Synchronization of Boolean networks with time delays

This paper studies analytically synchronization of two deterministic Boolean networks with time delays, which are unidirectionally coupled in the drive-response configuration. Necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi-tensor product of matrices. The synchronization conditions show that, for drive-response Boolean networks, different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena, and both kinds of time delays that ensure synchronization, if existent, are not unique. Examples are worked out to illustrate the present results.