Les Canards de Turing
Theodore Vo, Arjen Doelman, Tasso J. Kaper

TL;DR
This paper investigates spatially periodic canard solutions emerging from singular Turing bifurcations in reaction-diffusion systems with widely separated diffusivities, combining numerical simulations with geometric singular perturbation theory.
Contribution
It introduces a novel class of reversible folded singularities and analytically characterizes their role in the emergence of spatial canard patterns near Turing bifurcations.
Findings
Discovery of spatially periodic canard solutions near Turing bifurcations.
Identification of reversible folded saddle-node bifurcations of type II (RFSN-II).
Numerical evidence of these patterns acting as attractors in PDE simulations.
Abstract
In this article, we study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {\em singular Turing bifurcations}. The emergence of these spatially periodic canards asymptotically close to the Turing bifurcations, which are reversible 1:1 resonant Hopf bifurcations in the spatial ODE system, is an analog in spatial dynamics of the emergence of limit cycle canards in the canard explosions that occur asymptotically close to Hopf bifurcations in time-dependent ODEs. In the full PDE system, we show that for most parameter values under study the Turing bifurcation is sub-critical, and we present the results of some direct numerical simulations showing that several of the different types of spatial canard patterns are attractors in the prototypical PDE. To…
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Taxonomy
TopicsComputability, Logic, AI Algorithms
