Emulating the logistic map with totalistic cellular automata
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914371280568320 |
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| author | Bagnoli, Franco |
| author_facet | Bagnoli, Franco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Emulating the logistic map with totalistic cellular automata Bagnoli, Franco Cellular Automata and Lattice Gases Chaotic Dynamics 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. |
| title | Emulating the logistic map with totalistic cellular automata |
| topic | Cellular Automata and Lattice Gases Chaotic Dynamics |
| url | https://arxiv.org/abs/2512.04140 |