Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity
arXiv:2610.37990
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 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 counts the applications of the symmetry that maps the configuration onto itself after $\TG$ steps: $\TG$ and the type of belong to the pattern, 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 to while $\Tst$ takes values from 24 to 960, given in every case by the order of ; on a rectangular torus it takes the value 48, absent from all square tori. With more than runs at each of and 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.
arXiv comment27 pages, 20 figures