Bautin bifurcation with additive noise

Abstract In this paper, we consider stochastic dynamics of a two-dimensional differential equation with additive noise. When the strength noise is zero, undergoes Bautin bifurcation. We obtain main conclusions including existence and uniqueness solution, synchronization system property random equilibrium, where going through some processes like deducing stationary probability density calculating Lyapunov exponent. For better understanding effect under noise, make clear comparison between deterministic one precise numerical simulations to show slight changes at bifurcation point. Furthermore, take real model as an example present application our theoretical results.