Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

Abstract

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem $(H_L,X_R,P)$, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies $C_{\mathcal{K}}^2<\hat{I}{¼}_L\hat{I}{¼}_R$. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.