Bautin bifurcation with additive noise

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