Kolmogorov-Sinai entropy from recurrence times

TL;DR

This paper introduces two formulas to estimate the Kolmogorov-Sinai (KS) entropy from recurrence times in chaotic systems, enabling better analysis of complex dynamics from data.

Contribution

It proposes novel formulas to estimate KS entropy and its lower bound directly from recurrence times, bridging theoretical and experimental approaches.

Findings

One formula accurately estimates KS entropy using long return times.

A second formula provides a reliable lower bound using short return times.

Recurrence times contain sufficient information to characterize system dynamics.

Abstract

Observing how long a dynamical system takes to return to some state is one of the most simple ways to model and quantify its dynamics from data series. This work proposes two formulas to estimate the KS entropy and a lower bound of it, a sort of Shannon's entropy per unit of time, from the recurrence times of chaotic systems. One formula provides the KS entropy and is more theoretically oriented since one has to measure also the low probable very long returns. The other provides a lower bound for the KS entropy and is more experimentally oriented since one has to measure only the high probable short returns. These formulas are a consequence of the fact that the series of returns do contain the same information of the trajectory that generated it. That suggests that recurrence times might be valuable when making models of complex systems.