Application of Kronecker products to the analysis of systems with uniform linear coupling

In this paper, we show how the Kronecker product can be useful in simplifying the notation encountered in the analysis of linearly coupled systems where the cells are identical and the coupling is uniform. We give two applications where this is useful. First we use it in the analysis of Turing patterns in reaction-diffusion systems to obtain conditions for the coupling to destabilize the uniform equilibrium point. In the second application we use the Kronecker product to obtain simple sufficient conditions for an array of linearly coupled dynamical systems to synchronize. We discuss briefly extensions to additive nonlinear coupling