Researches On The Asymptotic Expansions Of Maximal Lyapunov Exponent And Moment Lyapunov Exponent For Several Kinds Of Random Dynamical Systems

Stochastic bifurcation?noise induced transition, is a special kind of nonlinear complicated phenomena which are different from the deterministic bifurcations and general chaos. The maximal Lyapunov exponent has been effectively employed as an important index in defining the stochastic bifurcation point for a random dynamical system in the probability 1 sense, which is called dynamic bifurcation point.Different from the concept of maximal Lyapunov exponent, moment Lyapunov exponent can give a more complete description of the properties of the stochastic stability for a random dynamical system, which includes both the sample stability boundary and the moment stability one, and as a result, the moment Lyapunov exponent leads to a more comprehensive understanding on the bifurcation actions for a random dynamical system. With no doubt, moment Lyapunov exponent of a random dynamical system is one of the most important indexes to define the stochastic bifurcations, which include D-bifurcation and P-bifurcation. On the basis of the theory of nonlinear random dynamical system, in the present thesis, the asympototic expansions of the maximal Lyapunov exponent and the moment Lyapunov exponent for two stochastic bifurcation systems are investigated respectively.In the paper , for a co-dimension two-bifurcation system that is on a three-dimensional central manifold and subjected to a parametric excitation by a white noise, the asymptotic expansion of the maximal Lyapunov exponent is evaluated. On the basis of the theory of singular boundaries of one-dimensional diffusion processes and via a perturbation method introduced by L. Arnold, we obtain the explicit asymptotic expressions of the maximal Lyapunov exponent for three cases, in which different forms of the coefficient matrix included in the noise excitation term are assumed.And then, the stability properties including the moment Lyapunov exponent, the maximal Lyapunov exponent and the stability in probability sense for a nonlinear Duffing-Van der Pol oscillator, that is excited parametrically by a real noise, are investigated in detail.