Impulsive Synchronization of Multilinks Delayed Coupled Complex Networks with Perturb Effects

1 Shiyan Vocational and Technical College, Hubei, Shiyan 442000, China 2 School of Economics, Huazhong University of Science and Technology, Hubei 430074, China 3 NOSTA, The Ministry of Science and Technology of China, GPO Box 2143, Beijing 100045, China 4 Computer School of Wuhan University, Wuhan 430079, China 5 Department of Mathematics and Finance, Yunyang Teachers’ College, Hubei 442000, China 6 College of Information Science and Technology, Donghua University, Shanghai 201620, China


Introduction
In the past few decades, the problem of control and synchronization of complex dynamical networks has been extensively investigated in various fields of science and engineering due to its many potential practical applications 1-8 .One important consideration in practical networks is the existence of time delays because obstructions to the transmission of signals are inevitable in a biological neural network, in an epidemiological model, in a communications network, or in an electrical power grid.Since recently, there are many studies on dynamical networks with time delays 9-16 .Moreover, the multidelayed coupling consists of providing more information about the dynamics in nodes to the other nodes in the network, such as the transportation network; we all know that the transmission

Problem Formulation
In 17 , the authors achieve synchronization between two complex networks with multilinks by designing effective controller.For simplicity, the complex network model is written in the following form:

2.1
where x i x 1 , x 2 , . . ., x n T ∈ R n , f : R n → R n standing for the activity of an individual subsystem is a vector value function.A l a l ij N×N ∈ R N×N l 0, 1, . . ., m − 1 is the lth subnetwork's topological structure.The definition of a l ij is that in the lth sub-network, if there exists a link from node i to j i / j , then a l ij / 0. Otherwise, a l ij 0 • τ l l 0, 1, . . ., m − 1 is time-delay of the lth subnetwork compared to the zero subnetwork τ 0 0 which is without time delayed.
Remark 2.1.In 17 , a l ii − N j 1,i / j a l ij is defined, we are not concerned whether the coupling matrix A l satisfies a l ii − N j 1,i / j a l ij in this paper.
In the paper, we have the following mathematical preliminaries.
Assumption 2.2.We assume that f x i t is Lipschitz continuous on x i t , that is, there exists a positive constant η > 0 such that

Synchronization Scheme
In this section, we will investigate impulsive synchronization of the complex networks with perturb functions.The multidelayed coupled complex network with perturb functions can be described by

3.1
We take the network given by 3.1 as the driving network and a response network with impulsive control scheme which is given by then the driving network 3.1 and the response network 3.2 can realize impulsive synchronization.
Proof.We choose a nonnegative function as e i s T e i s ds.

3.7
Then the differentiation of V along the trajectories of 3.4 is Mathematical Problems in Engineering 5

6 Mathematical Problems in Engineering
This implies that

3.9
On the other hand, when t t k , we have

3.12
In the same way, for t ∈ t 1 , t 2 , we have

3.13
In general for any t ∈ t k , t k 1 , one finds that Thus for all t ∈ t k , t k 1 , k 1, 2, . .., we have

3.15
From the assumptions given in the theorem When θ ≥ 1, from 21 , this implies that the origin in system 3.4 is globally asymptotically stable or the driving network is synchronized with the response network asymptotically for any initial conditions.This completes the proof.

Mathematical Problems in Engineering
Remark 3.2.Systems 3.1 -3.2 are the time-invariant complex networks.As discussed in 22-24 , systems 3.1 -3.2 are the time-varying complex networks, which is a more complicated research issue.Remark 3.3.Normally, it is difficult to control a complex networks by adding the controllers to all nodes, so it would be much better to use the pinning control method since the most complex networks have large number of nodes 25 .Regarding for the pinning control of the network systems 3.1 -3.2 , are the next research topic for us.
Remark 3.4.For the transportation network, we all know that the transmission speed is different among highway network, railway network and airline network.So we can use multilinks delayed to describe these networks 17 .Also impulsive control is an artificial control strategy which is cheaper to operate compared with other control strategy, so impulsive control method of the network systems 3.1 -3.2 should have potential applications.

Illustrative Example
It is well known that the Lorenz system families are typical chaotic systems and the L ü chaotic system is a member of the families which is known as 26 It is well known that the L ü attractor is bounded.Here we suppose that all nodes are running in the given bounded region.Our numerical analyses show that there exist constants M 1 25, M 2 30, M 3 45.
Satisfying y p , z p ≤ M p for 1 ≤ p ≤ 3. Therefore, one has

4.2
Obviously, P ≈ 52.9843.Thus the L ü system satisfies Assumption 2.2, η 117.5985.In the same way, it can be seen that the Chen system, the Lorenz system, the unified chaotic system and the Lorenz system families also satisfy Assumption 2.2.So, in the simulations, we select the L ü chaotic system as an example to show the effectiveness of the proposed method 27, 28 .According to Section 3, we show that the network with 4 nodes described by

4.3
In numerical simulation, let we choose τ 1 0.1, τ 2 0.2, and the gain matrixes B ik k 1, 2, . . .as a constant matrix, B ik B diag −0.7, −0.8, −0.9 , then ρ k 0.09.Let θ 1.1, from ln θρ k 2 η m r 1 α r−1 m H t k 1 −t k ≤ 0, then t k 1 −t k ≤ 0.0085, we let t k 1 −t k 0.008.All initial values are x i 1 1 0.5i, x i 2 2 0.7i, x i 3 2 0.8i, y i 1 1 − 0.6i, y i 2 2 − 0.8i, y i 3 3 − 0.8i. Figure 1 shows the variance of the synchronization errors.We introduce the quantity E t N i 1 y i t − x i t 2 /N 29 which is used to measure the quality of the control process.It is obvious that when E t no longer increases, two networks achieve synchronization.

Conclusion
This paper deals with the problem of impulsive synchronization of multilinks delayed coupled complex networks with perturb effects.On the basis of the comparison theory of impulsive differential system, the novel synchronization criteriion is derived and an impulsive controller is designed simultaneously.Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria.