ONSET OF OSCILLATORY INSTABILITIES UNDER STOCHASTIC MODULATION

We study the effect of external stochastic modulation on a system with O(2) symmetry that exhibits a Hopf or oscillatory instability in the absence of modulation. The study includes a random component in both t he control parameter of the bifurcation and in the modulation amplitud e. Stability boundaries are computed by either solving the stationary Fokker-Planck equation on the center manifold of the underlying determ inistic system whenever possible, or by direct numerical solution othe rwise. If the modulation amplitude has a stochastic component, the pri mary bifurcation is always to standing waves at a value of the control parameter that depends on the intensity of the fluctuations. More pre cisely, and to contrast our results with the case of a deterministic p eriodic forcing, the onset of instability in the standing-wave regime is shifted from its deterministic location, and the region of primary bifurcation to traveling waves disappears, yielding instead standing w aves at negative values of the control parameter.