Synchronization in an array of chaotic systems coupled via a directed graph

Most analytical results on the synchronization of coupled chaotic systems consider the case of reciprocal coupling, i.e., the coupling matrix is symmetric and the underlying topology is an undirected graph. We study synchronization in arrays of systems where the coupling is nonreciprocal. This corresponds to the case where the underlying topology can be expressed as a weighted directed graph. We show that several recently proposed definitions of the algebraic connectivity of directed graphs are useful in deriving sufficient conditions for synchronization. In particular, we show that an array synchronizes for sufficiently strong cooperative coupling if the coupling topology includes a spanning directed tree. This is an intuitive result since the existence of such a tree implies that there is a system which influences directly or indirectly all other systems and thus it is possible to make every system synchronize to it.