Lattice differential equations embedded into reaction–diffusion systems

Abstract

We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction–diffusion equations with spatially periodic diffusive fluxes. The result connects wave-pinning phenomena in lattice differential equations and in reaction–diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, where the infinite-dimensional manifold corresponds to an infinite-dimensional centre eigenspace.