Bifurcation in the presence of small noise

Abstract

The methods of deterministic bifurcation arc sensitive to the addition of small amounts of white noise. Thus, for example, in systems whose macroscopic (deterministic) description arises from an aggregation of microscopically fluctuating dynamics the predictions of deterministic bifurcation may be incorrect. Here we use Laplace's method of steepest descent to study bifurcation in the presence of small noise. Motivation for this study arises from Maxwell's equal-area rule for phase transitions in Van der Waals gases. The theory is then applied to the study of the dynamics of noisy, constrained or implicitly defined dynamical systems.