Density of first Poincar\'e returns, periodic orbits, and Kolmogorov-Sinai entropy

TL;DR

This paper presents a method to estimate the Kolmogorov-Sinai entropy from the density of first Poincaré returns using unstable periodic orbits, facilitating analysis of experimental chaotic systems.

Contribution

It introduces a novel approach linking periodic orbits and return time densities to compute entropy, with a practical method for numerical orbit calculation.

Findings

Density functions of Poincaré returns can estimate system entropy.

The method is applicable to experimental data due to easy measurement of return times.

Numerical techniques for unstable periodic orbit computation are developed.

Abstract

It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincar\'e returns. The close relation between periodic orbits and the Poincar\'e returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov-Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.