For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametricallyexcited bya real noise, a model of enhanced generalityis developed in the present paper byassuming the real noise to be a component of the output of a linear 0lter system—a zero-mean stationary Gaussian di4usion vectoral process, which conforms with the detailed balance condition. The strong mixing condition is removed in the present paper. To handle the complexities encountered in the present work, an asymptotic analysis approach and the eigenfunction expansion of the solution to the relevant FPK equation are employed in the construction of the asymptotic expansions of the invariant measure and the maximal Lyapunov exponents for the relevant system.

In this paper, the almost-sure stability condition for a co-dimension two-bifurcation system on a threedimensional central manifold, which is parametrically excited by a real noise, is investigated. A model of enhanced generality is developed by assuming the real noise as the first component of an output of a linear filter system—a zero-mean stationary Gaussian diffusion vectoral process, which conforms to the detailed balance condition. The strong mixing condition, which is the essential theoretic basis for the stochastic averaging method, is removed in the present study. To solve the complicated problem encountered in this work, the asymptotic analysis approach and the eigenfunction expansion of the solutions to the relevant Fokker-Planck equations are employed in the construction of the asymptotic expansions of the invariant measures and the maximal Lyapunov exponents for the relevant system.

This paper examines the almost-sure asymptotic stability condition of a linear multiplicative stochastic system, which is a linear part of a co-dimension two-bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by an ergodic real noise. The excitation is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process which is the output of a linear filter system, while both the detailed balance condition and the strong mixing condition are removed. Through a perturbation method and the spectrum representations of the Fokker Planck operator and its adjoint operator of the linear filter system, the explicit asymptotic expressions of the maximal Lyapunov exponent for three case studies, in which different forms of the coefficient matrix included in the noise excitation term are assumed, are obtained.

In this paper, we evaluate the maximal Lyapunov exponent for a co-dimension two bifurcation system, which is on a threedimensional central manifold and is subjected to a parametric excitation by a white noise. Through a perturbation method, we obtain the explicit asymptotic expressions of the maximal Lyapunove xponent for three cases, in which different forms of the coefficient matrix that are included in the noise excitation term are assumed.

在随机动力系统中的分岔──噪声导致的跃迁行为，是一种有别于确定性系统分岔与混沌的独特的非线性复杂现象．本文全面评述非线性随机系统的稳定性问题、离出问题、随机动力系统理论和随机分岔等各项研究的发展历史、基本的思想方法以及主要的研究成果．

This article examines a viscoelastic plate that is driven parametrically by a non-Guassian colored noise, which is simplified to an Ornstein-Uhlenbeck process based on the approximation method. To examine the moment stability property of the viscoelastic system, we use the stochastic averaging method, Girsanov theorem and Feynmann-Kac formula to derive the approximate analytic expansion of the moment Lyapunov exponent. Furthermore, the Monte Carlo simulation results for the original system are given to check the accuracy of the approximate analytic results. At the end of this paper, results are presented to show some quantitative pictures of the effects of the system parameters, noise parameters and viscoelastic parameters on the stability of the viscoelastic plate.