Global synchronization of complex networks ... - Semantic Scholar
Applied Mathematics and Computation 219 (2012) 3831–3839
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Global synchronization of complex networks perturbed by the Poisson noise Bo Song a,b, Ju H. Park a,⇑, Zheng-Guang Wu a,c, Ya Zhang d a
Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China c National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, Zhejiang 310027, PR China d School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221000, PR China b
a r t i c l e
i n f o
Keywords: Stochastic complex networks Poisson noises Synchronization
a b s t r a c t In this paper, the problem of stochastic synchronization analysis is investigated for complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper proposes a globally exponentially synchronization criterion in mean square for complex networks perturbed by the Poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction As is known to all, complex dynamical networks (CDNs) widely exist in the real world, including food-webs, ecosystems, metabolic pathways, the Internet, the World Wide Web, social networks and global economic markets [1,2]. Since the discoveries of the small-world feature [3] and the scale-free feature [4] of complex networks, the analysis and the control of the dynamical behaviors in complex networks have been extensively investigated in the past decades. As a significant collective behavior, the studies on the synchronization phenomena of complex dynamical networks have gained considerable research interests [5–12]. On the other hand, in the real world, complex networks are often subject to environmental disturbances; especially the signal transfer within complex networks is always affected by the stochastic perturbations. Therefore, in order to reflect more realistic dynamical behaviors, many researchers have recently investigated the synchronization problems of complex networks perturbed by stochastic noises. For instance, complex networks perturbed by Brown noises have been discussed in [13–16]. The synchronization problems of discrete-time stochastic complex networks with Brown noises were investigated in [13,14]. As to the continuous case, the global exponential synchronization problem for complex dynamical networks with nonidentical nodes and Brown perturbations was studied in [15]. And the synchronization control problem for the competitive complex networks with Brown noises was investigated in [16]. However, it is well known that in the real world, beside Brown noises, there is a very common but important kind of random noises: Poisson noises. Poisson noises can be widely found in various applications such as neurophysiology systems, storage systems, queueing systems, economic systems, and so on [17,18]. It should be pointed out that, unlike the Brown process whose almost all sample paths are continuous, the Poisson process N ðtÞ is a jump process and has the sample paths which are right-continuous and have left limits (i.e. càdlàg). Therefore, there is a great difference between the stochastic integral with respect to the Brown process and the one with respect to the Poisson process. As a result, the dynamical ⇑ Corresponding author. E-mail addresses: [email protected] (B. Song), [email protected] (J.H. Park), [email protected] (Z.-G. Wu), [email protected] (Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.012
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behaviors of the stochastic systems driven by the Poisson process are essential different from the stochastic systems driven by the Brown process. Thus, it is very important to investigate the dynamic behaviors, such as the synchronization phenomena, for complex networks perturbed by the Poisson process. However, to the best of our knowledge, there is still no paper to discuss the synchronization problem for this kind of systems. Motivated by above reasons, this paper investigates the synchronization problem for stochastic complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper presents a globally exponentially synchronization criterion in mean square for complex networks perturbed by the poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. Notation: Throughout the paper, unless otherwise specified, we will employ the following notation. Let ðX; F ; fF t gtP0 ; PÞ be a complete probability space with a natural filtration fF t gtP0 and EðÞ be the expectation operator with respect to the probability measure. If A is a vector or matrix, its transpose is denoted by AT . If P is a square matrix, P > 0 ( P < 0) means that is a symmetric positive (negative) definite matrix of appropriate dimensions while P P 0 ( P 6 0) is a symmetric positive (negative) semidefinite matrix. I stands for the identity matrix of appropriate dimensions. Let j j denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. L2 ðXÞ denotes the space of all random variables X with EjXj2 < 1, it is a Banach space with norm X Y X Y ¼ . kXk2 ¼ ðEjXj2 Þ1=2 . The symbol ‘⁄’ within a matrix represents the symmetric terms of the matrix, e.g. T Z Y Z 2. Problem formulation and preliminaries Consider the following complex dynamical networks consisting of N nodes perturbed by the Poisson noise:
"
# N X dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ;
i ¼ 1; 2; . . . ; N
ð1Þ
j¼1
where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞ; . . . ; xin ðtÞT 2 Rn is the state vector of the ith network at time t; A denotes a known connection matrix, B denotes the connection weight matrix; C 2 Rnn is the matrix describing the inner-coupling between the subsystems at time t; G ¼ ðg ij ÞNN is the out-coupling configuration matrix representing the coupling strength and the topological structure of the complex networks. ri ð; Þ : R Rn ! Rn is the noise intensity function vector and fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k > 0. And f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; . . . ; fn ðxin ðtÞÞÞT is an unknown but sectorbounded nonlinear function. The initial conditions associated with system (1) are given by
xi ð0Þ ¼ ui ;
i ¼ 1; 2; . . . ; N;
ð2Þ
where ui is the F 0 -measurable random variable and independent of fN ðtÞgtP0 such that Eðu Let
2 i Þ
< 1.
T xðtÞ ¼ x1 ðtÞT ; . . . ; xN ðtÞT ; T FðxðtÞÞ ¼ f ðx1 ðtÞÞT ; . . . ; f ðxN ðtÞÞT ; T rðt; xðtÞÞ ¼ r1 ðt; x1 ðtÞÞT ; . . . ; rN ðt; xN ðtÞÞT : With the Kronecker product ‘’ for matrices, system (1) can be rearranged as
dxðtÞ ¼ yðt; xðtÞÞdt þ rðt; xðtÞÞdN ðtÞ;
ð3Þ
where yðt; xðtÞÞ ¼ ðIN A þ G CÞxðtÞ þ ðIN BÞFðxðtÞÞ. In this paper, we will mainly be concerned with the globally exponentially synchronization criterion in mean square. For this, we need the infinitesimal operator D associated to Eq. (3) (see [18–21]).
DVðt; xÞ ¼
@Vðt; xÞ @Vðt; xÞ þ yðt; xÞ þ kðVðt; x þ rðt; xÞÞ Vðt; xÞÞ; @t @x
ð4Þ
where Vðt; xÞ is any non-negative function on R RnN and is continuously twice differentiable with respect to x and once differentiable with respect to t. Throughout this paper, the following assumptions, definitions and propositions are needed to prove our main results. Definition 1 [15]. The stochastic complex network (1) is globally exponentially synchronized in mean square, if there exist constants a > 0, c > 0, such that for all ui , uj , the following holds for t P 0:
Efjxi ðt; ui Þ xj ðt; uj Þj2 g 6 ceat ;
1 6 i < j 6 N:
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
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Assumption 1. For 8x; y 2 Rn , the nonlinear function f ðÞ is assumed to satisfy the following condition:
ðf ðxÞ f ðyÞ Uðx yÞÞT ðf ðxÞ f ðyÞ Vðx yÞÞ 6 0;
ð5Þ
where U and V are known constant real matrices. Assumption 2. The outer-coupling configuration matrix of the complex networks (1) satisfies
g ij ¼ g ji P 0; g ii ¼
N X
ði – jÞ;
g ij ;
i; j ¼ 1; 2; . . . ; N:
j¼1;j–i
Assumption 3. The noise intensity function vector ri : R Rn ! Rn satisfies the Lipschitz condition, i.e., there exists a constant matrix W of appropriate dimension such that
jri ðt; uÞ rj ðt; v Þj2 6 jWðu v Þj2
ð6Þ
for all i; j ¼ 1; 2; . . . ; N and u; v 2 R . n
Proposition 1 [15]. The Kronecker product has the following properties:
ðaAÞ B ¼ A ðaBÞ; ðA þ BÞ C ¼ A C þ B C; ðA BÞðC DÞ ¼ ðACÞ ðBDÞ; ðA BÞT ¼ AT BT : Proposition 2 [13]. Let U ¼ ðaij Þnn , P 2 Rmm , x ¼ ðxT1 ; xT2 ; . . . ; xTn ÞT , y ¼ ðyT1 ; yT2 ; . . . ; yTn ÞT , where xi ¼ ðxi1 ; xi2 ; . . . ; xim ÞT 2 Rm , yi ¼ ðyi1 ; yi2 ; . . . ; yim ÞT 2 Rm (i ¼ 1; 2; . . . ; n). If U ¼ U T and each row sum of U is equal to zero, then
X
xT ðU PÞy ¼
aij ðxi xj ÞT Pðyi yj Þ:
ð7Þ
16i<j6n
3. Main results We are in the position to present our main results of the globally exponentially synchronization criterion in mean square for the complex networks perturbed by the Poisson noise. Theorem 1. Under Assumptions 1–3, the dynamic system (1) is globally exponentially synchronized in mean square if there exist matrices P > 0 and scalars > 0, 1 > 0 such that the following LMI hold for all 1 6 i < j 6 N
0
N11
B @
N¼B B
N12
pffiffiffi kP
21 I
0
P
0
1
0 C C C < 0; P A
ð8Þ
I
where
N11 ¼ PA þ AT P Ng ij PC Ng ij CT P þ kW T W kP 1 U T V 1 V T U; N12 ¼ PB þ 1 U T þ 1 V T : Proof. Firstly, from (8), we can see that the matrix H ¼ diagðI; I; P1 ; IÞ is nonsingular. Thus we can have
0
N11
B B HT NH ¼ B @
N12
pffiffiffi kI
21 I
0
1
P
0
1
0 C C C < 0: I A I
ð9Þ
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
Then by the Schur complement, we can obtain
~¼ N
~ 11 N
N12
21 I
!
< 0;
ð10Þ
where
~ 11 ¼ N11 þ kðP1 1 IÞ1 : N Secondly, for the system (3), consider the following Lyapunov function:
Vðt; xÞ ¼ xðtÞT ðU PÞxðtÞ; where
0
N1 1 B 1 N 1 B U¼B @ 1 1
ð11Þ 1
1
1 C C C: A
N 1
Then, by the formula (4), the infinitesimal operator can be obtained
DVðt; xÞ ¼ 2xðtÞT ðU PÞyðt; xðtÞÞ þ kðxðtÞ þ rðt; xðtÞÞÞT ðU PÞðxðtÞ þ rðt; xðtÞÞÞ kxðtÞT ðU PÞxðtÞ:
ð12Þ
By Propositions 1 and 2, it is easy to obtain
X
DVðt; xÞ ¼
½2ðxi ðtÞ xj ðtÞÞT ðPA Ng ij PCÞðxi ðtÞ xj ðtÞÞ þ 2ðxi ðtÞ xj ðtÞÞT PBðf ðxi ðtÞÞ f ðxj ðtÞÞÞ
16i<j6N
T þ k xi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ P xi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ kðxi ðtÞ xj ðtÞÞT Pðxi ðtÞ xj ðtÞÞ:
ð13Þ
By (9), we have
P1 1 I > 0;
ð14Þ
I P > 0:
ð15Þ
Then, we can prove that
ðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT Pðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ 6 ðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ:
ð16Þ
In fact, using the matrix inversion formula, we can easily know that
ðP1 1 IÞ1 ¼ P þ PðI PÞ1 P:
ð17Þ
Let
!¼
ðP1 1 IÞ1
0
I
!
P
P
P
¼
PðI PÞ1 P
P
I P
! ð18Þ
:
Noticing that
U¼
P1
0
ðI PÞ1
I
! ð19Þ
is nonsingular, we can obtain that
UT !U ¼
0
0 I P
P 0;
ð20Þ
which denotes ! P 0. Thus, it follows that
T
xi ðtÞ xj ðtÞ
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ P
P P
!
xi ðtÞ xj ðtÞ
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
xi ðtÞ xj ðtÞ
¼
xi ðtÞ xj ðtÞ
T
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
ðP1 1 IÞ1
0
I
!
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
¼ ðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ ðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT Pðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ P 0;
ð21Þ
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
From (21), it is very easy to obtain (16). Using (16), we can have
X h
DVðt; xÞ 6
2ðxi ðtÞ xj ðtÞÞT ðPA Ng ij PCÞðxi ðtÞ xj ðtÞÞ þ 2ðxi ðtÞ xj ðtÞÞT PBðf ðxi ðtÞÞ f ðxj ðtÞÞÞ
16i<j6N
þ kðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ kðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ i kðxi ðtÞ xj ðtÞÞT Pðxi ðtÞ xj ðtÞÞ :
ð22Þ
From Assumption 3, it is clear that
ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ 6 ðxi ðtÞ xj ðtÞÞT W T Wðxi ðtÞ xj ðtÞÞ:
ð23Þ
From Assumption 1, it can be derived that
0 6 21 ðxi ðtÞ xj ðtÞÞT U T f ðxi ðtÞÞ f ðxj ðtÞÞ þ 21 ðf ðxi ðtÞÞ f ðxj ðtÞÞÞT V xi ðtÞ xj ðtÞ T 21 xi ðtÞ xj ðtÞ U T V xi ðtÞ xj ðtÞ 21 ðf ðxi ðtÞÞ f ðxj ðtÞÞÞT ðf ðxi ðtÞÞ f ðxj ðtÞÞÞ:
ð24Þ
Combining (22)–(24), we have
X
DVðt; xÞ 6
~ nij ; nTij N
ð25Þ
16i<j6N
where
nij ¼
xi ðtÞ xj ðtÞ f ðxi ðtÞÞ f ðxj ðtÞÞ
:
From (10), it is easy to prove that there exists a scalar c > 0 such that
DVðt; xÞÞ 6 c
X
ðxi ðtÞ xj ðtÞÞT ðxi ðtÞ xj ðtÞÞ:
ð26Þ
16i<j6N
Finally, using a method similar to that used to prove the stability of stochastic differential equations driven by the Poisson process in [18,20,21], we can prove that all the subsystems in (1) are globally asymptotically synchronized in the mean square. The proof is completed. h 0.25 x11(t)−x21(t) x11(t)−x31(t)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 1. State error of x11 ðtÞ xi1 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
Remark 1. In recent years, the synchronization problems for stochastic complex networks have been studied extensively in [13–16]. It should be pointed out that most of stochastic complex networks in above papers are perturbed by the Brown noises, and stochastic complex networks perturbed by the Poisson noise are still not investigated. Theorem 1 gives a globally exponentially synchronization criterion in mean square for complex networks perturbed by the Poisson noise by using the infinitesimal operator of stochastic differential equations driven by the Poisson process.
4. Numerical examples In this section, we present two simulation examples to illustrate the effectiveness of our approach. Example 1. Consider the following complex network consisting of three nodes.
" dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ
# 3 X g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ j¼1
for all i ¼ 1; 2; 3, where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞT 2 R2 is the state vector of the ith subsystem, fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k ¼ 6:5. Let
A¼
3
0
0
3
B¼
;
1
0:1
0:2
1
:
The out-coupling configuration matrices G and inner-coupling matrices C are chosen as
0
1
3
B G¼@ 1
2
2
1
1
2
C 1 A; 3
C¼
The noise intensity function vector
ri ðt; xi ðtÞÞ ¼
pffiffiffiffiffiffiffi 0:1 0
0:2
0
0:1 0:2
:
ri ð; Þ is of the following form:
!
0 pffiffiffiffiffiffiffi xi ðtÞ; 0:2
0.6 x12(t)−x22(t) x12(t)−x32(t)
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 2. State error of x12 ðtÞ xi2 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
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and the nonlinear function f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; f2 ðxi2 ðtÞÞÞT ¼ ðtanhðxi1 ðtÞÞ; tanhðxi2 ðtÞÞÞT . Thus, the matrices U; V; W in Assumptions 1 and 3 are
U¼
;
0 0 0 0
V¼
1 0
W¼
;
0 1
pffiffiffiffiffiffiffi 0:1 0
! 0 pffiffiffiffiffiffiffi : 0:2
According to Theorem 1, we can know that this complex network is globally exponentially synchronized in mean square. When we randomly choose the initial states in ½0; 1 ½0; 1, the synchronization errors are plotted in Figs. 1 and 2, which can confirm that the stochastic complex dynamical system (1) is globally exponentially synchronized in mean square. Example 2. Consider the following complex network consisting of three nodes.
"
# 3 X dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ j¼1
for all i ¼ 1; 2; 3, where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞT 2 R2 is the state vector of the ith subsystem, fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k ¼ 5. Let
A¼
1
0
0
1
;
B¼
2
0:1
5
3
:
The out-coupling configuration matrices G and inner-coupling matrices C are chosen as
0
1 3 1 2 B C G ¼ @ 1 2 1 A; 2 1 3
C¼
The noise intensity function vector
ri ðt; xi ðtÞÞ ¼
1 0 0 1
4 0 0
4
:
ri ð; Þ is of the following form:
xi ðtÞ;
1.2 x11(t)−x21(t) x11(t)−x31(t) 1
0.8
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 3. State error of x11 ðtÞ xi1 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
0.6 x12(t)−x22(t) x12(t)−x32(t) 0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time/seconds Fig. 4. State error of x12 ðtÞ xi2 ðtÞ, i ¼ 2; 3.
and the nonlinear function f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; f2 ðxi2 ðtÞÞÞT ¼ ðtanhðxi1 ðtÞÞ; tanhðxi2 ðtÞÞÞT . Thus, the matrices U; V; W in Assumptions 1 and 3 are
U¼
0 0 0 0
;
V¼
1 0 0 1
;
W¼
1 0 0 1
:
According to Theorem 1, we can know that this complex network is globally exponentially synchronized in mean square. When we randomly choose the initial states in ½0; 1 ½0; 1, the synchronization errors are plotted in Figs. 3 and 4, which can confirm that the stochastic complex dynamical system (1) is globally exponentially synchronized in mean square.
5. Conclusions This paper is concerned with the problem of stochastic synchronization analysis for complex networks perturbed by the Poisson noise. Using the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper gives a globally exponentially synchronization criterion in mean square. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. On the other hand, it is worth mentioning that there are still some important problems to solve for stochastic complex networks perturbed by the Poisson noise. (1) When the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize. Therefore, it is necessary to consider the control problem, such as the adaptive control and pinning control, for synchronization of stochastic complex networks perturbed by the Poisson noise in the future. (2) It has now been well realized that in spreading information through complex networks, there always exist time delays, which may decrease the quality of the system and even lead to oscillation, divergence, and instability. Accordingly, the synchronization problems for stochastic delayed complex networks perturbed by the Poisson noise should be studied in the future researches. Acknowledgements The work was supported by 2012 Yeungnam University Research Grant. Also, the work of B. Song was supported by the National Natural Science Foundation of China under Grants 61104221 and 61174029 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 10KJB120004.
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References [1] X.F. Wang, Complex networks: topology, dynamics and synchronization, International Journal of Bifurcation and Chaos 12 (2002) 885–916. [2] X.F. Wang, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49 (2002) 54–62. [3] D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. [4] A.L. Barbaasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. [5] C.W. Wu, Synchronization in arrays of coupled nonlinear systems: passivity, circle criterion, and observer design, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 48 (2001) 1257–1261. [6] C.W. Wu, Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 50 (2003) 294–297. [7] J. Lü, G. Chen, A time-varying complex dynamical network model and its controlled synchronization criteria, IEEE Transactions on Automatic Control 50 (2005) 841–846. [8] J. Lü, X. Yu, G. Chen, Chaos synchronization of general complex dynamical networks, Physica A 334 (2004) 281–302. [9] W. Yu, J. Cao, G. Chen, et al, Local synchronization of a complex network model, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics) 39 (2009) 230–241. [10] T. Lee, J.H. Park, D. Ji, Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control, Applied Mathematics and Computation 218 (2012) 6469–6481. [11] Q. Song, Synchronization analysis in an array of asymmetric neural networks with time-varying delays and nonlinear coupling, Applied Mathematics and Computation 216 (2010) 1605–1613. [12] Y.J. Zhang, S.Y. Xu, Y. Chu, etc, Robust global synchronization of complex networks with neutral-type delayed nodes, Applied Mathematics and Computation 216 (2010) 768–778. [13] Z. Wang, Y. Wang, Y. Liu, Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Transactions on Neural Networks 21 (2010) 11–25. [14] J.H. Park, S.M. Lee, H.Y. Jung, LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks, Journal of Optimization Theory and Applications 143 (2009) 357–367. [15] X. Yang, J. Cao, J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Transactions on Circuits and Systems-I: Regular Papers 59 (2012) 371–384. [16] W. Zhou, T. Wang, J. Mou, Synchronization control for the competitive complex networks with time delay and stochastic effects, Communications in Nonlinear Science and Numerical Simulation 17 (2012) 3417–3426. [17] H.J. Kushner, Stochastic Stability and Control, Academic Press, New York, 1967. [18] I.I. Gihman, A.V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. [19] I. Kolmanovsky, T. Maizenberg, Optimal constainment control for a class of stochastic systems perturbed by poisson and wiener Processes, IEEE Transactions on Automatic Control 47 (2002) 2041–2046. [20] M.A. Garcia, R.j. Griego, An elementary theory of stochastic differential equations driven by a poisson process, Communications in Statistics. Stochastic Models 10 (1994) 335–363. [21] Y. Ting, The stability theory of stochastic differential equations driven by a poisson process, Soochow Journal of Mathematics 25 (1999) 145–165.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Global synchronization of complex networks perturbed by the Poisson noise Bo Song a,b, Ju H. Park a,⇑, Zheng-Guang Wu a,c, Ya Zhang d a
Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou, Jiangsu 221116, PR China c National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, Zhejiang 310027, PR China d School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221000, PR China b
a r t i c l e
i n f o
Keywords: Stochastic complex networks Poisson noises Synchronization
a b s t r a c t In this paper, the problem of stochastic synchronization analysis is investigated for complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper proposes a globally exponentially synchronization criterion in mean square for complex networks perturbed by the Poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction As is known to all, complex dynamical networks (CDNs) widely exist in the real world, including food-webs, ecosystems, metabolic pathways, the Internet, the World Wide Web, social networks and global economic markets [1,2]. Since the discoveries of the small-world feature [3] and the scale-free feature [4] of complex networks, the analysis and the control of the dynamical behaviors in complex networks have been extensively investigated in the past decades. As a significant collective behavior, the studies on the synchronization phenomena of complex dynamical networks have gained considerable research interests [5–12]. On the other hand, in the real world, complex networks are often subject to environmental disturbances; especially the signal transfer within complex networks is always affected by the stochastic perturbations. Therefore, in order to reflect more realistic dynamical behaviors, many researchers have recently investigated the synchronization problems of complex networks perturbed by stochastic noises. For instance, complex networks perturbed by Brown noises have been discussed in [13–16]. The synchronization problems of discrete-time stochastic complex networks with Brown noises were investigated in [13,14]. As to the continuous case, the global exponential synchronization problem for complex dynamical networks with nonidentical nodes and Brown perturbations was studied in [15]. And the synchronization control problem for the competitive complex networks with Brown noises was investigated in [16]. However, it is well known that in the real world, beside Brown noises, there is a very common but important kind of random noises: Poisson noises. Poisson noises can be widely found in various applications such as neurophysiology systems, storage systems, queueing systems, economic systems, and so on [17,18]. It should be pointed out that, unlike the Brown process whose almost all sample paths are continuous, the Poisson process N ðtÞ is a jump process and has the sample paths which are right-continuous and have left limits (i.e. càdlàg). Therefore, there is a great difference between the stochastic integral with respect to the Brown process and the one with respect to the Poisson process. As a result, the dynamical ⇑ Corresponding author. E-mail addresses: [email protected] (B. Song), [email protected] (J.H. Park), [email protected] (Z.-G. Wu), [email protected] (Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.012
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behaviors of the stochastic systems driven by the Poisson process are essential different from the stochastic systems driven by the Brown process. Thus, it is very important to investigate the dynamic behaviors, such as the synchronization phenomena, for complex networks perturbed by the Poisson process. However, to the best of our knowledge, there is still no paper to discuss the synchronization problem for this kind of systems. Motivated by above reasons, this paper investigates the synchronization problem for stochastic complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper presents a globally exponentially synchronization criterion in mean square for complex networks perturbed by the poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. Notation: Throughout the paper, unless otherwise specified, we will employ the following notation. Let ðX; F ; fF t gtP0 ; PÞ be a complete probability space with a natural filtration fF t gtP0 and EðÞ be the expectation operator with respect to the probability measure. If A is a vector or matrix, its transpose is denoted by AT . If P is a square matrix, P > 0 ( P < 0) means that is a symmetric positive (negative) definite matrix of appropriate dimensions while P P 0 ( P 6 0) is a symmetric positive (negative) semidefinite matrix. I stands for the identity matrix of appropriate dimensions. Let j j denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. L2 ðXÞ denotes the space of all random variables X with EjXj2 < 1, it is a Banach space with norm X Y X Y ¼ . kXk2 ¼ ðEjXj2 Þ1=2 . The symbol ‘⁄’ within a matrix represents the symmetric terms of the matrix, e.g. T Z Y Z 2. Problem formulation and preliminaries Consider the following complex dynamical networks consisting of N nodes perturbed by the Poisson noise:
"
# N X dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ;
i ¼ 1; 2; . . . ; N
ð1Þ
j¼1
where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞ; . . . ; xin ðtÞT 2 Rn is the state vector of the ith network at time t; A denotes a known connection matrix, B denotes the connection weight matrix; C 2 Rnn is the matrix describing the inner-coupling between the subsystems at time t; G ¼ ðg ij ÞNN is the out-coupling configuration matrix representing the coupling strength and the topological structure of the complex networks. ri ð; Þ : R Rn ! Rn is the noise intensity function vector and fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k > 0. And f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; . . . ; fn ðxin ðtÞÞÞT is an unknown but sectorbounded nonlinear function. The initial conditions associated with system (1) are given by
xi ð0Þ ¼ ui ;
i ¼ 1; 2; . . . ; N;
ð2Þ
where ui is the F 0 -measurable random variable and independent of fN ðtÞgtP0 such that Eðu Let
2 i Þ
< 1.
T xðtÞ ¼ x1 ðtÞT ; . . . ; xN ðtÞT ; T FðxðtÞÞ ¼ f ðx1 ðtÞÞT ; . . . ; f ðxN ðtÞÞT ; T rðt; xðtÞÞ ¼ r1 ðt; x1 ðtÞÞT ; . . . ; rN ðt; xN ðtÞÞT : With the Kronecker product ‘’ for matrices, system (1) can be rearranged as
dxðtÞ ¼ yðt; xðtÞÞdt þ rðt; xðtÞÞdN ðtÞ;
ð3Þ
where yðt; xðtÞÞ ¼ ðIN A þ G CÞxðtÞ þ ðIN BÞFðxðtÞÞ. In this paper, we will mainly be concerned with the globally exponentially synchronization criterion in mean square. For this, we need the infinitesimal operator D associated to Eq. (3) (see [18–21]).
DVðt; xÞ ¼
@Vðt; xÞ @Vðt; xÞ þ yðt; xÞ þ kðVðt; x þ rðt; xÞÞ Vðt; xÞÞ; @t @x
ð4Þ
where Vðt; xÞ is any non-negative function on R RnN and is continuously twice differentiable with respect to x and once differentiable with respect to t. Throughout this paper, the following assumptions, definitions and propositions are needed to prove our main results. Definition 1 [15]. The stochastic complex network (1) is globally exponentially synchronized in mean square, if there exist constants a > 0, c > 0, such that for all ui , uj , the following holds for t P 0:
Efjxi ðt; ui Þ xj ðt; uj Þj2 g 6 ceat ;
1 6 i < j 6 N:
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
3833
Assumption 1. For 8x; y 2 Rn , the nonlinear function f ðÞ is assumed to satisfy the following condition:
ðf ðxÞ f ðyÞ Uðx yÞÞT ðf ðxÞ f ðyÞ Vðx yÞÞ 6 0;
ð5Þ
where U and V are known constant real matrices. Assumption 2. The outer-coupling configuration matrix of the complex networks (1) satisfies
g ij ¼ g ji P 0; g ii ¼
N X
ði – jÞ;
g ij ;
i; j ¼ 1; 2; . . . ; N:
j¼1;j–i
Assumption 3. The noise intensity function vector ri : R Rn ! Rn satisfies the Lipschitz condition, i.e., there exists a constant matrix W of appropriate dimension such that
jri ðt; uÞ rj ðt; v Þj2 6 jWðu v Þj2
ð6Þ
for all i; j ¼ 1; 2; . . . ; N and u; v 2 R . n
Proposition 1 [15]. The Kronecker product has the following properties:
ðaAÞ B ¼ A ðaBÞ; ðA þ BÞ C ¼ A C þ B C; ðA BÞðC DÞ ¼ ðACÞ ðBDÞ; ðA BÞT ¼ AT BT : Proposition 2 [13]. Let U ¼ ðaij Þnn , P 2 Rmm , x ¼ ðxT1 ; xT2 ; . . . ; xTn ÞT , y ¼ ðyT1 ; yT2 ; . . . ; yTn ÞT , where xi ¼ ðxi1 ; xi2 ; . . . ; xim ÞT 2 Rm , yi ¼ ðyi1 ; yi2 ; . . . ; yim ÞT 2 Rm (i ¼ 1; 2; . . . ; n). If U ¼ U T and each row sum of U is equal to zero, then
X
xT ðU PÞy ¼
aij ðxi xj ÞT Pðyi yj Þ:
ð7Þ
16i<j6n
3. Main results We are in the position to present our main results of the globally exponentially synchronization criterion in mean square for the complex networks perturbed by the Poisson noise. Theorem 1. Under Assumptions 1–3, the dynamic system (1) is globally exponentially synchronized in mean square if there exist matrices P > 0 and scalars > 0, 1 > 0 such that the following LMI hold for all 1 6 i < j 6 N
0
N11
B @
N¼B B
N12
pffiffiffi kP
21 I
0
P
0
1
0 C C C < 0; P A
ð8Þ
I
where
N11 ¼ PA þ AT P Ng ij PC Ng ij CT P þ kW T W kP 1 U T V 1 V T U; N12 ¼ PB þ 1 U T þ 1 V T : Proof. Firstly, from (8), we can see that the matrix H ¼ diagðI; I; P1 ; IÞ is nonsingular. Thus we can have
0
N11
B B HT NH ¼ B @
N12
pffiffiffi kI
21 I
0
1
P
0
1
0 C C C < 0: I A I
ð9Þ
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
Then by the Schur complement, we can obtain
~¼ N
~ 11 N
N12
21 I
!
< 0;
ð10Þ
where
~ 11 ¼ N11 þ kðP1 1 IÞ1 : N Secondly, for the system (3), consider the following Lyapunov function:
Vðt; xÞ ¼ xðtÞT ðU PÞxðtÞ; where
0
N1 1 B 1 N 1 B U¼B @ 1 1
ð11Þ 1
1
1 C C C: A
N 1
Then, by the formula (4), the infinitesimal operator can be obtained
DVðt; xÞ ¼ 2xðtÞT ðU PÞyðt; xðtÞÞ þ kðxðtÞ þ rðt; xðtÞÞÞT ðU PÞðxðtÞ þ rðt; xðtÞÞÞ kxðtÞT ðU PÞxðtÞ:
ð12Þ
By Propositions 1 and 2, it is easy to obtain
X
DVðt; xÞ ¼
½2ðxi ðtÞ xj ðtÞÞT ðPA Ng ij PCÞðxi ðtÞ xj ðtÞÞ þ 2ðxi ðtÞ xj ðtÞÞT PBðf ðxi ðtÞÞ f ðxj ðtÞÞÞ
16i<j6N
T þ k xi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ P xi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ kðxi ðtÞ xj ðtÞÞT Pðxi ðtÞ xj ðtÞÞ:
ð13Þ
By (9), we have
P1 1 I > 0;
ð14Þ
I P > 0:
ð15Þ
Then, we can prove that
ðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT Pðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ 6 ðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ:
ð16Þ
In fact, using the matrix inversion formula, we can easily know that
ðP1 1 IÞ1 ¼ P þ PðI PÞ1 P:
ð17Þ
Let
!¼
ðP1 1 IÞ1
0
I
!
P
P
P
¼
PðI PÞ1 P
P
I P
! ð18Þ
:
Noticing that
U¼
P1
0
ðI PÞ1
I
! ð19Þ
is nonsingular, we can obtain that
UT !U ¼
0
0 I P
P 0;
ð20Þ
which denotes ! P 0. Thus, it follows that
T
xi ðtÞ xj ðtÞ
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ P
P P
!
xi ðtÞ xj ðtÞ
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
xi ðtÞ xj ðtÞ
¼
xi ðtÞ xj ðtÞ
T
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
ðP1 1 IÞ1
0
I
!
ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞ
¼ ðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ ðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT Pðxi ðtÞ xj ðtÞ þ ri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ P 0;
ð21Þ
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B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
From (21), it is very easy to obtain (16). Using (16), we can have
X h
DVðt; xÞ 6
2ðxi ðtÞ xj ðtÞÞT ðPA Ng ij PCÞðxi ðtÞ xj ðtÞÞ þ 2ðxi ðtÞ xj ðtÞÞT PBðf ðxi ðtÞÞ f ðxj ðtÞÞÞ
16i<j6N
þ kðxi ðtÞ xj ðtÞÞT ðP1 1 IÞ1 ðxi ðtÞ xj ðtÞÞ þ kðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ i kðxi ðtÞ xj ðtÞÞT Pðxi ðtÞ xj ðtÞÞ :
ð22Þ
From Assumption 3, it is clear that
ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞT ðri ðt; xi ðtÞÞ rj ðt; xj ðtÞÞÞ 6 ðxi ðtÞ xj ðtÞÞT W T Wðxi ðtÞ xj ðtÞÞ:
ð23Þ
From Assumption 1, it can be derived that
0 6 21 ðxi ðtÞ xj ðtÞÞT U T f ðxi ðtÞÞ f ðxj ðtÞÞ þ 21 ðf ðxi ðtÞÞ f ðxj ðtÞÞÞT V xi ðtÞ xj ðtÞ T 21 xi ðtÞ xj ðtÞ U T V xi ðtÞ xj ðtÞ 21 ðf ðxi ðtÞÞ f ðxj ðtÞÞÞT ðf ðxi ðtÞÞ f ðxj ðtÞÞÞ:
ð24Þ
Combining (22)–(24), we have
X
DVðt; xÞ 6
~ nij ; nTij N
ð25Þ
16i<j6N
where
nij ¼
xi ðtÞ xj ðtÞ f ðxi ðtÞÞ f ðxj ðtÞÞ
:
From (10), it is easy to prove that there exists a scalar c > 0 such that
DVðt; xÞÞ 6 c
X
ðxi ðtÞ xj ðtÞÞT ðxi ðtÞ xj ðtÞÞ:
ð26Þ
16i<j6N
Finally, using a method similar to that used to prove the stability of stochastic differential equations driven by the Poisson process in [18,20,21], we can prove that all the subsystems in (1) are globally asymptotically synchronized in the mean square. The proof is completed. h 0.25 x11(t)−x21(t) x11(t)−x31(t)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 1. State error of x11 ðtÞ xi1 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
3836
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
Remark 1. In recent years, the synchronization problems for stochastic complex networks have been studied extensively in [13–16]. It should be pointed out that most of stochastic complex networks in above papers are perturbed by the Brown noises, and stochastic complex networks perturbed by the Poisson noise are still not investigated. Theorem 1 gives a globally exponentially synchronization criterion in mean square for complex networks perturbed by the Poisson noise by using the infinitesimal operator of stochastic differential equations driven by the Poisson process.
4. Numerical examples In this section, we present two simulation examples to illustrate the effectiveness of our approach. Example 1. Consider the following complex network consisting of three nodes.
" dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ
# 3 X g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ j¼1
for all i ¼ 1; 2; 3, where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞT 2 R2 is the state vector of the ith subsystem, fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k ¼ 6:5. Let
A¼
3
0
0
3
B¼
;
1
0:1
0:2
1
:
The out-coupling configuration matrices G and inner-coupling matrices C are chosen as
0
1
3
B G¼@ 1
2
2
1
1
2
C 1 A; 3
C¼
The noise intensity function vector
ri ðt; xi ðtÞÞ ¼
pffiffiffiffiffiffiffi 0:1 0
0:2
0
0:1 0:2
:
ri ð; Þ is of the following form:
!
0 pffiffiffiffiffiffiffi xi ðtÞ; 0:2
0.6 x12(t)−x22(t) x12(t)−x32(t)
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 2. State error of x12 ðtÞ xi2 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
3837
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
and the nonlinear function f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; f2 ðxi2 ðtÞÞÞT ¼ ðtanhðxi1 ðtÞÞ; tanhðxi2 ðtÞÞÞT . Thus, the matrices U; V; W in Assumptions 1 and 3 are
U¼
;
0 0 0 0
V¼
1 0
W¼
;
0 1
pffiffiffiffiffiffiffi 0:1 0
! 0 pffiffiffiffiffiffiffi : 0:2
According to Theorem 1, we can know that this complex network is globally exponentially synchronized in mean square. When we randomly choose the initial states in ½0; 1 ½0; 1, the synchronization errors are plotted in Figs. 1 and 2, which can confirm that the stochastic complex dynamical system (1) is globally exponentially synchronized in mean square. Example 2. Consider the following complex network consisting of three nodes.
"
# 3 X dxi ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ g ij Cxj ðtÞ dt þ ri ðt; xi ðtÞÞdN ðtÞ j¼1
for all i ¼ 1; 2; 3, where xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞT 2 R2 is the state vector of the ith subsystem, fN ðtÞgtP0 is a one-dimension fF t gtP0 adapted Poisson process with parameter k ¼ 5. Let
A¼
1
0
0
1
;
B¼
2
0:1
5
3
:
The out-coupling configuration matrices G and inner-coupling matrices C are chosen as
0
1 3 1 2 B C G ¼ @ 1 2 1 A; 2 1 3
C¼
The noise intensity function vector
ri ðt; xi ðtÞÞ ¼
1 0 0 1
4 0 0
4
:
ri ð; Þ is of the following form:
xi ðtÞ;
1.2 x11(t)−x21(t) x11(t)−x31(t) 1
0.8
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
2
2.5
3
time/seconds Fig. 3. State error of x11 ðtÞ xi1 ðtÞ, i ¼ 2; 3.
3.5
4
4.5
5
3838
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
0.6 x12(t)−x22(t) x12(t)−x32(t) 0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time/seconds Fig. 4. State error of x12 ðtÞ xi2 ðtÞ, i ¼ 2; 3.
and the nonlinear function f ðxi ðtÞÞ ¼ ðf1 ðxi1 ðtÞÞ; f2 ðxi2 ðtÞÞÞT ¼ ðtanhðxi1 ðtÞÞ; tanhðxi2 ðtÞÞÞT . Thus, the matrices U; V; W in Assumptions 1 and 3 are
U¼
0 0 0 0
;
V¼
1 0 0 1
;
W¼
1 0 0 1
:
According to Theorem 1, we can know that this complex network is globally exponentially synchronized in mean square. When we randomly choose the initial states in ½0; 1 ½0; 1, the synchronization errors are plotted in Figs. 3 and 4, which can confirm that the stochastic complex dynamical system (1) is globally exponentially synchronized in mean square.
5. Conclusions This paper is concerned with the problem of stochastic synchronization analysis for complex networks perturbed by the Poisson noise. Using the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper gives a globally exponentially synchronization criterion in mean square. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. On the other hand, it is worth mentioning that there are still some important problems to solve for stochastic complex networks perturbed by the Poisson noise. (1) When the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize. Therefore, it is necessary to consider the control problem, such as the adaptive control and pinning control, for synchronization of stochastic complex networks perturbed by the Poisson noise in the future. (2) It has now been well realized that in spreading information through complex networks, there always exist time delays, which may decrease the quality of the system and even lead to oscillation, divergence, and instability. Accordingly, the synchronization problems for stochastic delayed complex networks perturbed by the Poisson noise should be studied in the future researches. Acknowledgements The work was supported by 2012 Yeungnam University Research Grant. Also, the work of B. Song was supported by the National Natural Science Foundation of China under Grants 61104221 and 61174029 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 10KJB120004.
B. Song et al. / Applied Mathematics and Computation 219 (2012) 3831–3839
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