Synchronization on Directed Delayed Duplex Networks: From the Perspective of Coupled Delays

Synchronization on multi-layer networks are paid increasingly attention in recent years, and the interplay between network dynamics and structures in multi-layer networks remains open. Focusing on interactions between synchronization dynamics and network structures from the perspective of coupled delays, this paper investigates the intra-layer and inter-layer synchronization on delayed duplex networks without and with additional controllers via Lyapunov stability theory and Halanay inequality, presents the synchronization criteria which are not only applied for the delayed duplex networks with general intra-layer topologies but for the case of networks without coupling delays, and gives the upper bound of delay thresholds for the first time, characterizing the interaction between intra-layer (inter-layer) synchronization and inter-layer (intra-layer) connections. Further on, we analyze the influence of coupling strengthes, topological structures and control gains on the threshold of delay, and obtain some interesting results: 1) the network with single kind of coupling delays (either intra-layer or inter-layer) can admit larger threshold than that with both kinds of coupling delays, and the network with controllers can admit larger threshold than that without controllers. 2) For intra-layer (inter-layer) synchronization in the network with just inter-layer (intra-layer) delays, the threshold of inter-layer (intra-layer) delays is bounded by an upper bound determined by the topology and coupling strength of the inter-layer (intra-layer). Finally, extensive numerical simulations are provided to verify the present theoretical results.