Dynamical estimates of chaotic systems from Poincar\'e recurrences

TL;DR

This paper investigates the distribution of return times in chaotic systems, revealing deviations from exponential behavior due to deterministic dynamics, and introduces efficient methods for estimating Lyapunov exponents and correlation functions from data.

Contribution

It provides a novel analysis of return time distributions showing small power-law deviations and introduces simplified methods for estimating key dynamical quantities.

Findings

Return time distribution deviates from exponential with a small power-law correction.

New methods for faster Lyapunov exponent estimation using special unstable periodic orbits.

Approaches applicable to data from complex systems.

Abstract

We show that the probability distribution function that best fits the distribution of return times between two consecutive visits of a chaotic trajectory to finite size regions in phase space deviates from the exponential statistics by a small power-law term, a term that represents the deterministic manifestation of the dynamics, which can be easily experimentally detected and theoretically estimated. We also provide simpler and faster ways to calculate the positive Lyapunov exponents and the short-term correlation function by either realizing observations of higher probable returns or by calculating the eigenvalues of only one very especial unstable periodic orbit of low-period. Finally, we discuss how our approaches can be used to treat data coming from complex systems.