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.

We propose a new model for pattern formation in peeling of an adhesive tape
based on the equation of motion for the displacement of deformed adhesives in
the peel front. The spatiotemporal patterns obtained from the model are
consistent with those from previous models and experiments. Moreover, dynamical
and statistical properties of the patterns are investigated.

We perform a Koopman spectral analysis of elementary cellular automata (ECA).
By lifting the system dynamics using a one-hot representation of the system
state, we derive a matrix representation of the Koopman operator as a transpose
of the adjacency matrix of the state-transition network. The Koopman
eigenvalues are either zero or on the unit circle in the complex plane, and the
associated Koopman eigenfunctions can be explicitly constructed. From the
Koopman eigenvalues, we can judge the reversibility, determine the number of
connected components in the state-transition network, evaluate the periods of
asymptotic orbits, and derive the conserved quantities for each system. We
numerically calculate the Koopman eigenvalues of all rules of ECA on a
one-dimensional lattice of 13 cells with periodic boundary conditions. It is
shown that the spectral properties of the Koopman operator reflect Wolfram's
classification of ECA.