Third International Workshop on Advanced Computational Intelligence August 25-27,201 0 - Suzhou, Jiangsu, China

Adaptive Synchronization of Complex Dynamical Networks with Coupling Delays and Multi-links

Minghui Jiang, Shengle Fang and Xiaohong Wang

Abstract- The paper is mainly concerned with the issues of synchronization dynamics of complex dynamical networks with coupling delays and multi-links. A general model of complex networks with multi-links is formulated, which can well describe practical architectures of more realistic complex networks related to multi-links. Based on Lyapunov stability theory and adaptive control techniques, some simple controllers with updated feedback strength are introduced to make the network locally and globally synchronized. Furthermore, the effectiveness of theoretical results are verified by numerical simulations.

I. INTRODUCTION

C OMPLEX networks have become the focal research

topic within the nonlinear science and technology com­

munities such as physicists, biologists, social scientists and

control scientists in recent years. From a system-theoretic

point of view, a complex dynamical network can be con­

sidered as a large-scale system with special interconnections

among its dynamical nodes. In general, complex networks

consist of a large number of nodes and the links among

them, in which a node is fundamental cell with specific

activity. Since the work of small-world model and scale-free

networks are introduced, the modern complex network theory

is founded.

Recently, synchronization dynamics of complex networks

have been extensively studied since synchronization of dy­

namical systems has potential applications in many fields

including secure communication, parallel image processing,

neural networks, biological system, information science,

etc.(see [1], [2], [3], [4], [5], [6] and references cited therein).

Many control methods, such as pinning control[7], [8], de­

centralized control[9], [10], adaptive control[II], [12], robust

adaptive control[13], observer-based synchronization[14], are

used on synchronization of complex networks. In [15], [16],

the authors considered a dynamical network consisting of N identical nodes with diffusive couplings, in which each node

is an n-dimensional dynamical system as follows:

N Xi = AXi + f(Xi, t) + C L bijrxj, i = 1,2" , . , N. (1)

j=l

Minghui Jiang, Shengle Fang and Xiaohong Wang are with Institute of Nonlinear Complex Systems, China Three Gorges University. (email: {jmhI239}@sina.com).

This work was partially supported by NFS of China ( 60740430664), supported by the Scientific Innovation Team Project of Hubei Provincial Department of Education (T200809) and the Doctoral Start-up Funds of China Three Gorges University(0620060061).

978-1-4244-6337-4/10/$26.00 @2010 IEEE

Also, based on delayed complex network models, such as

N Xi = f(t, Xi(t), Xi(t-T)) + C LbijrXj, i = 1,2" " ,N,

j=l (2)

some synchronization criteria for both delay independent and

delay dependent stability of the synchronization manifold are

derived[17], [18].

Between two nodes in a complex network, multi-links

inevitably exist in real world networks. As a matter of fact, on

can find numerous examples in nature which are character­

ized by functional differential equations having multi-links in

the dynamical nodes. For instance, in a traffic transportation

network, there are maybe more than one road between two

cities; we can take bus, by railway, or by air traveling

from one city to another. Another example is that we can

communicate with our friends by telephone, by mailing, or

by e-mail. Since the links between two nodes are variable, the

speed of transmitting information from different ways may

be different and we can introduce time lags to solve this

problem. However, such complex networks with multi-links

are still relatively unexplored due to their complexity and the

absence of an appropriate simplification procedure. It is thus

imperative to further investigate delayed complex dynamical

networks with multi-links, which is more interesting and

meaningful.

Therefore, in this paper, we present a new adaptive ex­

ponential synchronization approach for the local and global

synchronization of complex dynamical networks with multi­

links. For models (1) or (2), most authors studied the fact

that all nodes with single link in the network achieved a

synchronous state, which was determined by s(t) = As(t) + f(s(t), t) or s(t) = f(t, s(t), s(t - T)). Rather than these

cases, the synchronous state in a network may be different

from the steady state determined by a single node. So in the

paper, we denote an average synchronous manifold s(t) = 11 2:!1 Xi(t). Then, s(t) may be an equilibrium, a periodic

state or a chaotic attractor.

Notations. Throughout this paper, I stands for the identity

matrix. The superscript "T" represents the transpose. For

Vx = (XbX2,'" ,Xnr E Rn, the norm Ilxll is defined

as Ilxll = (2:�=1 xD 2". For a matrix A, IIAII denotes the

spectral norm defined by IIAII = (AM(AT A))! and AM(A) denotes the maximal eigenvalue of matrix A.

The rest of paper is organized as follows. In Section 2, a

general model of complex delayed dynamical networks with

multi-links is presented, and then some necessary definitions

448

and preliminary results are given. The main results of design­

ing a proper controller making the networks synchronized are

presented in Section 3. In section 4, we employ a numerical

simulation for a network of coupled chaotic oscillators to

illustrate the theoretical results. Finally, some concluding

remarks are given in Section 5.

II. PROBLEM STATEMENTS

In this paper, we consider a general dynamical complex

networks consisting of N identical diffusively coupled nodes,

in which each nodes is an n dimensional dynamical unit. This

model can be described as follows:

N Xi(t) = f(Xi(t)) + Co L b(O)ijrOXj(t)

j=1 N N

+C1 L b(1)ijr1Xj(t -T1) + C2 L b(2)ij j=1 j=1

Xr2Xj(t -T2), i = 1, 2, ··· , N, (3)

where Xi(t) = (Xi! (t), Xi2(t), ·· . , Xin(t))T E Rn are the

state variables of node i, T1 > ° and T2 > ° are constant

delays; ci(i = 0, 1, 2) represent the coupling strength of the

semi-networks i; b(k)ij are the directed couplings from node

i to node j. The positive definite matrix r kERn x n (k = 0, 1, 2) is the inner connecting matrix of each node. In this

model (3), it is required that the elements of the coupling

matrices B(m) = (b(m)ij)NXN(m = 0, 1, 2) satisfy

N b(m)ii = - L b(m)ij < 0,

j=1,j#.i i = 1, 2, ··· ,N, m =0, 1, 2

and B(m)(m = 0, 1, 2) are not need to be symmetric.

The initial conditions associated with system (3) are of the

form

Xi(S) = <Pi(S), S E [-T, O], T = max{T1, T2}, (4)

where <Pi(S) is a continuous real-valued function for S E [-T, O].

Remark 1. In the network (3), the outer-coupling matrix

B(m)(m = 0, 1, 2) is not necessary to be symmetric and the

elements bij are not assumed to be only ° or 1. Moreover,

there is no any constraint on the inner-coupling matrix r. Extended by model (3), a general complex dynamical

works, in which there exist at most m links between two

nodes, can be displayed as follows:

N N Xi(t) = f(Xi(t)) + Co L b(O)ijrOXj(t) + C1 L b(1)ij

j=1 j=1 N

Xr1Xj(t -T1) + . . . + Cm-1 L b(m-1)ij j=1

xr m-1Xj(t -Tm-1), i = 1, 2, ··· , N. (5)

Definition 1. [2] Let Xi(tj to, Xo)(1 S i S N) be

a solution of the controlled network (3), where Xo =

(X�, xg, . . . , x'Jv) E RnxN. Assume that f : n x n --+ Rn and Ui : n x . . . x n --+ Rn(1 SiS N) are continuous,

n E Rn. If there is a nonempty subset E � n, with

x? E r(1 SiS N), such that Xi(tj to, Xo) E n for all

t � to, 1 SiS N, and

lim Ilxi(tjtO, Xo) -s(tjto, xo)11 = 0, (6) t-+oo where Xo E w and s(t) = s(tj to, xo) is a designed

synchronization manifold, then the controlled network (3)

is said to achieve asymptotical network synchronization and

Ex· . . x E is called the region of synchrony for the dynamical

network (3).

If (6) holds, we can immediately obtain that

limHoo Ilxi(t) -xj(t)11 = 0, i, j = 1, 2, ··· ,N. Define error vector as ei(t) = Xi(t) - s(t), i =

1, 2, · . . , N. Then we can give the definition of exponentially

synchronous.

Definition 2. The network (3) is said to be globally

exponentially synchronous if there exist constant M > ° and j..L > 0, such that for the initial conditions (4),

Ilei(t)11 < Mexp(-j..Lt) sup I I<pi(s)ll, -7::;8::;0

vt > 0, i = 1, 2, . . . , N. (7)

III. SYNCHRONIZATION ANALYSIS

In this section, we will introduce several useful adaptive

feedback synchronization criteria for the complex dynamical

network (3) with multi-links. Thus, we mainly consider the

controlled system as follows:

N N Xi(t) = f(Xi(t)) + Co L b(O)ijrOXj(t) + C1 L b(1)ij

j=1 j=1 N

Xr1Xj(t -T1) + C2 L b(2)ijr2Xj(t -T2) j=1

+Ui, i = 1, 2, ··· ,N, (8)

where ui(i = 1, 2, ··· , N) are the controllers to be designed.

Firstly, we will give a local synchronization result.

Assumption 1. Suppose there exists a nonnegative con­

stant a such that

I IDf(s(t), t)11 = I IJ(t)11 sat.

where J(t) is the Jacobian matrix of f(s(t), t) at s(t). Theorem 1. Suppose that Assumption 1 holds. Let the

controllers be Ui = -diei(i = 1, 2, ··· ,N) with the

following updating laws

di = 8illei(t)112exp(j..Lt), i = 1, 2, ··· ,N, (9)

where j..L and 8i are positive constants. Then the controlled

network (8) is locally exponentially synchronous. N Proof. Let x(t) = 11 L Xi(t), ei(t) = Xi(t) -x(t). De-i=1

note s(t) = x(t). Then, we can get the following dynamical

449

error systems:

where

N !(Xi(t), t) -f(Xi(t), t) + Co L b(O)ij

j=1 N

XrO(Xj(t) -X(t)) + Cl L b(1)ij j=1

N Xr1 (Xj(t -7t) -X(t -71)) + C2 L b(2)ij

j=1 Xr2(Xj(t -72) -X(t -72)) + Ui, (10)

N 1 '" -!if L..J !(Xi(t), t). Let hiO(t) =

i=1 Co L�1 b(O)ijrO(Xj(t) - X(t)), hi1(t - 71) Cl Lj=1 b(l)ijr1(Xj(t - 71) - X(t - 71)) and

hi2(t -72) = C2 Lf=1 b(2)ijr2(Xj(t -72) -x(t -72)). Linearizing the error system (10) at the origin, one can get

ei(t) = J(t)ei(t) + hiO(t) + hi1(t -71) + hi2(t -72), i = 1,2,··· ,N, (11)

where J(t) is the Jacobian matrix of !(Xi(t), t) at x = s(t). Then, we can construct the following Lyapunov function:

where

1 N VI (t) = 2 L ef(t)ei(t) exp(/-tt),

i=1

TT ( ) = ! � (di -d*)2 V4 t 2 � 8. . i=1 �

(12)

Calculating the time derivatives of V(t) along the trajec-

tories of system (11):

N Vi(t) = L [ef(t)ei(t)exp(/-tt) + �ef(t)ei(t)exp(/-tt)]

i=1 N L[ef(t)(J(t)ei(t) + hiO(t) + hi1(t) + hi2(t) i=1 +Ui) + �ef (t)ei(t)] exp(/-tt) N N

< L [alllei(t)112 + COb(O)illroll L Ilei(t)11 i=1 j=1

N xllej(t)11 + Clb(l)illrlll L Ilei(t)11

j=1 N

xllej(t -71)11 + C2b(2)illr211 L Ilei(t)11 j=1

xllej(t -72)11 + (�-di)llei(t)112] x exp(/-tt), (13)

. � Ll( T T V2(t) � 2 ei (t)ei(t) exp(/-t71) -ei (t -71) i=1 xei(t -71)) exp(/-tt) � Ll ( 2 2) < � 2 Ilei(t)11 exp(/-t7t} -llei(t -71)11 i=1 x exp(/-tt),

. � L2( T T V3(t) � 2 ei (t)ei(t) exp(/-t71) -ei (t -72) i=1 xei(t -72)) exp(/-tt)

(14)

� L2 ( 2 2) < � 2 Ilei(t)11 exp(/-t72) -llei(t -72)11 i=1 X exp(/-tt), (15)

and

N 1 . L "8-(di -d*)di i=1 �

N L(di -d*)llei(t)112 exp(/-tt). (16) i=1

Here in (13) b(m)i = max Ib(m)ijl, (m = 0,1,2; i 15,j5,N 1,2,· . . ,N). Take account into the fact 2ab ::; a2 + b2, we

further have

450

Vi(t) < t [allei(t)112 + eob(o)�"roll (Nllei(t)112 i=l + t Ilej(t)112) + C1b(1)�"r111 (Nllei(t)112 j=l

+ t Ilej(t -71)112) + C1b(2)�"r211 j=l

N (Nllei(t)112 + L Ilej(t -72)112) j=l

+(� -di)llei(t)112] exp(/-tt) � [( Neob(O)illroll NC1b(1)illr111 � a1 + 2 + 2 i=l

+ NC2b(�illr211 ) llei(t)112 + eoll�oll t b(O)j j=l

2 c111r111 � 2 xllei(t)11 + 2 � b(1)jllei(t-71)11 j=l

c211r211 � 2 (/-t ) + 2 � b(2)jllei(t -72)11 + '2 -di j=l

xllei(t)112] exp(/-tt). (17)

N N We choose L1 = c111r111 E b(l)j' L2 = c211r211 E b(2)j. j=l j=l Substituting (13)-(17) into V(t), we have

N 1 V(t) :::; L [a1 + "2 (Neob(O)illroll + NC1b(1)illr111 i=l

N +NC2b(2)illr211 + collroll L b(o)j

j=l +L1 exp(/-t71) + L2 exp(/-t72)) + � -di] x Ilei(t) 112 exp(/-tt).

Select suitable constants di(i = 1, 2, ··· , N) such that

a1 + � (Ncob(O)illroll + NC1b(1)illr111 N

+NC2b(2)illr211 + eollroll L b(o)j j=l

(18)

+L1 exp(/-t71) + L2 exp(/-t72) + /-t) < di. (19)

Therefore, we have V(t) :::; O. It follows that V(t) :::; V(O) for any t 2: O. Using the Lyapunov function (12), we have

1 1 "2llei(t)11

2 exp(/-tt) "2ef(t)ei(t) exp(/-tt) < V(t):::; V(O).

Also, if we take di (0) = di, one can get

V(O) = 1 N

10 "2 � [ef(o)ei(o) + L1 -7"1 e[(O)ei(O) x exp(/-t(O + 71))dO + L2 107"2 ef(O)ei(O) x exp(/-t(O + 72))dO] N < "2 (1 + L171 exp(/-t71) + L272 exp(/-t72) ) x sup I I¢i(S) I I, -7"�S�O

where 7 = max{7I, 72} and ei(s) = ¢i(S), S E [-7, 0] . Let

M = 1f(1 + L171exP(/-t71) + L272exp(/-t72)). Therefore, we obtain that

Ilei(t)ll:::; v'2Mexp(-�t) _:���o ll¢i(S) I I.

(20)

Thus, limHoo Ilei(t)11 = O. By definition 2, the controlled

network (8) is locally exponentially synchronous. The proof

is completed.

Based on Theorem 1, we can immediately obtain the

asymptotically synchronization result.

Corollary 1. Suppose that Assumption 1 holds. Let the

controllers be Ui = -diei(i = 1, 2, ··· , N) with the

following updating laws

451

where /-t and 8i are positive constants. Then the controlled

network (8) is locally asymptotically synchronous.

In the following, we will consider the global synchro­

nization of the network. Generally speaking, the dynamical

differential equations of nodes especially for chaotic nodes

Xi(t) = !(Xi(t), t) can be expressed in this way: Xi(t) = AXi(t) + g(Xi(t), t), where A E Rnxn is a constant matrix,

9 : n x R+ -+ Rn is a differentiable nonlinear function.

Then the controlled network (8) becomes:

N Xi(t) = AXi(t) + g(Xi(t)) + eo L b(O)ijrOXj(t)

j=l N N

+C1 L b(1)ijr1Xj(t -71) + C2 L b(2)ij j=l j=l

Xr2Xj(t -72) + Ui, i = 1, 2, ··· , N. (22)

Similarly, the error dynamical equation is as follows:

ei(t) = Aei(t) + g(Xi(t), s(t), t) + hi1(t -71) +hn(t -72) + Ui, (23)

where g(Xi(t), s(t), t) = g(Xi(t)) -g(s(t), t). Assumption 2. Suppose there exist a nonnegative constant

a2 such that

Theorem 2. Suppose that Assumption 1 holds. Let the

controllers be Ui = -diei(i = 1,2,···, N) with the

following updating laws

(25)

where /L and 8i are positive constants. Then the controlled

network (8) is globally exponentially synchronous.

Proof. Since A is a constant matrix, then there exists a

nonnegative constant 13 such that I IA I I ::; 13. The Lyapunove

function (12) can be used here and the rest procedure of the

proof for this theorem is very similar to that of Theorem 1,

and thus it is omitted.

IV. NUMERICAL SIMULATION

In this section, numerical simulations results are presented

to demonstrate the effectiveness of the proposed methods.

Consider a dynamical network consisting of 5 identical

chaotic Chen's systems. Here, the i-th node dynamic is

described by

a(Xi2 -XiI), (c -a)Xil -XilXi3 + CXi2, XilXi2 -bXi3·

(26)

When a = 3 5, b = 3, C = 28, Chen's system has a chaotic

attractor. Then the controlled network (8) is given by [ �il 1 Xi2 Xi3

or

[ a(xi2 -XiI) 1 N (C -a)Xil -XilXi3 + CXi2 + eo 2:=

XilXi2 -bXi3 j=1 N

Xb(O)ijfOXj(t) + Cl 2:= b(l)ijf1Xj(t -71) j=1

N +C2 2:= b(2)ijf2Xj(t -72)

j=1 x + Ui, i = 1,2,· . . , 5,

N N +eo 2:= b(O)ijfOXj(t) + Cl 2:= b(l)ijfl

j=1 j=1 N

XXj(t -71) + C2 2:= b(2)ijf2Xj(t -72) j=1

+Ui, i = 1,2,··· , 5.

(27)

(28)

For simplicity, we take fo = fl = f2 = E3x3, where E is an identity matrix. Let eo = 0. 2, Cl = 0. 3, C2 = 0. 4 and

time lags 71 = 0. 3, 72 = 0. 5. To be more generally, different kinds of coupling

matrices are chosen here. We take B(o) = (b(O)ijhx5, B(1) = (b(l)ijhx5 and B(2) = (b(2)ijhx5, respectively, as

follows:

-4 1 1 1 1

-2 1 0 0 1

1 -4

1 1 1 1

-2 1 0 0

o -1 o o 1

1 1

-4 1 1 0 1

-2 1 0

o o

-1 o 1

1 1 1

-4 1 0 0 1

-2 1

o o o

-1 1

1 1 1 1

-4 1 0 0 1

-2

11 That is, B(o) is a global symmetric coupling matrix, which

means any two different nodes are connected directly; B(1) is a nearest-neighbor symmetric coupling matrix, which a

ring formed in the network; B(2) is the coupling matrix of

the network with star symmetric coupling.

If we take d* = 20, condition (19) is satisfied and the

error system (10) can be numerically solved by matlab and

the synchronization error curves are show in Fig. 1 and Fig.2,

respectively. All the synchronization errors rapidly converge

to zero.

V. CONCLUSION

In this paper, a general model of a delayed complex

dynamical networks with multi-links has been formulated

and its local or global synchronization dynamics have been

studied based on adaptive control scheme and Lyapunove

stability theory. It is shown that our controllers are simple

and easy to realize in real complex networks. simulations are

given to verify theoretical results, too.

10.-----�----�----�----�--__,

-2

-4

-6

-BO'-----�----:------:------�------: Us

Fig. l. The synchronization errors ei 1

1,2" " ,5).

N XiI - -k 2:: Xil, (i

i=1

452

15r------r------r------r------r-----�

10

/w -5

-10

-15 L-____ � ______ � ____ � ______ � ____ ___' o

Us

N Fig. 2. The synchronization errors ei2 Xi2 - 11 L Xi2, (i

i=l 1,2,··· ,5).

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