Discrete Dynamics in Nature and Society

Volume 2011 (2011), Article ID 287670, 19 pages

http://dx.doi.org/10.1155/2011/287670

## Global Synchronization of Complex Networks with Discrete Time Delays on Time Scales

Department of Mathematics, Southeast University, Nanjing 210096, China

Received 4 April 2011; Accepted 9 June 2011

Academic Editor: Wei-Der Chang

Copyright © 2011 Quanxin Cheng and Jinde Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies the global synchronization problem for a class of complex networks with discrete time delays. By using the theory of calculus on time scales, the properties of Kronecker product, and Lyapunov method, some sufficient conditions are obtained to ensure the global synchronization of the complex networks with delays on time scales. These sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). The main contribution of the result is that the global synchronization problems with both discrete time and continuous time are unified under the same framework.

#### 1. Introduction

As well known, complex dynamical networks have been a subject of high importance and increasing interest within the science and technology communities. The synchronization is one of the most typical phenomena in complex networks, which is ubiquitous in the real world, such as secure communication, chaos generators design, and harmonic oscillation generation, ([1–8], and references cited therein).

During the past many years, the synchronization of complex networks has received increasing research attention. There are lots of the papers studying the continuous time and the discrete time dynamical systems. However, most of the investigations are restricted to the continuous or discrete systems, respectively, [9–22]. For avoiding this trouble, it is meaningful to study this problem on time scales which can unify the continuous and discrete dynamical systems under the unified framework.

The theory of time scale calculus was initiated by Hilger in 1988, developed, and consummated by Bohner and Peterson [23–25], which has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, population dynamics, biotechnology, economics, and so on [26, 27]. This novel and fascinating type of mathematics is more general and versatile than the traditional theories of differential and difference equations as it can, under one framework, mathematically describe continuous and discrete hybrid processes and hence is the optimal way forward for accurate and malleable mathematical modeling. The field of dynamic equations on time scales contains, links, and extends the classical theory of differential and difference equations.

However, to the best of our knowledge, there are few works investigating the synchronization problem of complex networks with delays on time scales.

* Notations.* Throughout this paper, and denote the n-dimensional Euclidean space and the set of all real matrices, respectively. is a time scale, which is an arbitrary nonempty closed subset of the real number with the topology and ordering inherited from , and assume that , and is unbounded above, that is, . Set . . means that matrix is real, symmetric, and positive definite. and denote the identity matrix and the zero matrix with compatible dimensions, respectively; and diag stands for a block-diagonal matrix. The superscript “T” stands for a matrix transposition. The Kronecker product of matrices and is a matrix in and denoted as . Let and denote the family of continuous functions from to with the norm , where is the Euclidean norm in .

The rest of this paper is organized as follows. In Section 2, some preliminaries on time scale are briefly outlined. In Section 3, by utilizing the approach of the Lyapunov functional method on time scale and the LMI [28], our main result for ensuring the global synchronization is derived. In Section 4, an example is given to illustrate the effectiveness of our main result. Finally, in Section 5, this paper is concluded.

#### 2. Preliminaries

In this paper, the global synchronization problem is investigated for a class of complex networks with discrete time delays which is described by the following dynamic equation on time scale : where , is the state vector of the th network at time . denotes a known connection matrix, and denote the connection weight matrices, is the matrix describing the innercoupling between the subsystems at time , is the outercoupling configuration matrix representing the coupling strength and the topological structure of the complex networks, and is the external input. The constants and stand for the constant time delays, and is an unknown but sector-bounded nonlinear function.

The initial conditions associated with system (2.1) are given by where is rd-continuous and the corresponding state trajectory is denoted as .

Throughout this paper, the following assumptions are needed.

*Assumption 2.1. *The outer-coupling configuration matrix of the complex networks (2.1) satisfies

*Assumption 2.2. *For all , the nonlinear function is assumed to satisfy the following sector-bounded condition:
where and are real constant matrices with being symmetric and positive definite.

*Assumption 2.3. *.

*Remark 2.4. *System (2.1) is a general model of a class of complex networks. Its one special case with continuous time system is the following:
for , and its another special case with discrete time system is the following:
for , where is the forward difference operator.

The continuous-time system (2.5) and the discrete-time system (2.6) are unified as system (2.1). The main objective of this paper is to study the synchronization problem of system (2.1) under the same framework.

In order to obtain the main results, some preliminary results are presented in this section.

*Definition 2.5 (see [23]). *A time scale is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from . The forward and backward jump operators and the graininess are defined, respectively, by . We put := and := , where denotes the empty set. A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if . If has a left-scattered maximum , then we define to be . Otherwise .

*Definition 2.6 (see [23]). *A function is called right-dense continuous provided it is continuous at right-dense point of and the left side limit exists (finite) at left-dense point of . The set of all right-dense continuous functions on is defined by .

*Definition 2.7 (see [23]). *For a function , the delta derivative of , , is the number (if it exists) with the property that, for a given , there exists a neighborhood of such that
for all .

If is right-scattered and is continuous at , then
If is right-dense, then
For all , one can get
If , then is not decreasing on time scale.

Lemma 2.8 (see [23]). *If and are two differentiable functions, then the product rule for the derivative of product is that
*

*Definition 2.9 (see [23]). *A function is called a delta-antiderivative of provided holds for all . In this case, we define the integral of by
and we have the following formula:

*Definition 2.10 (see [23]). *A function is called regressive provided

The addition “” is defined by . The set of all regressive functions on a time scale forms an Abelian group under the addition “”. The additive inverse in this group is denoted by . Then the subtraction on the set of regressive functions is defined by . It can be shown easily that . The set of all regressive and right-dense continuous functions will be denoted by .

We denote that . Obviously is the set of all positively regressive elements of . One can easily verify that if , then .

Lemma 2.11 (see [23]). *Assume is strictly increasing and is a time scale. If is an rd-continuous function and is differentiable with rd-continuous derivative, then for ,
*

*Definition 2.12. *Let be an -matrix-valued function on . We say that is rd-continuous on if each entry of is rd-continuous on , and the class of all such rd-continuous -matrix-valued function on is denoted, similar to the scalar case, by .

We say that is differentiable on provided each entry of A is differentiable on . In this case, we put
where . And denote that .

Lemma 2.13 (see [23]). *Suppose A and B are differentiable -matrix-valued functions. Then *(1)*;
*(2)*;
*(3)*. *

Let together with the Kronecker product “” for matrices, system (2.1) can be recasted into

*Definition 2.14. *The complex system (2.18) is said to be globally synchronized, if
hold for all .

Lemma 2.15 (see [29]). *The Kronecker product has the following properties: *(1)*,
*(2)*,
*(3)*,
*(4)*. *

In the mean of time scale, Schwarz inequality holds similarly.

Lemma 2.16. *Suppose are two rd-continuous functions, and . Then
*

*Remark 2.17. *One can prove Lemma 2.16 easily by using analysis knowledge. The proof is not given here for the purpose of space saving.

Corollary 2.18. *Suppose is an rd-continuous function and . Then
*

Lemma 2.19. *Assume . is a symmetric and positive semidefinite matrix, and is a vector function. If the integrations concerned are well defined, then the following inequality holds:
*

*Proof. *Since matrix is symmetric and positive semidefinite, there exists a reversible orthogonal matrix , such that
where are the eigenvalues of . Then, one has
By using Corollary 2.18, one can obtain
then
The proof is completed.

Lemma 2.20 (See [30]). *Let , , where and where . If and each row sum of is zero, then
*

#### 3. Main Results and Proofs

In this section, the main results for global synchronization criteria of the delayed complex networks on time scales are presented.

Theorem 3.1. *Suppose Assumptions 2.1 and 2.2 hold. The global synchronization of system (2.18) is achieved if there exist positive matrices , , , , , , and matrices , , , and positive scalars , , such that the following LMI holds for all :
**
where
**
with
*

*Proof. *Letting
system (2.18) becomes

Based on the theory of calculus on time scales, we choose the following Lyapunov functional candidate:
where
Note that . For any matrix with appropriate dimension, one obtains
Calculating the delta derivative along the trajectories of the network (2.1) (or (2.18)), one has
And synchronously,
where
Similarly, one has
where
Similarly, one has

By formula (2.18), for any matrices , the following equality is satisfied
which can be rewritten as
Similarly, for any matrices , one has

In addition, for any matrix , the following equality is always true:
that is,

Moreover, from Assumption 2.2, for , one obtains
where and .

Applying on both sides of the above inequality, the following formula can be obtained

Similarly, for , one has
and then
From (3.9)–(3.23), we have
where
and is as defined in (3.1).

From condition (3.1), it is guaranteed that all the subsystems in (2.1) are globally synchronized for any fixed time delays . The proof is completed.

Specially, in the case of system (2.5) with continuous time, the following corollary can be obtained.

Corollary 3.2. *Suppose Assumptions 2.1 and 2.2 hold. The global synchronization of system (2.5) is achieved if there exist positive matrices , , , , , , and matrices , , , and positive scalars , , such that the following LMI holds for all :
**
where
**
with
*

#### 4. A Numerical Example

In this part, a numerical example is given to verify the theoretical result.

Consider the following complex networks (4.1) with time delays on time scale : where is the state vector of the th subsystem. Choose the coupling matrix and the linking matrix as The other parameters are as follows:

The nonlinear function is given by , with . It is easy to see that the nonlinear functions satisfy all the assumptions. By using the Matlab LMI Toolbox, LMI (3.1) is feasible. According to Theorem 3.1, one concludes that the complex networks (4.1) with delays on time scale can achieve global synchronization. The synchronization errors of the complex networks (4.1) are plotted in Figure 1.

#### 5. Conclusions

In this paper, we have investigated the global synchronization of a kind of delayed complex networks on time scales. Utilizing the theory of calculus on time scales and the properties of Kronecker product, the synchronization conditions have been derived through a suitably Lyapunov functional. The obtained synchronization criterion which is expressed in the form of LMIs can be easily verified by the standard numerical software such as Matlab LMI toolbox. The obtained results are novel since there are few works about the synchronization of delayed complex networks on time scales. In addition, the approach utilized in this paper can be considered as a universal framework for the study of other complex systems on time scales.

#### References

- J. Cao, D. W. C. Ho, and X. Huang, “LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 7, pp. 1558–1572, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liang, J. Cao, and D. W. C. Ho, “Discrete-time bidirectional associative memory neural networks with variable delays,”
*Physics Letters A*, vol. 335, no. 2-3, pp. 226–234, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. H. Strogatz, “Exploring complex networks,”
*Nature*, vol. 410, no. 6825, pp. 268–276, 2001. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus - X. Nie and J. Cao, “Stability analysis for the generalized Cohen-Grossberg neural networks with inverse Lipschitz neuron activations,”
*Computers & Mathematics with Applications*, vol. 57, no. 9, pp. 1522–1536, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 17, no. 3, pp. 814–820, 2006. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus - Z. Wang, H. Shu, J. Fang, and X. Liu, “Robust stability for stochastic Hopfield neural networks with time delays,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 5, pp. 1119–1128, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Li and G. Chen, “Synchronization in general complex dynamical networks with coupling delays,”
*Physica A*, vol. 343, no. 1–4, pp. 263–278, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cao, H. X. Li, and D. W. C. Ho, “Synchronization criteria of Lur'e systems with time-delay feedback control,”
*Chaos, Solitons & Fractals*, vol. 23, no. 4, pp. 1285–1298, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, P. Li, and W. Wang, “Global synchronization in arrays of delayed neural networks with constant and delayed coupling,”
*Physics Letters A*, vol. 353, no. 4, pp. 318–325, 2006. View at Publisher · View at Google Scholar · View at Scopus - W. Lu and T. Chen, “Synchronization analysis of linearly coupled networks of discrete time systems,”
*Physica D*, vol. 198, no. 1-2, pp. 148–168, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Qiu and J. Cao, “Global synchronization of delay-coupled genetic oscillators,”
*Neurocomputing*, vol. 72, no. 16–18, pp. 3845–3850, 2009. View at Publisher · View at Google Scholar · View at Scopus - W. Yu and J. Cao, “Synchronization control of stochastic delayed neural networks,”
*Physica A*, vol. 373, pp. 252–260, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Liang, Z. Wang, Y. Liu, and X. Liu, “Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 38, no. 4, pp. 1073–1083, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus - J. Lü, X. Yu, and G. Chen, “Chaos synchronization of general complex dynamical networks,”
*Physica A*, vol. 334, no. 1-2, pp. 281–302, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - G. Chen, J. Zhou, and Z. Liu, “Global synchronization of coupled delayed neural networks and applications to chaotic CNN models,”
*International Journal of Bifurcation and Chaos*, vol. 14, no. 7, pp. 2229–2240, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao and J. Lu, “Adaptive synchronization of neural networks with or without time-varying delay,”
*Chaos*, vol. 16, no. 1, Article ID 013133, 6 pages, 2006. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH · View at MathSciNet - J. Cao, G. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with hybrid coupling,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 38, no. 2, pp. 488–498, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus - J. Liang and J. Cao, “Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays,”
*Physics Letters A*, vol. 314, no. 5-6, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. Lu and J. Cao, “Adaptive synchronization in tree-like dynamical networks,”
*Nonlinear Analysis: Real World Applications*, vol. 8, no. 4, pp. 1252–1260, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lu, D. W. C. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,”
*Automatica*, vol. 46, no. 7, pp. 1215–1221, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - D. Yue and H. Li, “Synchronization stability of continuous/discrete complex dynamical networks with interval time-varying delays,”
*Neurocomputing*, vol. 73, no. 4–6, pp. 809–819, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Xu, Y. Chen, and K. L. Teo, “Global exponential stability of impulsive discrete-time neural networks with time-varying delays,”
*Applied Mathematics and Computation*, vol. 217, no. 2, pp. 537–544, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales: An Introduction with Applications*, Birkhäuser, Boston, Mass, USA, 2001. - R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 1–26, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Bohner, A. Peterson, and D. O'Regan, Eds.,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003. - A. Chen and D. Du, “Global exponential stability of delayed BAM network on time scale,”
*Neurocomputing*, vol. 71, no. 16–18, pp. 3582–3588, 2008. View at Publisher · View at Google Scholar · View at Scopus - A. Chen and F. Chen, “Periodic solution to BAM neural network with delays on time scales,”
*Neurocomputing*, vol. 73, no. 1–3, pp. 274–282, 2009. View at Publisher · View at Google Scholar - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. - A. N. Langville and W. J. Stewart, “The Kronecker product and stochastic automata networks,”
*Journal of Computational and Applied Mathematics*, vol. 167, no. 2, pp. 429–447, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Cheng and J. Cao, “Global synchronization of complex networks with discrete time delays and stochastic disturbances,”
*Neural Computing & Applications*. In press. View at Publisher · View at Google Scholar