On Oscillations in Cellular Automata

TL;DR

This paper studies oscillatory behaviors in cellular automata, revealing that their predictability stems from weakly coupled oscillators rather than collective synchronization, with accuracy improving as system size increases.

Contribution

It introduces a mean-field approach to describe oscillations in cellular automata, emphasizing the weak coupling mechanism over synchronization.

Findings

Systems are highly predictable with periodic or quasiperiodic oscillations.

Global quantities become more precise as system size increases.

Oscillations are due to weakly coupled subsystems, not collective synchronization.

Abstract

We investigate cellular automata where some global quantity varies periodically or quasiperiodically with time. We find that these systems are highly predictable, and can be rather well described by a set of mean-field variables. We conclude that this is not a collective phenomenon - where different subsystems are supposed to synchronize - but rather like many very weakly coupled oscillators fluctuating around one exact frequency. The global quantity measured is a mean taken over all these subsystems, and gets more precise the larger the system is.