Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity
Abstract
Periodic attractors of spatially extended systems possess not one period but a hierarchy of periods, one for each level of observation and symmetry quotient, and this hierarchy separates how a pattern evolves from how it fits the finite domain. For a synchronous threshold model of cyclic dominance on an L× L torus, equivalent to the three-color cyclic cellular automaton of Fisch, Gravner and Griffeath, we measure for every attractor a \emph{tuple} of periods: that of the populations (Tpop), of the full configuration (Tst), and of the configuration up to color rotations (Tcol), lattice symmetries (Tgeo) and the full space--color group (TG). Equivariance implies exact divisibility relations and the factorization Tst=TG m, where m counts the applications of the symmetry h that maps the configuration onto itself after TG steps: TG and the type of h belong to the pattern, m to the lattice, and the attractors are relative periodic orbits whose full period emerges from the interaction between the intrinsic dynamics and the lattice symmetries. A family of attractors built around a pair of counter-rotating spirals, with Tpop=24 and TG=8, persists from L=9 to L=40 while Tst takes values from 24 to 960, given in every case by the order of h; on a rectangular 9×12 torus it takes the value 48, absent from all square tori. With more than 4.5×105 runs at each of L=8 and L=9 we estimate the basins of attraction: fixed points take 95--97\% of the configuration space, and no single period is enough to tell the oscillating attractors apart. The same analysis locates local diversity: it is balanced whenever the cycle has a color symmetry and, on large lattices, it is concentrated at the spiral cores.
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BibTeX
@misc{munoz2026symmetryresolved,
title = {{Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity}},
author = {Francisco J. Muñoz and Juan Carlos Nuño},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37990},
eprint = {2610.37990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37990},
}
RIS
TY - UNPB
TI - Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity
AU - Francisco J. Muñoz
AU - Juan Carlos Nuño
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37990
UR - https://arxiv.org/abs/2610.37990
AB - Periodic attractors of spatially extended systems possess not one period but a hierarchy of periods, one for each level of observation and symmetry quotient, and this hierarchy separates how a pattern evolves from how it fits the finite domain. For a synchronous threshold model of cyclic dominance on an L× L torus, equivalent to the three-color cyclic cellular automaton of Fisch, Gravner and Griffeath, we measure for every attractor a \emph{tuple} of periods: that of the populations (Tpop), of the full configuration (Tst), and of the configuration up to color rotations (Tcol), lattice symmetries (Tgeo) and the full space--color group (TG). Equivariance implies exact divisibility relations and the factorization Tst=TG m, where m counts the applications of the symmetry h that maps the configuration onto itself after TG steps: TG and the type of h belong to the pattern, m to the lattice, and the attractors are relative periodic orbits whose full period emerges from the interaction between the intrinsic dynamics and the lattice symmetries. A family of attractors built around a pair of counter-rotating spirals, with Tpop=24 and TG=8, persists from L=9 to L=40 while Tst takes values from 24 to 960, given in every case by the order of h; on a rectangular 9×12 torus it takes the value 48, absent from all square tori. With more than 4.5×10^5 runs at each of L=8 and L=9 we estimate the basins of attraction: fixed points take 95--97\% of the configuration space, and no single period is enough to tell the oscillating attractors apart. The same analysis locates local diversity: it is balanced whenever the cycle has a color symmetry and, on large lattices, it is concentrated at the spiral cores.
ER -