#### Abstract

This paper is devoted to exponential synchronization for complex dynamical networks with delay and impulsive effects. The coupling configuration matrix is assumed to be irreducible. By using impulsive differential inequality and the Kronecker product techniques, some criteria are obtained to guarantee the exponential synchronization for dynamical networks. We also extend the delay fractioning approach to the dynamical networks by constructing a Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical examples are given to demonstrate the theoretical results.

#### 1. Introduction

In the past two decades, complex dynamical networks have attracted lot of attention in different areas, such as physical science, engineering, mathematics, biology, and sociology [1–3]. The synchronization of all dynamical nodes is an important and interesting phenomena mostly because the synchronization can well explain many natural phenomena. Consequently, the synchronization has been actively investigated due to past physics and potential engineering applications. Recently, there has been an increasing interest in the investigation of synchronization of complex dynamical networks, then many synchronization results have been derived for complex dynamical networks [4–9].

Impulsive effects widely exist in the networks. Such systems are described by impulsive differential systems which have been used efficiently in modelling many practical problems that arise in the fields of engineering, physics, and science as well. So the theory of impulsive differential equations is also attracting much attention in recent years [10–13]. Correspondingly, based on the theory of impulsive differential equations, a lot of synchronization results of dynamical networks with impulsive effects have been obtained [13–20].

As is well known, two kinds of impulses in terms of synchronization in complex dynamical networks are considered. One is desynchronizing impulse, the other is synchronizing impulse. An impulsive sequence is said to be desynchronizing if the impulsive effect can suppress the synchronization of complex dynamical networks. An impulsive sequence is said to be synchronizing if a corresponding impulsive effect can enhance the synchronization of the complex dynamical networks. According to the previous literature, complex dynamical networks with delay and impulses can reach synchronization provided that delayed dynamical networks are synchronized. In this paper, by impulsive differential inequality [21], the Lyapunov functional method and the Kronecker product techniques, some sufficient conditions are derived for the globally exponential synchronization of dynamical networks. We also extend the delay fractioning method [22, 23] to dynamical networks by constructing Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical simulations are given to show that our derived criteria can easily be used to make judgements on synchronization for the delayed dynamical networks with impulsive effects and show that impulsive effects play an important role in the delay dynamical networks. The rest of this paper is organized as follows. In Section 2, the network model is presented, together with some definitions and lemmas. In Section 3, some synchronization criteria are derived for general dynamical networks with delay and impulsive effects. In Section 4, two numerical examples are given to demonstrate that our results are relevant to not only linear coupling but also delay and impulsive effects. Finally, some conclusions are given in Section 5.

*Notations. *Throughout this paper, the superscript represents the transpose. stands for the identity matrix of order . For , the norm is defined as . For matrix , and denote the maximum and minimum eigenvalues of matrix , respectively. For real symmetric matrices and , the notation (resp., ) means that the matrix is negative semidefinite (resp., negative definite). For a sequence satisfying , let , , .

#### 2. Model Description and Preliminaries

We consider a delayed complex dynamical network consisting of -coupled identical nodes. Each node is an -dimensional dynamical system composed of linear term and nonlinear term. The th node can be described as follows: where is the state vector of the th node at time , , , ; , , , , , .

The dynamical behavior of the dynamical network with delay can be described by the following linearly coupled systems: where is the inner coupling positive definite matrix between two connected nodes and , is the coupling strength, and is defined as follows: if there is a connection from node to node , then ; otherwise, .

In the process of signal transmission, due to the impulsive effects, the states are suddenly changed in the form of impulses at discrete times . That is, . Let . Thus, the dynamical network with delay and impulsive effects can be obtained by the following form: where are impulsive moments satisfying and , are the impulsive gains at for th unit, is the Laplacian matrix of the corresponding network.

By a solution of system (2.3), we mean a real function on such that for , and satisfies system (2.3) for , and is continuous everywhere except for some and left continuous at , and the right limit exists. Here, we always assume that system (2.3) has a unique solution.

*Remark 2.1. *If , the impulsive sequence is of synchronizing impulse, which may enhance the synchronization of the networks. But if , the impulsive sequence can suppress the synchronization, which is said to be desynchronizing impulse.

*Definition 2.2. *The dynamical networks (2.3) are said to be globally exponentially synchronized if there exist and such that for any initial values :
hold all , and for any .

*Definition 2.3. *For , , the Kronecker product between two matrices is defined by

*Assumption 2.4. *There exist constants such that and hold for any .

*Assumption 2.5. *The coupling configuration matrix is irreducible, and the real parts of the eigenvalues of are all negative except an eigenvalue 0 with multiplicity 1.

To derive our main results, we need the following lemmas.

Lemma 2.6 (see [24]). *If an irreducible matrix with nonnegative offdiagonal elements satisfies , then the following propositions are obtained:*(1)*if is an eigenvalue of and , then ;*(2)* has an eigenvalue 0 with multiplicity and the right eigenvector ;*(3)*suppose that satisfying is the normalized left eigenvector of A corresponding to eigenvalue 0. Then, hold for all ;*(4)*furthermore, if is symmetric, then we have for .*

Lemma 2.7 (see [21]). * Let , , , be constants and , , and assume that is a piece continuous nonnegative function satisfying:
**
If there exist such that for **
Then
**
where , , is an unique positive solution of .*

*Remark 2.8. *The condition of Lemma 2.7 does not need due to the effects , which implies that the above inequality is less conservative than the results in [25].

Lemma 2.9. *For any vectors , scalar , and positive definite matrix , the following inequality holds:
*

Lemma 2.10. *Let be a positive definite matrix, then for ,
*

#### 3. Synchronization Analysis

In this section, the globally exponential synchronization will be analyzed for delayed dynamical networks with impulsive effects. We assume that the network topology is strongly connected, then the corresponding Laplacian coupling matrix is irreducible.

Let , , and . Then, the delayed dynamical network (2.3) can be rewritten in the following Kronecker product form:

Suppose that is the left eigenvector of the configuration coupling matrix with respect to eigenvalue 0 satisfying . Since the coupling configuration matrix is irreducible, by Lemma 2.6, we can see that for . Let , , , and .

Theorem 3.1. *Suppose that Assumptions 2.4 and 2.5 hold. Also suppose that there exist a diagonal positive-definite matrix and scalars , , , , , such that**;**;** , ;**;**.**Then the complex dynamical networks (3.1) are exponentially synchronized, where , is defined to be the second largest eigenvalue of .*

* Proof. *We define a Lyapunov function . Since , we have for and . In view of , it follows that . Therefore, we can conclude that . Calculating the Dini derivative of along the trajectories of the systems (3.1), we have for :
By adding to (3.2) and noting that , we can obtain that
Since the matrix has the following property:
By Perron-Frobenius theorem (see [24]), we can arrange the eigenvalues of matrix as follows: . Applying matrix decomposition theory (see [24]), there exists unitary matrix , such that , where and with and .

Let , where , . Then we have . Thus, we have
In view of matrix is a zero row sum irreducible symmetric matrix with negative off-diagonal elements, we see that and . Hence by Lemma 2.10, we have
It follows from condition that
By Assumption 2.4 and Lemma 2.10, there exists such that
Similarly, we have the following estimation:
where . Substituting these into (3.3), we have for
For , we have
By Lemma 2.7, there exist such that
which implies that
Consequently, the complex dynamical network (3.1) can reach globally exponential synchronization.

*Remark 3.2. *When the impulsive effects are desynchronizing, that is, , the condition in Theorem 3.1 yields , which means that the delayed complex networks without impulsive effects of (2.2) is exponentially synchronized. But when the impulsive effects are synchronizing, that is, , we do not need the condition due to the effect of impulses.

Theorem 3.3. *Suppose that Assumptions 2.4 and 2.5 hold and . Also suppose that there exist a diagonal positive definite matrix and scalars , , , , such that**;**;** for all , ;** there exists a integer such that for all , and the discrete system:
* *is globally exponentially stable with decay , where
* *, , ; ** there exists a constant such that the average dwell time satisfies
* *where is the number of impulsive times of the impulsive sequence on the interval ;**.**Then the complex dynamical networks (3.1) are exponentially synchronized with decay rate .*

* Proof. *Consider a Lyapunov-Krasovskii functional:
with
Similar to the proof of Theorem 3.1, for , we get
For , we have
Then
Thus
By (3.11), for , we have
It follows from condition that there exists some such that
Then from (3.22), we have
Together with (3.22), (3.23) and the above inequality, we have
Set and , . Then
that is,
We consider the discrete system:
Then, by the comparison principle, we see that for
Note that the system (3.29) is globally exponential stable with decay , then there exists constant such that
where . From (3.17) and (3.22), we have
Furthermore, it follows that
By induction, there exists a constant , which is dependent on , , , , , such that
which yields that
From (3.31) and the above inequality, we see that for all ,
Therefore, by (3.17), (3.22), and (3.36), we conclude that for ,
where . For all , we obtain that . Then
which means that
This completes the proof of the theorem.

*Remark 3.4. *Theorem 3.3 presents a new delay-dependent exponential synchronization criterion for complex dynamical networks by using the Lyapunov-Krasovskii functional. Note that, for , , the proposed result demonstrates its superiority to Theorem 3.1, which will be well illustrated via an example in the next section.

Corollary 3.5. *Suppose that Assumptions 2.4 and 2.5 hold and and for all . If there exist positive definite matrix and scalars , , , , such that – and - of Theorem 3.3 hold, and condition of Theorem 3.3 is replaced by the following condition:** there exists a constant such that
* *where , then the complex dynamical networks (3.1) are exponentially synchronized with decay rate .*

* Proof. *Choose a Lyapunov-Krasovskii functional candidate as
with
By the proof of Theorem 3.3, for , we have
Note that , then there exists some such that
Thus
Then from condition , we obtain
for all . The remainder proof of the theorem is similar to Theorem 3.3.

*Remark 3.6. *By Corollary 3.5, under the case that for all , we see that the estimations of maximal time-delay and maximal dwell time as

*Remark 3.7. *By Corollary 3.5, if we take the impulsive gains as
then network (3.1) achieves exponential synchronization.

Corollary 3.8. *Suppose that Assumptions 2.4 and 2.5 hold and . If there exist positive definite matrix and scalars , , , , such that – and - of Theorem 3.3 hold and condition of Theorem 3.3 is replaced by one of the following two conditions:** there exists a constant such that for all , and the matrix satisfies the spectral radius condition for some * *where
* *, , , ; ** there exists a positive integer such that for all , and there exists a constant such that all roots of the characteristic polynomial:
* *satisfy that ,**then the complex dynamical networks (3.1) are exponentially synchronized.*

*Remark 3.9. *From Theorems 3.1 and 3.3, when the delayed network dynamics are desynchronizing and the impulsive effects are synchronizing, in order to ensure synchronization, it should be naturally assumed that the frequency of impulses should not be too low. Usually, we always use condition to ensure that the frequency of impulses should not be too low. Conversely, when the delayed network dynamics are synchronizing but the impulsive effects are desynchronizing, the impulses should not occur too frequently in order to guarantee synchronization. To ensure that the impulses do not occur too frequently, we always assume that .

#### 4. Examples and Simulations

In this section, some examples and numerical simulations are provided to illustrate our results.

*Example 4.1. *Consider the following delayed neural networks [26]:
where , , , , , and . These neural networks (4.1) are chaotic, and chaotic attractor is shown in Figure 1.

We consider the following linear coupled delayed networks:
where , , and .

Figure 2 shows the synchronization of networks of (4.3).

At last, we consider the following linear coupled delayed networks with impulsive effects: where . Letting and solving the LMIs in , in Theorem 3.1, we get that and . By Theorem 3.1, we see that the complex dynamical networks (4.3) are exponentially synchronized. Figure 3 shows the synchronization of networks with delay and impulsive effects.

*Example 4.2. *Consider the following neural networks with delay and impulse:
where , , ,
, . . Letting and solving the LMIs in , in Theorem 3.1, we get that and . We can verify that the synchronization criteria proposed by Theorem 3.1 are not satisfied. However, we conclude that the complex dynamical networks (4.4) are exponentially synchronized by Corollary 3.5. Figure 4 depicts the synchronization state variables , , and with impulsive effects. Figure 5 depicts the synchronization state variables , , and without impulsive effects.

*Remark 4.3. *In Example 4.2, if we take and , it is easy to see that the synchronization criteria proposed by Corollary 3.8 are not satisfied. However, we conclude that the networks (4.4) are exponentially synchronized by Theorem 3.1.

#### 5. Conclusions

In this paper, by establishing some lemmas of new impulsive differential inequality and by using the Lyapunov functional method and the Kronecker product techniques, exponential synchronization for impulsive dynamical networks with irreducible coupling matrix is derived. Some criteria are obtained not only relevant to delay but also to impulsive effects. In particular, the results can be extended to the case of one reducible coupling matrix , which implies that the network topology may be a weakly connected graph containing a rooted spanning tree.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60874088, the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271, and JSPS Innovation program under Grant CXZZ11_0132.