Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltonian systems

Abstract

A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is first reduced to an one-dimensional Itô stochastic differential equation for the averaged Hamiltonian by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Then the relationship between the qualitative behavior of the stationary probability density of the averaged Hamiltonian and the sample behaviors of the one-dimensional diffusion process of the averaged Hamiltonian near the two boundaries is established. Thus, the stochastic Hopf bifurcation of the original system is determined approximately by examining the sample behaviors of the averaged Hamiltonian near the two boundaries. Two examples are given to illustrate and test the proposed procedure.