LINEAR-STABILITY ANALYSIS OF BIFURCATIONS WITH A SPATIALLY PERIODIC, FLUCTUATING CONTROL PARAMETER

Multiplicative noise in spatially extended systems produces different effects depending upon whether the noise is spatially homogeneous or s patially varying. Whereas in previous work a stochastic distribution w as treated, here we consider the spatially periodic case, which is mor e amenable to an experimental approach, in particular in the electrica lly driven instabilities of nematic liquid crystals. We shall principa lly be interested in the threshold for the onset of symmetry breaking instabilities controlled by bifurcations in several stochastic partial differential equations. For the Ginzburg-Landau and Swift-Hohenberg e quations we calculate the behavior of the threshold for all moments to second order in the noise strength, allowing one to reconstruct the f ull probability distribution. For a system of two coupled equations wh ich mimics electroconvection in nematic liquid crystals (the ''one-dim ensional model''), we calculate the first two moments up to second ord er and estimate the threshold for convection. The general conclusion o f our work is that spatially periodic noise induces a reduction in the threshold similar to the stochastically distributed case. We propose that this reduction be independent of the periodicity of the noise to first order in the noise strength, the dependence on period appearing only at second order. This is in contrast to spatially homogeneous noi se where threshold shifts may be entirely absent.