Synchronization analysis of singular dynamical networks with unbounded time-delays

Abstract

This paper deals with singular dynamical networks with non-delay coupling and unbounded time-delay coupling simultaneously, where the coupling configuration matrices are symmetric with zero row sums and nonnegative off-diagonal entries. A sufficient condition of synchronization is derived based on the Lyapunov-Krasovskii functional method and matrix analysis technique. A numerical example shows that our proposed method is simple and convenient in computation.

Liu et al. Advances in Dierence Equations (2015) 2015:193 DOI 10.1186/s13662-015-0529-0

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Web End = Synchronization analysis of singular dynamical networks with unbounded time-delays

Song Liu*, Xiaoyan Li, Xian-Feng Zhou and Wei Jiang

*Correspondence: mailto:[email protected]

Web End [email protected] School of Mathematical Sciences, Anhui University, Hefei, 230601, P.R. China

Abstract

This paper deals with singular dynamical networks with non-delay coupling and unbounded time-delay coupling simultaneously, where the coupling conguration matrices are symmetric with zero row sums and nonnegative o-diagonal entries.

A sucient condition of synchronization is derived based on the Lyapunov-Krasovskii functional method and matrix analysis technique. A numerical example shows that our proposed method is simple and convenient in computation.

MSC: 34D06; 34D20

Keywords: synchronization; singular hybrid network; unbounded time-delay; matrix inequality

1 Introduction

In general, complex networks consist of a large number of nodes, in which every node is a fundamental cell with specic activity. In the past two decades, complex networks have attracted scholars increasing attention. Several famous network models, such as scale-free model [], small-world model [, ], which accurately characterize some important natural structures, have been presented. Synchronization is a universal phenomenon in various elds of science and society, and many signicant works have been obtained in []; Wu et al. investigate synchronization of an array of linearly coupled identical systems in []; research in [, ] shows that the structure of networks must have an inevitable eect on the ability and speed of synchronization. It is well known that time-delays widely exist in a large number of concrete systems, and coupled dynamical networks are often associated with time-delays due to the nite speeds of transmission and spreading as well as trac congestion. A lot of eorts have been made to study the synchronization of dynamical coupled systems with time-delays in [].

At the same time, we notice that a large number of practical networks, such as economic networks, power networks and so on, are singular dierential systems which are also named dierential-algebraic 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 []. In or-

2015 Liu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 2 of 9

der 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 []. Due to the eect of time-delays, coupled correlative terms will inevitably appear if the system is divided into two subsystems including a dierential subsystem and an algebraic subsystem, which makes the problems become more complicated. Recently, Xiong et al. [] have investigated synchronization of singular hybrid coupled networks without time-delays, the original system is divided into two subsystems including a dierential subsystem and an algebraic subsystem, the authors present a sucient condition of global synchronization, but the presented method cannot be applied to singular delayed networks. Koo et al. [] and Li et al. [] investigate synchronization of singular complex dynamical networks with time-varying bounded delays. Sucient 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 non-delay coupling and unbounded time-delay coupling simultaneously. Based on the Lyapunov-Krasovskii functional method, a simple sucient 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 []. 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. Rn denotes an n-dimensional Euclidean space, Rnn is the set of all n n real matrices, In stands for the identity matrix of order n, UT 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, max(A) and min(A) denote the maximal and minimal eigenvalue respectively. A = max(ATA) denotes the spectral norm of matrix A, A > B (or A B) means the symmetric matrix A B is positive denite (or positive semi-denite) and A 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 diusively coupled identical nodes, with full diagonal coupling, and each node is an n-dimensional dynamical oscillator which can be chaotic. The state equations of the network are described as

Exi(t) = Axi(t) + f xi(t), t + c

N

xj t i(t) , i = , , . . . , N, ()

where matrix E may be singular and < rank(E) = p < n, A Rnn is a constant matrix. xi = (xi, xi, . . . , xin)T Rn is the state vector of node i, f : Rn R Rn is a continuously dierentiable vector-valued function describing the dynamics of the nodes, ci > (i = , )

represent the coupling strength, the inner coupling link matrices are diagonal matrices, = diag{r, r, . . . , rn} with ri > ,

j= bij

= diag{r, r, . . . , rn} with ri > . The coupling time-delays i(t) > are dierentiable and

i(t) di < , i = , , . . . , N. The coupling conguration matrices B = (bij)NN and B = (bij)NN describe the topological structure of the

N

j=

bij xj(t)

+ c

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 3 of 9

network, in which bij (the entries bij are dened similarly) is dened as follows: if there is a connection from node i to node j (i = j), then bij > ; else bij = , the diagonal entries of matrix B are dened by

bii =

)x t (t) , ()

where x(t) = (xT(t), xT(t), . . . , xTN(t))T, x(t (t)) = (xT(t (t)), xT(t (t)), . . . , xTN(t N(t)))T, IN f (x(t), t) = (f T(x(t), t), f T(x(t), t), . . . , f T(xN(t), t))T.

To obtain our main results, the following lemmas will be used later.

Lemma . ([]) For any vectors x, y Rn and > , the inequality xTy xTx + yTy holds.

Lemma . ([]) Suppose that U and V are real symmetric matrices and U > , V ,

is a positive number. Then

U > V max VU < max U

3 Main results

In this section, the main results of this paper on asymptotical synchronization of singular delayed network () are derived. We rst introduce the following denition.

Denition . ([]) The singular delayed network () is said to achieve asymptotical synchronization if

xi(t) = s(t), i = , , . . . , N, as t , ()

where s(t) Rn is a synchronous solution of an isolated cell such that Es(t) = As(t) + f (s(t), t).

Remark . In [], the authors presented a sucient condition on the existence and uniqueness of solution of the system Ex(t) = Ax(t) + f (x(t), t).

N

j=,j =i

N

j=,j =i

bji, i = , , . . . , N.

We assume that network () is connected in the sense that there are no isolated clusters,i.e., matrices B and B are irreducible, hence the zero is an eigenvalue of B and B with multiplicity (see []). Furthermore, the eigenvalues can be arranged respectively as

= > N, = > N.

In virtue of the Kronecker product, system () can be written as

(IN E)x(t) = (IN A)x(t) + IN f x(t), t + c(B )x(t) + c( B

bij =

VU

< .

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 4 of 9

Theorem . Suppose that matrix B B is symmetric, then the synchronization state s(t) of the singular delayed network () is asymptotically stable if the linear time-varying singular delayed systems

Eyi = (A + J + ci )yi(t) + ci

yi t i(t) , i = , , . . . , N, ()

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

ei(t) = xi(t) s(t), i = , , . . . , N. ()

Then we obtain an error system in a compact form

(IN E)e(t) = IN (A + J) e(t) + c(B )e(t) + c( B

)e t (t) , ()

where e(t) = (eT(t), eT(t), . . . , eTN(t))T, e(t (t)) = (eT(t (t)), eT(t (t)), . . . , eTN(t N(t)))T.

Obviously, the dynamical network () will achieve asymptotical synchronization if the error system () is asymptotically stable about the zero solution.

Since matrix B B is symmetric, there exists an orthogonal matrix U RNN such that

UBUT = , U BUT = , ()

where = diag{, , . . . , N}, = diag{, , . . . , N} are both diagonal matrices.

Let y(t) = (UT In)e(t) = (yT(t), yT(t), . . . , yTN(t))T, we have

(IN E)y(t) = IN (A + J) y(t) + c( )y(t) + c(

)y t (t) . ()

yi t i(t) , i = , , . . . , N. ()

The proof is completed.

In order to make singular system () or () solvable with no impulse, we suppose that the following assumption holds.

Assumption There exist matrices Pi and positive-dene matrices Qi such that the following inequalities hold:

ETPi = PTiE , i = , , . . . , N, ()

(A + J)TP + PT(A + J) < ,

PTi(A + J + ci ) + (A + J + ci )TPi + iQi , i = , , . . . , N

Eyi(t) = (A + J + ci )yi(t) + ci

Namely,

()

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 5 of 9

and

Pi < diminQi, i = , , . . . , N, ()

where i = cr|i| (i = , , . . . , N), r = maxin ri.

Remark . The initial function space of dierential systems with unbounded delays is not completed. Let BC := {| : (, ] Rn, is bounded and continuous}, then (BC, ) is a Banach space. We also may dene a new complete initial function space.

Let x : [a, b] Rn, with the norm x := sups[a,b] |x(s)|. Denote C := {| : (, ] Rn, is continuous}, then C and BC are both linear spaces.

Let h C, h , and <

h(s) ds < . Denote Cnh := {|

h(s) ds < , C},

the norm of Cnh is dened as h :=

h(s) ds, hence Cnh is a linear subspace of

C and BC [squareimage] Cnh.

Lemma . ([]) (Cnh, h) is a Banach space.

Under condition (), it follows from [] that the pair (E, A + J + ci ) is regular and impulse free, hence the solution of Eq. () exists and is impulse free and unique on [t, )

for any admissible initial condition Cnh.

Theorem . Suppose that matrix B B is symmetric and Assumption holds, then the singular networks with unbounded coupled delays () will asymptotically synchronize in the sense of ().

Proof Construct the Lyapunov-Krasovskii functionals as:

Vi(t) = yTi(t)ETPiyi(t) + i

t

ti(t) yTi(s)Qiyi(s) ds. ()

We get the derivatives of Vi(t) along the trajectories of the ith equation () as follows:

Vi(t) = yTi(t)PTi (A + J + ci )yi(t) + ci

yi t i(t)

+ (A + J + ci )yi + ci

yi t i(t) T

Piyi(t) + iyTi(t)Qiyi(t) i

i(t) yT

i

t i(t) Qiyi t i(t)

= yTi(t) PTi (A + J + ci ) + (A + J + ci )TPi + iQi yi(t)

+ ciyTi(t)PTi

yi t i(t) i

i(t) yT

i

t i(t) Qiyi t i(t)

iyTi(t)Qiyi(t) + ciyTi(t)PTi

yi t i(t)

i( di)yTi t i(t) Qiyi t i(t) .

From Lemma ., we obtain

ciyTi(t)PTi

yi t i(t)

= ciyTi(t)PTi

Q

i Q

i

yi t i(t)

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 6 of 9

cr i( di)|

i|yTi(t)PTiQiPiyi(t) + icr|i|( di)yTi t i(t) Qiyi t i(t)

=

ii( di)yTi(t)PTiQiPiyi(t) +

ii( di)yTi t i(t) Qiyi t i(t) .

Then we get

Vi(t) iyTi(t)

i( di)PTiQiPi Qi

yi(t)

+ i( di)(i )yTi t i(t) Qiyi t i(t) .

From the denition of spectral norm, we know

max Q

i

di PTiQiPi

Q

i

=

di

max Q

i PTiQiPiQ

i

=

di

Q

i PiQ

i

di

Q

i

Pi

=

dimin(Qi) Pi .

Since PTiQiPi , from () we get

max Q

i

di PTiQiPi

Q

i

< .

Hence, there exist i > (i = , , . . . , N) such that

max Q

i

di PTiQiPi

Q

i

< i < .

Since PTiQiPi , from Lemma . we obtain

di PTiQiPi <

iQi.

Choose i < such that < ii < . Then

Mi := i

i( di)PTiQiPi Qi

< i

ii

Qi < ()

and

Ni := i( di)(i )Qi < . ()

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 7 of 9

Hence we obtain

Vi(t) yTi(t)Miyi(t) + yTi t i(t) Niyi t i(t) .

From () and (), we know Vi(t) (i = , , . . . , N) are negative denite. Therefore Eq. () is asymptotically stable about zero solution via the Lyapunov stability theory, then the singular delayed network () will achieve asymptotical synchronization.

Remark .

() If bij = (i.e., B = ) and i(t) = (t) is bounded, Eq. () is reduced to

Exi(t) = Axi(t) + f xi(t), t + c

N

j= bij

xj t (t) , i = , , . . . , N, ()

which is the network model presented in [, ]. () If bij = (i.e., B = ), Eq. () is reduced to

Exi(t) = Axi(t) + f xi(t), t + c

N

j=

bij xj(t), i = , , . . . , N, ()

which is the network model presented in [].

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 . In reality, it is dicult to compute matrices Pi (i = , , . . . , N) and Qi (i = , , . . . , N) for a general complex model with a large number N of nodes or with a large dimension from conditions () and (). It should be pointed out that () and () cannot be solved directly by the LMI toolbox of Matlab. However, if matrix E is positive semi-denite and matrix A is negative denite, one can easily choose positive denite matrices Pi and Qi satisfying conditions ()-() (see the following numerical example). Comparing with [], 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 . can be easily obtained. We consider the following singular complex network with six nodes (see Figure ) in which each node is connected to other nodes and which is described as

xi

xi

xi

=

xi

xi

xi

+

xi + xi xi

xi xi

, i = , , , , , ,

and the solution of the state equation can be written as

xi = ket k

et,

xi = ke t,

xi = k e

t,

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 8 of 9

Figure 1 Structure of the network.

which is asymptotically stable at s(t) = , where k, k are both constants. And the Jacobian is J = diag{, , }. For convenience, we assume coupled time-delays i(t) = .t, the coupling strength c = c = and the inner coupling matrices =

= I, the coupling

conguration matrices are

B = B =

,

and the eigenvalues are , , , , , .

One can choose Pi = I (i = , , , , , ), hence condition () holds. Matrices Qi can be chosen as

Qi =

, i = , , , , ,

and the minimal eigenvalue of Qi is , which shows that condition () is satised. Noting that di = ., hence condition () holds. Therefore the singular delayed network () will achieve asymptotical synchronization by Theorem ..

5 Conclusions

This paper investigates singular complex networks with non-delay coupling and unbounded time-delay coupling simultaneously. Based on the Lyapunov stability theory and matrix inequalities and singular system theory, a simple sucient 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 []. Finally, a simple example is given to illustrate the eectiveness of our theoretical results.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Liu et al. Advances in Dierence Equations (2015) 2015:193 Page 9 of 9

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).

Received: 25 January 2015 Accepted: 5 June 2015

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