Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata

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 is investigated. While standard DMD fails to reproduce the system dynamics and...

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Published inChaos (Woodbury, N.Y.) Vol. 34; no. 1
Main Authors Taga, Keisuke, Kato, Yuzuru, Yamazaki, Yoshihiro, Kawahara, Yoshinobu, Nakao, Hiroya
Format Journal Article
LanguageEnglish
Published United States 01.01.2024
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Summary: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 is 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.
ISSN:1089-7682
DOI:10.1063/5.0159069