Synchronization Of Complex Networks And Flocking Of Cucker-smale System

In the past decades,complex networks have been emerged and developed.In recent years,complex networks have rapidly been developed into a new field of research and attracted different scholars of various fields.Both the synchronization of complex networks and the flocking of Cucker-Smale systems are the typical behavior of complex networks.Both of them are the hot reaearch topics in recent years.However,distinuous coupling,delay and noise may affect the synchronization of systems.Moreover,the flocking behavior of Cucker-Smale systems may be influenced by nosie and a virtual leader.And the convergence speed of the system is also an important index for measuring the performance of the system.In order to investigate the synchronization of complex networks and the flocking behavior of Cucker-Smale systems further,the main contents of this paper are as follows:The inner synchronization of time-delayed complex dynamical networks with distinuous coupling is studied in this paper.By constructing suitable system model and using Lyapunov stability theory,the sufficient conditions for the networks synchronization are established and the upper bound estimation of the time delay was obtained.Numerical examples are provided to verify the effectiveness of the theoretical results.However,synchronization can occur between nodes in the network,and can occur between two networks.So the outer synchronization of time-delayed complex dynamical networks with periodic on-off coupling is studied in this paper.By using delay differential equation theory,sufficient conditions for the complete and generalized outer synchronization are provided.Numerical examples are provided to confirm the effectiveness of the theoretical results.Secondly,finite-time stochastic mixed outer synchronization of complex dynamical networks is investigated.The sufficient conditions for the mixed outer synchronization are provided.In most previous works of leader-following Cucker-Smale systems,the flocking is asymptotical and the convergence time is infinite.However,in the real world,for example,a flock of birds can return to an orderly flight at a finite time after being influenced by occasional interference.Because of this phenomenon,the finite-time flocking of leader-following Cucker-Smale systems is studied.Based on Lyapunov function approach,the sufficient conditions for finite-time flocking are established.It is shown that the convergence time decreases with the increasing of the coupling strength.Simulation examples are provided to confirm the theoretical results.Noise iscommon in the real world and may affect systems.Thus,we investigate finite-time flocking problem of leader-following Cucker-Smale systems with noise further.The sufficient conditions for occuring flocking are given.It is shown that flocking can occur with noises.Simulation examples are provided to confirm the theoretical results.