Les Canards de Turing

SIAM Journal on Applied Dynamical Systems, vol. 24, pp. 2618–2684

Abstract

Abstract. In this article, we study a prototypical 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 singular Turing bifurcations. We show that the small-amplitude, spatially periodic solutions that emerge from the Turing bifurcations form families of spatially periodic canards that oscillate about the homogeneous equilibrium. 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. We also find families of large-amplitude, spatially periodic canards. These have a “fast-slow” spatial structure, with segments of steep gradients and segments of gradual variation. In this prototypical PDE, we also show that for most parameter values under study the Turing bifurcation is sub-critical, and we present the results of numerical simulations showing that several types of spatial canard patterns are attractors in the PDE. To support the main numerical discoveries, we use the method of geometric desingularization and geometric singular perturbation theory on the spatial ODE system to demonstrate the existence of these families of spatially periodic canards. Crucially, in the singular limit, we study a novel class of reversible folded singularities of the spatial ODE system. 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 small-amplitude and large-amplitude 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 system 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.

Authors 3

1. Theodore Vo

   Monash University

   Affiliation as printed

   School of Mathematics, Monash University, Clayton, Victoria, 3800 Australia

2. Arjen Doelman Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, 2300 RA Leiden, The Netherlands

3. Tasso J. Kaper

   Boston University

   Affiliation as printed

   Department of Mathematics and Statistics, Boston University, Boston, MA 02215 USA