How additive noise generates a phantom attractor in a model with cubic nonlinearity

• Two-dimensional nonlinear system with cubic nonlinearity is studied.
• Additive noise generates a new phantom attractor.
• By averaging over the fast variable one-dimensional equation is derived.
• Phantom attractor appearance is analyzed by bifurcation analysis of this equation.

Two-dimensional nonlinear system forced by the additive noise is studied. We show that an increasing noise shifts random states and localizes them in a zone far from deterministic attractors. This phenomenon of the generation of the new “phantom” attractor is investigated on the base of probability density functions, mean values and variances of random states. We show that increasing noise results in the qualitative changes of the form of pdf, sharp shifts of mean values, and spikes of the variance. To clarify this phenomenon mathematically, we use the fast–slow decomposition and averaging over the fast variable. For the dynamics of the mean value of the slow variable, a deterministic equation is derived. It is shown that equilibria and the saddle-node bifurcation point of this deterministic equation well describe the stochastic phenomenon of “phantom” attractor in the initial two-dimensional stochastic system.