Dynamics of the Noise-Induced Phase Transition of the Verhulst Model

Langevin·type equation with a nonlinear drift term and a multiplicative noise term is treated. Dynamical aspects of noise-induced phase transition are studied with the aid of the method of asymptotic iteration (MAl) introduced in a previous paper. The temporal behavior of the first moment is solved exactly. It is shown that the critical slowing down does not occur at the so·called transition point. § 1. Introduction Recently 'noise-induced phase transition' has attracted many worker's interest.1)-3) Mostly, their attention is directed to the influence of the external noise on the stationary solution of the Fokker-Planck equation, and it has been shown that above a threshold value of the intensity of the external noise, the shape of the stationary solution suffers alteration. I In this paper we consider the dynamical aspects of noise-induced phase transition. We want to find whether the critical slowing down occurs or not near the noise-induced phase transition. We solve the moment equations corresponding to the Langevin equation under consideration with the aid of the method of asymptotic iteration (MAl) developed in the previous paper. 4) We pay attention only to the first moment, because the determination of its temporal development is sufficient to know whether the critical slowing down occurs or not. Now the stochastic differential equation (SDE) for our problem is given as (1.1) where B t is the Wiener process with =O and = (J2dt. Here p is a POSItIve integer. It is to be noted that in the cases p=l and p=2, SDE(1'1) represents the Verhulst modeI 1),5) and the kinetic Ising model 6) with a multiplicative noise respectively. As for the noise term we adopt the Stratonovich interpretation. According to the theorem of Wong and Zakai/) the SDE should be interpreted as a Stratonovich equation as long as the limit of a realistic noise can be considered as the white noise. Moreover Kabashima et al. B) have shown that their experimental results are in good agreement with the theoretical results based on the Stratonovich interpretation. They found a noise