Influence of noise on delayed bifurcations

We study the influence of noise on the delay experienced by a steady bifurcation point involving a zero or constant solution when the control parameter is swept in time starting in the vicinity of the stable constant solution. This study is realized by a direct analysis of the moment equations rather than via a Fokker-Planck equation with an emphasis on analytical results. We treat separately multiplicative and additive noise sources and the noise can be white or colored. Furthermore, we analyze the case where the noise is added during the sweep but is absent during the preparation of the system and compare it with the case where noise is already present during the preparation of the system. In general, noise reduces the delay even to the extent that the bifurcation point can occur before the static bifurcation point. However, when the initial condition is noisy the delay may increase beyond the deterministic value in the limit of strongly colored noise.