Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems
Bibliographic record
Abstract
Spatially localized two-dimensional spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software pde2path [H. Uecker, D. Wetzel, and J. D. Rademacher, Numer. Math. Theory Methods Appl., 7 (2014), pp. 58--106] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".