Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions
Bibliographic record
Abstract
For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter $\tau$ and the inhibitor diffusivity $D$. In the limit of large diffusivity $D={D_0/\nu}\gg 1$ with $D_0={\mathcal O}(1)$, $\nu\equiv {-1/\log\varepsilon}$, and $\varepsilon^2$ denoting the activator diffusivity, a leading-order-in-$\nu$ analysis shows that the linear stability of multispot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether $\tau={\mathcal O}(1)$ or $\tau\gg 1$. For $D_0 0$ is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any ${\mathcal O}(1)$ value of the reaction-time parameter $\tau$. This resolves a long-standing open problem in NLEP theory (see [J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Appl. Math. Sci. 189, Springer, 2014]). Instead, by deriving a new modified NLEP appropriate to the regime $\tau\gg 1$, we show for the range $D_0
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".