Graph and attractor equivalence between reaction–diffusion PDEs and their ODEs for small coupling

Establish that for the reaction–diffusion system ∂_t u_k = D_k Δ u_k + λ F_k(u_1,…,u_n) on a periodic spatial domain, with smooth vector field F and sufficiently small λ>0, if the ordinary differential equation x˙ = F(x) possesses a compact connected global attractor, then (i) the global attractor of the reaction–diffusion system consists solely of spatially constant functions, and (ii) the chain graph (whose nodes are maximal chain-recurrent sets and edges represent the downstream relation) of the reaction–diffusion system coincides with the chain graph of the ODE x˙ = F(x).

Background

The paper introduces the chain graph of a dynamical system, whose nodes are maximal chain-recurrent sets and where a directed edge A→B exists if some point in A is downstream from some point in B via ε-chains contained in a compact set. For the scalar Chafee–Infante equation with 0<λ<1 on S^1, the authors prove that the global attractor consists only of spatially constant functions and that the chain graph matches that of the corresponding ODE u˙=λu(1−u^2).

They then generalize to a system of reaction–diffusion equations ∂_t u_k = D_k Δ u_k + λ F_k(u) with periodic boundary conditions and propose a conjecture asserting that, for sufficiently small λ, if the ODE x˙=F(x) has a compact connected global attractor, then (a) the PDE’s global attractor comprises only constant-in-space functions and (b) the PDE’s chain graph is the same as the ODE’s chain graph.