Emulating the logistic map with totalistic cellular automata

Abstract

We investigate the conditions under which the mean-field formulation of a probabilistic, totalistic cellular automaton approximates the logistic equation. We show that this goal can be only fulfilled for an infinite-range neighborhood. We numerically study the corresponding one-dimensional implementation, showing that the mean-field description is obviously approached by shuffling the configuration at each time step, but also by rewiring a fraction of links, either at each time step, or using the same random sampling once and for all, in the spirit of the "small-world" mechanism. We show that it is possible to obtain a good approximation of the logistic behavior already with a fraction of rewired links different from one. We also show that there is a bifurcation cascade of the density as a function of the fraction of the rewired links, and that this scenario also holds for a deterministic, totalistic CA with the same basic symmetries of the probabilistic one.