The false Hopf bifurcation and noise sensitivity in bifurcations with symmetry

The effect of small external noise on the solution of certain bifurcation problems is discussed. Of particular interest are problems in which there is an invariant subspace with slow dynamics near the bifurcation point. As an example, equations modeling the instability of tall thin convective cells are described. The inclusion of any amount of external noise leads to an entirely incorrect bifurcation structure in the neighborhood of the initial bifurcation, with the 'true' solution being replaced by noisy periodic orbits of smaller amplitude.