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.

在各种控制技术中，变量替换控制是用于同步系统的最实用的控制之一。在本文中，我们研究了通过变量替换控制的两层网络的层间同步。我们关注三种情况：控制所有节点的拓扑相同，控制所有节点的拓扑不同以及控制部分节点的相同拓扑。理论和数值结果表明，无论耦合强度如何，控制响应层中的所有节点都可以导致层间同步。但是，对于在驱动层和响应层中具有不同拓扑的两层网络，完全连接的网络和星形网络具有最佳的驱动能力，环形网络紧随其后，而链条的性能最差。此外，当在响应层中控制部分节点时，是否发生层间同步取决于层大小，内部耦合矩阵，耦合强度以及在响应层中控制的节点数。提供的结果为通过变量替换控制同步两个网络提供了说明性和有用的提示。