The influence of noise on the correlation dimension of chaotic attractors

The present paper investigates the influence of noise on the correlation dimension Dc of chaotic attractors arising in discrete and continuous in time dynamical systems. Our numerical results indicate that the presence of noise leads to an increase of the correlation dimension. Assuming that the correlation dimension for a white noise is infinite, we prove, first, that the increase of the dimension of a chaotic attractor in a stochastic system is a generic property of the set of stochastic dynamical systems and, secondly, that the existence of a small correlation dimension in a time series implies that the deterministic part of its Wold decomposition is nonzero. We also present a collection of dynamical systems subject to noise which may be considered as models for predictions on the response of time series with a finite correlation dimension, as encountered in physical or numerical experiments.