https://doi.org/10.1140/epjs/s11734-026-02271-5
Regular Article
Emulating the logistic map with totalistic cellular automata
1
Department of Physics and Astronomy, University of Florence and CSDC, via G. Sansone, 1, 50019, Sesto Fiorentino, FI, Italy
2
INFN, Sect. Florence, via G. Sansone, 1, 50019, Firenze, Sesto Fiorentino, Italy
3
Espace-Dev, University of Perpignan via Domitia, 52 Avenue Paul Alduy, 66000, Perpignan, France
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
3
December
2025
Accepted:
13
March
2026
Published online:
27
March
2026
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.
© The Author(s) 2026
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
