Pergamon Chaos, Sdrrions & Fractds, Vol. 9, No. 3, pp. 343-361. 190X 0 1998 Published by Elsevier Science Ltd. All rights rezxved Printed in Great Britam 0960.0779198 $19.00 + ~I.00 PII: s0960-0779(97)00l20-3 The Influence of Noise on the Correlation Dimension of Chaotic Attractors JOHN ARGYRIS, IOANNIS ANDREADIS Institute for Computer Applications and University of Stuttgart, D-70569 Stuttgart, Germany and GEORGIOS PAVLOS, MICHALIS ATHANASIOU Department of Electrical Engineering, University of Thrace, Xanthi 67100, Greece (Accepted I4 May 1997) Abstract-The present paper investigates the influence of noise on the correlation dimension D, 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. 0 1998 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION We propose to study the influence of noise on the correlation dimension D, of chaotic attractors arising in discrete and continuous in time dynamical systems. We present also a classification of the number of mathematical schemes by which noise influences a dynamical system. When a noise interferes with the evolution of a dynamical system, it is called a dynamical noise. Such dynamical noise may take the form of an additive or multiplicative expression which illustrates the kind of parameters by which noise may appear in the equations of a dynamical system. The notion of a dynamical noise has already been considered in [l]. We refer also to [Z-9] for examples of an additive dynamical noise in discrete dynamical systems. The special case of multiplicative dynamical noise for the logistic map is treated in [lo, 111. We also consider the case of an output noise which does not influence the evolution of the dynamical system. We recall that the evolution of a dynamical system on R” is expressed through a functional vector x(t) for the continuous in time formulations or a map x, for the discrete in time formulations. Therefore, in the case of an output noise we study a new evolution X(t) (resp. X,) by applying the noise to x(t) (resp. x,J. The output noise is again divided into additive and multiplicative forms depending on how one introduces noise into the definition of X(t) (resp. X,). The additive output dynamical noise is discussed in the work of Hammel as a shadowing property of an orbit and implies a deviation from the true orbit due to noise of a discrete in time dynamical system [4,12, 13,9,14]. The output noise is also discussed in the bibliography on electrical circuit theory [15]. 343