<p>In this article, we study the canonical Brusselator partial differential equation (PDE) from the field of pattern formation in the limit in which the diffusivity of the activator is much smaller than that of the inhibitor. The PDE robustly exhibits a subcritical Turing bifurcation that, in this limit, is labeled as a <i>singular Turing bifurcation</i>. We show that families of spatially-periodic canard solutions emerge from this subcritical singular Turing bifurcation. Then, right after they emerge, the solutions lose their purely sinusoidal structure (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$e^{ik_Tx}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mi>i</mi> <msub> <mi>k</mi> <mi>T</mi> </msub> <mi>x</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k_T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> is the critical wavenumber at the Turing bifurcation) and gain a distinct multi-scale spatial structure. They consist of segments along which the components vary gradually in space, interspersed with short intervals on which the activator component exhibits pulses and steep gradients. The branches of these spatially-periodic canards undergo a saddle-node bifurcation. Some of the large-amplitude patterns on the upper branches are attractors of the PDE, and unstable patterns with small pulses that exist below the folds appear to guide the evolution of data to the attractors. We also analyze the spatial ordinary differential equations (ODEs) that govern time-independent solutions. We show that the spatial ODE system has a folded singularity known as a reversible folded saddle-node of type II (RFSN-II) point that coincides with the Turing bifurcation in the singular limit. We demonstrate that, for parameter values close to the Turing bifurcation, the true and faux canards of the RFSN-II point are responsible for generating the spatially-periodic canards, and for parameter values away from the Turing value there is a reversible folded saddle point whose true and faux canards generate the spatially-periodic canard solutions. In short, these solutions are new examples of “les canards de Turing", first discovered in the van der Pol PDE in Vo (SIAM J Appl Dyn Syst 24(4):2618–84, 2025). Overall, we identify the RFSN-II and RFS folded singularities and their canards as new selection mechanisms for subcritical bifurcations.</p>

Stable and Unstable Spatially-Periodic Canards Created in Singular Subcritical Turing Bifurcations in the Brusselator System

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2026年5月19日

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Robert Jencks,
Arjen Doelman,
Tasso J. Kaper,
Theodore Vo

摘要

In this article, we study the canonical Brusselator partial differential equation (PDE) from the field of pattern formation in the limit in which the diffusivity of the activator is much smaller than that of the inhibitor. The PDE robustly exhibits a subcritical Turing bifurcation that, in this limit, is labeled as a singular Turing bifurcation. We show that families of spatially-periodic canard solutions emerge from this subcritical singular Turing bifurcation. Then, right after they emerge, the solutions lose their purely sinusoidal structure ( $e^{ik_Tx}$ $e^{i k_{T} x}$ , where $k_T$ $k_{T}$ is the critical wavenumber at the Turing bifurcation) and gain a distinct multi-scale spatial structure. They consist of segments along which the components vary gradually in space, interspersed with short intervals on which the activator component exhibits pulses and steep gradients. The branches of these spatially-periodic canards undergo a saddle-node bifurcation. Some of the large-amplitude patterns on the upper branches are attractors of the PDE, and unstable patterns with small pulses that exist below the folds appear to guide the evolution of data to the attractors. We also analyze the spatial ordinary differential equations (ODEs) that govern time-independent solutions. We show that the spatial ODE system has a folded singularity known as a reversible folded saddle-node of type II (RFSN-II) point that coincides with the Turing bifurcation in the singular limit. We demonstrate that, for parameter values close to the Turing bifurcation, the true and faux canards of the RFSN-II point are responsible for generating the spatially-periodic canards, and for parameter values away from the Turing value there is a reversible folded saddle point whose true and faux canards generate the spatially-periodic canard solutions. In short, these solutions are new examples of “les canards de Turing", first discovered in the van der Pol PDE in Vo (SIAM J Appl Dyn Syst 24(4):2618–84, 2025). Overall, we identify the RFSN-II and RFS folded singularities and their canards as new selection mechanisms for subcritical bifurcations.