Hopf bifurcation in the presence of both parametric and external stochastic excitations

In this paper the effect of small parametric and external stochastic perturbations on a two-dimensional system exhibiting Hopf bifurcation is discussed in detail. It is shown that even small fluctuations in parameters can have an important effect when the system is in the neighborhood of bifurcation points. The Markov diffusion approximation is used to obtain analytical results relating to the statistical properties of the stochastic response. Both moment and sample stability conditions along with the stationary moments are obtained. For the nonlinear system, stationary and transient probability densities are found along with their stationary moments. It was found that in nonlinear systems, a shift of the bifurcation point takes place due to the presence of parametric noise.