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

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.