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Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes
Physics Letters A 374 (2010) 2539–2550
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Physics Letters A www.elsevier.com/locate/pla
Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes Shuiming Cai a,b , Qinbin He b,c , Junjun Hao b , Zengrong Liu a,b,∗ a b c
Department of Mathematics, Shanghai University, Shanghai 200444, China Institute of System Biology, Shanghai University, Shanghai 200444, China Department of Mathematics, Taizhou University, Linhai 317000, China
a r t i c l e
i n f o
Article history: Received 24 February 2010 Received in revised form 1 April 2010 Accepted 7 April 2010 Available online 10 April 2010 Communicated by A.R. Bishop Keywords: Exponential synchronization Complex dynamical networks Nonidentical time-delayed nodes Open-loop control Intermittent control Impulsive control
a b s t r a c t In this Letter, exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes is considered. Two effective control schemes are proposed to drive the network to synchronize globally exponentially onto any smooth goal dynamics. By applying open-loop control to all nodes and adding some intermittent controllers to partial nodes, some simple criteria for exponential synchronization of such network are established. Meanwhile, a pinning scheme deciding which nodes need to be pinned and a simply approximate formula for estimating the least number of pinned nodes are also provided. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network with nonidentical time-delayed dynamical nodes, and an estimate of the upper bound of impulsive intervals ensuring global exponential stability of the synchronization process is also given. Numerical simulations are presented finally to demonstrate the effectiveness of the theoretical results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In the past few decades, the control and synchronization of complex dynamical networks has been extensively investigated in various fields of science and engineering due to its many potential practical applications [1–4]. And many effective methods including feedback control [5–7], adaptive control [9,10], impulsive control [11–14] and intermittent control [15,16] have been proposed to solve the above problem. Among these approaches, the discontinuous control method, which includes impulsive control and intermittent control, has attracted much attention recently because it is practical and easily implemented in engineering fields. In some cases, using impulsive control or intermittent control is more effective and robust [17,18]. In recent years, several synchronization criteria for complex dynamical networks with or without time delays via impulsive control or intermittent control have been presented, see [11–16] and the references therein. It has been noticed that complex dynamical networks studied in the literature always adopt the assumption that all nodes dynamics are identical. However, this assumption of identical nodes is a highly unlikely circumstance for almost all complex dynamical networks in biology and engineering due to individuals inside a network usually have different physical parameters, etc. [19]. Taking a genetic network as an example, the genetic oscillators in the network, even in the same species, are usually nonidentical possibly due to heterogeneous nutrition conditions and fluctuated environments. When allowing the nodes of a complex network to be nonidentical, they will exhibit different dynamics and the synchronization schemes for networks with identical nodes will not work any more. Therefore, investigating new synchronization schemes for complex dynamical networks with nonidentical nodes is necessary and few results have been reported by now [19–23]. Recently, Song et al. [24] studied synchronization of complex dynamical networks with nonidentical nodes, and proposed two control schemes to synchronize the network onto any smooth goal dynamics by combining open-loop control with local adaptive feedback control and impulsive control, respectively. On the other hand, time delays are ubiquitous in natural and artificial systems. In much of the literature, time delays in the couplings are considered; however, the time delays in the dynamical nodes [25,26], which are more complex, are still relatively unexplored. In this Letter, we further investigate the problem of exponential synchronization for a general complex dynamical network models with nonidentical time-delayed dynamical nodes. Two effective control schemes are proposed to drive
*
Corresponding author at: Department of Mathematics, Shanghai University, Shanghai 200444, China. Tel.: +8621 66136109. E-mail address: [email protected] (Z. Liu).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.023
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the network to synchronize globally exponentially onto any smooth goal dynamics. Moreover, the developed techniques are applied to a complex network composed of the time-delayed Lorenz system and the time-delayed Chua oscillator, and numerical simulations are given to demonstrate the effectiveness of the theoretical results. The rest of this Letter is organized as follows. In Section 2, the synchronization problem of a general complex network with nonidentical time-delayed dynamical nodes is formulated, and some preliminaries are given. Section 3 is the main results. Some numerical results are provided in Section 4. Conclusions are drawn in Section 5. Throughout this Letter, the following notations will be used. Let Z + = {1, 2, 3, . . .}, · be the Euclidean norm, I N be an N-dimensional identity matrix, λmax (·) be the maximum eigenvalue of a symmetric matrix. X > (<)Y , where X and Y are symmetric matrices, means that X − Y is positive (negative) definite. P C ([−τ , 0], R n ) denotes the set of all functions of bounded variation and right-continuous on any compact subinterval of [−τ , 0]. 2. Model and preliminaries Consider a generally controlled complex network consisting of N nonidentical time-varying delays dynamical nodes with linearly diffusive couplings described by the following equations:
x˙ i (t ) = f i t , xi (t ), xi t − τi (t )
+c
N
b i j Γ x j (t ) + U i ,
i = 1, . . . , N ,
(1)
j =1
where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t )) ∈ R n is the state vector of the ith delayed dynamical node, f i : R × R n × R n is a continuously vector-valued function governing the evolution of the ith individual node xi (t ), the time delay τi (t ) may be unknown (constant or timevarying) but is bounded by a known constant, i.e., 0 τi (t ) τi , i = 1, . . . , N, and τ = max1i N τi . U i is the control action applied to node i, the positive constant c is the coupling strength, Γ ∈ R n×n is the inner connecting matrix, B = (b i j ) N × N is the coupling matrix
N
representing the underlying topology structure of the network and satisfies the following properties: j =1 b i j = 0, b i j 0, i = j, and rank( B ) = N − 1. It is worth mentioning that the coupling matrix B can be regarded as the Laplacian matrix of a weighted graph with a spanning tree, and B has an eigenvalue 0 with multiplicity 1, as indicated in [7,27]. Definition 1. Suppose that s(t ) ∈ R n is any smooth goal dynamics. The controlled complex delayed dynamical network (1) is said to be globally exponentially synchronized onto the homogeneous state s(t ) if its solution satisfies xi (t ) − s(t ) = O (exp(−λt )), λ > 0, i = 1, . . . , N, for any initial conditions. Remark 1. In the network (1), the coupling matrix B is not necessarily to be symmetric or irreducible. In addition, the inner connecting matrix Γ is not assumed to be symmetric or positive definite. Moreover, each dynamical node may have different time-varying delays and different node dynamics. Therefore, our network model can represent a large variety of complex dynamical networks. In particular, if all functions f i are the same: f i ≡ f , i = 1, . . . , N, and the synchronization state s(t ) is a solution of an isolated node, i.e., s˙ (t ) = f (t , s(t ), s(t − τ )), then the synchronization problem in (1) becomes the synchronization problem of a network with identical nodes discussed in much of the literature. Assumption 1. For the vector-valued function f i (t , x(t ), x(t − τi (t ))), suppose the uniform semi-Lipschitz condition with respect to the time t holds, i.e., for any x(t ), y (t ) ∈ R n , there exist positive constants θi > 0, γi > 0, i = 1, . . . , N, such that [28]
− f i t , y (t ), y t − τi (t ) x t − τi (t ) − y t − τi (t ) . θi x(t ) − y (t ) x(t ) − y (t ) + γi x t − τi (t ) − y t − τi (t )
x(t ) − y (t )
f i t , x(t ), x t − τi (t )
Lemma 1. (See [15].) Let ω : [t 0 − τ , ∞) → [0, ∞) be a continuous function such that
ω˙ (t ) −aω(t ) + b sup ω(s) t −τ st
is satisfied for t t 0 . If a > b > 0, then
ω(t )
sup
t 0 −τ θ t 0
ω(θ) exp−λ(t −t0 ) , t t 0 ,
where λ > 0 is the unique positive solution of the equation −a + λ + b expλτ = 0. Lemma 2. (See [15].) Let ω : [t 0 − τ , ∞) → [0, ∞) be a continuous function such that
ω˙ (t ) δ1 ω(t ) + δ2 sup ω(s) t −τ st
is satisfied for t t 0 . If δ1 > 0, δ2 > 0, then
ω(t )
sup
t 0 −τ θ t 0
ω(θ) exp(δ1 +δ2 )(t −t0 ) , t t 0 .
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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N −l
Lemma 3. (See [8,24].) Assume that Q = (q i j ) N × N
˜ E−D is symmetric. Let D = diag(d1 , . . . , dl , 0, . . . , 0), Q − D = S
S Ql
, and d = min1i l {di },
where 1 l < N, di > 0, i = 1, . . . , l, Q l is the minor matrix of Q by removing its first l(1 l < N ) row-column pairs, E and S are matrices with ˜ = diag(d1 , . . . , dl ). When d > λmax ( E − S Q −1 S ), Q − D < 0 is then equivalent to Q l < 0. appropriate dimensions, D l
R+
m+1
¯ i for each fixed (t , μ, μ ¯ 1, . . . , μ ¯ m) : ¯ 1, . . . , μ ¯ i −1 , μ ¯ i +1 , . . . , μ ¯ m ), × R × · · · × R → R be nondecreasing in μ Lemma 4. (See [33].) Let F (t , μ, μ i = 1, . . . , m, and I k (μ) : R → R be nondecreasing in μ. Suppose that u (t ), v (t ) ∈ P C ([−τ , ∞), R ) satisfy
D+ u (t ) F t , u (t ), u t − τ1 (t ) , . . . , u t − τm (t ) , −
u (tk ) I k u tk
,
k∈
D+ v (t ) > F t , v (t ), v t
−
v (tk ) I k v tk
,
− τ1 (t ) , . . . , v t − τm (t ) ,
k∈
t 0,
Z +.
t 0,
Z +.
Then u (t ) v (t ), for −τ t 0 implies u (t ) v (t ), for t 0. 3. Main results In this section, two synchronization schemes are developed for a network with nonidentical time-delayed dynamical nodes, and some sufficient conditions ensuring global exponential stability of the synchronization process are derived. 3.1. Open-loop controlled network with nonidentical time-delayed dynamical nodes This Letter aims to design appropriate controllers to drive the network (1) with nonidentical time-delayed dynamical nodes to synchronize globally exponentially onto any smooth goal dynamics s(t ). Considering open-loop control (also called entrainment control) provides a general approach to entrain a dynamical system to arbitrary goal dynamics [29,30], here we also apply open-loop control to all nodes in network (1) to achieve synchronization between the network (1) and the given smooth dynamics s(t ). The controlled dynamical network can be characterized by
x˙ i (t ) = f i t , xi (t ), xi t − τi (t )
+c
N
b i j Γ x j (t ) + s˙ (t ) − f i t , s(t ), s t − τi (t ) ,
(2)
j =1
where i = 1, . . . , N. Unfortunately, it was pointed out that open-loop control imposes some restrictions on initial conditions and goal dynamics, and cannot guarantee the stability of the control process [29,30]. Hence, it is necessary to develop effective approaches to ensure the open-loop controlled network (2) can be globally exponentially synchronized. In [31], the authors improved the open-loop control method by combining it with linear feedback control or with impulsive control to globally entrain a dynamical system to arbitrary goal dynamics. Furthermore, the authors investigated how to synchronize a complex network with nonidentical nodes onto any smooth goal dynamics by combing open-loop and adaptive feedback control or combing open-loop and impulsive control [24]. Consequently, we also consider introducing the discontinuous control method, intermittent control or impulsive control, to the open-loop controlled network (2) to ensure global exponential stability of the synchronization process. 3.2. Synchronization scheme combining open-loop and intermittent control In this subsection, we will stabilize the open-loop controlled network (2) via periodically intermittent control. Some intermittent controllers are added to partial nodes of the controlled network (2). Without loss of generality, suppose the first l(1 l < N ) nodes to be selected and pinned, then we have the following controlled dynamical network:
⎧ N ⎪ ⎪ τ ⎪ ⎪ x˙ i (t ) = f i i (t , xi , s) + c b i j Γ x j (t ) + s˙ (t ) + di (t ) xi (t ) − s(t ) , ⎪ ⎪ ⎨ j =1
N ⎪ ⎪ ⎪ τ ⎪ xi (t ) = f i i (t , xi , s) + c b i j Γ x j (t ) + s˙ (t ), ⎪ ⎪ ⎩
1 i l, (3)
l + 1 i N,
j =1
τ
where f i i (t , xi , s) = f i (t , xi (t ), xi (t − τi (t ))) − f i (t , si (t ), si (t − τi (t ))), s(t ) is any smooth dynamics, and di (t ) is the intermittent feedback control gain defined as follows:
di (t ) =
−di , 0,
t ∈ [nT, nT + δ),
1 i l,
t ∈ nT + δ, (n + 1)T ,
1 i l,
(4)
where di > 0 is positive constant called control gain, T is the control period, δ > 0 is called the control width (control duration), and n = 0, 1, 2, . . . . Hereafter, denote ρmin as the minimum eigenvalue of the matrix (Γ + Γ )/2. Assume that ρmin = 0 and Γ = ρ > 0. Let Bˆ s = ( Bˆ + Bˆ )/2 where Bˆ is a modified matrix of B via replacing the diagonal elements b ii by (ρmin /ρ )b ii , then Bˆ s is a symmetric irreducible matrix with nonnegative off-diagonal elements [27]. Note that generally Bˆ does not possess the property of zero row sums. Moreover, there does not exist a definite relationship between the eigenvalues of B and those of Bˆ for the general matrix G [10,15]. In the
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following, we will show that how to design suitable δ , T and di , i = 1, . . . , l, such that the controlled dynamical network (3) can realize globally exponential synchronization. The main results are stated as follows. Theorem 1. Suppose that τ δ . Under Assumption 1, if there exist positive constants a1 , a2 , and di , i = 1, . . . , l, such that (i) (θ + 12 a1 ) I N + c ρ Bˆ s − D < 0, (ii) γ − a1 < 0, (iii) θ I N + c ρ Bˆ s − 12 a2 I N 0, (iv) = λ(δ − τ ) − (a2 + γ )(T − δ) > 0, N −l
where θ = max1i N {θi }, γ = 2 max1i N {γi }, D = diag(d1 , . . . , dl , 0, . . . , 0), and λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Then the controlled dynamical network (3) is globally exponentially synchronized. Proof. Define synchronization error as e i (t ) = xi (t ) − s(t ) (i = 1, . . . , N ), according to the control law (4), then we can derive the following error dynamical system:
⎧ N ⎪ ⎪ ⎪ e˙ i (t ) = f˜ i t , e i (t ), e i t − τi (t ) + c b i j Γ e j (t ) − di e i (t ), nT t < nT + δ, 1 i l, ⎪ ⎪ ⎪ ⎪ ⎪ j = 1 ⎪ ⎪ ⎪ ⎪ N ⎨ e˙ i (t ) = f˜ i t , e i (t ), e i t − τi (t ) + c b i j Γ e j (t ), nT t < nT + δ, l + 1 i N , ⎪ ⎪ j = 1 ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ˜ ⎪ ˙ e ( t ) = f t , e ( t ), e t − τ ( t ) + c b i j Γ e j (t ), nT + δ t < (n + 1)T, 1 i N . i i i i ⎪ ⎩ i
(5)
j =1
where f˜ i (t , e i (t ), e i (t − τi (t ))) = f i (t , e i (t ) + s(t ), e i (t − τi (t )) + s(t − τi (t ))) − f i (t , s(t ), s(t − τi (t ))). Clearly, if the zero solution of the error system (5) is globally exponentially stable, then the controlled network (3) is globally exponentially synchronized. Construct the following Lyapunov candidate function:
1 e i (t )e i (t ). 2 N
V (t ) =
(6)
i =1
Then the derivative of V (t ) with respect to time t along the solutions of Eq. (5) can be calculated as follows: When nT t < nT + δ , for n = 0, 1, 2, . . . ,
V˙ (t ) =
N
˜ e i (t ) f i t , e i (t ), e i t − τi (t )
+c
i =1
N
θi e i (t )e i (t ) +
N
N
γi e i t − τi (t ) e i t − τi (t ) + c
b ii ρmin e i (t )e i (t ) −
i =1
+c
γ 2
N
l
bi j e i (t )Γ e j (t )
di e i (t )e i (t )
e i t − τi (t ) e i t − τi (t ) + c
i =1
b ii ρmin e i (t )e i (t ) −
i =1 N
N
i =1
e i (t )e i (t ) +
N
di e i (t )e i (t )
i =1 j =1, j =i
i =1 N
l i =1
N
i =1
+c
θ
bi j e i (t )Γ e j (t ) −
i =1 j =1
i =1
θ
N N
N N
b i j ρ e i (t )e j (t )
i =1 j =1, j =i l
di e i (t )e i (t )
i =1
e i (t )e i (t ) +
i =1
γ
sup
N
2 t −τ st i =1
ˆ e i (s)e i (s) + c ρ e (t )( B ⊗ I n )e (t ) −
1 1 γ = e (t ) θ + a1 I N + c ρ Bˆ s − D ⊗ I n e (t ) − a1 e (t )e (t ) +
2
−a1 V (t ) + γ
2
sup
t −τ st
l
di e i (t )e i (t )
i =1
sup e (s)e (s) 2 t −τ st
V (s),
(7)
where e (t ) = (e 1 (t ), e 2 (t ), . . . , e N (t )) . From condition (ii) and the above inequality (7), we can get by Lemma 1
V (t )
sup
nT−τ φnT
V (φ) exp −λ(t − nT) ,
nT t < nT + δ,
(8)
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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where λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Similarly, when nT + δ t < (n + 1)T, using condition (iii), one has
V˙ (t ) a2 V (t ) + γ
sup
t −τ st
V (s).
(9)
It follows from Lemma 2 that
V (t )
sup
nT+δ−τ φnT+δ
V (φ) exp (a2 + γ )(t − nT − δ) ,
nT + δ t < (n + 1)T.
(10)
Now, we estimate V (t ) based on Eqs. (8) and (10). By (8), noticing that δ τ and λ > 0, we can obtain the following:
V (δ) = τ where
sup
δ−τ sδ
V (s)
sup
V (φ) exp{−λt } =
sup
δ−τ sδ −τ φ0
sup
−τ φ0
V (φ) exp −λ(δ − τ ) =
sup
−τ φ0
V (φ) exp{− 1 }, (11)
1 = λ(δ − τ ). By (10), and noticing that (a2 + γ ) > 0, we have the following: V (T) = sup V (s) max sup V (s), sup V (s) τ T−τ sT
T−τ sδ
δsT
sup V (δ)τ exp (a2 + γ )(t − δ) = V (δ)τ exp (a2 + γ )(T − δ) = V (δ)τ exp{ 2 }, δsT
where
2 = (a2 + γ )(T − δ). Substituting (11) into (12) gives V (T) V (δ) exp{ 2 } sup V (φ) exp{− 1 + 2 }. τ τ
(12)
(13)
−τ φ0
Similarly, we have
V (2T) τ
sup
T−τ sT
V (s) exp{− 1 } exp{ 2 }
sup
−τ φ0
V (φ) exp −2( 1 − 2 ) .
(14)
By mathematical induction, we have
V (kT) τ
sup
−τ φ0
V (φ) exp −k( 1 − 2 ) ,
(15)
for any nonnegative integers k. Since for any t 0, there exists a nonnegative integers k, such that kT t < (k + 1)T, we can deduce the following estimation of V (t ) for any t 0 by (8), (10) and (15). For kT t < kT + δ ,
V (t )
sup
kT−τ skT
sup
−τ φ0
V (s) exp −λ(t − kT)
T
sup
kT+δ−τ skT+δ
sup
−τ φ0
−τ φ0
−τ φ0
V (φ) exp
V (φ) exp
δ
T
exp −
sup
−τ φ0
T
δ
exp −
T
−τ φ0
V (φ) exp −
T
sup
V (φ) exp −(k + 1)
sup
sup
V (φ) exp −k( 1 − 2 ) exp −λ(t − kT)
V (s) exp (a2 + γ )(t − kT − δ)
Therefore, for any t > 0, we have
V (t )
−τ φ0
t−δ V (φ) exp −
=
Similarly, for kT + δ t < (k + 1)T,
V (t )
sup
T
sup
−τ φ0
V (φ) exp{−k } (16)
t .
V (φ) exp −k( 1 − 2 ) exp{− 1 } exp{ 2 }
t .
(17)
t ,
t 0.
(18)
It follows from condition (iv) that the zero solution of the error dynamical system (5) is globally exponentially stable. The proof is thus completed. 2 Let Q = (θ + 12 a1 ) I N + c ρ Bˆ s , and (θ + 12 a1 ) I N + c ρ Bˆ s − D = Q − D =
˜ S E−D S Q l
where Q l is the minor matrix of Q by removing
˜ = diag(d1 , . . . , dl ). Let d = min1i l {di }. By its first l(1 l < N ) row-column pairs, E and S are matrices with appropriate dimensions, D Lemma 3, we know that Q − D < 0 is equivalent to Q l < 0 and d > λmax ( E − S Q l−1 S ). If d can be sufficiently large (one can simply
select di > λmax ( E − S Q l−1 S ), i = 1, . . . , l), then Q − D < 0 is equivalent to Q l < 0. Note that Q l = (θ + 12 a1 ) I N −l + c ρ Bˆ ls , where Bˆ ls is the
minor matrix of Bˆ s by removing its first l(1 l < N ) row-column pairs. Thus we can derive the following corollary.
Corollary 1. Suppose that τ δ . Under Assumption 1, if the control gain di can be sufficiently large and there exist positive constants a1 and a2 , such that
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(i) (θ + 12 a1 ) + c ρ λmax ( Bˆ ls ) < 0, (ii) γ − a1 < 0, (iii) θ I N + c ρ Bˆ s − 12 a2 I N 0, (iv) = λ(δ − τ ) − (a2 + γ )(T − δ) > 0, where θ = max1i N {θi },
γ = 2 max1iN {γi }, and λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Then the
controlled dynamical network (3) is globally exponentially synchronized. One can simply select di > λmax ( E − S Q l−1 S ), i = 1, . . . , l.
Remark 2. In [8], pinning synchronization of directed and undirected complex dynamical networks were investigated, and it was pointed out that the nodes whose out-degrees are bigger than their in-degrees should be specifically selected as pinned candidates. For the controlled network (3), we can first apply intermittent control to the nodes with zero in-degrees, and then continue to try others nodes in descending order based on their degree-difference defined as the difference between the out-degree of the nodes and the in-degree of the nodes (for those with the same degree-difference, in descending order according to their out-degrees). Moreover, from condition (i) in Corollary 1, we can see that the number of nodes should be selected and pinned to synchronize the controlled network (3) is at least l0 , a1 a1 where l0 satisfies λmax ( Bˆ ls −1 ) − 2θ+ and λmax ( Bˆ ls ) < − 2θ+ [8]. 2c ρ 2c ρ 0
0
3.3. Synchronization scheme combining open-loop and impulsive control In this subsection, impulsive control strategy will be adopted to stabilize the open-loop controlled network (2). Applying impulsive control to the open-loop controlled network (2), we have the following synchronization scheme for the complex network with nonidentical time-delayed dynamical nodes:
⎧ N ⎪ ⎪ ⎪ ˙ x ( t ) = f t , x ( t ), x t − τ ( t ) + c b i j Γ x j (t ) + s˙ (t ) − f i t , si (t ), si t − τi (t ) , ⎪ i i i i ⎨ i
t = tk ,
j =1
(19)
⎪ ⎪ x = xi tk+ − xi tk− = Ak xi tk− − s tk− , t = tk , k ∈ Z + , ⎪ ⎪ ⎩ i xi (s) = ϕi (s), −τ s 0, i = 1, . . . , N , t 0.
+ − + where the time sequence {tk }k+∞ =1 satisfy tk−1 < tk and limk→∞ tk = +∞, xi = xi (tk ) − xi (tk ) is the control law in which xi (tk ) = − n×n limt →t + xi (t ) and xi (tk ) = limt →t − xi (t ), A k ∈ R is the control gain matrix. Without loss of generality, we assume that limt →t + xi (t ) = k
k
k
xi (tk ), which means the solution xi (t ) is continuous from the right. The initial conditions of Eq. (19) are given by xi (t ) = ϕi (t ) ∈ P C ([−τ , 0], R n ). For convenience, we now make some assumptions about the time-delays set {τ1 (t ), . . . , τ N (t )}. Suppose that the time-delays set {τ1 (t ), . . . , τ N (t )} contains m0 different types time-delays, and T i = {li −1 + 1, . . . , li }, i ∈ {1, . . . , m0 } denotes the index set of the nodes m with the same time-delays, where l0 = 0, lm0 = N, and li −1 < li . Let γ T i = 2 max j ∈ T i {γ j } and γˆ = i =01 γ T i , then we have the following result: Theorem 2. Let Ω = supk∈ Z + {tk − tk−1 } < ∞. Under Assumption 1, if there exist positive constants a3 and d0 < 1 such that (i) θ I N + c ρ Bˆ s − 12 a3 I N 0, (ii) ( I n + A k ) ( I n + A k ) − d0 I n 0, k ∈ Z + , (iii)
ln d0
Ω
ˆ + a3 + dγ0 < 0,
then the controlled dynamical network (19) is globally exponentially synchronized. Proof. Let v i (t ) = xi (t ) − s(t ), i = 1, . . . , N. Considering that s(t ) is assumed to be smooth, we get s(tk ) = s(tk+ ) − s(tk− ) = 0. Then from (19), the error dynamical system is governed as follows:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
v˙ i (t ) = f˜ i t , v i (t ), v i t − τi (t )
−
v i (tk ) = ( I n + A k ) v i tk ,
+c
N
b i j Γ v j (t ),
t = tk ,
j =1
(20)
+
t = tk , k ∈ Z .
Let v (t ) = ( v 1 (t ), . . . , v N (t )) , consider the following Lyapunov function:
1 v i (t ) v i (t ). 2 N
V (t ) =
(21)
i =1
Taking the upper Dini derivative of V (t ) along the solution of Eq. (20), we get +
D V (t ) =
N i =1
N ˜ v i (t ) f i t , v i (t ), v i t − τi (t ) + c b i j Γ v j (t )
j =1
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
N
θi v i (t ) v i (t ) +
i =1
N
2545
ˆ γi v i t − τi (t ) v i t − τi (t ) + c ρ v (t )( B ⊗ I n ) v (t )
i =1
m0 v (t ) θ I N + c ρ Bˆ s ⊗ I n v (t ) + γ j v j t − τi (t ) v j t − τi (t ) i =1 j ∈ T i m0 γ Ti
a3 V (t ) +
2
i =1
v j t − τi (t ) v j t − τi (t ) a3 V (t ) +
j∈T i
m0
γ T i V t − τi (t ) , t = tk , k ∈ Z + .
(22)
i =1
When t = tk , k ∈ Z + , we have
V (tk ) =
1 1 − v i (tk ) v i (tk ) = v i tk ( I n + A k ) ( I n + A k ) v i tk− d0 V tk− . 2 2 N
N
i =1
i =1
(23)
> 0, let μ (t ) be an unique solution of the following impulsive delayed dynamical system ⎧ m0 ⎪ ⎪ ⎪ ˙ μ ( t ) = a μ ( t ) + γ T i μ t − τi (t ) + , t = tk , t 0, ⎪ 3 ⎪ ⎪ ⎨ i =1 − μ (tk ) = d0 μ tk , k ∈ Z + , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ (t ) = sup V (s), −τ t 0.
For any
(24)
−τ s0
−1
Let M 1 = d0 sup−τ s0 V (s), and
η = −(ln d0 )/Ω − a3 . In the following, we will prove that condition (iii) implies
μ (t ) < M 1 exp{−λ1t } +
t 0,
(25)
γ T i exp{λ1 τi } = 0.
(26)
1 ˆ (η − d− 0 γ )d0
,
where λ1 > 0 is an unique solution of 1 λ1 + (ln d0 )/Ω + a3 + d− 0
m0 i =1
m0
Denote H (λ1 ) = λ1 − η + d0−1
i =1
dH (λ1 ) dλ1
m0
γ T i exp{λ1 τi }. From condition (iii), we have η − d0−1 γˆ > 0, d0−1 (
i =1
γ T i ) > 0, and so H (0) < 0,
H (+∞) > 0 and > 0. Using the continuity and monotonicity of H (λ1 ), the equality (26) has an unique positive solution λ1 > 0. By the formula for the variation of parameters [32], we have
t
μ (t ) = W (t , 0)μ (0) +
W (t , s)
m0
γ μ s − τi (s) + ds,
Ti
t 0,
(27)
i =1
0
where W (t , s), t , s 0 is the Cauchy matrix of linear system
˙ t ) = a3 ψ(t ), t = tk , ψ( ψ(tk ) = d0 ψ tk− , k ∈ Z + .
(28)
According to the representation of the Cauchy matrix [32], we get the estimation of W (t , s) since 0 < d0 < 1 and tk − tk−1 Ω ,
W (t , s) =
[(t −s)/Ω−1]
d0 exp a3 (t − s) d0 s
1 = d− 0 exp
exp
−η −
ln d0
Ω
(t − s)
ln d0 1 (t − s) = d− (t − s) exp −η − 0 exp −η (t − s) , Ω Ω
ln d0
t s 0.
(29)
Substituting (29) into (27) gives −1
μ (t ) d0
sup V (s) exp{−ηt } +
−τ s0
t
−1
d0 0
t = M 1 exp{−ηt } +
1 exp −η(t − s) d− 0
0
−1
Since , λ1 , η − d0 γˆ > 0, and 0 < d0 < 1, we have
m 0 Ti exp −η(t − s) γ μ s − τi (s) + ds
i =1
m 0 i =1
γ μ s − τi (s) + ds, Ti
t 0.
(30)
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
μ (t ) sup V (s)/d0 < M 1 exp{−λ1t } + −τ s0
1 ˆ (η − d− 0 γ )d0
−τ t 0.
,
(31)
Now, we show that (25) is true, that is,
μ (t ) < M 1 exp{−λ1t } + If this is not true, from (31) and
1 ˆ (η − d− 0 γ )d0
(32)
μ (t ) ∈ P C ([−τ , ∞), R + ), then there must exists a t ∗ > 0 satisfying
μ t ∗ M 1 exp −λ1t ∗ + μ (t ) < M 1 exp{−λ1t } +
t 0.
,
1 ˆ (η − d− 0 γ )d0
1 ˆ (η − d− 0 γ )d0
,
(33)
−τ t < t ∗ .
(34)
,
By (26), (30), (34), we have
∗
μ t
M 1 exp −ηt
∗
t ∗ +
∗
−1
exp −η t − s d0
exp −ηt
t ∗ M1 +
−1
exp{η s} d0 M 1
m0
exp −ηt
γ
−1
M 1 + d0 M 1
m0
< M 1 exp −λ1 t
∗
+
exp −λ1 s − τi (s) +
t ∗ Ti
γ exp{λ1 τi }
i =1
Ti
i =1
0
∗
γ μ s − τi (s) + ds
Ti
i =1
0
∗
m0
exp (η − λ1 )s ds + 0
1 ˆ (η − d− 0 γ )d0
1 γˆ d− 1 0 + d− 0 ds 1 ˆ (η − d− γ ) d 0 0
η 1 ˆ (η − d− 0 γ )d0
!
!
t ∗ exp{η s} ds 0
(35)
. m
This contradicts (33), and so (25) holds. Denote F (t , V , V¯ 1 , . . . , V¯ m0 ) = a3 V + i =01 γ T i V¯ i , and I k ( V ) = d0 V . Then the function F and I k satisfy the monotonicity given in Lemma 4. Since V (t ) sup−τ s0 V (s) = μ (t ), for −τ t 0, it follows from Lemma 4 that
V (t ) μ (t ) < M 1 exp{−λ1 t } + Letting
1 ˆ (η − d− 0 γ )d0
,
t 0.
ε → 0+ , then we have V (t ) M 1 exp{−λ1 t }, t 0. This completes the proof of Theorem 2. 2
For simplicity, we consider the equidistant impulsive interval tk − tk−1 = t, and the impulsive control gain matrix A k ≡ a0 I n , k = 1, 2, . . . . Then, the following result can be obtained readily from Theorem 2. Corollary 2. Under Assumption 1, if the following condition holds
t < −
2(1 + a0 )2 ln |1 + a0 | 2λmax (θ I N + c ρ Bˆ s )(1 + a0 )2 + γˆ
,
−2 < a0 < 0,
(36)
then the controlled dynamical network (19) is globally exponentially synchronized. 4. Numerical simulations In this section, numerical simulations are given to verify the effectiveness of the proposed synchronization schemes for synchronizing a complex network with time-delayed dynamical nodes onto a chaotic trajectory and a periodic orbit. Consider a complex dynamical network consisting of two types nonidentical time-delayed chaotic nodes, which are described by timedelayed Lorenz system and time-delayed Chua oscillator. The network is given by
⎧ 200 ⎪ ⎪ ⎪ ⎪ x˙ i (t ) = f 1 t , xi (t ), xi t − τ1 (t ) + c b i j Γ x j (t ), ⎪ ⎪ ⎨ j =1 200 ⎪ ⎪ ⎪ ⎪ ˙ x ( t ) = f t , x ( t ), x t − τ ( t ) + c b i j Γ x j (t ), ⎪ 2 2 i i ⎪ ⎩ i
i = 1, . . . , 100, (37) i = 101, . . . , 200,
j =1
where Γ = diag(1, 1.1, 1), the coupling strength c = 35, B = (b i j )200×200 is a symmetrically diffusive coupling matrix with g i j = 0 or 1 ( j = i ). Here we assume that the network structure of Eq. (37) obeys the scale-free distribution of the BA model [34]. The parameters of the BA model are given by m0 = m = 5, N = 200. Moreover, the first 100 nodes and the other 100 nodes are described, respectively, by time-delayed Lorenz system and time-delayed Chua oscillator [35,36]:
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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Fig. 1. (Left) Chaotic behavior of the time-delayed Lorenz system (38); (Right) chaotic behavior of the time-delayed Chua oscillator (39).
= C 10 xi (t ) + g 11 xi (t ) + g 12 xi t − τ1 (t ) , f 2 t , xi (t ), xi t − τ2 (t ) = C 20 xi (t ) + g 21 xi (t ) + g 22 xi t − τ2 (t ) ,
f 1 t , xi (t ), xi t − τ1 (t )
(38) (39)
where xi (t ) = (xi1 (t ), xi2 (t ), xi3 (t )) , g 11 (xi ) = (0, −xi1 xi3 , xi1 xi2 ) , g 12 (xi ) = (0, σ0 xi2 , 0) , g 21 (xi ) = (− 12 α0 (m1 − m2 )(|xi1 + 1| − |xi1 − 1|), 0, 0) , g 22 (xi ) = (0, 0, −β0 0 sin( v 0 xi1 )) ,
⎡
−a0
a0
C 10 = ⎣ r0
σ0 − 1
0
0
⎤
0 ⎦,
⎡ C 20 = ⎣
−b0
0
⎤
−α0 (1 + m2 )
α0
0
1
−1
1 ⎦,
0
−β0 −ω0
(40)
and a0 = 10, b0 = 8/3, r0 = 28, σ0 = 5, α0 = 10, β0 = 19.53, ω0 = 0.1636, m1 = −1.4325, m2 = −0.7831, v 0 = 0.5, 0 = 0.2, τ1 (t ) = 0.1, and τ2 (t ) = 0.02. Fig. 1 shows the chaotic behaviors of the time-delayed Lorenz system (38) and the time-delayed Chua oscillator (39). Consider a smooth dynamics s(t ) = (s1 , s2 , s3 ) , which we aim to synchronize the network (37) onto, satisfies |si | M i0 , where M i0 > 0, i = 1, 2, 3. Let e i (t ) = xi (t ) − s(t ), using the inequality |xy | 12 (x2 + y 2 ), then it is easy to verify [15,24]
e i (t ) f 1 t , xi (t ), xi t − τ1 (t )
− f 1 t , s(t ), s t − τ1 (t )
= e i (t )C 10 e i (t ) + s2 e i1 (t )e i3 (t ) − s3 e i1 (t )e i2 (t ) + σ0 e i2 (t )e i2 t − τ1 (t ) −10 + κ1 38 + M 30 /2 + κ2 M 20 /2 e 2i1 (t ) + 4 + 38 + M 30 /(2κ1 ) + 5κ3 /2 e 2i2 (t ) + M 20 /(2κ2 ) − 8/3 e 2i3 (t ) + 5/(2κ3 )e 2i2 t − τ1 (t ) i = 1, . . . , 100, θ1 e i (t )e i (t ) + γ1 e i t − τ1 (t ) e i t − τ1 (t ) ,
and
− f 2 t , s(t ), s t − τ2 (t ) 2 1 e i (t ) C 20 + C 20 e i (t ) + α0 (m1 − m2 ) e i1 (t ) + β0 0 v 0 e i3 (t )e i1 t − τ2 (t ) 2 2 λmax (C˜ 20 )e i (t )e i (t ) + β0 0 v 0 /(2κ4 )e i1 t − τ2 (t ) i = 101, . . . , 200, θ2 e i (t )e i (t ) + γ2 e i t − τ2 (t ) e i t − τ2 (t ) ,
e i (t ) f 2 t , xi (t ), xi t − τ2 (t )
C 20 +C 20 where C˜ 20 = ( + diag(|α0 (m1 − m2 )|, 0, κ4 β0 20 v 0 )), and θ1 , θ2 , 2 κ1 , κ2 , κ3 , κ4 > 0.
(41)
γ1 , γ2 can be determined by choosing appropriate parameters
4.1. Synchronizing network onto a chaotic trajectory via open-loop and intermittent control To demonstrate the effectiveness of Corollary 1, we use the synchronization scheme combining open-loop and intermittent control to achieve synchronization between the network (37) and the solution of the following Rössler system
s˙ 1 (t ) = −(s2 + s3 ),
s˙ 2 (t ) = s1 + 0.2s2 ,
s˙ 3 (t ) = s3 (s1 − 5.7) + 0.2.
(42)
It is well known that the Rössler system exhibits chaotic behavior and its state is bounded. Simulation result shows that there exist constants M 10 = 11.3236, M 20 = 10.6547, M 3 = 21.8865 satisfying |si | M i0 , i = 1, 2, 3 [24,31]. Letting κ1 = 1.26, κ2 = 0.17, κ3 = 0.36, κ4 = 1, one has θ1 = 28.6709, γ1 = 6.9444, θ2 = 11.3931, and γ2 = 0.9765 based on (41). It follows that θ = maxi=1,2 {θi } = 28.6709, and γ = maxi=1,2 {γi } = 6.9444. In addition, the values of the parameters for the intermittent controllers are taken as T = 2.2, δ = 2.0. Since λmax ( Bˆ s ) = 0.9130, ρ = Γ = 1.1, then if we choose a1 = 43.5, a2 = 127.65, it is easy to verify that conditions (ii)–(iv) in Corollary 1 are satisfied. Consequently, condition (i) in Corollary 1 becomes
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
Fig. 2. λmax ( Bˆ ls ) versus the number of pinned nodes (l).
Fig. 3. Time evolution of synchronization error via open-loop and adaptive intermittent control.
2θ + a 1 λmax Bˆ ls < − = −1.3096.
(43)
2c ρ
Using the pinned-node selection scheme for the controlled network (3) in Remark 2, we rearrange network nodes. Choose l from 1 to 100, and depict λmax ( Bˆ ls ) in Fig. 2, which reveals that λmax ( Bˆ ls ) decreases with the increase of the number of pinned nodes (l). Especially,
s s when l = 18 and l = 19, we have λmax ( Bˆ 18 ) = −1.3050 and λmax ( Bˆ 19 ) = −1.3487. Therefore, we can apply open-loop control to all nodes and choose the first 19 rearranged nodes of the network (37) as pinned nodes to achieve synchronization between the network (37) and the solution s(t ) of Rössler system (42). In addition, in Corollary 1, the control gain is required to be large enough [di > λmax ( E − S Q l−1 S ), i = 1, . . . , l], but it may be much larger than the needed values. So we here adopt the adaptive control approach [9,10], and apply the following adaptive intermittent feedback control gain [15]:
di (t ) =
(t )e i (t ), −di , d˙ i = qi e i 0,
di (0) = 0, qi > 0, t ∈ [nT, nT + δ), 1 i 19,
t ∈ [nT + δ, (n + 1)T),
1 i 19.
(44)
Though we have not found a rigorous proof for the synchronization scheme with adaptive control gain (44), the method is very useful in this example [15]. The initial conditions of the numerical simulations are as follows: xi (0) = (−8 + 0.5i , −5 + 0.5i , −10 + 0.5i ) , s(0) = (1, 2, 3) , where 1 i 200, and q1 = · · · = q19 = 0.4. The synchronous errors e i (t ) are illustrated in Fig. 3, and the values for the control gains after synchronization satisfy di 114.71, 1 i 19, which illustrate the adaptive control approach can obtain a more applicable control gain.
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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Fig. 4. Time evolution of synchronization error via open-loop and impulsive control.
4.2. Controlling network onto a period orbit via open-loop and impulsive control Now, we use the synchronization scheme (19) which combines open-loop and impulsive control to achieve synchronization between the network (37) and a periodic orbit given by
s(t ) =
π0 cos(ωt ), 1.5π0 sin(1.5ωt ), 2π0 cos(2ωt ) ,
(45)
where π0 = 1, ω = 2. Obviously, the bounds of s(t ) are M 10 = 1, M 20 = 1.5, M 30 = 2, and s˙ (t ) = (−π0 ω sin(ωt ), 2.25π0 ω cos(1.5ωt ), −4π0 ω sin(2ωt )) . Choosing κ1 = 1.6, κ2 = 0.03, κ3 = 2.5, κ4 = 19, one has θ1 = 22.75, γ1 = 1, θ2 = 22.3507, and γ2 = 0.0514 based on (41). It follows that θ = 22.75, and γˆ = 2.103. By Corollary 2, we can derive the following condition ensuring the synchronization between the network (37) and the periodic orbit (45)
t < −
2(1 + a0 )2 ln |1 + a0 | 2λmax (θ I N + c ρ Bˆ s )(1 + a0 )2 + γˆ
=−
2(1 + a0 )2 ln |1 + a0 | 115.80(1 + a0 )2 + 2.103
,
−2 < a0 < 0.
Let the impulsive control gain a0 = −1.20 and the impulsive interval t = 0.015. Fig. 4 shows the evolution of the synchronization errors. We can clearly see that the network (37) is quickly synchronized onto the periodic orbit (45). 5. Conclusions In this Letter, the synchronization issue of a general complex network with nonidentical time-delayed dynamical nodes is discussed. Two control schemes are proposed to synchronize such dynamical network onto any smooth goal dynamics by combining open-loop control with local intermittent feedback control and impulsive control, respectively. Some sufficient conditions ensuring the global exponential stability of the synchronization process are derived. Both theoretical and numerical analysis illustrate the effectiveness of the proposed control methodologies. Acknowledgements We thank the reviewers for their helpful suggestions on the manuscript. This work was supported by the National Science Foundation of China (Grant Nos. 10832006, 10802043), and key disciplines of Shanghai Municipality (S30104), Shanghai Academic Discipline Project (J50101). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes Shuiming Cai a,b , Qinbin He b,c , Junjun Hao b , Zengrong Liu a,b,∗ a b c
Department of Mathematics, Shanghai University, Shanghai 200444, China Institute of System Biology, Shanghai University, Shanghai 200444, China Department of Mathematics, Taizhou University, Linhai 317000, China
a r t i c l e
i n f o
Article history: Received 24 February 2010 Received in revised form 1 April 2010 Accepted 7 April 2010 Available online 10 April 2010 Communicated by A.R. Bishop Keywords: Exponential synchronization Complex dynamical networks Nonidentical time-delayed nodes Open-loop control Intermittent control Impulsive control
a b s t r a c t In this Letter, exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes is considered. Two effective control schemes are proposed to drive the network to synchronize globally exponentially onto any smooth goal dynamics. By applying open-loop control to all nodes and adding some intermittent controllers to partial nodes, some simple criteria for exponential synchronization of such network are established. Meanwhile, a pinning scheme deciding which nodes need to be pinned and a simply approximate formula for estimating the least number of pinned nodes are also provided. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network with nonidentical time-delayed dynamical nodes, and an estimate of the upper bound of impulsive intervals ensuring global exponential stability of the synchronization process is also given. Numerical simulations are presented finally to demonstrate the effectiveness of the theoretical results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In the past few decades, the control and synchronization of complex dynamical networks has been extensively investigated in various fields of science and engineering due to its many potential practical applications [1–4]. And many effective methods including feedback control [5–7], adaptive control [9,10], impulsive control [11–14] and intermittent control [15,16] have been proposed to solve the above problem. Among these approaches, the discontinuous control method, which includes impulsive control and intermittent control, has attracted much attention recently because it is practical and easily implemented in engineering fields. In some cases, using impulsive control or intermittent control is more effective and robust [17,18]. In recent years, several synchronization criteria for complex dynamical networks with or without time delays via impulsive control or intermittent control have been presented, see [11–16] and the references therein. It has been noticed that complex dynamical networks studied in the literature always adopt the assumption that all nodes dynamics are identical. However, this assumption of identical nodes is a highly unlikely circumstance for almost all complex dynamical networks in biology and engineering due to individuals inside a network usually have different physical parameters, etc. [19]. Taking a genetic network as an example, the genetic oscillators in the network, even in the same species, are usually nonidentical possibly due to heterogeneous nutrition conditions and fluctuated environments. When allowing the nodes of a complex network to be nonidentical, they will exhibit different dynamics and the synchronization schemes for networks with identical nodes will not work any more. Therefore, investigating new synchronization schemes for complex dynamical networks with nonidentical nodes is necessary and few results have been reported by now [19–23]. Recently, Song et al. [24] studied synchronization of complex dynamical networks with nonidentical nodes, and proposed two control schemes to synchronize the network onto any smooth goal dynamics by combining open-loop control with local adaptive feedback control and impulsive control, respectively. On the other hand, time delays are ubiquitous in natural and artificial systems. In much of the literature, time delays in the couplings are considered; however, the time delays in the dynamical nodes [25,26], which are more complex, are still relatively unexplored. In this Letter, we further investigate the problem of exponential synchronization for a general complex dynamical network models with nonidentical time-delayed dynamical nodes. Two effective control schemes are proposed to drive
*
Corresponding author at: Department of Mathematics, Shanghai University, Shanghai 200444, China. Tel.: +8621 66136109. E-mail address: [email protected] (Z. Liu).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.023
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
the network to synchronize globally exponentially onto any smooth goal dynamics. Moreover, the developed techniques are applied to a complex network composed of the time-delayed Lorenz system and the time-delayed Chua oscillator, and numerical simulations are given to demonstrate the effectiveness of the theoretical results. The rest of this Letter is organized as follows. In Section 2, the synchronization problem of a general complex network with nonidentical time-delayed dynamical nodes is formulated, and some preliminaries are given. Section 3 is the main results. Some numerical results are provided in Section 4. Conclusions are drawn in Section 5. Throughout this Letter, the following notations will be used. Let Z + = {1, 2, 3, . . .}, · be the Euclidean norm, I N be an N-dimensional identity matrix, λmax (·) be the maximum eigenvalue of a symmetric matrix. X > (<)Y , where X and Y are symmetric matrices, means that X − Y is positive (negative) definite. P C ([−τ , 0], R n ) denotes the set of all functions of bounded variation and right-continuous on any compact subinterval of [−τ , 0]. 2. Model and preliminaries Consider a generally controlled complex network consisting of N nonidentical time-varying delays dynamical nodes with linearly diffusive couplings described by the following equations:
x˙ i (t ) = f i t , xi (t ), xi t − τi (t )
+c
N
b i j Γ x j (t ) + U i ,
i = 1, . . . , N ,
(1)
j =1
where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t )) ∈ R n is the state vector of the ith delayed dynamical node, f i : R × R n × R n is a continuously vector-valued function governing the evolution of the ith individual node xi (t ), the time delay τi (t ) may be unknown (constant or timevarying) but is bounded by a known constant, i.e., 0 τi (t ) τi , i = 1, . . . , N, and τ = max1i N τi . U i is the control action applied to node i, the positive constant c is the coupling strength, Γ ∈ R n×n is the inner connecting matrix, B = (b i j ) N × N is the coupling matrix
N
representing the underlying topology structure of the network and satisfies the following properties: j =1 b i j = 0, b i j 0, i = j, and rank( B ) = N − 1. It is worth mentioning that the coupling matrix B can be regarded as the Laplacian matrix of a weighted graph with a spanning tree, and B has an eigenvalue 0 with multiplicity 1, as indicated in [7,27]. Definition 1. Suppose that s(t ) ∈ R n is any smooth goal dynamics. The controlled complex delayed dynamical network (1) is said to be globally exponentially synchronized onto the homogeneous state s(t ) if its solution satisfies xi (t ) − s(t ) = O (exp(−λt )), λ > 0, i = 1, . . . , N, for any initial conditions. Remark 1. In the network (1), the coupling matrix B is not necessarily to be symmetric or irreducible. In addition, the inner connecting matrix Γ is not assumed to be symmetric or positive definite. Moreover, each dynamical node may have different time-varying delays and different node dynamics. Therefore, our network model can represent a large variety of complex dynamical networks. In particular, if all functions f i are the same: f i ≡ f , i = 1, . . . , N, and the synchronization state s(t ) is a solution of an isolated node, i.e., s˙ (t ) = f (t , s(t ), s(t − τ )), then the synchronization problem in (1) becomes the synchronization problem of a network with identical nodes discussed in much of the literature. Assumption 1. For the vector-valued function f i (t , x(t ), x(t − τi (t ))), suppose the uniform semi-Lipschitz condition with respect to the time t holds, i.e., for any x(t ), y (t ) ∈ R n , there exist positive constants θi > 0, γi > 0, i = 1, . . . , N, such that [28]
− f i t , y (t ), y t − τi (t ) x t − τi (t ) − y t − τi (t ) . θi x(t ) − y (t ) x(t ) − y (t ) + γi x t − τi (t ) − y t − τi (t )
x(t ) − y (t )
f i t , x(t ), x t − τi (t )
Lemma 1. (See [15].) Let ω : [t 0 − τ , ∞) → [0, ∞) be a continuous function such that
ω˙ (t ) −aω(t ) + b sup ω(s) t −τ st
is satisfied for t t 0 . If a > b > 0, then
ω(t )
sup
t 0 −τ θ t 0
ω(θ) exp−λ(t −t0 ) , t t 0 ,
where λ > 0 is the unique positive solution of the equation −a + λ + b expλτ = 0. Lemma 2. (See [15].) Let ω : [t 0 − τ , ∞) → [0, ∞) be a continuous function such that
ω˙ (t ) δ1 ω(t ) + δ2 sup ω(s) t −τ st
is satisfied for t t 0 . If δ1 > 0, δ2 > 0, then
ω(t )
sup
t 0 −τ θ t 0
ω(θ) exp(δ1 +δ2 )(t −t0 ) , t t 0 .
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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N −l
Lemma 3. (See [8,24].) Assume that Q = (q i j ) N × N
˜ E−D is symmetric. Let D = diag(d1 , . . . , dl , 0, . . . , 0), Q − D = S
S Ql
, and d = min1i l {di },
where 1 l < N, di > 0, i = 1, . . . , l, Q l is the minor matrix of Q by removing its first l(1 l < N ) row-column pairs, E and S are matrices with ˜ = diag(d1 , . . . , dl ). When d > λmax ( E − S Q −1 S ), Q − D < 0 is then equivalent to Q l < 0. appropriate dimensions, D l
R+
m+1
¯ i for each fixed (t , μ, μ ¯ 1, . . . , μ ¯ m) : ¯ 1, . . . , μ ¯ i −1 , μ ¯ i +1 , . . . , μ ¯ m ), × R × · · · × R → R be nondecreasing in μ Lemma 4. (See [33].) Let F (t , μ, μ i = 1, . . . , m, and I k (μ) : R → R be nondecreasing in μ. Suppose that u (t ), v (t ) ∈ P C ([−τ , ∞), R ) satisfy
D+ u (t ) F t , u (t ), u t − τ1 (t ) , . . . , u t − τm (t ) , −
u (tk ) I k u tk
,
k∈
D+ v (t ) > F t , v (t ), v t
−
v (tk ) I k v tk
,
− τ1 (t ) , . . . , v t − τm (t ) ,
k∈
t 0,
Z +.
t 0,
Z +.
Then u (t ) v (t ), for −τ t 0 implies u (t ) v (t ), for t 0. 3. Main results In this section, two synchronization schemes are developed for a network with nonidentical time-delayed dynamical nodes, and some sufficient conditions ensuring global exponential stability of the synchronization process are derived. 3.1. Open-loop controlled network with nonidentical time-delayed dynamical nodes This Letter aims to design appropriate controllers to drive the network (1) with nonidentical time-delayed dynamical nodes to synchronize globally exponentially onto any smooth goal dynamics s(t ). Considering open-loop control (also called entrainment control) provides a general approach to entrain a dynamical system to arbitrary goal dynamics [29,30], here we also apply open-loop control to all nodes in network (1) to achieve synchronization between the network (1) and the given smooth dynamics s(t ). The controlled dynamical network can be characterized by
x˙ i (t ) = f i t , xi (t ), xi t − τi (t )
+c
N
b i j Γ x j (t ) + s˙ (t ) − f i t , s(t ), s t − τi (t ) ,
(2)
j =1
where i = 1, . . . , N. Unfortunately, it was pointed out that open-loop control imposes some restrictions on initial conditions and goal dynamics, and cannot guarantee the stability of the control process [29,30]. Hence, it is necessary to develop effective approaches to ensure the open-loop controlled network (2) can be globally exponentially synchronized. In [31], the authors improved the open-loop control method by combining it with linear feedback control or with impulsive control to globally entrain a dynamical system to arbitrary goal dynamics. Furthermore, the authors investigated how to synchronize a complex network with nonidentical nodes onto any smooth goal dynamics by combing open-loop and adaptive feedback control or combing open-loop and impulsive control [24]. Consequently, we also consider introducing the discontinuous control method, intermittent control or impulsive control, to the open-loop controlled network (2) to ensure global exponential stability of the synchronization process. 3.2. Synchronization scheme combining open-loop and intermittent control In this subsection, we will stabilize the open-loop controlled network (2) via periodically intermittent control. Some intermittent controllers are added to partial nodes of the controlled network (2). Without loss of generality, suppose the first l(1 l < N ) nodes to be selected and pinned, then we have the following controlled dynamical network:
⎧ N ⎪ ⎪ τ ⎪ ⎪ x˙ i (t ) = f i i (t , xi , s) + c b i j Γ x j (t ) + s˙ (t ) + di (t ) xi (t ) − s(t ) , ⎪ ⎪ ⎨ j =1
N ⎪ ⎪ ⎪ τ ⎪ xi (t ) = f i i (t , xi , s) + c b i j Γ x j (t ) + s˙ (t ), ⎪ ⎪ ⎩
1 i l, (3)
l + 1 i N,
j =1
τ
where f i i (t , xi , s) = f i (t , xi (t ), xi (t − τi (t ))) − f i (t , si (t ), si (t − τi (t ))), s(t ) is any smooth dynamics, and di (t ) is the intermittent feedback control gain defined as follows:
di (t ) =
−di , 0,
t ∈ [nT, nT + δ),
1 i l,
t ∈ nT + δ, (n + 1)T ,
1 i l,
(4)
where di > 0 is positive constant called control gain, T is the control period, δ > 0 is called the control width (control duration), and n = 0, 1, 2, . . . . Hereafter, denote ρmin as the minimum eigenvalue of the matrix (Γ + Γ )/2. Assume that ρmin = 0 and Γ = ρ > 0. Let Bˆ s = ( Bˆ + Bˆ )/2 where Bˆ is a modified matrix of B via replacing the diagonal elements b ii by (ρmin /ρ )b ii , then Bˆ s is a symmetric irreducible matrix with nonnegative off-diagonal elements [27]. Note that generally Bˆ does not possess the property of zero row sums. Moreover, there does not exist a definite relationship between the eigenvalues of B and those of Bˆ for the general matrix G [10,15]. In the
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following, we will show that how to design suitable δ , T and di , i = 1, . . . , l, such that the controlled dynamical network (3) can realize globally exponential synchronization. The main results are stated as follows. Theorem 1. Suppose that τ δ . Under Assumption 1, if there exist positive constants a1 , a2 , and di , i = 1, . . . , l, such that (i) (θ + 12 a1 ) I N + c ρ Bˆ s − D < 0, (ii) γ − a1 < 0, (iii) θ I N + c ρ Bˆ s − 12 a2 I N 0, (iv) = λ(δ − τ ) − (a2 + γ )(T − δ) > 0, N −l
where θ = max1i N {θi }, γ = 2 max1i N {γi }, D = diag(d1 , . . . , dl , 0, . . . , 0), and λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Then the controlled dynamical network (3) is globally exponentially synchronized. Proof. Define synchronization error as e i (t ) = xi (t ) − s(t ) (i = 1, . . . , N ), according to the control law (4), then we can derive the following error dynamical system:
⎧ N ⎪ ⎪ ⎪ e˙ i (t ) = f˜ i t , e i (t ), e i t − τi (t ) + c b i j Γ e j (t ) − di e i (t ), nT t < nT + δ, 1 i l, ⎪ ⎪ ⎪ ⎪ ⎪ j = 1 ⎪ ⎪ ⎪ ⎪ N ⎨ e˙ i (t ) = f˜ i t , e i (t ), e i t − τi (t ) + c b i j Γ e j (t ), nT t < nT + δ, l + 1 i N , ⎪ ⎪ j = 1 ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ˜ ⎪ ˙ e ( t ) = f t , e ( t ), e t − τ ( t ) + c b i j Γ e j (t ), nT + δ t < (n + 1)T, 1 i N . i i i i ⎪ ⎩ i
(5)
j =1
where f˜ i (t , e i (t ), e i (t − τi (t ))) = f i (t , e i (t ) + s(t ), e i (t − τi (t )) + s(t − τi (t ))) − f i (t , s(t ), s(t − τi (t ))). Clearly, if the zero solution of the error system (5) is globally exponentially stable, then the controlled network (3) is globally exponentially synchronized. Construct the following Lyapunov candidate function:
1 e i (t )e i (t ). 2 N
V (t ) =
(6)
i =1
Then the derivative of V (t ) with respect to time t along the solutions of Eq. (5) can be calculated as follows: When nT t < nT + δ , for n = 0, 1, 2, . . . ,
V˙ (t ) =
N
˜ e i (t ) f i t , e i (t ), e i t − τi (t )
+c
i =1
N
θi e i (t )e i (t ) +
N
N
γi e i t − τi (t ) e i t − τi (t ) + c
b ii ρmin e i (t )e i (t ) −
i =1
+c
γ 2
N
l
bi j e i (t )Γ e j (t )
di e i (t )e i (t )
e i t − τi (t ) e i t − τi (t ) + c
i =1
b ii ρmin e i (t )e i (t ) −
i =1 N
N
i =1
e i (t )e i (t ) +
N
di e i (t )e i (t )
i =1 j =1, j =i
i =1 N
l i =1
N
i =1
+c
θ
bi j e i (t )Γ e j (t ) −
i =1 j =1
i =1
θ
N N
N N
b i j ρ e i (t )e j (t )
i =1 j =1, j =i l
di e i (t )e i (t )
i =1
e i (t )e i (t ) +
i =1
γ
sup
N
2 t −τ st i =1
ˆ e i (s)e i (s) + c ρ e (t )( B ⊗ I n )e (t ) −
1 1 γ = e (t ) θ + a1 I N + c ρ Bˆ s − D ⊗ I n e (t ) − a1 e (t )e (t ) +
2
−a1 V (t ) + γ
2
sup
t −τ st
l
di e i (t )e i (t )
i =1
sup e (s)e (s) 2 t −τ st
V (s),
(7)
where e (t ) = (e 1 (t ), e 2 (t ), . . . , e N (t )) . From condition (ii) and the above inequality (7), we can get by Lemma 1
V (t )
sup
nT−τ φnT
V (φ) exp −λ(t − nT) ,
nT t < nT + δ,
(8)
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
2543
where λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Similarly, when nT + δ t < (n + 1)T, using condition (iii), one has
V˙ (t ) a2 V (t ) + γ
sup
t −τ st
V (s).
(9)
It follows from Lemma 2 that
V (t )
sup
nT+δ−τ φnT+δ
V (φ) exp (a2 + γ )(t − nT − δ) ,
nT + δ t < (n + 1)T.
(10)
Now, we estimate V (t ) based on Eqs. (8) and (10). By (8), noticing that δ τ and λ > 0, we can obtain the following:
V (δ) = τ where
sup
δ−τ sδ
V (s)
sup
V (φ) exp{−λt } =
sup
δ−τ sδ −τ φ0
sup
−τ φ0
V (φ) exp −λ(δ − τ ) =
sup
−τ φ0
V (φ) exp{− 1 }, (11)
1 = λ(δ − τ ). By (10), and noticing that (a2 + γ ) > 0, we have the following: V (T) = sup V (s) max sup V (s), sup V (s) τ T−τ sT
T−τ sδ
δsT
sup V (δ)τ exp (a2 + γ )(t − δ) = V (δ)τ exp (a2 + γ )(T − δ) = V (δ)τ exp{ 2 }, δsT
where
2 = (a2 + γ )(T − δ). Substituting (11) into (12) gives V (T) V (δ) exp{ 2 } sup V (φ) exp{− 1 + 2 }. τ τ
(12)
(13)
−τ φ0
Similarly, we have
V (2T) τ
sup
T−τ sT
V (s) exp{− 1 } exp{ 2 }
sup
−τ φ0
V (φ) exp −2( 1 − 2 ) .
(14)
By mathematical induction, we have
V (kT) τ
sup
−τ φ0
V (φ) exp −k( 1 − 2 ) ,
(15)
for any nonnegative integers k. Since for any t 0, there exists a nonnegative integers k, such that kT t < (k + 1)T, we can deduce the following estimation of V (t ) for any t 0 by (8), (10) and (15). For kT t < kT + δ ,
V (t )
sup
kT−τ skT
sup
−τ φ0
V (s) exp −λ(t − kT)
T
sup
kT+δ−τ skT+δ
sup
−τ φ0
−τ φ0
−τ φ0
V (φ) exp
V (φ) exp
δ
T
exp −
sup
−τ φ0
T
δ
exp −
T
−τ φ0
V (φ) exp −
T
sup
V (φ) exp −(k + 1)
sup
sup
V (φ) exp −k( 1 − 2 ) exp −λ(t − kT)
V (s) exp (a2 + γ )(t − kT − δ)
Therefore, for any t > 0, we have
V (t )
−τ φ0
t−δ V (φ) exp −
=
Similarly, for kT + δ t < (k + 1)T,
V (t )
sup
T
sup
−τ φ0
V (φ) exp{−k } (16)
t .
V (φ) exp −k( 1 − 2 ) exp{− 1 } exp{ 2 }
t .
(17)
t ,
t 0.
(18)
It follows from condition (iv) that the zero solution of the error dynamical system (5) is globally exponentially stable. The proof is thus completed. 2 Let Q = (θ + 12 a1 ) I N + c ρ Bˆ s , and (θ + 12 a1 ) I N + c ρ Bˆ s − D = Q − D =
˜ S E−D S Q l
where Q l is the minor matrix of Q by removing
˜ = diag(d1 , . . . , dl ). Let d = min1i l {di }. By its first l(1 l < N ) row-column pairs, E and S are matrices with appropriate dimensions, D Lemma 3, we know that Q − D < 0 is equivalent to Q l < 0 and d > λmax ( E − S Q l−1 S ). If d can be sufficiently large (one can simply
select di > λmax ( E − S Q l−1 S ), i = 1, . . . , l), then Q − D < 0 is equivalent to Q l < 0. Note that Q l = (θ + 12 a1 ) I N −l + c ρ Bˆ ls , where Bˆ ls is the
minor matrix of Bˆ s by removing its first l(1 l < N ) row-column pairs. Thus we can derive the following corollary.
Corollary 1. Suppose that τ δ . Under Assumption 1, if the control gain di can be sufficiently large and there exist positive constants a1 and a2 , such that
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
(i) (θ + 12 a1 ) + c ρ λmax ( Bˆ ls ) < 0, (ii) γ − a1 < 0, (iii) θ I N + c ρ Bˆ s − 12 a2 I N 0, (iv) = λ(δ − τ ) − (a2 + γ )(T − δ) > 0, where θ = max1i N {θi },
γ = 2 max1iN {γi }, and λ > 0 is the unique positive solution of the equation −a1 + λ + γ exp{λτ } = 0. Then the
controlled dynamical network (3) is globally exponentially synchronized. One can simply select di > λmax ( E − S Q l−1 S ), i = 1, . . . , l.
Remark 2. In [8], pinning synchronization of directed and undirected complex dynamical networks were investigated, and it was pointed out that the nodes whose out-degrees are bigger than their in-degrees should be specifically selected as pinned candidates. For the controlled network (3), we can first apply intermittent control to the nodes with zero in-degrees, and then continue to try others nodes in descending order based on their degree-difference defined as the difference between the out-degree of the nodes and the in-degree of the nodes (for those with the same degree-difference, in descending order according to their out-degrees). Moreover, from condition (i) in Corollary 1, we can see that the number of nodes should be selected and pinned to synchronize the controlled network (3) is at least l0 , a1 a1 where l0 satisfies λmax ( Bˆ ls −1 ) − 2θ+ and λmax ( Bˆ ls ) < − 2θ+ [8]. 2c ρ 2c ρ 0
0
3.3. Synchronization scheme combining open-loop and impulsive control In this subsection, impulsive control strategy will be adopted to stabilize the open-loop controlled network (2). Applying impulsive control to the open-loop controlled network (2), we have the following synchronization scheme for the complex network with nonidentical time-delayed dynamical nodes:
⎧ N ⎪ ⎪ ⎪ ˙ x ( t ) = f t , x ( t ), x t − τ ( t ) + c b i j Γ x j (t ) + s˙ (t ) − f i t , si (t ), si t − τi (t ) , ⎪ i i i i ⎨ i
t = tk ,
j =1
(19)
⎪ ⎪ x = xi tk+ − xi tk− = Ak xi tk− − s tk− , t = tk , k ∈ Z + , ⎪ ⎪ ⎩ i xi (s) = ϕi (s), −τ s 0, i = 1, . . . , N , t 0.
+ − + where the time sequence {tk }k+∞ =1 satisfy tk−1 < tk and limk→∞ tk = +∞, xi = xi (tk ) − xi (tk ) is the control law in which xi (tk ) = − n×n limt →t + xi (t ) and xi (tk ) = limt →t − xi (t ), A k ∈ R is the control gain matrix. Without loss of generality, we assume that limt →t + xi (t ) = k
k
k
xi (tk ), which means the solution xi (t ) is continuous from the right. The initial conditions of Eq. (19) are given by xi (t ) = ϕi (t ) ∈ P C ([−τ , 0], R n ). For convenience, we now make some assumptions about the time-delays set {τ1 (t ), . . . , τ N (t )}. Suppose that the time-delays set {τ1 (t ), . . . , τ N (t )} contains m0 different types time-delays, and T i = {li −1 + 1, . . . , li }, i ∈ {1, . . . , m0 } denotes the index set of the nodes m with the same time-delays, where l0 = 0, lm0 = N, and li −1 < li . Let γ T i = 2 max j ∈ T i {γ j } and γˆ = i =01 γ T i , then we have the following result: Theorem 2. Let Ω = supk∈ Z + {tk − tk−1 } < ∞. Under Assumption 1, if there exist positive constants a3 and d0 < 1 such that (i) θ I N + c ρ Bˆ s − 12 a3 I N 0, (ii) ( I n + A k ) ( I n + A k ) − d0 I n 0, k ∈ Z + , (iii)
ln d0
Ω
ˆ + a3 + dγ0 < 0,
then the controlled dynamical network (19) is globally exponentially synchronized. Proof. Let v i (t ) = xi (t ) − s(t ), i = 1, . . . , N. Considering that s(t ) is assumed to be smooth, we get s(tk ) = s(tk+ ) − s(tk− ) = 0. Then from (19), the error dynamical system is governed as follows:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
v˙ i (t ) = f˜ i t , v i (t ), v i t − τi (t )
−
v i (tk ) = ( I n + A k ) v i tk ,
+c
N
b i j Γ v j (t ),
t = tk ,
j =1
(20)
+
t = tk , k ∈ Z .
Let v (t ) = ( v 1 (t ), . . . , v N (t )) , consider the following Lyapunov function:
1 v i (t ) v i (t ). 2 N
V (t ) =
(21)
i =1
Taking the upper Dini derivative of V (t ) along the solution of Eq. (20), we get +
D V (t ) =
N i =1
N ˜ v i (t ) f i t , v i (t ), v i t − τi (t ) + c b i j Γ v j (t )
j =1
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
N
θi v i (t ) v i (t ) +
i =1
N
2545
ˆ γi v i t − τi (t ) v i t − τi (t ) + c ρ v (t )( B ⊗ I n ) v (t )
i =1
m0 v (t ) θ I N + c ρ Bˆ s ⊗ I n v (t ) + γ j v j t − τi (t ) v j t − τi (t ) i =1 j ∈ T i m0 γ Ti
a3 V (t ) +
2
i =1
v j t − τi (t ) v j t − τi (t ) a3 V (t ) +
j∈T i
m0
γ T i V t − τi (t ) , t = tk , k ∈ Z + .
(22)
i =1
When t = tk , k ∈ Z + , we have
V (tk ) =
1 1 − v i (tk ) v i (tk ) = v i tk ( I n + A k ) ( I n + A k ) v i tk− d0 V tk− . 2 2 N
N
i =1
i =1
(23)
> 0, let μ (t ) be an unique solution of the following impulsive delayed dynamical system ⎧ m0 ⎪ ⎪ ⎪ ˙ μ ( t ) = a μ ( t ) + γ T i μ t − τi (t ) + , t = tk , t 0, ⎪ 3 ⎪ ⎪ ⎨ i =1 − μ (tk ) = d0 μ tk , k ∈ Z + , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ (t ) = sup V (s), −τ t 0.
For any
(24)
−τ s0
−1
Let M 1 = d0 sup−τ s0 V (s), and
η = −(ln d0 )/Ω − a3 . In the following, we will prove that condition (iii) implies
μ (t ) < M 1 exp{−λ1t } +
t 0,
(25)
γ T i exp{λ1 τi } = 0.
(26)
1 ˆ (η − d− 0 γ )d0
,
where λ1 > 0 is an unique solution of 1 λ1 + (ln d0 )/Ω + a3 + d− 0
m0 i =1
m0
Denote H (λ1 ) = λ1 − η + d0−1
i =1
dH (λ1 ) dλ1
m0
γ T i exp{λ1 τi }. From condition (iii), we have η − d0−1 γˆ > 0, d0−1 (
i =1
γ T i ) > 0, and so H (0) < 0,
H (+∞) > 0 and > 0. Using the continuity and monotonicity of H (λ1 ), the equality (26) has an unique positive solution λ1 > 0. By the formula for the variation of parameters [32], we have
t
μ (t ) = W (t , 0)μ (0) +
W (t , s)
m0
γ μ s − τi (s) + ds,
Ti
t 0,
(27)
i =1
0
where W (t , s), t , s 0 is the Cauchy matrix of linear system
˙ t ) = a3 ψ(t ), t = tk , ψ( ψ(tk ) = d0 ψ tk− , k ∈ Z + .
(28)
According to the representation of the Cauchy matrix [32], we get the estimation of W (t , s) since 0 < d0 < 1 and tk − tk−1 Ω ,
W (t , s) =
[(t −s)/Ω−1]
d0 exp a3 (t − s) d0 s
1 = d− 0 exp
exp
−η −
ln d0
Ω
(t − s)
ln d0 1 (t − s) = d− (t − s) exp −η − 0 exp −η (t − s) , Ω Ω
ln d0
t s 0.
(29)
Substituting (29) into (27) gives −1
μ (t ) d0
sup V (s) exp{−ηt } +
−τ s0
t
−1
d0 0
t = M 1 exp{−ηt } +
1 exp −η(t − s) d− 0
0
−1
Since , λ1 , η − d0 γˆ > 0, and 0 < d0 < 1, we have
m 0 Ti exp −η(t − s) γ μ s − τi (s) + ds
i =1
m 0 i =1
γ μ s − τi (s) + ds, Ti
t 0.
(30)
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
μ (t ) sup V (s)/d0 < M 1 exp{−λ1t } + −τ s0
1 ˆ (η − d− 0 γ )d0
−τ t 0.
,
(31)
Now, we show that (25) is true, that is,
μ (t ) < M 1 exp{−λ1t } + If this is not true, from (31) and
1 ˆ (η − d− 0 γ )d0
(32)
μ (t ) ∈ P C ([−τ , ∞), R + ), then there must exists a t ∗ > 0 satisfying
μ t ∗ M 1 exp −λ1t ∗ + μ (t ) < M 1 exp{−λ1t } +
t 0.
,
1 ˆ (η − d− 0 γ )d0
1 ˆ (η − d− 0 γ )d0
,
(33)
−τ t < t ∗ .
(34)
,
By (26), (30), (34), we have
∗
μ t
M 1 exp −ηt
∗
t ∗ +
∗
−1
exp −η t − s d0
exp −ηt
t ∗ M1 +
−1
exp{η s} d0 M 1
m0
exp −ηt
γ
−1
M 1 + d0 M 1
m0
< M 1 exp −λ1 t
∗
+
exp −λ1 s − τi (s) +
t ∗ Ti
γ exp{λ1 τi }
i =1
Ti
i =1
0
∗
γ μ s − τi (s) + ds
Ti
i =1
0
∗
m0
exp (η − λ1 )s ds + 0
1 ˆ (η − d− 0 γ )d0
1 γˆ d− 1 0 + d− 0 ds 1 ˆ (η − d− γ ) d 0 0
η 1 ˆ (η − d− 0 γ )d0
!
!
t ∗ exp{η s} ds 0
(35)
. m
This contradicts (33), and so (25) holds. Denote F (t , V , V¯ 1 , . . . , V¯ m0 ) = a3 V + i =01 γ T i V¯ i , and I k ( V ) = d0 V . Then the function F and I k satisfy the monotonicity given in Lemma 4. Since V (t ) sup−τ s0 V (s) = μ (t ), for −τ t 0, it follows from Lemma 4 that
V (t ) μ (t ) < M 1 exp{−λ1 t } + Letting
1 ˆ (η − d− 0 γ )d0
,
t 0.
ε → 0+ , then we have V (t ) M 1 exp{−λ1 t }, t 0. This completes the proof of Theorem 2. 2
For simplicity, we consider the equidistant impulsive interval tk − tk−1 = t, and the impulsive control gain matrix A k ≡ a0 I n , k = 1, 2, . . . . Then, the following result can be obtained readily from Theorem 2. Corollary 2. Under Assumption 1, if the following condition holds
t < −
2(1 + a0 )2 ln |1 + a0 | 2λmax (θ I N + c ρ Bˆ s )(1 + a0 )2 + γˆ
,
−2 < a0 < 0,
(36)
then the controlled dynamical network (19) is globally exponentially synchronized. 4. Numerical simulations In this section, numerical simulations are given to verify the effectiveness of the proposed synchronization schemes for synchronizing a complex network with time-delayed dynamical nodes onto a chaotic trajectory and a periodic orbit. Consider a complex dynamical network consisting of two types nonidentical time-delayed chaotic nodes, which are described by timedelayed Lorenz system and time-delayed Chua oscillator. The network is given by
⎧ 200 ⎪ ⎪ ⎪ ⎪ x˙ i (t ) = f 1 t , xi (t ), xi t − τ1 (t ) + c b i j Γ x j (t ), ⎪ ⎪ ⎨ j =1 200 ⎪ ⎪ ⎪ ⎪ ˙ x ( t ) = f t , x ( t ), x t − τ ( t ) + c b i j Γ x j (t ), ⎪ 2 2 i i ⎪ ⎩ i
i = 1, . . . , 100, (37) i = 101, . . . , 200,
j =1
where Γ = diag(1, 1.1, 1), the coupling strength c = 35, B = (b i j )200×200 is a symmetrically diffusive coupling matrix with g i j = 0 or 1 ( j = i ). Here we assume that the network structure of Eq. (37) obeys the scale-free distribution of the BA model [34]. The parameters of the BA model are given by m0 = m = 5, N = 200. Moreover, the first 100 nodes and the other 100 nodes are described, respectively, by time-delayed Lorenz system and time-delayed Chua oscillator [35,36]:
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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Fig. 1. (Left) Chaotic behavior of the time-delayed Lorenz system (38); (Right) chaotic behavior of the time-delayed Chua oscillator (39).
= C 10 xi (t ) + g 11 xi (t ) + g 12 xi t − τ1 (t ) , f 2 t , xi (t ), xi t − τ2 (t ) = C 20 xi (t ) + g 21 xi (t ) + g 22 xi t − τ2 (t ) ,
f 1 t , xi (t ), xi t − τ1 (t )
(38) (39)
where xi (t ) = (xi1 (t ), xi2 (t ), xi3 (t )) , g 11 (xi ) = (0, −xi1 xi3 , xi1 xi2 ) , g 12 (xi ) = (0, σ0 xi2 , 0) , g 21 (xi ) = (− 12 α0 (m1 − m2 )(|xi1 + 1| − |xi1 − 1|), 0, 0) , g 22 (xi ) = (0, 0, −β0 0 sin( v 0 xi1 )) ,
⎡
−a0
a0
C 10 = ⎣ r0
σ0 − 1
0
0
⎤
0 ⎦,
⎡ C 20 = ⎣
−b0
0
⎤
−α0 (1 + m2 )
α0
0
1
−1
1 ⎦,
0
−β0 −ω0
(40)
and a0 = 10, b0 = 8/3, r0 = 28, σ0 = 5, α0 = 10, β0 = 19.53, ω0 = 0.1636, m1 = −1.4325, m2 = −0.7831, v 0 = 0.5, 0 = 0.2, τ1 (t ) = 0.1, and τ2 (t ) = 0.02. Fig. 1 shows the chaotic behaviors of the time-delayed Lorenz system (38) and the time-delayed Chua oscillator (39). Consider a smooth dynamics s(t ) = (s1 , s2 , s3 ) , which we aim to synchronize the network (37) onto, satisfies |si | M i0 , where M i0 > 0, i = 1, 2, 3. Let e i (t ) = xi (t ) − s(t ), using the inequality |xy | 12 (x2 + y 2 ), then it is easy to verify [15,24]
e i (t ) f 1 t , xi (t ), xi t − τ1 (t )
− f 1 t , s(t ), s t − τ1 (t )
= e i (t )C 10 e i (t ) + s2 e i1 (t )e i3 (t ) − s3 e i1 (t )e i2 (t ) + σ0 e i2 (t )e i2 t − τ1 (t ) −10 + κ1 38 + M 30 /2 + κ2 M 20 /2 e 2i1 (t ) + 4 + 38 + M 30 /(2κ1 ) + 5κ3 /2 e 2i2 (t ) + M 20 /(2κ2 ) − 8/3 e 2i3 (t ) + 5/(2κ3 )e 2i2 t − τ1 (t ) i = 1, . . . , 100, θ1 e i (t )e i (t ) + γ1 e i t − τ1 (t ) e i t − τ1 (t ) ,
and
− f 2 t , s(t ), s t − τ2 (t ) 2 1 e i (t ) C 20 + C 20 e i (t ) + α0 (m1 − m2 ) e i1 (t ) + β0 0 v 0 e i3 (t )e i1 t − τ2 (t ) 2 2 λmax (C˜ 20 )e i (t )e i (t ) + β0 0 v 0 /(2κ4 )e i1 t − τ2 (t ) i = 101, . . . , 200, θ2 e i (t )e i (t ) + γ2 e i t − τ2 (t ) e i t − τ2 (t ) ,
e i (t ) f 2 t , xi (t ), xi t − τ2 (t )
C 20 +C 20 where C˜ 20 = ( + diag(|α0 (m1 − m2 )|, 0, κ4 β0 20 v 0 )), and θ1 , θ2 , 2 κ1 , κ2 , κ3 , κ4 > 0.
(41)
γ1 , γ2 can be determined by choosing appropriate parameters
4.1. Synchronizing network onto a chaotic trajectory via open-loop and intermittent control To demonstrate the effectiveness of Corollary 1, we use the synchronization scheme combining open-loop and intermittent control to achieve synchronization between the network (37) and the solution of the following Rössler system
s˙ 1 (t ) = −(s2 + s3 ),
s˙ 2 (t ) = s1 + 0.2s2 ,
s˙ 3 (t ) = s3 (s1 − 5.7) + 0.2.
(42)
It is well known that the Rössler system exhibits chaotic behavior and its state is bounded. Simulation result shows that there exist constants M 10 = 11.3236, M 20 = 10.6547, M 3 = 21.8865 satisfying |si | M i0 , i = 1, 2, 3 [24,31]. Letting κ1 = 1.26, κ2 = 0.17, κ3 = 0.36, κ4 = 1, one has θ1 = 28.6709, γ1 = 6.9444, θ2 = 11.3931, and γ2 = 0.9765 based on (41). It follows that θ = maxi=1,2 {θi } = 28.6709, and γ = maxi=1,2 {γi } = 6.9444. In addition, the values of the parameters for the intermittent controllers are taken as T = 2.2, δ = 2.0. Since λmax ( Bˆ s ) = 0.9130, ρ = Γ = 1.1, then if we choose a1 = 43.5, a2 = 127.65, it is easy to verify that conditions (ii)–(iv) in Corollary 1 are satisfied. Consequently, condition (i) in Corollary 1 becomes
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S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
Fig. 2. λmax ( Bˆ ls ) versus the number of pinned nodes (l).
Fig. 3. Time evolution of synchronization error via open-loop and adaptive intermittent control.
2θ + a 1 λmax Bˆ ls < − = −1.3096.
(43)
2c ρ
Using the pinned-node selection scheme for the controlled network (3) in Remark 2, we rearrange network nodes. Choose l from 1 to 100, and depict λmax ( Bˆ ls ) in Fig. 2, which reveals that λmax ( Bˆ ls ) decreases with the increase of the number of pinned nodes (l). Especially,
s s when l = 18 and l = 19, we have λmax ( Bˆ 18 ) = −1.3050 and λmax ( Bˆ 19 ) = −1.3487. Therefore, we can apply open-loop control to all nodes and choose the first 19 rearranged nodes of the network (37) as pinned nodes to achieve synchronization between the network (37) and the solution s(t ) of Rössler system (42). In addition, in Corollary 1, the control gain is required to be large enough [di > λmax ( E − S Q l−1 S ), i = 1, . . . , l], but it may be much larger than the needed values. So we here adopt the adaptive control approach [9,10], and apply the following adaptive intermittent feedback control gain [15]:
di (t ) =
(t )e i (t ), −di , d˙ i = qi e i 0,
di (0) = 0, qi > 0, t ∈ [nT, nT + δ), 1 i 19,
t ∈ [nT + δ, (n + 1)T),
1 i 19.
(44)
Though we have not found a rigorous proof for the synchronization scheme with adaptive control gain (44), the method is very useful in this example [15]. The initial conditions of the numerical simulations are as follows: xi (0) = (−8 + 0.5i , −5 + 0.5i , −10 + 0.5i ) , s(0) = (1, 2, 3) , where 1 i 200, and q1 = · · · = q19 = 0.4. The synchronous errors e i (t ) are illustrated in Fig. 3, and the values for the control gains after synchronization satisfy di 114.71, 1 i 19, which illustrate the adaptive control approach can obtain a more applicable control gain.
S. Cai et al. / Physics Letters A 374 (2010) 2539–2550
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Fig. 4. Time evolution of synchronization error via open-loop and impulsive control.
4.2. Controlling network onto a period orbit via open-loop and impulsive control Now, we use the synchronization scheme (19) which combines open-loop and impulsive control to achieve synchronization between the network (37) and a periodic orbit given by
s(t ) =
π0 cos(ωt ), 1.5π0 sin(1.5ωt ), 2π0 cos(2ωt ) ,
(45)
where π0 = 1, ω = 2. Obviously, the bounds of s(t ) are M 10 = 1, M 20 = 1.5, M 30 = 2, and s˙ (t ) = (−π0 ω sin(ωt ), 2.25π0 ω cos(1.5ωt ), −4π0 ω sin(2ωt )) . Choosing κ1 = 1.6, κ2 = 0.03, κ3 = 2.5, κ4 = 19, one has θ1 = 22.75, γ1 = 1, θ2 = 22.3507, and γ2 = 0.0514 based on (41). It follows that θ = 22.75, and γˆ = 2.103. By Corollary 2, we can derive the following condition ensuring the synchronization between the network (37) and the periodic orbit (45)
t < −
2(1 + a0 )2 ln |1 + a0 | 2λmax (θ I N + c ρ Bˆ s )(1 + a0 )2 + γˆ
=−
2(1 + a0 )2 ln |1 + a0 | 115.80(1 + a0 )2 + 2.103
,
−2 < a0 < 0.
Let the impulsive control gain a0 = −1.20 and the impulsive interval t = 0.015. Fig. 4 shows the evolution of the synchronization errors. We can clearly see that the network (37) is quickly synchronized onto the periodic orbit (45). 5. Conclusions In this Letter, the synchronization issue of a general complex network with nonidentical time-delayed dynamical nodes is discussed. Two control schemes are proposed to synchronize such dynamical network onto any smooth goal dynamics by combining open-loop control with local intermittent feedback control and impulsive control, respectively. Some sufficient conditions ensuring the global exponential stability of the synchronization process are derived. Both theoretical and numerical analysis illustrate the effectiveness of the proposed control methodologies. Acknowledgements We thank the reviewers for their helpful suggestions on the manuscript. This work was supported by the National Science Foundation of China (Grant Nos. 10832006, 10802043), and key disciplines of Shanghai Municipality (S30104), Shanghai Academic Discipline Project (J50101). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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