The Maximal Lyapunov Exponent for a Three-Dimensional Stochastic System

Abstract

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.