Koopman spectral analysis of elementary cellular automata

Abstract

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.

arXiv comment29 pages, 9 figures

References in corpus (7)

• On Dynamic Mode Decomposition: Theory and Applications
• Applied Koopmanism
• Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics
• Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
• Damage spreading and Lyapunov exponents in cellular automata
• Koopman analysis of Burgers equation
• Spectral Analysis of the Koopman Operator for Partial Differential Equations