Inter-layer synchronization in two-layer networks via variable substitution control

Abstract

Among various control techniques, variable substitution control is one of the most practical one for synchronizing systems. In this paper, we study inter-layer synchronization of two-layer networks via variable substitution control. We focus on three cases: identical topologies with all nodes being controlled, different topologies with all nodes being controlled, and identical topologies with part of nodes being controlled. Theoretical and numerical results show that, controlling all nodes in the response layer can lead to inter-layer synchronization whatever the coupling strength is. However, for two-layer networks with non-identical topologies in the drive and response layers, the fully-connected network and the star network have the best driving ability, the ring network follows, and the chain has the worst. Furthermore, when partial nodes are controlled in the response layer, whether inter-layer synchronization occurs depends on the layer size, the inner-coupling matrix, the coupling strength as well as the number of nodes controlled in the response layer. The provided results offer illustrative and useful hints for synchronizing two networks via variable substitution control.