Interlayer Synchronization In The Multilayer Networks With The Centrally Symmetric Structure

Complex networks are very common in real life,such as the Internet,social networks and transportation networks.In many practical problems,networks interact with each other,so the interaction between multilayer complex networks has attracted a lot of attention.The synchronization of networks is a widespread phenomenon in nature and has been a major concern of scholars.With the development of multilayer complex network theory,the generalized synchronization of multilayer networks has also received extensive attention.Most of the research on generalized synchronization of multilayer networks is based on the theory of Lyapunov function,By using auxiliary system method and linear matrix inequality method,a certain number of nodes are selected and controlled externally to to achieve inter-layer synchronization.The main problems of this method are as follows: Large-scale complex networks have many nodes and connected edges,so it is difficult to choose the nodes to apply control,and the control strategy is also complicated.In this paper,we study the multilayer network with a centrosymmetric structure,which is easy to construct and has practical application value.The details are as follows.(1)Firstly,we introduce complex networks in real-life scenarios,such as the Internet,social networks and transportation networks,and introduce multilayer network and its mathematical definition.The theoretical properties of complex networks,common network models,research status of synchronization problems and dynamical systems with time lags and their properties are introduced.After that,the synchronization concept and research methods of single-layer complex networks,intra-layer synchronization and inter-layer synchronization of multilayer networks are introduced.Finally,the definition and properties of the centrosymmetric matrix used in this paper are introduced.(2)For the two layer networks with the same topology,we get the centrosymmetric structure by adjusting the intra-layer and inter-layer coupling between two layers.The synchronization problem of this multilayer network is studied;the conditions that guarantee the inter-synchronization and anti-inter-synchronization are obtained.In addition,according to the linear stability theorem,the stability of the equilibriums and the existence of Hopf bifurcation are investigated.The inter-synchronization problems are then investigated for the equilibrium and the periodic solutions.Finally,a two-layer network is constructed for numerical simulation to verify the theoretical results.(3)Considering the inter-synchronization problems of the multi-layer delayed networks with centrosymmetric structure.The stability of the equilibriums and the existence of the Hopf bifurcation of the system are then studied.Based on the properties of the central symmetric matrix,the conditions of interlayer synchronization and antiphase synchronization of the equilibrium and periodic solutions are then obtained.The numerical experiments are then used to verify the theoretical results.