Comment on "Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems"

TL;DR

This paper clarifies that differences in mean properties between chaotic and unstable periodic orbits are due to improper averaging, and when correct weights are used, the discrepancy vanishes, emphasizing the importance of natural measure in chaotic systems.

Contribution

It demonstrates that accounting for the natural measure resolves apparent discrepancies between chaotic and periodic orbit averages.

Findings

Discrepancies are artifacts of improper averaging.

Correct weighting by natural measure aligns averages.

Proper analysis removes the supposed differences.

Abstract

The recent paper claims that mean characteristics of chaotic orbits differ from the corresponding values averaged over the set of unstable periodic orbits, embedded in the chaotic attractor. We demonstrate that the alleged discrepancy is an artifact of the improper averaging: Since the natural measure is non-uniformly distributed over the attractor, different periodic orbits make different contributions into the time averages. As soon as the corresponding weights are accounted for, the discrepancy disappears.