ON THE CONCEPT OF COMPLEXITY OF RANDOM DYNAMICAL-SYSTEMS

We show how to introduce a characterization the ''complexity'' of rand om dynamical systems. More precisely we propose a suitable indicator o f complexity in terms of the average number of bits per time unit nece ssary to specify the sequence generated by these systems. This indicat or of complexity, which can be extracted from real experimental data, turns out to be very natural in the context of information theory. For dynamical systems with random perturbations, it coincides with the ra te K of divergence of nearby trajectories evolving under two different noise realizations. In presence of strong dynamical intermittency, th e value of K is very different from the standard Lyapunov exponent chi (sigma) computed through the consideration of two nearby trajectories evolving under the same realization of the random perturbation. Howeve r, the former is much more relevant than the latter from a physical po int of view as illustrated by some numerical examples of noisy and ran dom maps.