Singular Turing bifurcations and spatial canard solutions in nonlinear reaction-diffusion systems

Abstract

We study a general class of nonlinear reaction-diffusion equations that model pattern-forming systems. The class includes the Gierer-Meinhardt PDE, Brusselator model, and van der Pol PDE, as well as the Gray-Scott, Klausmeier, Lengyel-Epstein, Schnakenberg PDEs and others of activator-inhibitor type. In the limit in which the activator diffusivity is much smaller than that of the inhibitor, these PDEs exhibit singular Turing bifurcations, which have only recently begun to receive attention. We analytically establish that the spatially-periodic solutions that emerge in both the sub-critical and super-critical cases of singular Turing bifurcations are spatially-periodic canard solutions. These canard patterns are new types of spatially-periodic solutions that --just beyond the Turing point-- have fast-slow structure in space, rather than the classical sinusoidal profile. In addition, their amplitude grows more rapidly than the classical square root growth for Turing patterns. Indeed, even for parameter values that differ by one part in a hundred from the Turing point, they can have $\mathcal{O}(1)$ amplitude. We also establish the existence of general spatially-dependent canard solutions with fast-slow structure. Our analysis focuses on the spatial ODEs that govern the time-independent solutions. We show that these ODEs have a reversible folded saddle-node singularity of type II asymptotically close to the singular Turing point, that reversible folded saddles occur for parameters away from it, and that the true and faux canards of these folded singularities are the mechanisms responsible for creating the spatial canard solutions in the general class of PDEs.