Emulating the logistic map with totalistic cellular automata

TL;DR

This work analyzes when the mean-field dynamics of a totalistic cellular automaton can reproduce the logistic map. By deriving the mean-field update and the corresponding transition probabilities, it shows that exact equivalence requires infinite-range neighborhoods, with finite ranges yielding a bounded parameter regime $a_M(R)=4\frac{R-1}{R}$. The study then demonstrates that introducing small-world rewiring (quenched) or randomization of neighborhoods can recover logistic-like density dynamics for finite $R$, with about 50% rewiring sufficing to closely approximate the logistic behavior. The findings highlight a practical pathway to induce mean-field-like chaos and bifurcation structure in low-dimensional CA models via network topology, offering a bridge between local interactions and global logistic dynamics.

Abstract

We investigate the conditions under which the mean-field formulation of a totalistic cellular automaton can approximate the logistic equation. We obtain that this can be obtained only for infinite-range neighborhood. We then performed simulation of one-dimensional cellular automata, showing that this mean-field approximation is clearly obtained by shuffling the configuration or choosing at random the neighbors, but also rewiring a fraction of links, in the spirit of the small-world mechanism. We show that it is possible to obtain a good approximation of the logistic behavior with a fraction of rewiring link of 50% or more.

Figures (5)

• Figure 1: Bifurcation diagram of the logistic map
• Figure 2: Adjacency matrix for $N=100$, $R=5$, $p=0.1$. Yellow dots marks connected neighbors.
• Figure 3: Plot of the bifurcation diagrams of $x$ for $0\le a\le a_M$, $T=1000$ steps and lattice size $N=10000$, after a transient of $100000$ steps. The vertical line marks the value of $a_M$. The plots are the same for the shuffling, annealed and quenched procedure with $p=1$. (a) $R=3$; (b) $R=10$; (c) $R=20$ and (c) $R=100$.
• Figure 4: Plot of the logistic map for $a=a_M=3.96$ and the return map $x'\equiv(t+1)$ vs $x\equiv(t)$ for $R=100$, $T=1000$ steps and lattice size $N=10000$, after a transient of $100000$ steps. (a) $p=0.1$; (b) $p=0.2$; (c) $p=0.5$ and (c) $p=1.0$.
• Figure 5: (a) The bifurcation diagram of $x$ vs rewiring probability $p$. (b) The average variance of the configuration $\mathrm{var}(x)=\overline{x(1-x)}$ vs rewiring probability $p$. Parameters: $R=100$, $a=a_M=3.96$, $T=1000$ steps and lattice size $N=10000$, after a transient of $100000$ steps.