Research Article | Open Access
Global Synchronization of Complex Networks with Discrete Time Delays on Time Scales
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.
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 , 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.
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 ). 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 ). 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 ). 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 ). If and are two differentiable functions, then the product rule for the derivative of product is that
Definition 2.9 (see ). 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 ). 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 ). 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 ). 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 ). 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 ). 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
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