referring to the stability in the second response moments (mean square stability). The stochastic perturbation is the output of a linear filter excited by a stationary Gaussian white noise stochastic process, and the hypothesis of weak excitation is not formulated. The first assumption allows using Itô׳s stochastic differential calculus. The moment equations of the response are written till the second order by means of Itô׳s differential rule. They form an infinite hierarchy, that is the equations for the moments of order r contain moments of order larger than r. In order to close the hierarchy, the cumulant neglect closure method is applied herein. The moment equation set is closed by neglecting the cumulants of third order. Nonlinear moment equations result: they are linearized, and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations. Several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second-order parametric excitation. A first-order excitation, an Ornstein–Uhlenbeck process, does not cause stochastic resonance, while a second-order excitation does. The first result was not known previously.