Asymptotic synchronization in coupled Boolean and probabilistic Boolean networks with delays

This paper addresses the delay output synchronization in a coupled system of a Boolean network and a probabilistic Boolean network with state delays. We model the system as a Markov chain-based augmented system, simplifying the synchronization problem to a delay set stability issue. Analysis of the Markov chain’s transition matrix through positive recurrent closed sets yields synchronization criteria, both in probability and in probability one, based on Boolean matrix characteristics and matrix splitting techniques. These methods avoid the cumulative multiplication of high-dimensional matrices and reduce the complexity of the computation. Besides, the study progresses to explore asymptotic output synchronization, employing these methods. To demonstrate the practicality and effectiveness of the proposed methodology, several illustrative examples are provided.