Guaranteeing Spatial Uniformity in Diffusively-Coupled Systems

TL;DR

This paper introduces a new, relaxed condition using Jacobian matrices and graph Laplacian eigenvalues to guarantee spatial uniformity in diffusively-coupled systems, applicable to both ODE and PDE models.

Contribution

It provides a novel, verifiable condition for spatial uniformity that accommodates diverse diffusion terms and interconnection structures in coupled systems.

Findings

Condition verified via linear matrix inequalities.

Applicable to both compartmental ODE models and reaction-diffusion PDEs.

Ensures spatial uniformity under broad coupling configurations.

Abstract

We present a condition that guarantees spatially uniformity in the solution trajectories of a diffusively-coupled compartmental ODE model, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments. Each set of like components has its own weighted undirected graph describing the topology of the interconnection between compartments. The condition makes use of the Jacobian matrix to describe the dynamics of each compartment as well as the Laplacian eigenvalues of each of the graphs. We discuss linear matrix inequalities that can be used to verify the condition guaranteeing spatial uniformity, and apply the result to a coupled oscillator network. Next we turn to reaction-diffusion PDEs with Neumann boundary conditions, and derive an analogous condition guaranteeing spatial uniformity of solutions. The paper contributes a relaxed condition to check spatial uniformity that allows individual components to have their own specific diffusion terms and interconnection structures.