Hopf Bifurcation of Compound Stochastic van der Pol System

Abstract

Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strengthδand noise intensityσon stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increasedδcan relocate the critical value of bifurcation parameter forward while increasedσmakes it backward and the influence ofδis more sensitive thanσ. The results are verified by numerical simulations.