Characteristic functional approach to multiplicative fractal noise with application to environmental fluctuations in nonlinear chemical systems far from equilibrium: A temporal analog of Anderson localization

We compute the characteristic functional of a nonlinear transformation of a set of random variables. The equation is applied to the study of fractal s tochastic processes with multiplicative noise. For illustration we investig ate two types of chemical systems subjected to environmental fluctuations. We analyze the interaction between the intrinsic chemical fluctuations and environmental noise and evaluate the stochastic properties of the system in the presence of multiplicative noise. For an autocatalytic reaction subjec ted to external noise and with Gaussian and generally non-Markovian multipl icative external noise the averaged joint probability density of concentrat ions displays long tails of the negative power law type, which are independ ent of the detailed properties of the noise. We also study the influence of multiplicative noise on a nonlinear chemical system with a stable limit cy cle. In the presence of multiplicative noise the limit cycle collapses and the average trajectory in the composition space becomes a spiral. This effe ct is a temporal analog of Anderson localization. We evaluate the damping f actors and localization times for different types of noise ranging from noi se with infinite memory (static disorder) and noise with long memory (dynam ic disorder with fractal noise) to Markovian noise (dynamic disorder with s hort memory). (C) 2001 Published by Elsevier Science B.V.