Les Canards de Turing

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 support the numerical discoveries, we use geometric desingularization and geometric singular perturbation theory to demonstrate the existence of these families of spatially periodic canards. Crucially, in the singular limit, we study a novel class of {\em reversible folded singularities}. In particular, there are two reversible folded saddle-node bifurcations of type II (RFSN-II), each occurring asymptotically close to a Turing bifurcation. We derive analytical formulas for these singularities and show that their canards play key roles in the observed families of spatially periodic canard solutions. Then, for an interval of values of the bifurcation parameter further below the Turing bifurcation and RFSN-II point, the spatial ODE also has spatially periodic canard patterns, however these are created by a reversible folded saddle (instead of the RFSN-II). It also turns out that there is an interesting scale invariance, so that some components of some spatial canards exhibit nearly self-similar dynamics.