Linear Stability Analysis for Bifurcations in Spatially Extended Systems with Fluctuating Control Parameter

Abstract

We study the threshold in systems which exhibit a symmetry breaking instability, described, e.g., by Ginzburg-Landau or Swift-Hohenberg equations, with the control parameter fluctuating in space and time. Because of the long-tail property of the probability distributions all the moments of the linearized equations have different thresholds and none of them coincides with the threshold of the nonlinear equations, where the long tails are suppressed. We introduce a method to obtain the threshold of the full nonlinear system from the stability exponents of the first and second moments of the linearized equation.