Planar bifurcation subject to multiplicative noise: role of symmetry.

The effect of multiplicative noise on a system described by two modes close to a bifurcation point is investigated. The bifurcation is assumed stationary and noise acts as random coupling between these modes. An analytic formula that predicts the onset of instability is derived, and the domain of existence of on-off intermittency is calculated based on an eigenvalue problem. This approach, confirmed by numerical simulations of the Langevin equations, allows quantifying the possible effects of the noise. The stability and the on-off behavior are shown to be very sensitive to deviations of the deterministic system from the case where both modes grow with equal rate and the system displays a continuous symmetry associated to rotation in phase space. In general, a noise term that breaks this continuous symmetry will increase the domain of instability of the system while a noise term that preserves the symmetry reduces the domain of instability.