Global exponential synchronization of chaotic networks with non-delayed and double-delayed coupling

Guoliang Cai Yihong Du Nonlinear Scientific Research Center

Faculty of Science, Jiangsu University Zhenjiang, P.R. China

e-mail: glcai@ujs.edu.cn, 474885457@qq.com

Xianbin Wu Junior College

Zhejiang Wanli University Ningbo, Zhejiang, P.R. China e-mail: wxb3210@zwu.edu.cn

Abstract— Base on the Derivation method and the Halanay inequality technique, the global exponential synchronization of chaotic networks with non-delayed and double-delayed coupling is derived under two different conditions in the paper, respectively. An illustrative example is presented to verify the effectiveness of the synchronization scheme.

Keywords- chaotic networks; global exponential synchronization; Halanay inequality

I. INTRODUCTION Complex dynamical networks are becoming increasingly

important in contemporary society both in science and technology. A complex network is a large set of interconnected communicating and interacting nodes where a node is a fundamental unit having specific contents and exhibiting dynamical behavior, typically. In fact, many systems in science and technology can be modeled as complex networks, and most well-known examples are: power grids, communication networks, internet, World Wide Web, metabolic systems, food webs, etc. In turn, the analysis and control of dynamical behaviors in complex networks have become a very hot topic in various disciplines.

Synchronization in complex networks, one of the most important controlling activity to excite the collective behavior of complex dynamical networks, has received much of the focus in recent years since Pecora and Carroll [1] introduced a method for synchronization two identical chaotic systems with different initial conditions. Especially in recent decades, as the Internet and the World Wide Web are continuously expanding over our world, all things in our world are connected much more closely than before. As a result, some new types of synchronization have appeared in the literatures, such as lag synchronization [2], projective synchronization [3-4], impulsive synchronization [5], cluster synchronization [6], mixed synchronization [7], and so on.

It has been noticed that some papers discussed the global synchronization based on the Lyapunov function approach, but the obtained conditions are either difficult to be verified or reformulated to be in a conservative form such as a linear matrix inequality (LMI) [8-9]. Furthermore, in general, the structure of a network is partially known or even completely unknown, which cause it is very difficult to achieve the expected network synchronization in terms of the centralized control method. Therefore, in this paper, by applying the

Halanay Inequality Lemma and designing some unknown and decentralized controllers, some simple delay-independent criteria are derived to ensure the global exponential synchronization of different complex networks.

The remainder of the paper is organized as follows: In Section 2, network mode and preliminaries are introduced. In Section 3, the global exponential synchronization of the complex dynamical networks with non-delayed and double-delayed coupling is investigated and some criteria for synchronization are obtained. In Section 4, an example are given to show the effectiveness of the proposed method. The paper is concluded in Section 5.

II. NETWORK MODEL AND PRELIMINARIES In general, a complex dynamical network consisting of N

identical double-delayed nodes and n-N non-delayed nodes coupling can be described as follows:

1 21 1

1

( ) ( ( )) ( ( )) ( ), 1,2, ,

( ) ( ( )) ( ), 1, 2, ,

r r

i i i ij j j ij jj j

n

i i r ij jj r

x t f x t a g x t b x t i r

x t k x t c x t i r r n

= =

= +

= + − − − =

= + = + +

(1)

where ( )ix t ( 1,2, , )i n denotes the state variable of the chaotic system, 1 2( ) ( ( ), ( ), ( ))T

r rx t x t x t x t , r is an integer and 1 r n , 1 2 0 . Functions ( )if and ( )ig : R R are continuous, and (0) (0) 0i if g . 0 is a constant,

( ) [ , ]rik C R R and (0) 0ik . ( ) , ( )ij r r ij r rA a B b and

( ) ( )( )ij n r n rC c are real matrixes, which denote the strength of neuron interconnections. The initial values with (1) are:

1 2( ) ( ) ([ ,0], ), { , }, 1,2, , .i ix t t C R max i n Consider the response coupled complex dynamical

network with control as follows:

1 21 1

1

( ) ( ( )) ( ( )) ( ) ( ), 1, 2, ,

( ) ( ( )) ( ), 1, 2, ,

r r

i i i ij j j ij j ij j

n

i i r ij jj r

z t f z t a g z t b z t t i r

z t k z t c z t i r r n

= =

= +

= + − − − + =

= + = + +

(2)

where ( )iz t ( 1,2, , )i n denotes the state variable of the response system, ( )i t indicates the external control input that will be appropriately designed for a control objective, and 1 2( ) ( ( ), ( ), ( ))T

r rz t z t z t z t . The initial values with (2) are:

1 2( ) ( ) ([ ,0], ), { , }, 1, 2, , .i iz t t C R max i n And, at

2012 Fifth International Conference on Information and Computing Science

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2012 Fifth International Conference on Information and Computing Science

2160-7443/12 $26.00 © 2012 IEEE

DOI 10.1109/ICIC.2012.26

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2012 Fifth International Conference on Information and Computing Science

2160-7443/12 $26.00 © 2012 IEEE

DOI 10.1109/ICIC.2012.26

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least, there exists a constant (1 )i i r such that ( ) ( )i it t , for [ ,0]t .

Let ( ) ( ) ( )i i ie t x t z t= − , the error dynamical system of (1) and (2) is

1 21 1

1

( ) ( ( )) ( ( )) ( ) ( ), 1, 2, ,

( ) ( ( )) ( ), 1, 2, ,

r r

i i i ij j j ij j ij j

n

i i r ij jj r

e t F e t a G e t b e t t i r

e t K e t c e t i r r n

= =

= +

= + − − − − =

= + = + + (3)

where ))(())(())(( tzftxfteF iiiiii −= , ))(())(())(( tzgtxgteG iiiiii −= ,

))(())(())(( txktxkteK ririri −= , and Trr tetetete ))(,),(),(()( 21= .

Assumption 1 There exist nonnegative iL and iL for 1,2,i r such that

( ) ( ) ,i i if x f z L x z− ≤ − ( ) ( ) ,i i ig x g z L x z− ≤ −

for ,x z R , and let 1 1,i r i i r iL max L L max L . Definition 1 System (3) is said to be globally

exponentially stable if there exist constants 0M and 0 such that

01 1

( ) ( sup | ( ) ( ) |) , 0,r r

ti i i

si i

e t M s s e t where 1 2 1 2( ) ( ( ), ( ), ( )) , ( ) ( ( ), ( ), ( )) .T T

n ns s s s s s s s Moreover, the constant is defined as the exponential synchronization rate.

Lemma 1 (Cheng et al. [10]) For a r r× real symmetric matrix Ω Ω is positive definite if and only if all its eigenvalues are positive. Moreover, for rx R∀ ∈ , the inequality holds:

2 2min max( ) || || ( ) || || .Tx x x xλ λΩ ≤ Ω ≤ Ω

Lemma 2 (Halanay Inequality Lemma [11]) Let 0τ ≥ be a constant, ( )V t be a non-negative continuous function defined for [ , )τ− ∞ which satisfies

( ) ( ) ( sup ( ))t s t

V t pV t q V sτ− ≤ ≤

≤ − + for 0t > , where p and q are constants. If 0p q> > , then

0( ) ( sup ( )) , 0,t

sV t V s e tδ

τ

−

− ≤ ≤≤ >

where δ is a unique positive root of the equation = exp( )p qδ δτ− .

Lemma 3 ([11]) If System (3) is globally exponentially stable about its partial variables ( )re t , where

1 2( ) ( ( ), ( ), , ( ))Tr re t e t e t e t= , the constant matrix D is Hurwitz,

and

|| ( ( )) || || ( ) || ,r rK e t e t βζ≤ (4) for some constants 0ζ > and 0β > , then system (3) is also globally exponentially stable about its partial variables

( )n re t− , where 1 2( ) ( ( ), ( ), , ( )) .T

n r n n ne t e t e t e t− + += i.e., the

zero solution ( ) 0ne t = , where 1 2( ) ( ( ), ( ), , ( ))Tn ne t e t e t e t= of

system (3) is globally exponentially stable.

III. EXPONENTIAL SYNCHRONIZATION OF DELAYED CHAOTIC NETWORKS

Theorem 1 Under Assumption 1, system (3) is globally exponentially stable if the following conditions are satisfied:

1) The control input vector ( )i tμ are designed as ( ) ( ), 1,2, , .i i it e t i rμ ξ= = (5)

where iξ are constants such that

1 10, 1,2, .

r r

i i ji i jij j

L a L b i rξ γ= =

− − − > = (6)

2) Matrix D is Hurwitz, and (4). Proof Choose the following Lyapunov function as

follows: 1

1

2

2

1 1

1

( ) ( | ( ) | | | | ( ) |

| ( ) | ), 0.

r r tt si ij j jt

i j

r t sij jt

j

V t e e t a L e e e s ds

b e e e s ds t

μτμ μ

τ

μτ μ

τγ

−= =

−=

= +

+ >

(7)

Let )()( teety it

iμ= , ,,,2,1 ri = [ ,t τ∈ − ∞ .

1

1

2

2

1 1

1

( ) ( ( ) | | ( )

( ) ), 0.

r r t

i ij j jti j

r t

ij jtj

V t y t a L e y s ds

b e y s ds t

μτ

τ

μτ

τγ

−= =

−=

= +

+ >

(8)

It is easy to verify that ( )V t is a non-negative function over [ , )τ− ∞ and radically unbounded, i.e., ( )V t → ∞ as | ( ) |ie t → ∞ 1, 2, ,i r= . Calculating the upper right derivative of ( )V t , we obtain

1 1

2 2

11 1 1

21 1

( ) ( ( ) ( ) ( )

( ) ( )

r r r

i ij j j ij j ji j j

r r

ij i ij jj j

D V t D y t a L e y t a L e y t

b e y t b e y t

where +D denotes the upper right derivative. Obviously, 0 is the zero solution of system (3). With

condition (5) and Assumption 1, the following inequalities can be obtained from the first equation of system (3)

11

21

( ) ( ) ( ) ( )

( ) 1, 2, , ,

r

i i i i i j ij jj

r

ij jj

D e t e t L e t L a e t

b e t i r We have

1 2

1 1 1

( ) ( ( ) ( ) | | ( ) | | ( ))r r r

i i i ij j j ij ji j j

D V t L y t a L e y t b e y tμτ μτξ μ γ+

= = =

≤ − − − + +

1 2

1 1 1( | | | | ) ( )

r r r

i i ji i ji ii j j

L a L e b e y tμτ μτξ μ γ= = =

= − − − − − (9) Define continuous functions )~(μih are as follows

.,2,1,~~)~(1

~

1

~21 riebeLaLh

r

jji

r

jijiiiii =−−−−=

==

τμτμ γμξμ

By applying (6), we have

.,2,1,0~)0(11

ribLaLhr

jji

r

jijiiii =>−−−=

==

γξ

According to the continuity of )(⋅ih on ),0[ ∞ , there exist a constant 0>μ such that

.,2,1,0~)(1

~

1

~21 riebeLaLh

r

jji

r

jijiiiii =≥−−−−=

==

τμτμ γμξμ (10)

Because of (10), 0)( ≤+ tVD ( 0>t ), which implies that ),0()( VtV ≤ 0>t .

In terms of Eq. (8), we get

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1 2

1 2

0 0

1 1 1 1( ) ( (0) | | ( ) ( ) )

r r r r

i i ij j j ij ji i j j

y t y a L e y s ds b e y s dsμτ μτ

τ τγ

− −= = = =

≤ + +

1 2

1 2

0 0

1 1 1

( (0) | | ( ) ( ) )r r r

i i ji i ji ii j j

y L a e y s ds b e y s dsμτ μτ

τ τγ

− −= = =

= + +

1 21 2

[ ,0]1 1 1

(1 | | )( sup ( ))r r r

i ji ji isi j j

L a e b e y sμτ μτ

ττ γ τ

∈ −= = =

≤ + ⋅ + ⋅

where 1 2max( , ).τ τ τ∈ i.e.,

1 2[ ,0]1 1

( ) ( sup ( ) ( ) ), max( , ).r r

ti i i

si i

e t e M s sμ

τφ ϕ τ τ τ−

∈ −= =

≤ ⋅ ⋅ − ∈

where 1 21 21 1 1

max 1 | | 0.r r

i ji jii r j j

M L a e b eμτ μττ γ τ≤ ≤ = =

= + ⋅ + ⋅ >

Therefore, system (3) is globally exponentially stable about its partial variables ( )re t . Combining Lemma 3, it is easy to conclude that the zero solution ( ) 0ne t = of system (3) is globally exponentially stable.

The proof is completed. Designing another different control input vector, we can

establish different synchronous conditions. Considering the following control input vector

,,2,1),()(1

ritetr

jjiji ==

=ωμ

i.e.,

)(~)(~ tet rr ⋅= ωμ where ),,2,1,( rjiij =ω are constants, and .)( rrij ×= ωω Then, system (3) can be rewritten as

+=−−−−+=

−− ).())(()(),()())(())(()( 21

teCteKte

teteBteAGteFte

rnrrn

rrrrr ωτγτ

(11)

Theorem 2 Under Assumption 1, system (11) is globally exponentially stable if the following conditions are satisfied:

1) Matrix B and matrix ω are real symmetric and positive, and satisfy that

min ( ) || || || ||, || || || ||L L A B L A Bλ ω γ γ> + − > (12) 2) Matrix D is Hurwitz, and (4).

Proof Choose the following Lyapunov function: 21 1( ) ( ) ( ) || ( ) ||

2 2Tr r rV t e t e t e t= = , 0t > .

It is easy to verify that ( )V t is a non-negative function over [ , )τ− ∞ and radically unbounded, i.e., ( )V t → ∞ as | ( ) |re t → ∞ .

Under Assumption 1, we gain 2 2 2

1 1|| ( ( )) || || ( ) ||r rG e t L e tτ τ− ≤ − , and 22 ||)(||||))((|| teLteF rr ≤ (13) The time derivative of ( )V t along the trajectory of (11) is

1 2( ) ( ) ( ( )) ( ) ( ( )) ( ) ( )

( ) ( )

T T Tr r r r r r

Tr r

V t e t F e t e t AG e t e t Be t

e t e t

τ γ τω

= + − − −

−

In terms of Lemma 1 and (13), we obtain 1 2( ) ( ) ( ( )) ( ) ( ( ) ( ) ( )

( ) ( )

T T Tr r r r r r

Tr r

V t e t F e t e t AG e t e t Be t

e t e t

τ γ τω

= + − − −

−

2 2min 1

2

( ) || ( ) || || ( ) || || || || ( ) || || ( ) |||| || || ( ) || || ( ) ||

r r r r

r r

e t L e t L A e t e tB e t e t

λ ω τγ τ

≤ − + + ⋅ ⋅ −− ⋅ ⋅ −

2 2 2min

2 2 21 2

1( ( )) || ( ) || || ( ) || || || (|| ( ) ||2

1|| ( ) || ) || || (|| ( ) || || ( ) || )2

r r r

r r r

e t L e t L A e t

e t B e t e t

λ ω

τ γ τ

≤ − + + ⋅

+ − − ⋅ + −

2min

2 21 2

1 (2 ( ) 2 || || || ||) || ( ) ||21 1|| || || ( ) || || || || ( ) ||2 2

r

r r

L L A B e t

L A e t B e t

λ ω γ

τ γ τ

= − − − +

+ ⋅ − − ⋅ −

min(2 ( ) 2 2 || || || ||) ( )

( || || || || ( sup ( )).t s t

L L A B V t

L A B V Sτ

λ ω γγ

− ≤ ≤

≤ − − − +

+ −

Condition (12) denotes that min2 ( ) || || 2 || || || || || || 0B L L A L A Bλ ω γ γ+ − − > − >

According to Lemma 2, we have

0( ) ( sup ( )) , 0,t

sV t V s e tδ

τ

−

− ≤ ≤≤ >

i.e.,

[ ,0]

1|| ( ) || sup || ( ) ( ) || exp( )2r r r

se t s s t

τφ ϕ δ

∈ −≤ − ⋅ − , 0t >

Therefore, system (11) is globally exponentially stable about its partial variables ( )re t . Combining Lemma 3, it is easy to conclude that the zero solution ( ) 0ne t = of system (11) is globally and exponentially stable.

The proof is completed. Remark 1 In Theorem 1, under certain conditions, it is

easy to be verified the global exponential synchronization base on the Derivation method, but it is difficult to find V (t).

Remark 2 In Theorem 2, by designing different controllers and applying the Halanay Inequality Lemma, it is more economic and timesaving to obtain V (t), but it is difficult to be verified.

IV. EXAMPLE In this section, we give a numerical example to verify the

effectiveness of our results. Consider the following system:

2 2

1 21 1

3 3 2 3

( ) ( ( )) ( ( )) ( ), 1, 2,

( ) ( ( )) ( ).

i i i ij j j ij jj j

x t f x t a g x t b x t i

x t k x t Cx t= =

= + − − − =

= + (14)

where ( ( )) tan ( ( )),i i if x t h x t= 1 1( ( )) tan ( ( )), 1,2.i i ig x t h x t i− = − =

1 2

3 11 04 10 , , 1, 1, =0.5 1

52 8 0 143 9

A B C and

3 2 1( ( )) ( )k x t x t .

Evidently, condition (4) is satisfied with1=1 =2 , and

matrix C is Hurwitz. The system satisfies assumption S1 with 1, 1, 2.i iL L i

The response coupled complex dynamical network

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2 2

1 21 1

3 3 2 3

( ) ( ( )) ( ( )) ( ) ( ), 1, 2,

( ) ( ( )) ( ).

i i i ij j j ij j ij j

z t f z t a g z t b z t t i

z t k z t Cz t= =

= + − − − + =

= +(15)

Let ( ) ( ) ( )i i ie t x t z t= − , 1, 2,3i the error dynamical

system of (14) and (15) is 2 2

1 21 1

3 3 2 3

( ) ( ( )) ( ( )) ( ) ( ), 1, 2,

( ) ( ( )) ( ).

i i i ij j j ij j ij j

e t F e t a G e t b e t t i

e t K e t Ce t= =

= + − − − − =

= + (16)

Model (16) can be rewritten as the following matrix from 2 2 2 1 2 2 2

3 2 3

( ) ( ( )) ( ( )) ( ) ( ),( ) ( ( )) ( ).

e t F e t AG e t Be t e t

e t K e t Ce t

= + − − − −

= + (17)

If the control input vector is designed as ( ) ( ), 1,2,i i it e t iμ ξ= =

where 1 2=4.5, =3.5. Then, we have 2 2

1 1 1 1 11 1

41= 0,90j j

j j

L a L bξ γ= =

− − − > and

2 2

2 2 2 2 21 1

13= 0.20j j

j j

L a L bξ γ= =

− − − > Therefore, all conditions in Theorem 1 are satisfied.

System (16) is globally exponentially stable about its partial variables 3 ( )=0e t . The initial condition of the system (14) and (15) are 1 2 3( ( ), ( ), ( )) (0.4,0.2,0.6)T Tx t x t x t and

1 2 3( ( ), ( ), ( )) (0.3,0.35,0.5)T Tz t z t z t , respectively, for -1 0t .

In addition, let 2

1

( ) ( ), 1, 2,i ij jj

t e t i where

122 ,

1 22

=

and other condition is defined above. Then min ( )=1.5λ ω and min ( ) ( || || || ||) 1.5 1 1.2 1 0.3 0,L L A Bλ ω γ− + − = − − + = >

|| || || || =1.2 1 0.2 0.L A Bγ− − = > Hence, the conditions in Theorem 2 are satisfied.

System (17) is globally exponentially stable about its partial variables 3 ( )=0e t . The initial condition of the system (14) and (15) are 1 2 3( ( ), ( ), ( )) (0.4,0.2,0.6)T Tx t x t x t and

1 2 3( ( ), ( ), ( )) (0.3,0.35,0.5)T Tz t z t z t , respectively, for -1 0t .

V. CONCLUSIONS In this paper, the global exponential synchronization of

the double delayed chaotic networks has been investigated effectively. Based on Derivation method and the Halanay inequality technique, it is successfully used to guarantee the global stability of the error system, which includes server

delay-independent conditions. The method presented in this paper is much general and effectiveness compared with someone. Consequently, it is a more comprehensive way to apply to the practice. In the future, we will further study the global exponential synchronization of the multi-delayed chaotic networks.

ACKNOWLEDGMENT This work was supported by the National Nature Science

foundation of China (Nos. 70571030, 90610031), the Society Science Foundation from Ministry of Education of China (Nos. 12YJAZH002, 08JA790057), the Advanced Talents’ Foundation of Jiangsu University (No. 07JDG054), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Student Research Foundation of Jiangsu University (No. 10A147).

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