An Adaptive Algorithm for Synchronization in Diffusively Coupled Systems

TL;DR

This paper introduces an adaptive algorithm that ensures synchronization in diffusively coupled systems, including compartmental ODE models and reaction-diffusion PDEs, by dynamically adjusting link weights based on agent differences.

Contribution

The paper develops a novel adaptive synchronization algorithm applicable to both ODE and PDE systems with diffusive coupling, ensuring spatial homogenization.

Findings

Algorithm guarantees synchronization in compartmental systems.

Algorithm ensures spatial homogenization in reaction-diffusion PDEs.

Numerical example demonstrates effectiveness of the method.

Abstract

We present an adaptive algorithm that guarantees synchronization in diffusively coupled systems. We first consider compartmental systems of ODEs, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments. Each set of like components may have its own weighted undirected graph describing the topology of the interconnection between compartments. The link weights are updated adaptively according to the magnitude of the difference between neighboring agents connected by the link. We next consider reaction-diffusion PDEs with Neumann boundary conditions, and derive an analogous algorithm guaranteeing spatial homogenization of solutions. We provide a numerical example demonstrating the results.