Study On Impulsive Synchronization Of Time-delayed Dynamic Complex Networks

In recent years,the synchronization of complex networks have been focused in various fields.Due to the extensive existence of complex networks,the mathematical models,such as the fileds of biology,sociology,finance,communication and power electronics are all classified as complex networks.It is well known that,as long as the communication between the system and the group,under the influence of the signal transmission rate or the internal and external environment,the acquisition of information is almost time-delayed.In oreder to be more closer to reality,the time lags situations must be taken into account,when study complex networks.Nowadays,the type of time-delays which can be divided into unchanged time-delays and variable time-delays,in which the variable time-delays is more general,is discrete time-delayed and continuous distributed time-delayed.Because of their complexity,complex networks sometimes exhibit chaotic characteristics.The realization of chaos is a deterministic system that can be described,but it shows uncertainty.Chaos has the property of non-convergence,non-divergence,and extreme dependence on initial values.The synchronization of the chaos complex networks also are hot research.In addition,for the synchronization of different networks with different initial values(also known as a system to track another system),adding controllers to the network is the focus research.There are many control methods,in which impulse control is gradually applied to complex network because of its low cost and reducing noise interference.At the same time,like a delay event in a complex network itself,as a controller,affected by physical communication equipment,etc.,time-delays of control signal is more real.How to use the time-delayed impulse controller to control the tracking synchronization between different time-delayed complex networks is a very significant issue.The content of the analysis discussed in this paper is also using the time-delayed impulse control method to promote the synchronization between two complex networks with same structure,different initial values.Based on Lyapunov stability theorems andfunctions,Lipschitz conditions,mathematical inequalities,and differential impulse inequalities,the different synchronization conditions under different situations are obtained.Firstly,this paper studies the complex network with varible discrete time-delays and uses the time-delayed impulse controller to control the synchronization of two networks.In the application of the impulse controller,considering the internal delay of the controller due to various factors,there is randomness in the appearance of the situation of time-delays,that is,there may be a delay at some moments,or it may not exist at some times.Therefore,a random delay is introduced in the impulse controller.The analysis shows that under the influence of this controller,the synchronization conditions of the varible discrete time-delays complex network.Secondly,for a class of varible discrete time-delays complex networks with a kind of random special cases,two networks arrive at lag synchronization via a delay impulse controller.The internal time delays of the network,due to the environmental impacts,in the non-ideal condition,the delay will appear,in the ideal environment,the situations of delays does not exist.Therefore,for the two complex networks with random delay as the object of study is very realistic,but in this paper,the two networks have the same random sequence,so it belongs to a special case study.On the other hand,hysteresis synchronization exists widely in life and production.for example,when an aircraft‘a'tracks another aircraft‘b',it is‘a'reaches that a time delay state of‘b'.In this paper,the hysteresis synchronization condition of the network is obtained by the time-delayed impulsive control method.Finally,for a class of two complex networks with varible discrete time-delays and varible continuous distribution time-delays,synchronization is achieved by a time-delayed impulse controller.Discrete time delays are transient,however,in some cases,such as when there are many parallel channels in the signal,the delay will be continuous.Therefore,it is necessary to study complex networks with continuous distributed time delays.In recent years,synchronization of complex networks with discrete time-delays and continuous distributed delays has benn focused researches in various fields.In this paper,the situation of time-delays is introduced into the impulsivecontroller,and a class of impulsive differential equation inequalities are extended.The criteria for the synchronization between complex networks with mixed time-delays under the time-delayed impulse control is obtained.