Approximate localised dihedral patterns near a turing instability
Bibliographic record
Abstract
Abstract Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of radially-symmetric patterns. Our analysis covers localised planar patterns equipped with a wide range of dihedral symmetries, thereby avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gröbner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton–Kantorovich procedure to prove the existence of localised patches with 6 m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".