On the influence of noise on the largest Lyapunov exponent and on the geometric structure of attractors

In this paper we present an overview of a classification of alternative mathematical schemes which determine possible impositions of noise on dynamic systems either of a continuous or discrete formulation in time; see [1]. When a noise interferes with the evolution of a dynamic system it is called a dynamic noise. Such a dynamic noise may take the form of an additive or multiplicative expression which illustrate the kind of parameters by which noise may enter into the equations of a dynamic system. We also consider the case of an output noise, i.e. a noise which does not influence the evolution of a dynamic system. The output noise is again divided into additive and multiplicative forms, depending on how it is introduced into the formulation of the system. We present some numerical investigations concerning the influence of noise on i) the correlation dimension, ii) the largest Lyapunov exponent and iii) the geometric structure of the Henon and Lorenz attractors. We also recall, a method of constructing models for effecting predictions of time series with a finite correlation dimension as obtained in [1]. (C) 1998 Elsevier Science Ltd. All rights reserved.