New hybrid adaptive control for function projective synchronization of complex dynamical network with mixed time-varying and coupling delays

Transcription

1 ew hybrid adaptive control for function projective synchronization of complex dynamical network with mixed time-varying and coupling delays THOGCHAI BOTMART Khon Kaen University Department of Mathematics 13 Moo 16, Muang District, Khon Kaen Thailand WAJAREE WEERA University of Phayao Department of Mathematics 19 Moo, Maeka, Muang, Phayao Thailand Abstract: This paper investigates function projective synchronization (FPS) for complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix need not be symmetric or irreducible. The designed controller ensures that the FPS of delayed complex dynamical network are proposed via hybrid error adaptive control, which contains error term, time-varying delay error term and distributed time-varying delay error term. Based on the construction of improved Lyapunov-Krasovskii functional is combined with Leibniz-ewton formula and the technique of dealing with some integral terms, we prove the stability of the closed-loop system and the convergence of the error system. umerical example is included to show the effectiveness of the proposed hybrid adaptive control scheme. Key Words: hybrid adaptive control; function projective synchronization; complex dynamical network; mixed time-varying delay; mixed coupling delay 1 Introduction Complex dynamical network, as an interesting subject, has been thoroughly investigated for decades. These networks show very complicated behavior and can be used to model and explain many complex systems in nature such as computer networks [5], the world wide web [6], food webs [7], cellular and metabolic networks [8], social networks [17], electrical power grids [1] etc. The concept of chaos synchronization is making two or more chaotic systems oscillate in a synchronized manner. There are several schemes which can be used to achieve chaos synchronization of chaos complex network, for example time-delay feedback control [8], intermittent control [8], adaptive control [9, 3], active control [18], nonlinear feedback control [31 33], sampled-data control [33]. The traditional method to synchronize a complex network is to add a controller to each of the network nodes to tame the dynamics to approach a desired synchronization trajectory. Authors in [34], investigated adaptive control strategy for complex delayed dynamical networks with time-varying coupling strength and time-varying delayed. Shi et al. [7], introduced FPS for complex dynamical networks with state coupling, which it is not necessary for the coupling matrix to satisfy symmetric or nonnegative criteria. FPS is investigated via adaptive feedback control and pinning control with adaptive coupling strength. In the last decade, the synchronization of complex dynamic networks has attracted much attention of researchers in this field [18,, 3, 7, 34]. Because the synchronization of complex dynamical networks can well explain many natural phenomena observed and is one of the important dynamical mechanisms for creating order in complex dynamical networks, the synchronization of coupled dynamical networks has come be a focal point in the study of nonlinear science. Wang and Chen introduced a uniform dynamical network model and also investigated its synchronization [37, 38]. They have shown that the synchronizability of a scale-free dynamical network is robust against random removal of nodes, and yet is fragile to specific removal of the most highly connected nodes [38]. Authors in [35, 36] investigated synchronization of general complex dynamical network models with coupling delays. Wang [39] introduced several synchronization criteria for both delayindependent and delay-dependent asymptotical stability. Li [4] investigated synchronization of complex ISS: X 45 Volume 1, 16

2 networks with time-varying couplings, the stability criteria were obtained by using Lyapunov-Krasovskii function method and subspace projection method. Function projective synchronization (FPS), which is the generalization of projective synchronization (PS), is one of the important synchronization methods that have been widely investigated to obtain faster communication with its proportional feature. FPS of general complex networks was investigated which means that the nodes of complex networks could be synchronize up to an equilibrium point or periodic orbit with a desired scaling function. FPS has attracted the interest of many researchers in various fields [15, 18,, 1, 3, 5]. Very recently, FPS has been investigated in a two-cell quantum-c chaotic oscillator, [5, 6]. In [1], the authors just considered the FPS in drive-response dynamical networks (DRDs) with coupled partially linear chaotic systems by assuming that the node dynamics are identical and using a simple control law. Furthermore, In [15], investigated the problem of FPS in DRDs with nonidentical nodes by the adaptive open-plusclosed-loop method. Ref. [], investigated FPS in DRDs with uncertain parameters and disturbances. In [3], a hybrid feedback control method was proposed for achieving FPS in CDs with constant time delay and time-varying coupling delay. In [18], the authors studied projective synchronization by using active control approach. These synchronization methods or ideas can be applied to the synchronization of complex network. This paper, inspired by the above discussions, we shall investigate function projective synchronization (FPS) for complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix need not be symmetric or irreducible. The designed controller ensures that the FPS of delayed complex dynamical network are proposed via hybrid error adaptive control, which contains error term, timevarying delay error term and distributed time-varying delay error term. Based on the construction of improved Lyapunov-Krasovskii functional is combined with Leibniz-ewton formula and the technique of dealing with some integral terms, we prove the stability of the closed-loop system and the convergence of the error system. umerical example is included to show the effectiveness of the proposed hybrid adaptive control scheme. otation R n is the n-dimensional Euclidean space; R m n denotes the set of m n real matrices; I n represents the n-dimensional identity matrix; λ(a) denotes the set of all eigenvalues of A; λ max (A) = max{reλ;λ λ(a)}; C([,t],R n ) denotes the set of allr n -valued continuous functions on [,t]; L ([,t],r m ) denotes the set of all the R m - valued square integrable functions on [, t]; The notation X (respectively, X > ) means that X is positive semidefinite (respectively, positive definite); [ diag( ) ] denotes [ a b lock ] diagonal matrix; X Y X Y stands for Z Y T. Matrix dimensions, if not explicitly stated, are assumed to be com- Z patible for algebraic operations. Problem statements and preliminaries Consider a complex dynamical network consisting of identical coupled nodes, with each node being an n-dimensional nonlinear dynamical system ẋ i = f(x i,x i (t h), x i (s)ds) +c 1 a ij G 1 x j +c b ij G x j (t h) (1) +c 3 c ij G 3 x j (s)ds+u i, t, i = 1,,...,, x i = φ i,t [ τ max,],τ max = max{h,k}, where x i = (x i1,x i,...,x in ) T R n is the state vector of ith node; U i R m are the control input of the node i; the constant c 1,c,c 3 > are the coupling strength; G 1 = (g 1ij ) n n, G = (g ij ) n n, G 3 = (g 3ij ) n n R n n are a constant inner-coupling matrix; A = (a ij ), B = (b ij ), C = (c ij ) R are the outercoupling matrix of the network, in which a ij, b ij are defined as follows: if there are a connection between node i and node j (j = i), then a ij >, b ij >, c ij > ; otherwise, a ij = a ji =, b ij = b ji =, c ij = c ji = (j i), and the diagonal elements of ISS: X 46 Volume 1, 16

3 matrix A, B and C are defined by a ii = b ii = c ii =,i j,i j,i j a ij = b ij = c ij =,i j,i j,i j a ji, b ji, () c ji,i = 1,,...,. Definition 1. The network (1) with time delay is said to achieve function projective synchronization if there exists a continuously differentiable scaling function matrix α such that lim e i = lim x i αs,i = 1,,..., t t where. stands for the Euclidean vector norm and s R n can be an equilibrium point, or a (quasi-)periodic orbit, or an orbit of a chaotic attractor, which satisfies ṡ = f(s, s(t h), 1 s(θ)dθ). To investigate the stability of the synchronized states (1), we set the synchronization error e i in the form e i = x i αs, i = 1,...,, where α is a n-order real diagonal matrix, i.e. α =diag(α 1,α,...,n) and α i is a continuously bounded differentiable function. Then, substituting it into complex dynamical network (1), it is easy to get the following: ė i = ẋ i αs αṡ = f(x i,x i (t h), αf(s,s(t h), x i (s)ds) s(θ) dθ) +c 1 a ij G 1 e j (3) +c b ij G e j (t h) +c 3 c ij G 3 e j (s)ds αs+u i, i = 1,...,. The initial condition function φ i denotes a continuous vector-valued initial function of t [ τ max,], τ max = max{h,k}. In the rest of this paper, we need the following assumption and some lemmas: Assumption. The time-varying delay functions h is differential function andk with satisfy the condition h h, k k and ḣ h < 1. Lemma 3. [1] (Cauchy inequality) For any symmetric positive definite matrix M n n andx,y R n we have ±x T y x T x+y T 1 y. Lemma 4. [1] For any constant symmetric matrix M R m m, M = M T >, γ >, vector function ω : [,γ] R m such that the integrations concerned are well defined, then γ ( γ γ ) TM ( γ ) ω T (s)ds ω(s)ds ω T (s)mω(s)ds. Lemma 5. []. Let c R and A, B, C, D, be matrices with appropriate dimensions. Then (i) c(a B) = (ca) B = A (cb), (ii) (A B) T = A T B T, (iii) (A B)(C D) = (AC) (BD), (iv) A B C = (A B) C = A (B C). 3 Main results In this section, we will give some sufficient condition for function projective synchronization of complex dynamical network with discrete and distributed time-varying delays and mixed- coupling delays (1) via hybrid adaptive control In order to stabilize the origin of delayed complex dynamical network (1) by means of the hybrid adaptive control U i such as where U i = u i1 +u i,i = 1,,...,, (4) u i1 = αs, u i = c 1 d i1 G 1 e i c d i G e i (t h) c 3 d i3 G 3 e i (θ)dθ, and the updating laws are i1 = q i1 e T i G 1 e i, i = q i e T i G e i (t h), (5) i3 = q i3 e T [ i G 3 e i (θ)dθ ], ISS: X 47 Volume 1, 16

4 where q i1,q i and q i3 are positive constants and s is a solution of an isolated node, satisfying ṡ = f(s,s(t h), s(θ)dθ). The controller in (4),u i1 is the nonlinear feedback control andu i is the hybrid adaptive linear feedback control. Then, substituting it into complex dynamical network (6), it is easy to get the following: ė i = ẋ i αs αṡ = f(x i,x i (t h), αf(s,s(t h), +c 1 a ij G 1 e j +c b ij G e j (t h) +c 3 c ij G 3 e j (s)ds x i (s)ds) c 1 d i1 e i c d i e i (t h) c 3 d i3 s(θ) dθ) e i (θ)dθ,i = 1,...,, (6) i1 = q i1 e T i G 1 e i,i = 1,..., i = q i e T i G e i (t h),i = 1,..., i3 = q i3 e T [ i G 3 e i (θ)dθ ],i = 1,...,. Let us set 1. J = f (s,s(t h), s(ξ)dξ) R n n is the Jacobian of f(x,x(t h), x(s)ds) at s with the derivative off(x,x(t h), x, x(s)ds) respect to. J h = f (s,s(t h), s(ξ)dξ) R n n is the Jacobian of f(x,x(t h), x(s)ds) at s(t h) with the derivative off(x,x(t h), x(s)ds) respect tox(t h), 3. J k1 = f (s,s(t h), s(ξ)dξ) R n n is the Jacobian of f(x,x(t h), x(s)ds) at s(ξ)dξ with the derivative off(x,x(t h), x(s)ds) respect to x(s)ds and δ = τ = η = 1 ( ε1 +c ε +c d λ min (I G ) ε 5 ), 1 ( ε +c 3 ε 4 +c 3 d 3 λ min (I G 3 ) ε 6), 1 ( λ(i J) λ min (I G 1 ) c +c 1 λ(a) λ(g1 )+ G ) (1 β) λ(i + c 3k λ(i G 3 )+ 1 ε 1 λ(i J h J T h ) + 1 ε λ(i J k J T k ) + c ε 3 λ(bb T ) λ(g G T )+ c 3 ε 4 λ(cc T ) λ(g 3 G T 3 ) + c d ε 5 λ(i G G T )+ c 3d 3 ε 6 λ(i G 3 G T 3) ). Theorem 6. For some given synchronization scaling function α, the complex dynamical networks (1) with time-varying delay satisfying Assumption. and target system can realize function projective synchronization by the adaptive control law as shown in (4) if there exist positive constants ε i, i = 1,,...,5 and by taking appropriate d 1, d and d 3 such that d 1 η c 1 >, (7) d 1 ε 5 ( ε 1 c +ε λ min (I G )) >, (8) d 3 1 ε 6 ( ε c 3 +ε 4 λ min (I G 3 )) >. (9) Then the controlled complex dynamical networks (1) is function projective synchronization. Proof. Since f(.) is continuous differentiable, it is easy to know that the origin of the nonlinear system (6) is an asymptotically stable equilibrium point if it is an asymptotically stable equilibrium point of the following linear time-varying delays systems ė i = Je i +J h e i (t h) +J k1 e i (s)ds +c 1 a ij G 1 e j (1) +c b ij G e j (t h) +c 3 c ij G 3 e j (s)ds ISS: X 48 Volume 1, 16

5 c 1 d i1 e i c d i e i (t h) c 3 d i3 e i (θ)dθ,i = 1,...,, i1 = q i1 e T i G 1 e i,i = 1,..., i = q i e T i G e i (t h),i = 1,..., i3 = q i3 e T [ i G 3 e i (θ)dθ ],i = 1,...,. Construct the the following Lyapunov-Krasovskii functional candidate: V = 1 e T i e i c (1 β) c 3k + 1 t h c q i (d i d ) k t+s c 1 q i1 (d i1 d 1) e T i (s)g e i (s)ds e T i (θ)g 3e i (θ)dθds c 3 q i3 (d i3 d 3). (11) By taking the derivative of V along the trajectories of system (6), we have the following: V e T i Je i + e T i J h e i (t h) + e T i J k e i (s)ds +c 1 e T i a ij G 1 e j +c +c 3 + c (1 β) e T i b ij G e j (t h) e T i c ijg 3 t h e T i G e i e j (s)ds c c d c 1 d 1 + c 3k c 3k c 3 d 3 e T i (t h)g e i (t h) e T i G e i (t h) e T i G 1e i e T i G 3 e i e T i G 3 e T i (s)g 3e i (s)ds e i (s)ds (1) Let e = (e 1,...,e ) R n, e(t h) = (e 1 (t h),...,e (t h)) R n, t e(s)ds = (e 1(s),e (s),...,e (s))ds R n. we have V e T (I J)e +e T (I J h )e(t h) +c 1 e T (A G 1 )e +c e T (B G )e(t h) c 1 d 1 et (I G 1 )e +c 3 e T (C G 3 ) c d e T (I G )e(t h) c 3 d 3 et (I G 3 ) + c (1 β) et (I G )e c et (t h)(i G )e(t h) + c 3k et (I G 3 )e +e T (I J k ) c 3k e T (s)(i G 3 )e(s)ds. (13) Applying Lemma 3., Lemma 4. and Lemma5. gives e T (I J h )e(t h) 1 ε 1 e T (I J h J T h )e + ε 1 et (t h)e(t h), (14) ISS: X 49 Volume 1, 16

6 e T (I J k ) 1 e T (I J k Jk T )e (15) ε + ε ( (i) ) T ( e T (s)ds c e T (B G )e(t h) c ε 3 e T (BB T G G T )e (i) ) e T (s)ds + c ε 3 et (t h)e(t h), (16) c 3 e T (C G 3 ) c 3 e T (CC T G 3 G T ε 3)e (17) 4 + c ( 3ε ) 4 T ( e T t ) (s)ds e T (s)ds (i) (i) c d et (I G )e(t h) c d ε 5 e T (I G G T )e + c d ε 5 e T (t h)e(t h), (18) c 3 d 3 et (I G 3 ) c 3d 3 e T (I G 3 G T ε 3)e (19) 6 + c 3d 3 ε ( ) 6 T ( ) e T (s)ds e T (s)ds. (i) Hence, according to(13) -(14) we have (i) V e T ( I J+c 1 (A G 1 ) c 1 d 1 (I c G 1 ))+ (1 β) (I G ) + c 3k (I G 3 )+ 1 ε 1 (I J h J T h ) + 1 ε (I J k J T k ) + c ε 3 (BB T G G T ) + c 3 ε 4 (CC T G 3 G T 3) + c d ε 5 (I G G T ) + c d 3 ε 6 (I G 3 G T 3) ) e + c ε 3 (BB T G G T ) e T (t h) ( c (I G ) +δ(i G ) ) e(t h) ( ) T( c 3 (I G 3 ) τ(i G 3 ) )( ) (η c 1 d 1 )et (I G 1 )e e T (t h)(( c δ) (I G ))e(t h) ( T(( c 3 e(s)ds) τ)(i G 3 ) ( ). () It is obviously that there exists sufficiently large positive constant d 1,d and d 3 such that d 1 η c 1 >,(1) d 1 ε 5 ( ε 1 c +ε λ min (I G )) >,() d 3 1 ε 6 ( ε c 3 +ε 4 λ min (I G 3 )) >.(3) We can choosed 1,d d 3 satisfying (1), () and (3), respectively. Since G 1, G and G 3 are positive definite diagonal matrix, we know that V and V = if and only if ξ =. Hence, the set W = {ξ =, d 1i = d 1, d i = d, d 3i = d 3 } is the invariant set contained in W 1 = {ξ = : V = } for system (6). According to LaSalle invariance principle [3] and Lyapunov stability theory, for any initial condition, every solution of system (6) approaches W as t, which indicates that e i, i = 1,,..., this means that the function projective synchronization between the delayed complex dynamical networks (1) and the reference nodes is achieved under hybrid adaptive control (4). The proof is this completed. 4 umerical Example In this section, we present example to illustrate the effectiveness and the reduced conservatism of our result. Example 4.1 We first consider the perturbed Chua s circuit system with mixed time-varying delays is used as uncoupled node in the network (1) to show the effectiveness of the proposed control scheme. The perturbed Chua s circuit system with mixed time-varying ISS: X 41 Volume 1, 16

7 delays is given by [1] ( ẋ 1 = p x (t h) 1 ( )) x x 1 ẋ = x 1 sx +x 3 (t h) (4) ẋ 3 = qx +r 1 x 1(s)ds where p, q, r and s are real positive constants. It is well known that the system (4) exhibits chaotic behavior with the parameters p, q, r and s are chosen as p = 7, q = 1 7, r =.7 and s = 1.5, the initial condition function φ = [.65cost,.3cost,.cost] T, the timevarying delay functions h =.1 +.1sin t and k =.1cos t is shown in Figure 1. It is stable at the equilibrium point s =, s(t h) =, s(θ)dθ = and Jacobian matrices are 1 J = x i, α s J k1 = x i1, α 1 s 1.5, J h = 1 drive system response network 1. x i3, α 3 s , Figure 1: Chaotic behavior of the perturbed Chua s circuit system with mixed time-varying delays (4) The parameters are selected as follows: the coupling strength c 1 =.4, c =.3, c 3 =.5, the time-varying scaling function matrix α =diag(.6sin( π 15 ),.7sin(π 15 ),.75sin(π 15 )), the inner-coupling matrix are 3 G 1 = 3, G =, 3 G 3 = 1 1 1, and The coupling configuration matrices are given respectively as follows: A = , B = , C = Solution: From the conditions (7)-(9) of Theorem 6 and there exist positive constants ε 1 =.86, ε =.75, ε 3 =.9, ε 4 = 1., ε 5 = 1.1, ε 6 =.7, one can check that the last three conditions in Theorem 6. are satisfied. From the conditions of Theorem 6, we can obtain d 1 > , d >.777 and d 3 >.9643 The numerical simulations are carried out using the explicit Runge-Kutta-like method (dde45), interpolation and extrapolation by spline of the third order. Figure. shows the function projective synchronization errors between the states of isolate node αs and node x i, where e ij = x ij α j s j for i = 1,...,8,j = 1,,3 without hybrid adaptive control. Figure 3. shows the function projective synchronization errors between the states of isolate node αs and node x i, where e ij = x ij α j s j fori = 1,...,8,j = 1,,3 with hybrid adaptive control. We see that the synchronization errors converge to zero under the above conditions. ISS: X 411 Volume 1, 16

8 e i1 =x i1 α 1 s 1 e i =x i α s e i3 =x i3 α 3 s Figure : Shows the function projective synchronization errors between the states of isolate node αs and nodex i, wheree ij = x ij α j s j for i = 1,...,8,j = 1,,3 without adaptive controller 5 Conclusion This paper investigated function projective synchronization (FPS) for complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix need not be symmetric or irreducible. The designed controller ensures that the FPS of delayed complex dynamical network are proposed via hybrid error adaptive control, which contains error term, time-varying delay error term and distributed time-varying delay error term. By using new methods to deal with asymmetric coupling matrix and a new class of Lyapunov-Krasovskii functional, improved PFS criteria are obtained. Simulation results have been given to illustrate the effectiveness of the proposed method. Acknowledgements: The authors sincerely thank the anonymous reviewers and the editors for their valuable comments that have led to the present improved version of the original manuscript. The first author was financial supported by the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), Khon Kaen University (grant number : MRG5889). e i1 =x i1 α 1 s 1 e i =x i α s e i3 =x i3 α 3 s Figure 3: Shows the function projective synchronization errors between the states of isolate node αs and nodex i, wheree ij = x ij α j s j for i = 1,...,8,j = 1,,3 with adaptive controller References: [1] K. Gu, V.L. Kharitonov and J.Chen, Stability of time-delay system, Boston: Birkhauser, 3. [] M. MacDuffeec, The theory of matrices, Dover publications, ew York, 4. [3] J.H. Hale, Theory of Functional Differential Equations, Springer-Verlag, ew York, [4] J.H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford University Press: U.K., [5] M. Faloutsos, P. Faloutsos, C. Faloutsos, On power-law relationships of the Internet topology, Comput. Commun. Rev., 9, 1999, pp [6] R. Albert, H. Jeong, A.L. BarabVasi, Diameter of the world wide web, ature., 41, 1999, pp [7] R.J. Williams,. D. Martinez, Simple rules yield complex food webs, ature., 44,, [8] H. Jeong, B. Tombor, R. Albert, Z. Oltvai, A.L. BarabVasi, The large-scale organization of metabolic network, ature., 47,, pp [9] S. Wassrman, K. Faust, Social etwork Analysis, Cambridge University Press, Cambridge, ISS: X 41 Volume 1, 16

9 [1] S.H. Strogatz, Exploring complex networks, ature., 41, 1, pp [11] Q. Zhang, J. Lu, C.K. Tse, Adaptive Feedback Synchronization of a General Complex Dynamical etwork With Delayed odes, IEEETrans. Circuits Syst. II., 55, 8, pp [1] T. Botmart, iamsup P. Adaptive control and synchronization of the perturbed Chuas system. Mathematics and Computers in Simulation, 75, 7, pp [13] H. Dai, G. Si, Y. Zhang, Adaptive generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimensions, onlinear Dyn, 74, 13, pp [14] S.K. Agrawal, S. Das, Function projective synchronization between four dimensional chaotic systems with uncertain parameters using modified adaptive control method, Journal of Process Control, 4, 14, pp [15] H. Du, Function projective synchronization in complex dynamical networks with or without external disturbances via error feedback control, eurocomputing, 173, 16, pp [16] G. A. mahbashi, M. S. M. oorani, S. A. Bakar and M. M. A. Sawalha, Adaptive projective lag synchronization of uncertain complex dynamical networks with delay coupling, Advances in Difference Equations, 356, 15, 356, pp [17] S. Wang, S. Zheng, B. Zhang, H. Cao, Modified function projective lag synchronization of uncertain complex networks with time-varying coupling strength, Optik, 17, 16, pp [18] C.F. Feng, Projective synchronization between two different time-delayed chaotic systems using active control approach, onlinear Dyn., 6, 1, pp [19] P.C. Rao, Z.Y. Wu, M. Liu, Adaptive projective synchronization of dynamical networks with distributed time delays, onlinear Dyn. 67, 1, pp [] L. Wang, Z. Yuan, X. Chen, Z. Zhou, Adaptive function projective synchronization of uncertain complex dynamical networks with disturbance, Commun. onlinear Sci. umer. Simul., 16, 11, pp [1] R. Zhang, Y. Yang, Z. Xu, M. Hu, Function projective synchronization in driveresponse dynamical network, Physics Letters A, 374, 1, pp [] S. Wang, S. Zheng, Adaptive function projective synchronization of uncertain complex dynamical networks with disturbance, China Phys. B (13) 753. [3] H. Du, P. Shi,. L?, Function projective synchronization in complex dynamical networks with time delay via hybrid feedback control, onlinear Anal. RWA, 14, 13, pp [4] X. Wu, H. Lu, Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes, Commun. onlinear Sci. umer. Simul., 17, 1, pp [5] K. S. Sudheer, M. Sabir, Adaptive function projective synchronization of two-cell Quantum- C chaotic oscillators with uncertain parameters, Physics Letters A, 373, 9, pp [6] W. Yang, J. Sun, Function projective synchronization of two-cell quantum-c chaotic oscillators by nonlinear adaptive controller, Physics Letters A, 374, 1, pp [7] L. Shi, H. Zhu, S. Zhong, K. Shi, J. Cheng, J, Function projective synchronization of complex networks with asymmetric coupling via adaptive and pinning feedback control, ISA Transactions, In Press, Available online 6 July 16. [8] T. Botmart, P.iamsup, Exponential Synchronization of Complex Dynamical etwork with Mixed Time-varying and Hybrid Coupling Delays via Intermittent Control, Advances in Difference Equation. 116, 14, pp [9] J. Lu J. Cao D.W.C. Ho, Adaptive stabilization and synchronization for chaotic Lur e systems with time-varying delay, IEEE Trans Circuits Syst I-Regul Pap, 55, 8, pp [3] Z. Wu X. Xu G. Chen X. Fu, Adaptive synchronization and pinning control of colored networks, Chaos, :43137, 1, pp [31] B. Cui, X. Lou, Synchronization of chaotic recurrent neural networks with time-varying delays using nonlinear feedback control, Chaos Solitons Fractals, 39, 9, pp [3] L. Shi, H. Zhu, S. Zhong, Y. Zeng, J. Cheng, Synchronization for time-varying complex networks based on control. J Comput Appl Math, 31, 16, pp [33] T.H. Lee, Z, Wu, J.H. Park, Synchronization of a complex dynamical network with coupling timevarying delays via sampled-data control. Appl Math Comput, 19, 1, pp [34] X. Guo, J. Li, A new synchronization algorithm for delayed complex dynamical networks via adaptive control approach, Commun onlinear Sci umer Simulat, 17, 1, pp ISS: X 413 Volume 1, 16

10 [35] H. Gao, J. Lam, G. Chen, ew criteria for synchronization stability of general complex dynamical networks with coupling delays, Phys. Lett. A. 36, 6, pp [36] J. Zhou, L Xiang, Z. Liu, Z, Global synchronization in general complex delayed dynamical networks and its applications, Phys. Lett. A. 385, 7, pp [37] X.F. Wang, G. Chen, Synchronization Scale-free dynamical networks: robustness and fragility, IEEETrans. Circuits Syst. I., 49,, pp [38] X.F. Wang, G. Chen, Synchronization small-world dynamical networks, Int.J.Bifurcat.Chaos., 1, [39] L. Wang, H.P. Dai, X.Y. Sun, Synchronization criteria for a generalized complex delayed dynamical network model, Phys. Lett. A., 383, 7, pp [4] P. Li, Z. Yi, Synchronization analysis of delayed complex networks with time-varying couplings, Phys. Lett. A. 387, 8, ISS: X 414 Volume 1, 16