Stochastic stability and the moment Lyapunov exponent for a gyro-pendulum system driven by a bounded noise

Abstract

The stochastic stability of a gyro-pendulum system parametrically excited by a real noise is investigated by the moment Lyapunov exponent in the paper. Using the spherical polar and non-singular linear stochastic transformations and combining these with Khasminskii's method, the diffusion process and the eigenvalue problem of the moment Lyapunov exponent are obtained. Then, applying the perturbation method and Fourier cosine series expansion, we derive an infinite-order matrix whose leading eigenvalue is the second-order expansion g2(p) of the moment Lyapunov exponent. Thus, an infinite sequence for g2(p) is constructed, and its convergence is numerically verified. Finally, the influences of the system and noise parameters on stochastic stability are given such that the stochastic stability is strengthened with the increased drift coefficient and the diffusion coefficient has the opposite effect; among the system parameters, only the increase in k and A0 strengthens moment stability.