The maximal lyapunov exponent for a three-dimensional stochastic system

The investigation of the maximal Lyapunov exponent for a random dynamicalsystem is the primary research focus in the field of stochastic bifurcation.This is mainly attributed to the fact that for a random dynamical system,Lyapunov exponent is analogous to the real part of the eigenvalue, whichcharacterizes the exponential rate of change of the response. Therefore, thealmost-sure stability of the stationary solution of a random dynamicalsystem depends on the sign of its maximal Lyapunov exponent.The system considered in this paper is a typical co-dimension twobifurcation system that is on a three-dimensional center manifold. Based onthe perturbation method of Arnold and the singularity theory forthe one-dimensional diffusion process, the maximal Lyapunovexponents for the co-dimensional two bifurcation system subjected toparametric excitation by a white noise of small intensity are evaluated. Itis well known that the expression of the maximal Lyapunov exponent dependson the form of coefficient matrix B that is included in the noiseexcitation term, and the singularities of the diffusion behaviors of thephase diffusion process take place as a consequence of a general form ofmatrix B. As an extension of the work of Liu and Liew10, thepresent study attempts to investigate in detail the general conditions thatlead to the coexistence of the two kinds of singular boundaries of theone-dimensional phase diffusion process and then the conditions of theexistence of the stationary solutions to the relevant FPK equation. And thenthe analytical expressions of the invariant measure are given. Via theanalysis on one special kind of matrix B, the comparisons betweenthe analytical solutions of the invariant measure and the relevant MonteCarlo simulations are given, and then the P-bifurcation point for theone-dimensional phase diffusion process is determined. Finally, for themaximal Lyapunov exponent, the explicit asymptotic expression is obtainedfor the case of one special kind of matrix B, andfor the case of a general matrix B. In addition, the numericalresults are given.