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Updated: Sep 19, 2025

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Computing chaotic time-averages from few periodic or non-periodic orbits.

Joshua L Pughe-Sanford1, Sam Quinn1, Teodor Balabanski1

  • 1School of Physics, Georgia Institute of Technology, 837 State St NW, Atlanta, Georgia 30332, USA.

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Summary

This study introduces a data-driven method for approximating temporal averages in chaotic systems. The new approach accurately predicts averages using fewer reference states than traditional methods, improving chaotic system analysis.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Dynamical Systems

Background:

  • Temporal averages in chaotic systems are often approximated using reference states like unstable periodic orbits.
  • Traditional methods like periodic orbit theory and Markov models have limitations when assumptions are violated or libraries are incomplete.

Purpose of the Study:

  • To develop a data-driven approach for computing weights to approximate temporal averages in chaotic systems.
  • To offer an alternative to periodic orbit theory and Markov models, especially for high-dimensional systems.

Main Methods:

  • A novel data-driven method was developed to compute weights for approximating temporal averages.
  • This method utilizes a variety of reference states, including periodic orbits and non-periodic trajectory segments.
  • The approach allows for a reduced-order statistical description of chaotic systems.

Main Results:

  • The data-driven approach accurately approximates temporal averages using a weighted sum of averages over diverse reference states.
  • This method significantly outperforms existing approaches based on periodic orbit theory and Markov models in terms of accuracy.
  • The approach requires substantially fewer reference states compared to traditional methods.

Conclusions:

  • The developed data-driven method provides a superior and more efficient way to approximate temporal averages in chaotic systems.
  • This technique is particularly valuable for applications involving high-dimensional chaotic systems due to its accuracy and reduced state requirements.