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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

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.