Intralayer Synchronization Of Multiplex Dynamical Networks Via Pinning Control

Because of the interactions between real-world networks,the research of single-layered networks cannot meet practical needs.In recent years,the concept of multiplex networks has been proposed and becomes popular.Multiplex networks can better describe the interactions between real-world networks.Synchronization is an universal phenomenon in real-world networks.The research on the synchronization of multiplex networks is of great significance,which is attracting the attention of researchers from different fields.Since the real-world networks usually contain a large number of nodes and their topological structures are complex,it is more practical and meaningful to realize synchronization of multiplex networks via pinning control.Most of the existing works about pinning control are based on single-layered complex networks.This thesis investigates the intralayer synchronization of multiplex networks via pinning control,which are based on two-layered networks with undirected connections.The two-layered networks are with one-to-one interlayer couplings.The topology of each layer can be different,and the individual dynamics of nodes in different layers can also be different.The topology of the multiplex network is represented by the supra-Laplacian matrix.The innovation points of the thesis are as follows:1)Describing the network models,synchronization states and the concept of intralayer synchronization under pinning control in this thesis.Combining pinning control with adaptive control and linear feedback control respectively,this thesis considers two cases: adding controllers to the nodes of one layer and adding controllers to all the layers.Using the Lyapunov stability theory,we obtain several sufficient conditions for two-layered networks to realize the intralayer synchronization via pinning control for each case.By analyzing the characteristics of the supra-Laplacian matrix,this thesis finds that when adding controllers to the all nodes of one layer,or global controlling one layer and pinning control the other layer,the greater the interlayer coupling strength is,the more conducive to realize the intralayer synchronization of the two-layered networks via pinning control.In addition,when the interlayer coupling strength does not exceed the intralayer coupling strength,increasing the interlayer coupling strength is conducive to the intralayer synchronization of the two-layered networks.Finally,numerical simulations are given to demonstrate the correctness of the theoretical results.2)Combining pinning control with impulsive control,this thesis considers two cases: adding impulsive controllers to all the layers simultaneously at the same impulsive instants to realize the intralayer synchronization of the two-layered networks via pinning control;and adding impulsive controllers to each layer network at different impulsive instants.Using the Lyapunov stability theory and impulsive control theory,this thesis obtains several synchronization criteria for multiplex networks.Based on these synchronization criteria,reasonable algorithms are designed to realize the intralayer synchronization of multiplex networks.Meanwhile,numerical simulations are presented to verify the validity and correctness of the theoretical results.Furthermore,we find that a larger interlayer coupling strength is beneficial for realizing the intralayer synchronization of multiplex dynamical network via pinning impulsive control.