Synchronization of Finite-Field Networks With Time Delays

In this paper, finite-field networks (FFNs) with time delays are investigated. We analyze the dynamics of FFNs with time delays in view of linear recursion theory. It is shown that delayed FFNs behave in a pattern similar to non-delayed FFNs in the sense that any state sequence finally reaches a periodic behavior within finite steps. In addition, a criterion assuring purely periodic behaviors is given. For delayed FFNs, an algebra-theoretic criterion for synchronization is proposed for the first time which imposes constraints on the network matrix tuple and the interaction weights. As an application, we study the synchronization problem of delayed FFNs with a tree-structured interaction topology and accordingly provide a criterion of a simple form. Inspired by the existing results, for the tree case, we prove that a given delayed FFN solves a consensus problem if and only if the same does the corresponding non-delayed FFN. As a comparison, examples are shown to illustrate similar conclusion is invalid for the general synchronization problems since the consensus is included in the synchronization. In the end, we identify the specific periodic behavior of synchronized FFNs satisfying the proposed sufficient condition.