Coexistence of dynamics for two-dimensional cellular automata

This paper is concerned with the study of six rules from the family of square Boolean cellular automata (CAs) having a neighborhood consisting of four peripheral neighbors and with periodic boundary conditions. Based on intensive computations, we are able to conclude, with statistical support, that these rules have a common feature: they all show coexistence of dynamics, in the sense that as the size n of the side of the square increases with fixed parity, the relative size of the basin of attraction of the homogeneous final state tends to a constant value that is neither zero nor one. It is also statistically shown that the values of the constant levels - one for n odd and the other for n even - can be considered as equal for five of the rules, while for the sixth rule these values are one-half of the others. Some results obtained for the one-dimensional CAs with four peripheral neighbors are also reported, to support our claim that with periodic boundary conditions, the coexistence of dynamics can only appear for automata with dimension higher than one.This work was partially supported by FEDER funds through Programa Operacional Factores de Competitividade – COMPETE and by national funds through FCT – Fundação para a Ciência e Tecnologia within the remits of projects PEst-OE/MAT/UI0013/2014 and FCOMP-01-0124-FEDER-037268 (PEst-C/EGE/UI3182/2013)