Synchronization of Two-layered Dynamical Networks with On-off Inter-layer Coupling

This work investigates the synchronization problems of a two-layered network, where the involved nodes are of delay nonlinearity. By using a generalized delay differential inequality, some sufficient conditions are established for reaching synchronization between the two layers. It is revealed that when the inter-layer coupling strength is time-varying, the time average of the smallest eigenvalue of an induced matrix plays a key role in reaching synchronization. Meanwhile, the largest admissible delay is also estimated. Numerical examples are also given to demonstrate the validity of the theoretical results.