Research note · 5 September 2026

Periodic Gray–Scott waves with cyclic time symmetry

This note describes the computation of periodic Gray–Scott solutions for which wallpaper-group operations act with prescribed time offsets. It extends the spatial symmetry setting illustrated by SymSim to constraints on the full space–time field. The governing reaction–diffusion equations remain unchanged. [1]

∂u/∂t = D_u Δu − uv² + F(1 − u) ∂v/∂t = D_v Δv + uv² − (F + k)v

Let u and v denote the concentrations. The coefficients are constant, and the spatial domain is a flat torus with periodic boundary conditions. For q = (u, v), the additional constraints are

q(x, t + T) = q(x, t), q(gx, t + τ_gT) = q(x, t). g is a spatial group operation; τ_g ∈ ℝ/ℤ is its phase offset as a fraction of the period T.

The offsets define a character with finite cyclic image: τ_gh = τ_g + τ_h (mod 1). Cyclic colour permutations are represented by time translations acting on both concentrations, without exchanging the species. The formulation introduces no additional reaction term, forcing, or time reversal. The display palette is independent of this group-theoretic use of colour. [2]

For the 442 case g96, with short colour signature ⁴4⁴4²2 in the correspondence catalog, the canonical rotation R(x, y) = (−y, x) has τ = ¼. The condition is q(Rx, t + T/4) = q(x, t). The displayed generator α uses R⁻¹ and τ = ¾, giving an equivalent constraint. Case g97 has the same permutation orders and the opposite directed phase. [2]

Domain length L. For 442, L is the side length of the periodic square cell in the model’s spatial units. It specifies the translation cell; the rotational fundamental domain is smaller. For 632, L is the length of each basis vector of the triangular lattice, whose cell is a rhombus.

With N nodes along each lattice direction, the grid spacing is h = L/N (denoted dx in the implementation). In the example below, h = 256/48 ≈ 5.33. The discrete diffusion operator has coefficient D/h². In normalized square coordinates ξ = x/L, the continuum coefficient is D/L². Doubling L at fixed D reduces this coefficient by a factor of four and changes the admissible physical wavelengths. Increasing N at fixed L refines the discretization of the same physical problem. Varying L changes the domain and therefore the problem; display magnification and tile count do not affect it. [3]

Parameters and numerical controls
Symbol	Meaning	Example g96
F	Feed/replenishment rate; also contributes to v removal.	0.004
k	Additional v removal rate; total linear loss is F + k.	0.020
D_u, D_v	Diffusion coefficients for the two species.	0.16, 0.08
L	Physical translation-cell side length.	256
T	Orbit period, solved along with the field, in model time units.	370.975586
N, M	N² spatial nodes; M saved phases per period. M does not set the integration timestep.	48, 128

Diffusion normalization. In the SymSim shader, DiffR and DiffG multiply unscaled neighbor differences; the corresponding coefficients in this implementation are D_u/h² and D_v/h². The example uses the five-point stencil. The optional SymSim nine-point weights (0.8 axial, 0.2 diagonal, −4 centre) have a leading continuum factor of 1.2. Matching continuum diffusion across these stencils requires dividing D by 1.2 in the nine-point version, or multiplying h by √1.2. Agreement in F and k alone is insufficient to reproduce a preset. [1] [3]

Periodic-orbit computation. For g96, the shooting equation is Φ_T/4(q₀) = q₀ ∘ R⁻¹, where Φ denotes numerical Gray–Scott evolution. The unknowns are the initial field and T, with a phase condition fixing the time origin. Hopf-mode initial guesses and continuation generate candidates; GPU batching accelerates the independent trajectories required for Newton correction. SymSim presets provide additional possible initial guesses. The computed examples are concentration waves; a periodic U-skate glider has not been established. [4]

Numerical verification. Independent, unprojected forward integration checks the return residual, the full trajectory, and every prescribed phase relation, followed by timestep refinement. The catalog additionally requires every rotation with nonzero time offset to exceed a prescribed same-time symmetry-error threshold in each concentration channel throughout interpolated playback. The results are numerical periodic orbits of the discretized equations, accepted within finite tolerances. They do not establish continuum existence, stability, or completeness. Failure of the search does not establish nonexistence. [5]

The illustrated g96 orbit has an independently measured return residual of approximately 5.3 × 10⁻⁹ RMS across u and v. Refining N from 24 to 48 at fixed L changes its period by approximately 1%. Discrete-orbit closure and spatial discretization error are therefore assessed separately. The eight-second playback represents T ≈ 370.98 model time units. [6]

The catalog organizes precomputed fields by symmetry, admitted parameter set, and solution. Palette, tiling, and playback speed affect only the presentation.

View the illustrated orbit →