2509.04835
Stable and unstable spatially-periodic canards created in singular subcritical Turing bifurcations in the Brusselator system
Robert Jencks, Arjen Doelman, Tasso J. Kaper, Theodore Vo
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:57 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper and the candidate solution agree on the spatial ODE formulation, the reduced/desingularized slow flow, the identification and type of the folded singularities (RFSN-II at B=1 and RFS for B>1), the existence and explicit characterization of the singular canards via a Hamiltonian first integral, the transversality that ensures persistence of a symmetric maximal canard for B>1+O(ε), and the geometric construction of spatially periodic canards by concatenating slow segments on S_s^ε with fast layer homoclinics. The only substantive technical discrepancy is the blow-up scaling: the paper uses the weights U=r^2, P=r^3, V=r^4, Q=r^3, ε=r^2, and the unfolding B=r^4, whereas the model states a different ‘standard’ FSN-II scaling. This mismatch does not alter the core conclusions but should be corrected for precision.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} strong field \textbf{Justification:} The manuscript delivers a coherent geometric mechanism for spatially periodic canards in the Brusselator, anchored in a careful analysis of the reduced flow, folded singularities (RFSN-II/RFS), and blow-up, with compelling numerical support. The results extend the emergent paradigm of ‘Turing’s ducks’ beyond the van der Pol system. Minor clarifications on blow-up scalings and the asymptotic link to the reversible 1:1 normal-form branch would make an already strong paper even clearer.