Chaté-Manneville Automata

  

Figure: The density of states in Chaté-Manneville automata runs through cycles rather than approaching an equilibrium value. One of the simplest is the binary totalistic rule 33 in three dimensions with von Neumann neighborhoods. Left: a plane cross section of the evolution of a random initial configuration for nine generations. Right: the return map, which does not exactly follow the mean field density curve. In the background the diagonal can be seen as well as the mean field curve with an unstable equilibrium density of about 0.1.

When statistical methods began to be applied to the evolution of cellular automata, it was understood that there was a difference between periodic configurations for the automaton and periodicities in the statistical behavior of the automaton.

The long time behavior of a single configuration is one thing; it may repeat sooner or later exhibiting periodic behavior, although if the evolution consists of a neverending transient, it may still have convergent statistical properties. But it is another thing to examine statistical behavior averaged over all configurations, or all the members of a class of configurations. It has generally been expected that those averages would converge, and there was even an article published arguing to that effect. Shortly thereafter Hugues Chaté and Paul Manneville [71,59] discovered some automata which, although they didn't exactly follow the return map of iteration theory, didn't tend toward a long term average either. The most prominent of them, discovered shortly after their original announcement by Jan Hemmingsson, was only three dimensional and followed a period of three in the density.