On the concept of complexity in random dynamical systems

TL;DR

This paper introduces a new measure of complexity for random dynamical systems based on information theory, which aligns with the divergence rate of trajectories and can be empirically determined, providing insights beyond traditional Lyapunov exponents.

Contribution

It defines a complexity measure K for random systems, relates it to information theory, and demonstrates its practical relevance through numerical examples, especially under intermittency.

Findings

K measures the information rate of sequences generated by random systems.

K differs significantly from Lyapunov exponents in intermittent regimes.

K can be estimated from experimental data, offering a practical tool.

Abstract

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 random dynamical system, 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 presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same 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.