Concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In dynamical systems with small random perturbations, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In the presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent lambda(sigma) computed considering two nearby trajectories evolving under the same realization of the randomness. However, the former is much more relevant than the latter from a physical point of view, as illustrated by some numerical computations for noisy maps and sandpile models.