Pinning Synchronization of Nonlinearly Coupled Complex Dynamical Networks on Time Scales
In this paper, we study the synchronization problem for nonlinearly coupled complex dynamical networks on time scales. To achieve synchronization for nonlinearly coupled complex dynamical networks on time scales, a pinning control strategy is designed. Some pinning synchronization criteria are established for nonlinearly coupled complex dynamical networks on time scales, which guarantee the whole network can be pinned to some desired state. The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Moreover, two numerical examples are given for illustration and verification of the obtained results.
Complex networks are an important part of our daily life and in nature, due to many systems in the real world which can be modeled by complex dynamical networks, such as the Internet, World Wide Web, and food webs . There have been a lot of researches on complex networks and neural networks (see, for example, [2–8] and references therein). Synchronization of complex dynamical network has been a hot topic in the past decades [9–16]. When considering synchronization problem of complex dynamical network, the controlled synchronization problem of complex dynamical network is significant. In recent years, the pinning synchronization of complex dynamical networks, which means the network to achieve desired synchronization by applying control to a small fraction of network nodes, has become a topic of great interest; see [17–34]. In particular, Liu and Chen  investigated the global synchronization for nonlinearly coupled complex networks. Further, they investigated pinning synchronization for continuous-time nonlinearly coupled networks; see . In , a pinning control scheme was developed for continuous-time nonlinearly coupled complex dynamical network, while the results were extended to discrete-time case. In , the synchronization of continuous-time dynamical networks with nonlinearly coupling function was considered.
In real life, the time domains do not always match the known continuous-time intervals or discrete integer time domains. From practical point of view, it is important to study complex dynamic networks on general time domains. This is the starting point of the present investigation. Recently, synchronization of complex dynamical networks on time scales has attracted considerable attention [35–43], which contains not only synchronization of continuous-time and discrete-time complex dynamical networks but also some continuous-time intervals accompanying some discrete moments.
Motivated by the aforementioned discussions, the synchronization of nonlinearly coupled complex dynamical networks on time scales by applying pinning control scheme will be investigated. The objective in this paper is driving the whole network to some desired state by pinning control strategy. By investigating pinning controlled networks on time scales, some sufficient conditions are presented to guarantee the realization of pinning synchronization for nonlinearly coupled complex dynamical networks on time scales. The main contributions of this paper are listed as follows: (i)The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Therefore, the obtained results include continuous-time and discrete-time nonlinearly coupled complex dynamical networks as special cases. Moreover, the model investigated in this paper is more general. Our results can be applied to investigate pinning synchronization of nonlinearly coupled complex dynamical networks on a mixed time domain(ii)Linearly coupled complex dynamical networks on time scale are included as a special case of the present work
The rest of this paper is organized as follows. Some foundational knowledge about time scales and some notations and supporting lemmas are simply outlined in Section 2. In Section 3, the pinning synchronization problem of nonlinearly coupled complex dynamical networks on time scales is formulated. In Section 4, the main theorems and some corollaries are established. In Section 5, two numerical examples are given to verify the effectiveness of our results. Finally, conclusions are provided in Section 6.
In this section, we will present some foundational knowledge about time scales and some notations and lemmas which are needed later.
2.1. Foundational Knowledge on Time Scales
Throughout this paper, , , and denote the sets of positive integers, integers, and real numbers, respectively. A time scale is defined as a nonempty closed subset of and denoted by .
Definition 1 (see [44, 45]). Let . Define the forward jump operator by , while the backward jump operator is defined by . In this definition, we put and , where denotes the empty set. If , we say that is right-scattered, while if , we say that is left-scattered. Also, if and , then is called right-dense, and if and , then is called left-dense. The graininess function is defined by . The set is derived from the time scale as follows: if has a left-scattered maximum , then . Otherwise, .
Definition 2 (see ). Let . Define the function by , i.e.,
Definition 3 (see [44, 45]). Assume that and , then is called -differentiable at the point if there exists such that for any given , there is an open neighborhood of the point such that In this case, is called the -derivative of at the point and we denote it by . Moreover, we say that is -differentiable (or in short: differentiable) on provided exists for all . The function is called the -derivative of on . If , , then for any , the integral is defined as follows:
Remark 4. If , then , is the usual derivative, and is the usual integral. If , then is the forward shift, is the usual forward difference and .
Lemma 5 (see [44, 45]). If are differentiable at , then
Lemma 6 (see [44, 45]). If is differentiable at , then
Definition 7 (see [44, 45]). A function is called -continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of these functions is denoted by .
Definition 8 (see [44, 45]). We say that a function is regressive (positive regressive) provided holds. The set of all regressive (positive regressive) and -continuous functions is denoted by .
Definition 9 (see ). If , then we define the exponential function by where the cylinder transformation is defined by where Log is the principal logarithm function.
Lemma 10 (see ). Let and . Then, implies
Lemma 11 (see ). For fixed , if and , then as .
Let be an -matrix-valued function on . We say that is differentiable on provided each entry of is differentiable on . In this case, we put , where .
Lemma 12 (see ). If a matrix-valued function is differentiable at , then
Lemma 13 (see ). Suppose that and are differentiable -matrix-valued functions on . Then, (1)(2), where is a constant(3)
2.2. Notations and Supporting Lemmas
For each interval of , is denoted by . denotes the -dimensional Euclidean space with the Euclidean norm . denotes the set of all real matrices. Let be the -dimensional identity matrix, indicate the diagonal matrix with diagonal entries to , be the transpose of matrix , and and represent the minimum eigenvalue and the maximum eigenvalue of a real symmetric matrix. For a symmetric matrix , write if is positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively). For square matrices and , the notation means that is a positive semidefinite (negative semidefinite) matrix. The symbol denotes the Kronecker product.
Lemma 14 (see ). For matrices , , , and with appropriate dimensions, we have the following properties: (1), where is a constant(2)(3)(4)If and are symmetric, then is symmetric(5)For square matrices and , every eigenvalue of arises as a product of eigenvalues of and
Lemma 15 (see ). Let , , , where and , where . If , and each row sum of is zero, then
Lemma 16 (see . Suppose is a real symmetric and irreducible matrix, in which and nonzero matrix satisfies . Let . Then, all the eigenvalues of are less than 0.
Lemma 17 (see ). If are symmetric, , (-dimensional zero vector) and , then (1) is symmetric for (2) is positive definite is negative definite(3) is positive definite there exists a positive definite matrix such that , that is, is a real square root of (4)(5)(6)if is the eigenvalue of , then is the eigenvalue of ,
3. Problem Formulations
Throughout the rest of the paper, let be a time scale with and . In this section, the nonlinearly coupled complex dynamical network on the time scale will be introduced.
In general, the dynamic for each isolated (uncoupled) node of the dynamical network can be described as where , is the -derivative of on , is continuous and of such a nature that existence and uniqueness of solutions to dynamic equation (10) subject to as well as their dependence on initial values is guaranteed.
Suppose that the dynamical network consists of identical nodes, with each node being an -dimensional dynamical system. Then, the nonlinearly coupled dynamical network can be described by where is the state vector of the th node at time ; the constant represents the coupling strength of network; the nonlinearly coupled function is continuous, which satisfies standard assumptions on existence and uniqueness of solutions to dynamic equation (11) subject to as well as their dependence on initial values; the coupling configuration matrix represents the topological structure of the complex network and is defined as follows: if there exists a connection between the node and the node (), then ; otherwise, , and Suppose that network (11) is connected in the sense of having no isolated clusters, which means that the coupling configuration matrix is irreducible.
Suppose that is a solution of the uncoupled system (10). In order to synchronize the network (11) to the objective state , we will design a pinning control scheme, if the network (11) cannot synchronize to the objective state without control. Without loss of generality, we add the controllers to the first nodes. Hence, we have the pinning-controlled network as follows: with the feedback controllers given by where Define the matrix:
Definition 18 (see ). The network (12) is said to be synchronized by pinning control, if We get the following error dynamical network by letting :
It follows from Lemma 16 that the symmetric matrix is negative definite, and so the maximal eigenvalue . From Lemma 17((1)–(3)), it is easy to prove that the matrix is symmetric positive definite.
4. Pinning Synchronization Criteria for Nonlinearly Coupled Complex Dynamical Networks on Time Scales
In order to derive the synchronization criteria for the pinning-controlled network (12), we make the following assumptions:
(A1) (see ). The function is assumed to satisfy Lipschitz condition, that is, there exists a constant such that
(A2) (see [9, 21, 49]). There exists a constant , such that
(A3). There exists a constant matrix , such that
(A4) (see ). The function is assumed to satisfy Lipschitz condition, that is, there exists a constant such that
We have the following theorems and corollaries, which give some sufficient conditions to guarantee synchronization of the network (12) by pinning control.
Theorem 19. Suppose that assumptions (A1), (A2), (A3), and (A4) hold. If there exists with and such that the following condition is satisfied: then, the network (12) is synchronized by pinning control.
Proof. Consider the Lyapunov function By Lemmas 5, 6, 12, and 13, calculating the -derivative of along the trajectories of the error dynamical network (15), one has
By assumption (A1), the following inequalities can be obtained: where , , , , , , , and
We introduce the notations
By assumption (A2) and Lemma 15, we have
By assumption (A2), we get
By assumption (A3) and Lemma 15, we have
By assumption (A3), we get
By calculating, one has
Hence, by Lemmas 14((1), (2), (4), (5)) and 17 (4) and assumption (A4), we have
In view of (22), (24), (25), (26), (27), (29), and Lemma 14((1), (2)), one can get
From (20), we have
By Lemma 10, it implies that
Since and , as . Thus, the network (12) is synchronized by pinning control.
Theorem 20. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold. Then, the network (12) is synchronized by pinning control, if there exists a constant with such that holds.
Proof. Lemma 11 yields . Choose the Lyapunov function Note that The rest proof of the theorem is analogy with the proof of Theorem 19.
Theorem 21. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and is symmetric in assumption (A3). Then, the network (12) is synchronized by pinning control, if and
Proof. Choose the Lyapunov function Taking into account the inequality (34), by analogy with the proof of Theorem 19, one has Hence, by Lemma 17(4), the following estimate holds: By Lemma 10, we have Since and it follows from Lemma 11 that . Therefore, as . This completes the proof.
Corollary 22. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and is symmetric in assumption (A3). If inequality