Sensitivity of long periodic orbits of chaotic systems

TL;DR

This paper investigates whether sensitivities of long periodic orbits in chaotic systems can reliably predict the response of the chaotic state to parameter changes, finding convergence in some quantities but not in sensitivities.

Contribution

It provides a comprehensive analysis of long periodic orbits in chaotic systems and assesses their effectiveness as proxies for chaotic responses to parameter perturbations.

Findings

Period averages and Floquet exponents converge with orbit length.

Sensitivity values do not necessarily match the chaotic state response.

Long orbits can approximate some properties of chaotic systems.

Abstract

The properties of long, numerically-determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.