Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata

Keisuke Taga; Yuzuru Kato; Yoshihiro Yamazaki; Yoshinobu Kawahara; Hiroya Nakao

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arXiv · 2312.01690

Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata

Keisuke Taga·Yuzuru Kato·Yoshihiro Yamazaki·Yoshinobu Kawahara·Hiroya Nakao

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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.

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Keisuke Taga, Yuzuru Kato, Yoshihiro Yamazaki, Yoshinobu Kawahara, Hiroya Nakao. 2023-12-04. Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata. https://arxiv.org/abs/2312.01690

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