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Synchronization analysis of singular dynamical networks with unbounded timedelays
Advances in Difference Equations volume 2015, Article number: 193 (2015)
Abstract
This paper deals with singular dynamical networks with nondelay coupling and unbounded timedelay coupling simultaneously, where the coupling configuration matrices are symmetric with zero row sums and nonnegative offdiagonal entries. A sufficient condition of synchronization is derived based on the LyapunovKrasovskii functional method and matrix analysis technique. A numerical example shows that our proposed method is simple and convenient in computation.
1 Introduction
In general, complex networks consist of a large number of nodes, in which every node is a fundamental cell with specific activity. In the past two decades, complex networks have attracted scholars’ increasing attention. Several famous network models, such as scalefree model [1], smallworld model [2, 3], which accurately characterize some important natural structures, have been presented. Synchronization is a universal phenomenon in various fields of science and society, and many significant works have been obtained in [4–8]; Wu et al. investigate synchronization of an array of linearly coupled identical systems in [9–11]; research in [12, 13] shows that the structure of networks must have an inevitable effect on the ability and speed of synchronization. It is well known that timedelays widely exist in a large number of concrete systems, and coupled dynamical networks are often associated with timedelays due to the finite speeds of transmission and spreading as well as traffic congestion. A lot of efforts have been made to study the synchronization of dynamical coupled systems with timedelays in [14–18].
At the same time, we notice that a large number of practical networks, such as economic networks, power networks and so on, are singular differential systems which are also named differentialalgebraic systems or descriptor systems. Singular systems have some particular complex properties which need not be considered in normal systems. In singular systems, impulse behavior may appear (if the index is greater than one) and initial value problem may also be unsolvable or have more than one solution, regularity is closely related to the solution behavior of the corresponding singular systems [19]. In order to make a singular system solvable with no impulse behavior and unique solution, the system must be regular and the initial condition must be compatible, which can be acquired similarly to the method presented in [20]. Due to the effect of timedelays, coupled correlative terms will inevitably appear if the system is divided into two subsystems including a differential subsystem and an algebraic subsystem, which makes the problems become more complicated. Recently, Xiong et al. [21] have investigated synchronization of singular hybrid coupled networks without timedelays, the original system is divided into two subsystems including a differential subsystem and an algebraic subsystem, the authors present a sufficient condition of global synchronization, but the presented method cannot be applied to singular delayed networks. Koo et al. [22] and Li et al. [23] investigate synchronization of singular complex dynamical networks with timevarying bounded delays. Sufficient conditions for synchronization in terms of LMIs (linear matrix inequalities) are obtained, respectively. Motivated by this research, in this paper we study synchronization problem of singular dynamical networks with nondelay coupling and unbounded timedelay coupling simultaneously. Based on the LyapunovKrasovskii functional method, a simple sufficient condition of synchronization is derived by using matrix analysis and matrix inequality technique. Our presented method can also be applied to more general dynamical networks including the networks presented in [21–23]. Finally, a numerical example shows that our presented method is simple and convenient in computation.
Notation: The notation used throughout this paper is fairly standard. \(R^{n}\) denotes an ndimensional Euclidean space, \(R^{n\times n}\) is the set of all \(n\times n\) real matrices, \(I_{n}\) stands for the identity matrix of order n, \(U^{T}\) means the transpose of a real matrix or vector U, \(\x\\) denotes the Euclidean norm of a real vector x. For a real matrix A, \(\lambda_{\mathrm{max}}(A)\) and \(\lambda_{\mathrm{min}}(A)\) denote the maximal and minimal eigenvalue respectively. \(\A\=\sqrt{\lambda_{\mathrm{max}}(A^{T}A)}\) denotes the spectral norm of matrix A, \(A>B\) (or \(A\ge B\)) means the symmetric matrix \(AB\) is positive definite (or positive semidefinite) and \(A\otimes B\) denotes the Kronecker product between matrix A and B.
2 Preliminaries
In this section, we now introduce some notations and preliminaries. Consider the singular delayed network consisting of N linearly and diffusively coupled identical nodes, with full diagonal coupling, and each node is an ndimensional dynamical oscillator which can be chaotic. The state equations of the network are described as
where matrix E may be singular and \(0<\operatorname{rank}(E)=p<n\), \(A\in R^{n\times n}\) is a constant matrix. \(x_{i}=(x_{i1},x_{i2},\ldots,x_{in})^{T} \in R^{n}\) is the state vector of node i, \(f:R^{n}\times R\rightarrow R^{n}\) is a continuously differentiable vectorvalued function describing the dynamics of the nodes, \(c_{i}>0\) (\(i=1,2\)) represent the coupling strength, the inner coupling link matrices are diagonal matrices, \(\Gamma=\operatorname{diag}\{r_{1},r_{2},\ldots,r_{n}\}\) with \(r_{i}>0\), \(\hat{\Gamma}=\operatorname{diag}\{\hat{r}_{1},\hat{r}_{2},\ldots,\hat{r}_{n}\}\) with \(\hat{r}_{i}>0\). The coupling timedelays \(\tau_{i}(t)> 0\) are differentiable and \(\dot{\tau}_{i}(t)\le d_{i} <1\), \(i=1,2,\ldots,N\). The coupling configuration matrices \(B=(b_{ij})_{N\times N}\) and \(\hat{B}=(\hat{b}_{ij})_{N\times N}\) describe the topological structure of the network, in which \(b_{ij}\) (the entries \(\hat{b}_{ij}\) are defined similarly) is defined as follows: if there is a connection from node i to node j (\(i\neq j\)), then \(b_{ij}>0\); else \(b_{ij}=0\), the diagonal entries of matrix B are defined by
We assume that network (1) is connected in the sense that there are no isolated clusters, i.e., matrices B and \(\hat{B}\) are irreducible, hence the zero is an eigenvalue of B and \(\hat{B}\) with multiplicity 1 (see [18]). Furthermore, the eigenvalues can be arranged respectively as
In virtue of the Kronecker product, system (1) can be written as
where \(x(t)=(x_{1}^{T}(t),x_{2}^{T}(t),\ldots,x_{N}^{T}(t))^{T}\), \(x(t\tau(t))=(x_{1}^{T}(t\tau_{1}(t)),x_{2}^{T}(t\tau_{2}(t)),\ldots,x_{N}^{T}(t\tau _{N}(t)))^{T}\), \(I_{N}\otimes f(x(t),t)=(f^{T}(x_{1}(t),t),f^{T}(x_{2}(t),t),\ldots,f^{T}(x_{N}(t),t))^{T}\).
To obtain our main results, the following lemmas will be used later.
Lemma 2.1
([20])
For any vectors \(x,y\in R^{n}\) and \(\varepsilon> 0\), the inequality \(2x^{T}y\leq\varepsilon x^{T}x+\frac{1}{\varepsilon} y^{T}y\) holds.
Lemma 2.2
([24])
Suppose that U and V are real symmetric matrices and \(U >0\), \(V\ge0\), α is a positive number. Then
3 Main results
In this section, the main results of this paper on asymptotical synchronization of singular delayed network (1) are derived. We first introduce the following definition.
Definition 3.1
([21])
The singular delayed network (1) is said to achieve asymptotical synchronization if
where \(s(t)\in R^{n}\) is a synchronous solution of an isolated cell such that \(E\dot{s}(t)=As(t)+f(s(t),t)\).
Remark 3.1
In [25], the authors presented a sufficient condition on the existence and uniqueness of solution of the system \(E\dot{x}(t)=Ax(t)+f(x(t),t)\).
Theorem 3.1
Suppose that matrix \(B\hat{B}\) is symmetric, then the synchronization state \(s(t)\) of the singular delayed network (1) is asymptotically stable if the linear timevarying singular delayed systems
are asymptotically stable about their zero solutions, where \(s(t)\) is an asymptotical stable solution of an isolated cell, J is the Jacobian matrix of \(f(x(t),t)\) at \(s(t)\).
Proof
To investigate the stability of the synchronous solution \(s(t)\), let
Then we obtain an error system in a compact form
where \(e(t)=(e_{1}^{T}(t),e_{2}^{T}(t),\ldots,e_{N}^{T}(t))^{T}\), \(e(t\tau(t))=(e_{1}^{T}(t\tau_{1}(t)),e_{2}^{T}(t\tau_{2}(t)),\ldots,e_{N}^{T}(t\tau _{N}(t)))^{T}\).
Obviously, the dynamical network (2) will achieve asymptotical synchronization if the error system (6) is asymptotically stable about the zero solution.
Since matrix \(B\hat{B}\) is symmetric, there exists an orthogonal matrix \(U\in R^{N\times N}\) such that
where \(\Lambda_{1}=\operatorname{diag}\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\}\), \(\Lambda_{2}=\operatorname{diag}\{\mu_{1},\mu_{2},\ldots,\mu_{N}\}\) are both diagonal matrices.
Let \(y(t)=(U^{T}\otimes I_{n})e(t)=(y_{1}^{T}(t),y_{2}^{T}(t),\ldots,y_{N}^{T}(t))^{T}\), we have
Namely,
The proof is completed. □
In order to make singular system (1) or (4) solvable with no impulse, we suppose that the following assumption holds.
Assumption 1
There exist matrices \(P_{i}\) and positivedefine matrices \(Q_{i}\) such that the following inequalities hold:
and
where \(\beta_{i}=c_{2}\hat{r}\mu_{i}\) (\(i=2,3,\ldots,N\)), \(\hat{r}= \max_{1\le i\le n} \hat{r}_{i}\).
Remark 3.2
The initial function space of differential systems with unbounded delays is not completed. Let \(\mathit{BC}:=\{\phi \phi:(\infty ,0] \rightarrow R^{n} , \phi \mbox{ is bounded and continuous}\}\), then \((\mathit{BC},\\cdot\)\) is a Banach space. We also may define a new complete initial function space.
Let \(x: [a,b]\rightarrow R^{n}\), with the norm \(\x\:=\sup_{s\in[a,b]}x(s)\). Denote \(C := \{\phi \phi: (\infty,0] \rightarrow R^{n} , \phi\mbox{ is continuous}\}\), then C and BC are both linear spaces.
Let \(h\in C\), \(h\geq0\), and \(0<\int^{0}_{\infty}h(s)\, ds<\infty\). Denote \(C^{n}_{h}:= \{\phi \int^{0}_{\infty}h(s)\\phi\\, ds< \infty, \phi\in C \}\), the norm of \(\phi\in C^{n}_{h}\) is defined as \(\\phi\_{h}:= \int^{0}_{\infty}h(s)\\phi\\, ds\), hence \(C^{n}_{h}\) is a linear subspace of C and \(\mathit{BC} \sqsubset C^{n}_{h}\).
Lemma 3.1
([26])
\((C^{n}_{h}, \\cdot\_{h})\) is a Banach space.
Under condition (11), it follows from [27] that the pair \((E, A+J+c_{1}\lambda_{i} \Gamma)\) is regular and impulse free, hence the solution of Eq. (4) exists and is impulse free and unique on \([t_{0},\infty)\) for any admissible initial condition \(\phi \in C^{n}_{h} \).
Theorem 3.2
Suppose that matrix \(B\hat{B}\) is symmetric and Assumption 1 holds, then the singular networks with unbounded coupled delays (4) will asymptotically synchronize in the sense of (3).
Proof
Construct the LyapunovKrasovskii functionals as:
We get the derivatives of \(V_{i}(t)\) along the trajectories of the ith equation (4) as follows:
From Lemma 2.1, we obtain
Then we get
From the definition of spectral norm, we know
Since \(P_{i}^{T}Q_{i}^{1}P_{i}\geq0\), from (12) we get
Hence, there exist \(\eta_{i}>0\) (\(i=1,2,\ldots,N\)) such that
Since \(P_{i}^{T}Q_{i}^{1}P_{i}\ge0\), from Lemma 2.2 we obtain
Choose \(\alpha_{i}<1\) such that \(0<\frac{\eta_{i}}{\alpha_{i}}<1\). Then
and
Hence we obtain
From (14) and (15), we know \(\dot{V}_{i}(t)\) (\(i=1,2,\ldots,N\)) are negative definite. Therefore Eq. (4) is asymptotically stable about zero solution via the Lyapunov stability theory, then the singular delayed network (1) will achieve asymptotical synchronization. □
Remark 3.3

(1)
If \(b_{ij}=0\) (i.e., \(B=0\)) and \(\tau_{i}(t)=\tau (t)\) is bounded, Eq. (1) is reduced to
$$ E\dot{x}_{i}(t)=Ax_{i}(t)+f\bigl(x_{i}(t),t \bigr) +c_{2}\sum_{j=1}^{N} \hat{b}_{ij}\hat{\Gamma} x_{j}\bigl(t\tau(t)\bigr),\quad i=1,2,\ldots ,N, $$(16) 
(2)
If \(\hat{b}_{ij}=0\) (i.e., \(\hat{B}=0\)), Eq. (1) is reduced to
$$ E\dot{x}_{i}(t)=Ax_{i}(t)+f\bigl(x_{i}(t),t \bigr) +c_{1}\sum_{j=1}^{N}b_{ij} \Gamma x_{j}(t) ,\quad i=1,2,\ldots,N, $$(17)which is the network model presented in [21].
Hence, the model investigated in this paper may characterize many natural dynamical networks and our proposed method can also be applied to more general dynamical networks.
Remark 3.4
In reality, it is difficult to compute matrices \(P_{i}\) (\(i =1, 2, \ldots , N\)) and \(Q_{i}\) (\(i=2, 3,\ldots , N\)) for a general complex model with a large number N of nodes or with a large dimension from conditions (10) and (11). It should be pointed out that (10) and (11) cannot be solved directly by the LMI toolbox of Matlab. However, if matrix E is positive semidefinite and matrix A is negative definite, one can easily choose positive definite matrices \(P_{i}\) and \(Q_{i}\) satisfying conditions (10)(12) (see the following numerical example). Comparing with [21–23], our proposed method is simple and convenient in computation.
4 An illustrative example
In this section, a simple example is given to illustrate theoretical results and the presented conditions in Theorem 3.2 can be easily obtained. We consider the following singular complex network with six nodes (see Figure 1) in which each node is connected to other nodes and which is described as
and the solution of the state equation can be written as
which is asymptotically stable at \(s(t)=0\), where \(k_{1}\), \(k_{2}\) are both constants. And the Jacobian is \(J=\operatorname{diag}\{1,1,2\}\). For convenience, we assume coupled timedelays \(\tau_{i}(t)=0.4t\), the coupling strength \(c_{1}=c_{2}=1\) and the inner coupling matrices \(\Gamma=\hat{\Gamma}=I_{3}\), the coupling configuration matrices are
and the eigenvalues are 0, −6, −6, −6, −6, −6.
One can choose \(P_{i}=I_{3}\) (\(i=1,2,3,4,5,6\)), hence condition (10) holds. Matrices \(Q_{i}\) can be chosen as
and the minimal eigenvalue of \(Q_{i}\) is \(\frac{4}{3}\), which shows that condition (11) is satisfied. Noting that \(d_{i}=0.4\), hence condition (12) holds. Therefore the singular delayed network (1) will achieve asymptotical synchronization by Theorem 3.2.
5 Conclusions
This paper investigates singular complex networks with nondelay coupling and unbounded timedelay coupling simultaneously. Based on the Lyapunov stability theory and matrix inequalities and singular system theory, a simple sufficient condition of synchronization is derived, which can be easily realized and is simple and convenient in computation. The proposed method also can be applied to more general complex networks comparing with [21–23]. Finally, a simple example is given to illustrate the effectiveness of our theoretical results.
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (11371027, 11326115, 11471015), Research Fund for the Doctoral Program of Higher Education of China (20133401120013), Program of College Natural Science of Anhui Province (KJ2013A032, KJ2011A020), Doctoral Starting Fund of Anhui University (023033190181), Young Scientist Fund of Anhui University (023033050055), Young Outstanding Teacher Fund of Anhui University (023033010264).
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Liu, S., Li, X., Zhou, XF. et al. Synchronization analysis of singular dynamical networks with unbounded timedelays. Adv Differ Equ 2015, 193 (2015). https://doi.org/10.1186/s1366201505290
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DOI: https://doi.org/10.1186/s1366201505290