dorsal/arxiv
View SchemaSimple and Superlattice Turing Patterns in Reaction-Diffusion Systems: Bifurcation, Bistability, and Parameter Collapse
| Authors | Stephen L. Judd, Mary Silber |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9807002 |
| URL | https://arxiv.org/abs/patt-sol/9807002 |
Abstract
This paper investigates the competition between both simple (e.g. stripes, hexagons) and ``superlattice'' (super squares, super hexagons) Turing patterns in two-component reaction-diffusion systems. ``Superlattice'' patterns are formed from eight or twelve Fourier modes, and feature structure at two different length scales. Using perturbation theory, we derive simple analytical expressions for the bifurcation equation coefficients on both rhombic and hexagonal lattices. These expressions show that, no matter how complicated the reaction kinectics, the nonlinear reaction terms reduce to just four effective terms within the bifurcation equation coefficients. Moreover, at the hexagonal degeneracy -- when the quadratic term in the hexagonal bifurcation equation disappears -- the number of effective system parameters drops to two, allowing a complete characterization of the possible bifurcation results at this degeneracy. The general results are then applied to specific model equations, to investigate the stability of different patterns within those models.
{
"annotation_id": "ddc713bf-4df7-4742-9842-e0e6b71b20f8",
"date_created": "2026-03-02T18:00:29.403000Z",
"date_modified": "2026-03-02T18:00:29.403000Z",
"file_hash": "8c17aa8274240ea63d4b35d41f5050030d3a7540e9770b94f9a1bca0904b63b3",
"private": false,
"record": {
"abstract": "This paper investigates the competition between both simple (e.g. stripes,\nhexagons) and ``superlattice\u0027\u0027 (super squares, super hexagons) Turing patterns\nin two-component reaction-diffusion systems. ``Superlattice\u0027\u0027 patterns are\nformed from eight or twelve Fourier modes, and feature structure at two\ndifferent length scales. Using perturbation theory, we derive simple analytical\nexpressions for the bifurcation equation coefficients on both rhombic and\nhexagonal lattices. These expressions show that, no matter how complicated the\nreaction kinectics, the nonlinear reaction terms reduce to just four effective\nterms within the bifurcation equation coefficients. Moreover, at the hexagonal\ndegeneracy -- when the quadratic term in the hexagonal bifurcation equation\ndisappears -- the number of effective system parameters drops to two, allowing\na complete characterization of the possible bifurcation results at this\ndegeneracy. The general results are then applied to specific model equations,\nto investigate the stability of different patterns within those models.",
"arxiv_id": "patt-sol/9807002",
"authors": [
"Stephen L. Judd",
"Mary Silber"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Simple and Superlattice Turing Patterns in Reaction-Diffusion Systems: Bifurcation, Bistability, and Parameter Collapse",
"url": "https://arxiv.org/abs/patt-sol/9807002"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "b6cd48e2-cb19-4fab-afd4-21416c334233",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}