Consider a system of mn ODEs
where s are uniformly Lipschitz continuous, are matrices. Various notions of synchronization are defined, which means that the trajectories of all the cells approach each other. Sufficient conditions for asymptotic synchronization of linearly coupled identical dynamic systems are proved and discussed for various coupling configurations using the main results of Lyapunov’s direct method. The considered coupling configurations are described by symmetric matrices, irreducible matrices, normal matrices, circulant matrices, nonnegative matrices including their perturbations. Sufficient conditions for additive nonlinear coupling to make the synchronized state asymptotically stable are given.