Guaranteeing Spatial Uniformity in Diffusively-Coupled Systems

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

An Adaptive Algorithm for Synchronization in Diffusively Coupled Systems

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