Global dynamics of the stochastic Rabinovich system

Abstract

In this paper, the global dynamics of the deterministic Rabinovich system and the dynamics of the stochastic Rabinovich are analyzed and compared. First, the important feature of the deterministic Rabinovich system is briefly recalled and analyzed; then the globally exponentially attractive set and positive invariant set of the system are given. Second, by using the theory of stochastic differential equation and the Lyapunov function, the asymptotic behavior and the globally exponentially attractive set of the Ito-type stochastic Rabinovich system and the Stratonovich-type stochastic Rabinovich system are discussed. Third, to illustrate the stochastic effects clearly, simulations for the deterministic case and corresponding stochastic case are performed, respectively. And the unstable results are numerically verified through the Heidelberg and Welch test by R project. The obtaining results show that the stability and the globally exponentially attractive set of the Rabinovich system occur change significantly under stochastic disturbance. Even under the same parameter conditions, the dynamics of the different-type stochastic differential equation has marked differences. Further, from the dynamical and phenomenological point, the stochastic pitchfork bifurcation of the Stratonovich-type system is analyzed. Results show the position where the stochastic bifurcation at the equilibrium point occurs will change as the intensity of the white noise change.