Anti-synchronization Between Coupled Networks with Two Active Forms

Commun. Theor. Phys. 55 (211) 835 84 Vol. 55, No. 5, May 15, 211 Anti-synchronization Between Coupled Networks with Two Active Forms WU Yong-Qing ( ï), 1 SUN Wei-Gang (êå ), 2, and LI Shan-Shan (Ó ) 3 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 73, China 2 School of Science, Hangzhou Dianzi University, Hangzhou 3118, China 3 Department of Basic Theory, Shandong Institute of P. E. and Sports, Jinan 2512, China (Received August 26, 21; revised manuscript received November 1, 21) Abstract This paper studies anti-synchronization and its control between two coupled networks with nonlinear signal s connection and the inter-network actions. If anti-synchronization does not exist between two such networks, adaptive controllers are designed to anti-synchronize them. Different node dynamics and nonidentical topological structures are considered and useful criteria for anti-synchronization between two networks are given. Numerical examples are presented to show the efficiency of our derived results. PACS numbers: 5.45.Xt Key words: complex networks, adaptive controller, anti-synchronization 1 Introduction Presently investigation the properties of complex networks [1] has been reported widely, ranging from the human brain to epidemiology, WWW, social groups etc. Specially the problem of synchronization and its control [2 3] inside a network is a hot topic. Inner synchronization, i.e., synchronizing all the nodes in a network, has been studied in Refs. [4 8], and many references cited therein. Recently the authors studied network synchronization of two groups, [9] as it is a collective behavior within each network. Using an open-plus-closed-loop controller, Li et al. theoretically and numerically demonstrated the possibility of outer synchronization [1] between two networks having the same topological structures. Later through the adaptive controllers, (generalized) synchronization between two networks are studied in Refs. [11 17], which could deal with more complicated cases, such as different node dynamics, nonidentical topological structures, or time-varying delays. As another type of synchronization, anti-synchronization can be characterized by vanishing of the sum of relevant state variables. It is a noticeable phenomenon in chaotic systems [18 19] that has potential application in practice fields such as secure communications, lasers, nonlinear circuits and so on. To the best of our knowledge, the anti-synchronization between two networks is seldom reported in literatures. It is known that the actions between two coupled networks are colorful. In reality, if the communicated information between them is so sufficient that synchronization can be easily achieved, therefore the controller is not necessary. In this paper, we consider anti-synchronization between two coupled networks and its control. A general network models between two networks is firstly introduced, and we mainly consider two active forms, including the nonlinear signal s connection and the internetwork reciprocity. Detailed theoretical analysis of antisynchronization between two networks with different node dynamics and nonidentical topological structures is technically difficult and has not been attempted. Here our aims is to design some appropriate controllers to make these two networks anti-synchronize. The remainder of this paper is organized as follows. In Sec. 2, we present a general model of two coupled networks and introduce two active forms. Anti-synchronization analysis between two networks with adaptive controllers are derived in Sec. 3. In Sec. 4, numerical examples are shown, finally conclusions and discussion are included in Sec. 5. 2 Network Models The coupled equations of two networks can generally be expressed as follows: ẋ i (t) = f(x i (t)) a ij Γx j (t) C YX (Y, X), i = 1, 2,..., N, ẏ i (t) = g(y i (t)) b ij Γy j (t) C XY (X, Y ), i = 1, 2,..., N, (1) where x i, y i R n, i = 1, 2,...,N, f( ), g( ): R n R n are continuously differentiable functions, which determine the dynamical behavior of the nodes in networks X and Supported by the National Natural Science Foundation of China under Grant No. 1872119 and Research Foundation of Hangzhou Dianzi University under Grant No. KYF756132 Corresponding author, E-mail: wgsun999@yahoo.com.cn c 211 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

836 Communications in Theoretical Physics Vol. 55 Y respectively. Γ R n n is a constant -1 matrix linking coupled variables. For simplicity, one often assumes that Γ = diag (r 1, r 2,...,r n ) is a diagonal matrix. A = (a ij ) N N and B = (b ij ) N N represent the coupling configurations of both networks, whose entries a ij and b ij are defined as follows: if there is a connection between node i and node j (j i), then set a ij = 1 and b ij = 1, otherwise a ij =, b ij = (j i); the matrices A and B can be symmetric or asymmetric, each line sum of A and B is equal to zero. C XY (X, Y )(C YX (Y, X)) represents the interaction from network X(Y) to network Y(X). There are lots of active forms between two networks, for instance, communicated by signals, special nodes or bidirectional actions. Here we focus on the nonlinear signal s connection and bidirectional actions. In this paper, the notation denotes the Euclidean norm, I N denotes an identity matrix of order N, and we have the following assumptions, lemma and definition. Assumption 1 (A1). Assume that the functions f(x), g(x) are Lipschitz continuous, i.e., there exist positive constants L 1, L 2, satisfying f(y) f(x) L 1 y x, g(y) g(x) L 2 y x, where y and x are time-varying vectors. Assumption 2 (A2). The functions f(x), g(x) are odd functions of x, i.e., f( x) = f(x), g( x) = g(x) for any x R n. Lemma 1 (Barbalat Lemma [2] ). If function φ(t) is uniformly continuous, and lim t φ(τ) dτ is bounded, then φ(t) when t. Definition 1 If lim x i(t)y i (t) =, i = 1, 2,..., N, t we call that network X and network Y achieve the antisynchronization. 2.1 Two Coupled Networks with Nonlinear Signals In this subsection, we choose C XY (X, Y ) = (1 θ)[f(y i (t)) g(x i (t))] and C YX (Y, X) = θ[f(y i (t)) g(x i (t))], where θ 1 represents the nonlinear coupling parameter between these two networks. If there does not exist such an active form, which makes antisynchronization happen, the control strategies are often applied. Then Eq. (1) reads as ẋ i (t) = f(x i (t)) a ij Γx j (t) θ[f(y i (t)) g(x i (t))], i = 1, 2,..., N, ẏ i (t) = g(y i (t)) b ij Γy j (t) (1 θ)[f(y i (t)) g(x i (t))] U i, i = 1, 2,...,N, (2) where U i is the controller for node i to be designed. 2.2 Two Coupled Networks with Reciprocity In this subsection, we change the forms of C XY (X, Y ) and C YX (Y, X), choose C XY (X, Y ) = C YX (Y, X) =,j i,j i d ij Γ[x j (t) y i (t)] c ij Γ[y j (t) x i (t)], the network equations are written as: ẋ i (t) = f(x i (t)) a ij Γx j (t),j i i = 1, 2,..., N, ẏ i (t) = g(y i (t)),j i c ij Γ[y j (t) x i (t)], b ij Γy j (t) d ij Γ[x j (t) y i (t)], i = 1, 2,..., N, (3) where C is an N N dimensional coupling matrix, whose entries (c ij ) represent the intensity of the direct interaction from i in network X to j in network Y, analogously the entries of (d ij ) are same defined. Naturally if antisynchronization does not happen under some appropriate coupling matrices A, B, C, D and it is necessary, we should adopt the control skills to make it appear, i.e., consider the following equations, ẋ i (t) = f(x i (t)) a ij Γx j (t),j i i = 1, 2,..., N, ẏ i (t) = g(y i (t)),j i c ij Γ[y j (t) x i (t)], b ij Γy j (t) d ij Γ[x j (t) y i (t)] U i, i = 1, 2,..., N, (4) where U i is the controller for node i to be designed. 3 Synchronization Criteria In this section we study network models (2) and (4) with different node dynamics and nonidentical topological structures if anti-synchronization does not exist between them, and the main results are summarized in the following two theorems. Theorem 1 Suppose that assumptions (A1) and (A2) hold. If anti-synchronization is achieved under the nonlinear connected action, let U i = ; otherwise, the net-

No. 5 Communications in Theoretical Physics 837 work (2) can realize anti-synchronization with the following adaptive control scheme: U i = â ij Γx j (t) ˆbij Γy j (t) E i e i, i = 1, 2,..., N, (5) where e i = x i y i, Ė i = ε i e i 2, ε i are arbitrary positive constants, and â ij = e T i Γx j, ˆbij = e T i Γy j. Proof Let ã ij = a ij â ij, b ij = b ij ˆb ij, then ã ij = â ij = e T i Γx j, bij = ˆbij = e T i Γy j. The error system is written as ė i = g(y i (t)) g(x i (t)) f(y i (t)) f(x i (t)) bij Γy j ã ij Γx j E i e i. (6) With the assumptions (A1) and (A2), we get f(x i ) f(y i ) = f(y i ) f( x i ) L 1 y i ( x i ) = L 1 e i, g(x i ) g(y i ) = g(y i ) g( x i ) L 2 y i ( x i ) = L 2 e i, where i = 1, 2,..., N. Choose the Lyapunov candidate as, V (t) = 1 e T i e i 1 1 (E i 2 2 ε ˆL) 2 i 1 2 where ˆL L 1 L 2 1. Then we get b2 ij 1 2 ã 2 ij, (7) 1 V (t) = e T i ė i (E i ε ˆL)Ėi bij bij ã ij ã ij i = e T i [g(y i(t)) g(x i (t)) f(y i (t)) f(x i (t)) bij Γy j ã ij Γx j E i e i ] (E i ˆL) e i 2 bij ( e T i Γy j ) ã ij ( e T i Γx j ) e T i (L 1 L 2 )e i e T b i ij Γy j e T i ãijγx j e T i E ie i (E i ˆL) e i 2 e T i Γy j b ij e T i Γx jã ij = (L 1 L 2 ) e i 2 E i e i 2 (E i ˆL) e i 2 = where e = (e T 1, e T 2,..., e T N) T R nn. Thus it follows that e 2 V. Since V, one has e 2 dτ (L 1 L 2 ˆL) e i 2 e 2, V dτ = V () V (t) V () <. From Barbalat Lemma, we have lim e 2 =, which implies anti-synchronization can be realized in network (2) t under this adaptive controllers. Next we derive an anti-synchronous theorem for network model (4). Theorem 2 Suppose that assumptions (A1) and (A2) hold. If the coupling matrices A, B, C, D make network (4) anti-synchronize, then take U i =, or else U i = f(y i (t)) g(x i (t)) p ij Γy j q ij Γ(x j y i ) H i e i, (8),j i where p ij = e T i Γy j, q ij = e T i Γ(x j y i ), Ḣ i = µ i e i 2, µ i >, then anti-synchronization between networks (4) can be realized under the controllers U i, i = 1, 2,...,N. Proof The error system reads as ė i = g(y i (t)) f(x i (t)) (b ij Γy j a ij Γx j ) [d ij Γ(x j y i ) c ij Γ(y j x i )] U i. (9) Construct a Lyapunov function candidate in the form of V (t) = 1 e T i 2 e i 1 (b ij p ij a ij ) 2 1 2 2,j i,j i (d ij q ij c ij ) 2 1 2 1 µ i (H i H) 2, where H is a sufficiently large positive constant, which is to be determined. By differentiating the function V (t) along

838 Communications in Theoretical Physics Vol. 55 the trajectories of the error system (9), we obtain V (t) = = = e T i e i (b ij p ij a ij ) p ij,j i (d ij q ij c ij ) q ij e T i [f(y i (t)) f(x i (t)) g(y i (t)) g(x i (t)) (b ij Γy j a ij Γx j ),j i,j i c ij Γ(y j x i ) p ij Γy j,j i (d ij q ij c ij )[ e T i Γ(x j y i )] q ij Γ(x j y i ) H i e i ] e T i [f(y i (t)) f(x i (t)) g(y i (t)) g(x i (t)) H i e i ] e T i a ij Γe j,j i e T i c ij Γ(e j e i ) (L 1 L 2 H) e i 2 e T i a ij Γe j 1 µ i (H i H)(µ i e i 2 ) (H i H) e i 2,j i 1 µ i (H i H)Ḣi,j i e T i c ij Γ(e j e i ) = e T Me, d ij Γ(x j y i ) (b ij p ij a ij )( e T i Γy j ) where e = (e T 1, et 2,..., et N )T R nn, M = (S S T )/2, S = (L 1 L 2 H)I nn (A C) Γ, the symbol denotes the Kronecker product and C = ( c ij ) N N is defined as follows: c ij = c ij (j i); c ii = c ij (1 i N).,j i We can select a suitable positive constant H to guarantee that M is a positive definite matrix. Hence, we get V e T Me, and λ min (M) e 2 e T Me V, where λ min (M) is the minimal eigenvalue of M. Since V, we obtain λ min (M) e 2 dτ V dτ = V () V (t) V () <. By Barbalat Lemma, we have lim λ min (M) e 2 =, i.e., t lim t e 2 =, thus the anti-synchronization between networks (4) can be achieved. 4 Numerical Examples In this section, numerical examples are given to show our results derived in Sec. 3. We choose the Chua s systems as our network node dynamics. The Chua system [21] is described by, where ẋ i1 = α(x i2 x i1 φ(x i1 )), ẋ i2 = x i1 x i2 x i3, ẋ i3 = βx i2, (1) φ(x i1 ) = bx i1 1 2 (a b)[ x i1 1 x i1 1 ], (11) and a, b, α, β are parameters. For simplicity, we choose f( ), g( ) as Chua s systems with different parameters. Network X: a = 1.27, b =.68, α = 1., β = 14.87; while network Y: a = 1.39, b =.75, α = 1., β = 18.6. The inner-coupling matrix Γ is an identity matrix. Obviously assumptions (A1) and (A2) are satisfied. In the numerical simulation, the initial values of state vectors X, Y and control E i (), H i () are randomly chosen in (, 1). Let e(t) = N [(y i1 (t) x i1 (t)) 2 (y i2 (t) x i2 (t)) 2 (y i3 (t) x i3 (t)) 2 ] be the 2-norm of the total anti-synchronization errors at time t, for t (, ). Example 1 Consider the network (2). The network size N is taken as 1 and configuration matrices of networks X and Y are given as follows,

No. 5 Communications in Theoretical Physics 839 3 1 1 1 4 1 1 1 1 1 3 1 1 1 4 1 1 1 1 3 1 1 A =, 1 1 1 5 1 1 1 1 1 4 1 1 1 1 4 1 1 1 1 3 (12) 1 1 1 1 4 2 1 1 4 1 1 1 1 4 1 1 1 1 1 3 1 1 1 1 4 1 1 B =. 1 1 5 1 1 1 1 1 3 1 1 1 1 1 5 1 1 1 1 4 1 (13) 1 1 1 3 Fig. 1 Anti-synchronization error between networks (2) by using the adaptive controllers (5) with ε i = 2, i = 1,..., 1. does not happen for arbitrary value of θ in [, 1]. Now we take the value of θ is.5, and use the adaptive controllers proposed in Theorem 1 to make these two networks antisynchronize. Figure 1 plots the anti-synchronization error for ε i = 2, i = 1,...,1. The trajectories of adaptive feedback gains E i are also shown in Fig. 2. Example 2 Consider the network model (4). Let configuration matrices A and B be Eqs. (12) and (13) respectively. Here, C and D are 1 1 matrices with random entries, chosen from a uniform distribution on the interval (,.4). Anti-synchronization also does not exist between networks (4) for U i =, and the adaptive controllers in Theorem 2 are applied. The anti-synchronization between networks (4) is shown in Fig. 3 with µ i = 2, i = 1,...,1. Figure 4 displays the evolution of adaptive feedback gains H i. Fig. 2 Evolution of adaptive feedback gains E i for ε i = 2, i = 1,..., 1. If the controllers U i =, i = 1, 2,..., 1, through the numerical simulation, we find that anti-synchronization Fig. 3 Anti-synchronization error between networks (4) by using the adaptive controllers (8) with µ i = 2, i = 1,..., 1.

84 Communications in Theoretical Physics Vol. 55 Fig. 4 Evolution of adaptive feedback gains H i for µ i = 2, i = 1,..., 1. 5 Conclusion In this paper, anti-synchronization between two networks with two actions has been studied. On condition that two networks are ultimately irrelated and synchronization is necessary, we should use the controllers to realize it. If synchronization happens under some internal actions between them, the controllers are not necessary. Based on this principle, firstly we propose two kinds of active forms, and then put forward two adaptive controllers to realize anti-synchronization. We do not confine that these two networks are of same node dynamics and identical topological structures. Numerical simulation shows that the network model (4) is more difficult to achieve the anti-synchronization than network model (2). It will not have escaped the reader that this paper only focuses on anti-synchronization, not study the inner synchronization inside each network. How to realize the inner synchronization and anti-synchronization simultaneously is interesting. We hope that such work will appear elsewhere. References [1] S. Boccaletti, V. Latora, Y. Morenoy, M. Chavez, and D.U. Hwang, Phys. Rep. 424 (26) 175. [2] T.P. Chen, X.W. Liu, and W.L. Lu, IEEE Trans. Circuits Syst. I 54 (27) 1317. [3] F. Sorrentino, M. Bernardo, F. Garofalo, and G. Chen, Phys. Rev. E 75 (27) 4613. [4] J. Lü, X. Yu, and G. Chen, Physica A 334 (24) 281. [5] Q. Wang, Z. Duan, G. Chen, and Z. Feng, Physica A 387 (28) 5616. [6] C.P. Li, W.G. Sun, and J. Kurths, Physica A 361 (26) 24; W.G. Sun, C.X. Xu, C.P. Li, and J.Q. Fang, Commun. Theor. Phys. 47 (27) 173; W.G. Sun, Y. Chen, C.P. Li, and J.Q. Fang, Commun. Theor. Phys. 48 (27) 871; Q.C. Wu, X.C. Fu, and W.G. Sun, Commun. Theor. Phys. 53 (21) 2. [7] W. Kinzel, A. Englert, G. Reents, M. Zigzag, and I. Kanter, Phys. Rev. E 79 (29) 5627. [8] J.H. Li, Commun. Theor. Phys. 49 (28) 665. [9] F. Sorrentino and E. Ott, Phys. Rev. E 76 (27) 56114. [1] C.P. Li, W.G. Sun, and J. Kurths, Phys. Rev. E 76 (27) 4624; C.P. Li, C.X. Xu, W.G. Sun, J. Xu, and J. Kurths, Chaos 19 (29) 1316. [11] Y. Li, Z. Liu, and J. Zhang, Chin. Phys. Lett. 25 (28) 874. [12] R. Li, Z. Duan, and G. Chen, J. Phys. A: Math. Theor. 41 (28) 38513. [13] H. Tang, L. Chen, J. Lu, and C.K. Tse, Physica A 387 (28) 5623. [14] M. Sun, C. Zeng, and L. Tian, Chin. Phys. Lett. 26 (29) 151; J. Chen, L. Jiao, J. Wu, and X. Wang, Chin. Phys. Lett. 26 (29) 655. [15] S. Zheng, Q. Bi, and G. Cai, Phys. Lett. A 373 (29) 1553. [16] X. Wu, W. Zheng, and J. Zhou, Chaos 19 (29) 1319. [17] W.G. Sun, R.B. Wang, W.X. Wang, and J.T. Cao, Cogn. Neurodyn. 4 (21) 225. [18] C.M. Kim, S. Rim, W.H. Kye, J.W. Ryu, and Y.J. Park, Phys. Lett. A 32 (23) 39. [19] R.H. Li, W. Xu, and S. Li, Chaos, Solitons & Fractals 4 (29) 1288. [2] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, ed. T. Kailath, Prentice Hall, New Jersey (1989) p. 19. [21] L.O. Chua, M. Itoh, L. Kocarev, and K. Eckert, J. Circ. Syst. Comput. 3 (1993) 93.