Intra-layer synchronization in duplex networks**Project supported in part by the National Natural Science Foundation of China (Grant Nos. 61573004 and 11501221), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-YX301), the Program for New Century Excellent Talents in Fujian Province University in 2016, and the Project of Education and Scientific

Abstract

This paper explores the intra-layer synchronization in duplex networks with different topologies within layers and different inner coupling patterns between, within, and across layers. Based on the Lyapunov stability method, we prove theoretically that the duplex network can achieve intra-layer synchronization under some appropriate conditions, and give the thresholds of coupling strength within layers for different types of inner coupling matrices across layers. Interestingly, for a certain class of coupling matrices across layers, it needs larger coupling strength within layers to ensure the intra-layer synchronization when the coupling strength across layers become larger, intuitively opposing the fact that the intra-layer synchronization is seemly independent of the coupling strength across layers. Finally, numerical simulations further verify the theoretical results.