Emulating the logistic map with totalistic cellular automata

TL;DR

This paper explores how totalistic cellular automata can emulate the logistic map, demonstrating that mean-field approximations are accurate with infinite-range neighborhoods and certain rewiring strategies, revealing bifurcation phenomena.

Contribution

It shows that the logistic map can be approximated by totalistic cellular automata through specific neighborhood configurations and rewiring mechanisms, extending understanding of CA dynamics.

Findings

Mean-field approximation is accurate only with infinite-range neighborhoods.

Rewiring links can induce logistic-like bifurcations in CA.

Good logistic approximation achieved with partial rewiring, not necessarily full.

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.