Explosive Synchronization And Generalized Synchronization With Time-varying Delay Via Pinning Impulsive Controllers Of Complex Networks

In the past two decades,complex networks have flourished with the introduction of the small-world network model and scale-free network model.They play an important role in modern network research and provide a basis for the development of complex networks.Synchronization,as a common phenomenon in nature,is a significant and interesting direction of complex networks,among which explosive synchronization and generalized synchronization have become a hot research field.However,most of the explosive synchronization is studied based on the Kuramoto oscillator model,and few of them focus on the coupled chaotic systems.In addition,in the real world,the delay is ubiquitous,and even time-varying.Thus,in this thesis,we study the explosive synchronization of networks coupled with chaotic systems and the generalized synchronization of drive-response networks with time-varying delays.The main contents and innovations points of this thesis are listed as follows:(1)Previous studies have shown that when the node dynamics is Kuramoto model,the explosive synchronization occurs if the network satisfies two conditions: the network structure must be heterogeneous and node dynamics must be are positively correlated with topological structure.In this thesis,unified chaotic system and Rossler system are used as node dynamics to discuss whether these two conditions play a decisive role in the emergence of explosive synchronization.The results show when the node dynamics are Lorenz system and classical Lü system,it is not necessary to satisfy the heterogeneity of the topology structure,nor the positive correlation between the dynamics and the topology structure,and the network will display obvious explosive synchronization,or even cluster explosive synchronization.However,for the generalized Lü system and Chen system,regardless of whether the network structure is heterogeneous or homogeneous,even if the positive correlation between the dynamics and the topology structure is satisfied,there is no obvious explosive synchronization phenomenon in the network.When the node dynamics is R?ssler system,the process of network tending to synchronization is continuous.Moreover,once the entire network is completely synchronized,the synchronization state of the network will not be destroyed even if the coupling strength is reduced.Finally,this thesis attempts to give some explanations from the dynamical behavior of chaotic systems and provides some ideas for studying the explosive synchronization of complex networks.(2)Generalized synchronization,which is weaker than complete synchronization,plays an important role in many networked systems.By taking the auxiliary system approach,we investigate generalized synchronization of drive-response networks with time-varying delays.Based on the Lyapunov stability theory and the pinning impulsive controllers,the two sufficient conditions for guaranteeing generalized synchronization of dynamical networks with time-varying coupling delays and node delays can be derived strictly.The pinning impulsive controllers in the thesis are used by pinning a small fraction of all nodes,low cost and easy to operate.Finally,some numerical simulations are used to verify the theoretical results.The results show that the proposed control scheme and theoretical results are effective and feasible.