Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata
Abstract
We apply Dynamic Mode Decomposition (DMD) to Elementary Cellular Automata (ECA). Three types of DMD methods are considered and the reproducibility of the system dynamics and Koopman eigenvalues from observed time series are investigated. While standard DMD fails to reproduce the system dynamics and Koopman eigenvalues associated with a given periodic orbit in some cases, Hankel DMD with delay-embedded time series improves reproducibility. However, Hankel DMD can still fail to reproduce all the Koopman eigenvalues in specific cases. We propose an Extended DMD method for ECA that uses nonlinearly transformed time series with discretized Walsh functions and show that it can completely reproduce the dynamics and Koopman eigenvalues. Linear-algebraic backgrounds for the reproducibility of the system dynamics and Koopman eigenvalues are also discussed.
Keywords
Cite
@article{arxiv.2312.01690,
title = {Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata},
author = {Keisuke Taga and Yuzuru Kato and Yoshihiro Yamazaki and Yoshinobu Kawahara and Hiroya Nakao},
journal= {arXiv preprint arXiv:2312.01690},
year = {2023}
}
Comments
20 pages, 10 figures