DYNAMICAL BIFURCATION WITH NOISE

It was shown by A. Neishtadt that dynamical bifurcation, in which the control parameter is varied with a small but finite speed epsilon, is characterized by a delay in bifurcation, here denoted lambda(j) and de pending on epsilon. Here we study dynamical bifurcation, in the framew ork and with the language of Landau theory of phase transitions, in th e presence of a Gaussian noise of strength sigma. By numerical experim ents at fixed epsilon = epsilon(0), we study the dependence of lambda( j) on sigma for order parameters of dimension less than or equal to 3; an exact scaling relation satisfied by the equations permits us to ob tain for this the behavior for general epsilon. We find that in the sm all-noise regime lambda(j)(sigma) similar or equal to a sigma((-b)), w hile in the strong-noise regime lambda(j)(sigma) similar or equal to c e((-d)); we also measure the parameters in these formulas.