Coarsening and Bifurcations in Wide-Range Two-Dimensional Totalistic Cellular Automata

TL;DR

This paper studies two-dimensional totalistic cellular automata with variable interaction ranges, revealing complex behaviors like bifurcations and coarsening that differ from mean-field predictions.

Contribution

It demonstrates how local interactions lead to rich dynamics such as stable patterns and bifurcations, contrasting with mean-field approximations.

Findings

Majority vote model exhibits absorbing states and bifurcations based on initial density.

Coarsening process halts with clusters of definite curvature radius.

Frustrated majority vote model shows stable density patterns and bifurcations at critical interaction ranges.

Abstract

We investigate Boolean, totalistic cellular automata with a majority or frustrated majority vote rule, and an interaction range of variable span. These two models show a behavior which differs from the mean-field one. The majority vote model is characterized by the presence of absorbing states, and there is a related bifurcation according to the initial density, in agreement with the mean-field approximation. For initial density equal to $0.5$, however, the dynamics is dominated by a coarsening process, which stops when clusters with a definite curvature radius are established. For the frustrated majority vote model, the mean-field approximation gives chaotic oscillations or a limit cycle. Instead, we observe active patterns, with stable density. Above a certain critical value for the interacting radius there is a bifurcation of the asymptotic density as a function of the initial one.