Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits

TL;DR

This paper introduces a data-driven method for accurately approximating time averages in chaotic systems using fewer reference states, outperforming traditional periodic orbit theory and Markov models especially in high-dimensional contexts.

Contribution

The authors develop an alternative, empirical approach to compute weights for chaotic averages, effective with both periodic and non-periodic orbits, reducing the number of states needed.

Findings

Proposed method outperforms traditional approaches in accuracy.

Requires fewer reference states for high-dimensional systems.

Effective for both periodic and non-periodic trajectory segments.

Abstract

For appropriately chosen weights, temporal averages in chaotic systems can be approximated as a weighted sum of averages over reference states, such as unstable periodic orbits. Under strict assumptions, such as completeness of the orbit library, these weights can be formally derived using periodic orbit theory. When these assumptions are violated, weights can be obtained empirically using a Markov partition of the chaotic set. Here, we describe an alternative, data-driven approach to computing weights that allows for an accurate approximation of temporal averages from a variety of reference states, including both periodic orbits and non-periodic trajectory segments embedded within the chaotic set. For a broad class of observables, we demonstrate that weights computed with the proposed method significantly outperform those derived from periodic orbit theory or Markov models, achieving superior accuracy while requiring far fewer states -- two critical properties for applications to high-dimensional chaotic systems.