Synchronization of multi-layer networks: From node-to-node synchronization to complete synchronization

Multi-layer networks, which incorporate multiple subsystems and different kinds of interactions, are recently believed to have a stronger ability in modelling various real-world systems than traditional single-layer complex networks. Motivated by this observation, we try to show how to achieve node-to-node synchronization and complete synchronization in multi-layer networks under directed switching communication topologies, where the nodes have Lur'e type dynamics. First, by using the multiple Lyapunov stability theory, we show that node-to-node synchronization in two-layer networks can be achieved by choosing sufficiently large coupling strength as well as suitable feedback gain matrix if the average dwell time is bounded below by some positive constants. The result is then extended to the case of multiple layers. Furthermore, if the inner communication topologies of each layer have a directed spanning tree with a common root, we show complete synchronization in the considered network can be achieved if the control parameters and the average dwell time are appropriately chosen. Finally, the theoretical results are validated through performing numerical simulations on Chua's circuit with double scroll attractors.