ADIABATIC REDUCTION NEAR A BIFURCATION IN STOCHASTICALLY MODULATED SYSTEMS

We reexamine the procedure of adiabatic elimination of fast relaxing v ariables near a bifurcation point when some of the parameters of the s ystem are stochastically modulated. Approximate stationary solutions o f the Fokker-Planck equation are obtained near threshold for the pitch fork and transcritical bifurcations. Correlations between fast variabl es and random modulation may shift the effective bifurcation point by an amount proportional to the intensity of the fluctuations. We also f ind that fluctuations of the fast variables above threshold are not al ways Gaussian and centered around the (deterministic) center manifold as was previously believed. Numerical solutions obtained for a few ill ustrative examples support these conclusions.