Nonlinear Dyn (2014) 77:481–489 DOI 10.1007/s11071-014-1311-7
ORIGINAL PAPER
Generalized outer synchronization between two uncertain dynamical networks Weigang Sun · Shixing Li
Received: 12 August 2013 / Accepted: 12 February 2014 / Published online: 2 March 2014 © Springer Science+Business Media Dordrecht 2014
Abstract This paper investigates generalized outer synchronization between two uncertain dynamical networks with a novel feature that the couplings of each network are unknown functions. With nonlinear control schemes, two sufficient criteria for generalized outer synchronization with or without time delay are obtained by Lyapunov stability theory and Barbalat’s lemma. Our results are valid for many studies of the couplings inside each network being linear or nonlinear. Finally, numerical simulations are given to verify the effectiveness of the control schemes. Keywords Complex dynamical network · Outer synchronization · Uncertain network
1 Introduction Synchronization and its control inside a network have been widely studied due to their great applications in many fields such as physics, biology, and engineerW. Sun (B) Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou 310018, China e-mail:
[email protected] S. Li School of Mathematics and Statistics, Zhejiang University of Finance & Economics, Hangzhou 310018, China
ing [1]. The main reason is the born of small-world and scale-free networks [2,3]. Node dynamics and network connected topology are two ingredients constituting complex dynamical networks, in which the issues focus on the interplay between the complexity in the overall topology and the local dynamical properties of the coupled nodes. Given a complex dynamical network, we refer to network synchronization as “inner synchronization” when synchronization happens inside a network. Generally speaking, the methodology on studying the inner synchronization is decoupling the networked systems, and studying the low dimensional systems by the master stability function, or using the linear matrix inequality toolbox [4]. Synchronization may not happen inside a network with unappropriate topological structures and node dynamics. In that case, some controlling (e.g., the pinning, adaptive, and impulsive control) methods are explored to realize synchronization [5–7] and many references cited therein. Further, for an uncertain network (with some unknown information), the inner synchronization has attracted considerable attention with the aim of identifying the uncertain parameters and topological connections of a network, which is helpful for designing a good network. In Li and Chen [8], pioneered in considering the robust adaptive synchronization inside a network with unknown couplings but bounded nonlinear functions. Afterwards, synchronization inside an uncertain network with delayed couplings has been investigated in the literature [9–14]. In addition, the adaptive and impulsive control methods
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are employed to achieve synchronization in an uncertain network [15–17]. Different from the concept of inner synchronization occurring inside a network, “outer synchronization” is referred as synchronization happening between two or more coupled networks as a generalization of synchronization between two chaotic systems [18]. In [19], we first theoretically and numerically demonstrated the outer synchronization between two unidirectionally networks with identical topological structures. Recently, robust global outer synchronization between two networks was studied in [20]. For different topological structures between two networks, Tang et al. [21] realized the outer synchronization by the adaptive control method. To deal with the different dimensions of node dynamics between two networks, Wu et al. [22,23] achieved generalized outer synchronization by designing the controllers or using open-plusclosed-loop (OPCL) method. The effect of noise on the outer synchronization was investigated in [24,25]. Besides, Sun et al. [26] studied mixed inner-outer synchronization between two networks with time delay. In Asheghan et al. [27], studied the outer synchronization between two networks with fractional-order dynamics using OPCL method. In the above-mentioned studies, the node dynamics and connected topologies between two networks are given. Recently, Wu and Lu [28] studied the outer synchronization between two uncertain networks with an unknown parameter vector of node dynamics, while the network connections within each network are known. To the best of our knowledge, few theoretical results involve the generalized outer synchronization between two networks with unknown topological connections. Inspired by the above discussions, we first propose a general network model between two uncertain networks with or without time delay under the appropriate controllers, and investigate generalized outer synchronization between them. By the adaptive control technique, two simple and effective control schemes are designed to guarantee generalized outer synchronization between two uncertain networks. We then obtain two theorems on the generalized outer synchronization. The unknown functions inside each network hold for the linear and nonlinear couplings. Compared to complete outer synchronization, the generalized outer synchronization between two uncertain networks may have a much wider application range in the engineering field. In addition, our method can be applied to study
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generalized outer synchronization with some uncertain parameters in network models. The rest of this paper is organized as follows. Two uncertain network models with both nondelay and delay are introduced and some necessary assumptions and Barbalat’s lemma are provided in Sect. 2. Two criteria on generalized outer synchronization with adaptive controllers are obtained in Sect. 3. Section 4 gives numerical examples to verify the effectiveness of the proposed controllers. Finally, conclusions are included in Sect. 5. Notations Throughout this paper, some necessary notations are first introduced. T denotes the transpose of a matrix or a vector. In is an identity matrix of size n. ξ indicates the 2-norm of a vector ξ , i.e., ξ 2 = ξ T ξ . The notation A > 0(A < 0) means that the matrix A is positive (negative) definite, that is, the eigenvalues of A are positive (negative).
2 Model presentation and preliminaries Based on the dynamical models inside a network, the coupled equations between two networks can be expressed as follows: x˙i (t) = f 1 (xi (t)) +
N1
ai j 1 x j (t) + CY X (Y, X ),
j=1
i = 1, 2, . . . , N1 , y˙i (t) = f 2 (yi (t)) +
N2
(1) bi j 2 y j (t) + C X Y (X, Y ),
j=1
i = 1, 2, . . . , N2 , where xi = (xi1 , xi2 , . . . , xin 1 )T ∈ R n 1 is an n 1 dimensional state vector and yi ∈ R n 2 is defined same. The node dynamical equations are x˙i (t) = f 1 (xi (t)), i = 1, . . . , N1 and y˙i (t) = f 2 (yi (t)), i = 1, . . . , N2 , respectively. N1 (N2 ) is the total number of the networks X (Y ). The functions f 1 (·) : R n 1 → R n 1 , f 2 (·) : R n 2 → R n 2 are continuously differential. The matrices A = (ai j ) N1 ×N1 and B = (bi j ) N2 ×N2 denote the coupling configurations of both networks, whose entries ai j (bi j ) are defined as follows: if there is a link between node i and node j ( j = i), then set ai j (bi j ) > 0, otherwise ai j (bi j ) = 0 ( j = i); the matrices A and B can be symmetric or asymmetric, but satisfying the sum of each row being zero.
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1 ∈ R n 1 ×n 1 , 2 ∈ R n 2 ×n 2 are the inner-coupling matrices. The goal in this paper is to study generalized outer synchronization between two uncertain networks. According to the original definition of the outer synchronization proposed in [19], we need N1 = N2 = N . The term C X Y (X, Y )(CY X (Y, X )) often represents the interactions from network X (Y ) to network Y (X ). For i) example, in [19], C X Y (X, Y ) = (H − ∂ f∂(x xi )(yi (t) − xi (t)), i = 1, 2, . . . , N , and CY X (X, Y ) = 0, where H is a constant Hurwitz matrix. If the coupling topologies inside each network are unknown functions, then the systems (1) become x˙i (t) = f 1 (xi (t)) + h i (x1 , . . . , x N ) + CY X (Y, X ), y˙i (t) = f 2 (yi (t)) + gi (y1 , . . . , y N ) + C X Y (X, Y ), (2)
where h i : R n 1 ×N → R n 1 , gi : R n 2 ×N → R n 2 . In reality, the interactions between two networks are colorful, such as through the nonlinear signals and mutual connections [29]. When the communicated information between them is not so sufficient that outer synchronization can be easily achieved, the controllers u i ∈ R n 2 are added, then Eq. (2) reads as x˙i (t) = f 1 (xi (t)) + h i (x1 , . . . , x N ),
(y − x)T ( f (y, t) − f (x, t)) ≤ θ (y − x)T P(y − x), where P is a positive definite matrix. Here x and y are time-varying vectors. Assumption 2 There exist nonnegative constants γi j (i, j = 1, 2, . . . , N ) such that gi (y1 , . . . , y N ) − gi (φ1 (x1 ), . . . , φ N (x N )) = g¯i (y1 , . . . , y N , φ1 (x1 ), . . . , φ N (x N )) ≤
N
γi j y j (t) − φ j (x j (t)).
j=1
i = 1, 2, . . . , N , i = 1, 2, . . . , N ,
Assumption 1 For any x, y ∈ R n , there exists a positive constant θ such that
(3)
i = 1, 2, . . . , N ,
Lemma 1 (Barbalat Lemma [30]). If afunction f (t) t is uniformly continuous, and limt→∞ 0 | f (s)|ds is bounded, then f (t) → 0 when t → ∞.
3 Controllers designed for generalized outer synchronization 3.1 Generalized outer synchronization of networks (3) With the network model (3) and the definition (5) previously, we obtain the following theorem. Theorem 1 Suppose that Assumptions 1 and 2 hold. Generalized outer synchronization between networks (3) can be achieved under the following control laws:
y˙i (t) = f 2 (yi (t)) + gi (y1 , . . . , y N ) + u i , i = 1, 2, . . . , N . In addition, there exist a lot of studies [9–14] in an uncertain network with delayed couplings, the systems (3) are rewritten as follows,
u i = Dφi (xi ) f 1 (xi ) − f 2 (φi (xi )) − di ei − gi (φ1 (x1 ), . . . , φ N (x N )) + Dφi (xi )h i (x1 , . . . , x N ), i = 1, 2, . . . , N ,
x˙i (t) = f 1 (xi (t)) + h i (x1 (t − τ ), . . . , x N (t − τ )),
(6)
i = 1, 2, . . . , N , y˙i (t) = f 2 (yi (t))+gi (y1 (t −τ ), . . . , y N (t −τ ))+u i , i = 1, 2, . . . , N ,
(4)
where τ is time delay. Definition 1 Let φi : R n 1 → R n 2 (i = 1, 2, . . . , N ) be continuously differentiable vector maps. Generalized outer synchronization between networks (3) or networks (4) is achieved if lim yi (t) − φi (xi (t)) = 0, i = 1, 2, . . . , N .
t→∞
(5)
where Dφi (xi ) are the Jacobian matrices of the maps φi (xi ), ei = yi − φi (xi ), d˙i = ki eiT ei , ki > 0, i = 1, 2, . . . , N , are positive constants. Proof By the control laws (6), we obtain the synchronization error of networks (3), which is described by e˙i = y˙i − Dφi (xi ) · x˙i = f 2 (yi ) + gi (y1 , . . . , y N ) + u i − Dφi (xi )( f 1 (xi ) + h i (x1 , . . . , x N )) = f 2 (yi ) + gi (y1 , . . . , y N ) + Dφi (xi ) f 1 (xi ) − f 2 (φi (xi )) − di ei − gi (φ1 (x1 ), . . . , φ N (x N ))
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+Dφi (xi )h i (x1 , . . . , x N ) − Dφi (xi )( f 1 (xi )
Further, t
+h i (x1 , . . . , x N )) = f 2 (yi ) − f 2 (φi (xi )) + g¯i (t) − di ei .
t λmin (M)e(s) ds ≤ − 2
(7) 0
0
−V (t) ≤ V (0) < +∞.
Choose a Lyapunov candidate function as follows: V (t) =
1 T 1 (di − dˆi )2 ei ei + , 2 2 ki N
N
i=1
i=1
where dˆi (i = 1, 2, . . . , N ) are sufficiently large positive constants. Calculating its derivative along the trajectories of (7) gives V˙ (t) =
N
e˙iT ei +
i=1
=
N
N (di − dˆi )d˙i ki i=1
Using Barbalat’s lemma, we have limt→∞ e(t)2 = 0, i.e., limt→∞ ei (t) = 0, i = 1, 2, . . . , N , which means that generalized outer synchronization between networks (3) is realized. Theorem 1 is proved completely. Remark 1 When the couplings of networks X and N Y are linear, that is, h i = j=1 ai j 1 x j , gi = N j=1 bi j 2 y j . For this widely studied model [19–23], we derive Assumption 2 as follows: gi (y1 , . . . , y N ) − gi (φ1 (x1 ), . . . , φ N (x N ))
eiT [ f 2 (yi ) − f 2 (φi (xi )) + g¯ i (t) − di ei ]
i=1
+
=
N (di − dˆi )eiT ei
N
≤
θ eiT Pei −
N N
N
di eiT ei
i=1
γi j ei T e j +
i=1 j=1
N (di − dˆi )eiT ei i=1
ˆ ≤ e T (t)(θ P + ϒ − D)e(t) = e T (t)Qe(t), where e(t) = (e1 (t), e2 (t), . . . , e N (t))T , , Dˆ = diag{dˆ1 , dˆ2 , . . . , dˆN } and Q = ϒ = (γi j ) N ×N ϒ+ϒ T θP + − Dˆ .
N
bi j 2 φ j (x j )
j=1
|bi j |2 y j − φ j (x j ) =
N
V˙ (t) ≤ e T (t)Qe(t) = −e T (t)Me(t) ≤ 0 and 0 ≤ λmin (M)e(t)2 ≤ e T (t)Me(t) ≤ −V˙ (t), where λmin (M) is the minimal eigenvalue of the positive definite matrix M.
γi j e j ,
j=1
where γi j = |bi j |2 . Then, Assumption 2 is valid for the linearly coupled dynamical networks. In addition, Assumption 2 also holds for some nonlinear and nonuniform coupled systems. For example, N N a q(x (t)), and g = take h i = j i j=1 i j j=1 bi j q(y j (t)). If the nonlinear function q(·) satisfies the Lipschitz condition with a constant L, we obtain the following inequality, gi (y1 , . . . , y N ) − gi (φ1 (x1 ), . . . , φ N (x N )) =
N
bi j q(y j ) − q(φ j (x j ))
j=1
2
Let M = −Q, then we can choose the positive constants dˆi (i = 1, 2, . . . , N ) to make M > 0, it follows that:
123
N
bi j 2 y j −
j=1
i=1
+
N j=1
i=1
≤
V˙ (s)ds = V (0)
≤
N j=1
L|bi j |y j − φ j (x j ) =
N
γi j e j ,
j=1
where γi j = L|bi j |. Hence, Assumption 2 holds for the nonlinearly coupled dynamical networks. 3.2 Generalized outer synchronization of networks (4) with delayed couplings Theorem 2 Suppose that Assumptions 1 and 2 hold. Networks (4) achieve generalized outer synchronization under the following control laws:
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u i = Dφi (xi ) f 1 (xi ) − f 2 (φi (xi ))
Applying Assumptions 1 and 2, and (9), we obtain
− di ei − gi (φ1 (x1 (t − τ )), . . . , φ N (x N (t − τ ))) + Dφi (xi )h i (x1 (t − τ ), . . . , x N (t − τ )), i = 1, 2, . . . , N ,
V˙ (t) =
+
where Dφi (xi ) are the Jacobian matrices of the maps φi (xi ), d˙i = ki eiT ei , ki > 0, i = 1, 2, . . . , N , are positive constants.
= f 2 (yi ) + gi (y1 (t − τ ), . . . , y N (t − τ )) + u i −Dφi (xi )( f 1 (xi ) + h i (x1 (t − τ ), . . . , x N (t − τ ))) = f 2 (yi ) − f 2 (φi (xi )) + g¯i (t − τ ) − di ei .
Select a Lyapunov candidate function as follows: 1 (di − di∗ )2 1 T ei ei + 2 2 ki N
N
i=1
+
1 2
N
i=1
t g¯iT (s)g¯i (s)ds,
i=1t−τ
where di∗ (i = 1, 2, . . . , N ) are sufficiently large positive constants. Then, V˙ (t) =
N
e˙iT ei +
i=1
+
1 2
N
i=1
1 T (g¯i (t)g¯i (t) − g¯iT (t − τ )g¯i (t − τ )) 2 N
+
i=1
≤
N
θeiT P ei −
+
N i=1
N
(di − di∗ )eiT ei +
i=1
1 T ei ei 2 N
di eiT ei +
i=1
1 2
N
g¯iT (t)g¯i (t)
i=1
1 1 ≤ e T (t) θ P + I N + N γ 2 − D ∗ e(t), 2 2
where D ∗ = diag{d1∗ , d2∗ , . . . , d N∗ }. Let W = (D ∗ − θ P − 21 I N − 21 N γ 2 ), then we can properly choose the positive constants di∗ (i = 1, 2, . . . , N ) to make W > 0. According to the proof of Theorem 1 and Barbalat’s lemma, we have ei (t) → 0(i = 1, 2, · · · , N ) as t → ∞. Hence, generalized outer synchronization between uncertain networks (4) is achieved under the adaptive controllers (8) and the update laws d˙i = ki eiT ei , i = 1, 2, . . . , N . Remark 2 For the uncertain networks (4) with time delay, when the unknown functions are linear or nonlinear couplings, Assumption 2 also holds.
(di − di∗ )eiT ei
i=1 N
N (di − di∗ )eiT ei
i=1
e˙i = y˙i − Dφi (xi ) · x˙i
V (t) =
eiT [ f 2 (yi ) − f 2 (φi (xi ))+ g¯i (t − τ )−di ei ]
i=1
(8)
Proof The synchronization error of the dynamical networks (4) is
N
(g¯iT (t)g¯i (t) − g¯iT (t − τ )g¯i (t − τ )). 4 Numerical simulations
i=1
From Assumption 2, we obtain the following inequality, that is, ⎛ ⎞ N N N N 1 ⎝ 1 T g¯i (t)g¯i (t) ≤ γi j e j T · γi j e j ⎠ 2 2 i=1
i=1
≤
j=1
j=1
1 N γ 2 e T (t) e(t), 2
where γ = max{γi j |i, j = 1, 2, . . . , N }, and is a matrix of size N with all elements being one. For any vectors x, y ∈ R n , the following inequality holds: 2x T y ≤ x T x + y T y.
(9)
In this section, we provide two examples to verify the effectiveness of the control schemes obtained in the preceding section. The Lorenz system is represented by ⎧ ⎨ x˙1 = α(x2 − x1 ), x˙2 = σ x1 − x2 − x1 x3 , ⎩ x˙3 = x1 x2 − βx3 , where the parameters α = 10, σ = 28, β = 8/3, the Lorenz system has a chaotic attractor [31]. Using Theorem 1 in [32], there exist positive constants Mi (i = 1, 2, 3), such that the chaotic attractor (x1 , x2 , x3 ) in the Lorenz system satisfies |xi | < Mi , i = 1, 2, 3. Then, we have
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(y(t) − x(t))T ( f (y(t), t) − f (x(t), t))
y˙i (t) = f 2 (yi (t)) +
= −10(y1 (t) − x1 (t))2 − (y2 (t) − x2 (t))2 8 − (y3 (t) − x3 (t))2 3 + (38 − y3 )(y1 (t) − x1 (t))(y2 (t) − x2 (t))
i = 1, 2, . . . , 50.
≤ θ (y(t) − x(t))T P(y(t) − x(t)). When the values of M2 , M3 are given, and P is an identity matrix, we can obtain the value of θ = 3 3M2 −16 }, that is, Assumption max{ 18+M22 +M3 , 36+M 2 , 6 1 is satisfied. As is known, there exist many four-dimensional (4D) hyperchaotic systems [22], such as the hyperchaotic Lorenz systems, hyperchaotic Rössler systems, hyperchaotic Chua’s circuit and hyperchaotic Lü systems. The hyperchaotic Lü system was presented in [33], which is described by ⎧ x˙1 = a(x2 − x1 ) + x4 , ⎪ ⎪ ⎨ x˙2 = cx2 − x1 x3 , (10) x ˙ = x1 x2 − bx3 , ⎪ ⎪ ⎩ 3 x˙4 = x1 x3 + d x4 . When a = 36, b = 3, c = 20, the system (10) has a periodic orbit for −1.03 ≤ d ≤ −0.46, a chaotic attractor for −0.46 < d ≤ −0.35, and a hyperchaotic attractor for −0.35 < d ≤ 1.30. 4.1 Generalized outer synchronization of networks (3) In this subsection, we consider the hyperchaotic Lü system with a hyperchaotic attractor as node dynamics with the parameters a = 36, b = 3, c = 20, d = 0.3 in network (11), and the Lorenz system with a chaotic attractor as node dynamics with α = 10, σ = 28, β = 8/3 in network (12). When the unknown functions h i , gi (i = 1, 2, . . . , 50) are linear couplings, networks (3) become 50
ai j 1 x j (t),
j=1
i = 1, 2, . . . , 50,
123
bi j 2 y j (t) + u i ,
j=1
+ y2 (t)(y1 (t) − x1 (t))(y3 (t) − x3 (t)) 18 + M2 + M3 )(y1 (t) − x1 (t))2 ≤( 2 36 + M3 (y2 (t) − x2 (t))2 + 2 3M2 − 16 (y3 (t) − x3 (t))2 + 6
x˙i (t) = f 1 (xi (t)) +
50
(11)
(12)
The connection structures A(B) in networks (11) and (12) are employed as the scale-free topologies, which are generated by the schemes proposed in [3]. Besides, A is obtained by the parameters m 0 = m = 3, N = 50, and B by m 0 = m = 5, N = 50. The elements of ai j are defined as follows: if there is a link between nodes i and j ( j = i), then ai j = a ji = 1, otherwise ai j = a ji = 0. The elements of B are defined as same as A, and they both satisfy the zerorow-sum property. 1 , 2 are taken as the identity matrices. We choose the maps φi as 2 T φi (xi ) = φ(xi ) = (xi1 , xi2 + xi4 , xi3 ) ,
i = 1, 2, . . . , 50. Then, ⎞ 1 0 0 0 Dφ(xi ) = ⎝ 0 1 0 1 ⎠ . 0 0 2xi3 0 ⎛
The controllers u i (i = 1, 2, . . . , 50) are then designed according to the control laws (6). Select the initial values xi (0) = (0.01i, −0.02i, 0.03i, −0.04i)T in network (11) and yi (0) = (−0.02i, 0.03i, 0.02i)T in network (12) for i = 1, 2, . . . , 50. In addition, we choose the initial values of the feedback strengths as di (0) = (−0.1i, 0.1i)T , i = 1, 2, . . . , 25. Let N E(t) = i=1 yi (t) − φi (x i (t)) denote the synchronization error between two networks. Figures 1 and 2 show evolutions of the generalized outer synchronization error E(t) and the feedback strengths di (t). When the controllers (6) are employed with ki = 0.1, i = 1, 2, . . . , 50, generalized outer synchronization between networks (11) and (12) with scalefree topologies is achieved.
4.2 Generalized outer synchronization of networks (4) with time delay In this subsection, we consider the effect of time delay on the generalized outer synchronization. When the delay is put into the networks (11) and (12), they read as
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Fig. 1 Generalized outer synchronization error E(t) between networks (11) and (12) with ki = 0.1, i = 1, 2, . . . , 50
Fig. 3 Generalized outer synchronization error E(t) between networks (13) and (14) with ki = 10, i = 1, 2, . . . , 50 and τ = 0.1, 0.4
dynamics with α = 10, σ = 28, β = 8/3 in network (14). The maps φi (xi ) are defined as φi (xi ) = φ(xi ) = (0.01xi1 , xi2 + xi4 , xi3 )T , i = 1, 2, · · · , 50. Therefore, ⎛
⎞ 0.01 0 0 0 Dφ(xi ) = ⎝ 0 1 0 1 ⎠ . 0 0 1 0
Fig. 2 Evolution of the feedback strengths di (t) with ki = 0.1, i = 1, 2, . . . , 50
x˙i (t) = f 1 (xi (t)) +
50
ai j 1
j=1
x j (t − τ ), i = 1, 2, . . . , 50, y˙i (t) = f 2 (yi (t)) +
50
(13)
bi j 2 y j (t − τ ) + u i ,
j=1
i = 1, 2, . . . , 50.
(14)
In the numerics, we also choose the hyperchaotic Lü system with a chaotic attractor as node dynamics with a = 36, b = 3, c = 20, d = −0.4 in network (13), and the Lorenz system with a chaotic attractor as node
With the above expressions, we design the controllers u i (i = 1, 2, . . . , 50) according to the control scheme (8). The topologies of A and B are set as same as those in the previous subsection. For i = 1, 2, . . . , 50, the initial values are xi (0) = (−0.2i, 0.1i, 0.3i, −0.1i)T in network (13) and yi (0) = (0.3i, −0.22i, 0.2i)T in network (14). The initial values of di (t) are chosen as di (0) = (−0.01i, 0.01i)T , i = 1, 2, . . . , 25. Under the controllers (8) with the update laws d˙i = ki eiT ei , ki = 10, i = 1, 2, . . . , 50, generalized outer synchronization between networks (13) and (14) is achieved. In Fig. 3, we plot the synchronization errors with delays τ = 0.1, 0.4, which shows that the delay influences the speed of generalized outer synchronization. The trajectories of the updated feedback strengths di (t)(i = 1, 2, . . . , 50) with τ = 0.1 are shown in Fig. 4. Compared to the networks without delay, the delayed networks need a large adaptive gain ki to achieve generalized outer synchronization.
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Fig. 4 Trajectories of the feedback strengths di (t) with ki = 10, i = 1, 2, . . . , 50 and τ = 0.1
5 Conclusions Synchronization of an uncertain dynamical network has attracted increasing attention, however, investigation on outer synchronization between two uncertain dynamical networks is at the initial stage. Based on the synchronization analysis of an uncertain dynamical network, we investigate generalized outer synchronization between two uncertain networks. To realize the generalized outer synchronization, we design the appropriate adaptive controllers. By Barbalat’s lemma and the adaptive controllers, we obtain two criteria on the generalized outer synchronization. Our results are generalizations of the studies on outer synchronization between two certain networks. Finally, we provide numerical examples to verify the efficiency of the control schemes and observe that the delay influences the speed of the outer synchronization. Acknowledgments The authors are very grateful to the reviewers for their valuable comments and suggestions. This work was supported by National Natural Science Foundation of China (Nos. 61203155 and 11232005), and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ12F03003.
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