Inter-layer Synchronization In Two-layer Networks Via PC Control

Chaos refers to that there is a seemingly irregular movement form in the real world.Due to the sensitivity of the initial values and unpredictability of long-term development trend of chaos,chaotic control and synchronization become the key of chaotic application.From which chaotic synchronization was noticed by people,to research objects that are couples by two or three systems,then to the synchronization of complex network the concepts of synchronizatiotn have been developed continuously.Scholars have achieved fruitful research results over the years.With the deepening of network research,people found that there are certain relationships among many real networks.Therefore,multi-layer network has become a hot topic in network research in recent years.This paper focuses on the inter-layer synchronization of two-layer networks via drive-response method and drive-feedback method.We apply the PC method to the two-layer network with Lorenz system as node dynamics and studies the inter-layer synchronization.Then we provide a theoretical proof of inter-layer synchronization in two-layer network with identical topologies when controlling all nodes in response layer.And it is verified by numerical simulation as well.In addition,the numerical simulations are also used to study the following two cases and come to some useful conclusions:the first case is that the topologies of two layers are different and controlling all nodes in the response layer;the second one is that the topologies of two layers are identical,but only controlling part of the nodes in the response layer.In addition,we combine drive and feedback methods to study inter-layer synchro-nization with other chaotic systems of Lorenz system family,Chen system,Lu system and unified chaotic system,as node dynamics respectively,and obtain some sufficient conditions.