Global Synchronization of Discrete-Time Dynamical Network With a Directed Graph

We investigate synchronization of linearly coupled map lattices with asymmetric and irreducible coupling matrices. In terms of graph terminology, the coupling matrix represents a directed graph. In case the uncoupled map satisfies Lipschitz conditions, a criterion of global synchronization of the linearly coupled map lattices is derived. With this criterion, we investigate how synchronizability depends on the coupling matrix as well as graph topology. Furthermore, we also prove that a directed graph can synchronize some chaotic map if and only if the graph contains a spanning tree