Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 871 876 c International Academic Publishers Vol. 48, No. 5, November 15, 2007 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems SUN Wei-Gang, 1 CHEN Yan, 1 LI Chang-Pin, 1, and FANG Jin-Qing 2 1 Department of Mathematics, Shanghai University, Shanghai 200444, China 2 China Institute of Atomic Energy, Beijing 102413, China (Received October 27, 2006; Revised January 15, 2007) Abstract Synchronization and bifurcation analysis in coupled networks of discrete-time systems are investigated in the present paper. We mainly focus on some special coupling matrix, i.e., the sum of each row equals a nonzero constant u and the network connection is directed. A result that the network can reach a new synchronous state, which is not the asymptotic limit set determined by the node state equation, is derived. It is interesting that the network exhibits bifurcation if we regard the constant u as a bifurcation parameter at the synchronous state. Numerical simulations are given to show the efficiency of our derived conclusions. PACS numbers: 05.45.Ra, 05.45.Xt Key words: complex dynamical networks, synchronization, bifurcation 1 Introduction Presently our knowledge on complex networks is experiencing rapid growth. Complex networks have attracted more and more attention. Among all kinds of complex networks, the random graph, small-world effect, and scalefree characteristics are most noticeable. Many authors investigated the properties of complex networks, such as the average path length, clustering coefficient, degree distribution and betweenness etc., and brought forward various models based on the small-world effect [1,2] and the scalefree characteristics. [3,4] In order to understand dynamical behavior of network well, we can introduce the dynamics into their nodes, and make them from static state to dynamical one. Amongst all studies of complex networks, one significant and interesting topic is their synchronizations. In the previous studies on synchronization of complex networks, most authors considered the sum of each row of the coupling matrix (symmetric or asymmetric) being zero, [5 10] we observe a phenomenon that the synchronous state is determined by node state function itself, somewhat contradictive with the real world. In most situations, we think they should synchronize a compromised state, such as in many kinds of negotiations. We have ever investigated the synchronization and bifurcation in network models of continuous-time systems with the symmetrical coupling matrix of being non-zero row sum, and derived the results that the bifurcation can appear in networks, [11] In this paper, we also do not confine the sum of each row to be zero, but to equal a nonzero constant u, and consider the network being asymmetrical configuration. In this kind of coupled networks of discrete-time systems, we conclude that the network can synchronize to a new state, not to that of limit set determined by the original node state function. Interestingly, at the synchronous state, the networks exhibit bifurcation if we regard the constant u as a bifurcation parameter. The layout of the rest paper is organized as follows. In Sec. 2, we analyze the stability of the networks. Numerical examples are given in Sec. 3, and some remarks are included in Sec. 4. 2 Model Presentation and Synchronization Analysis We consider the following networks model, N x i (t + 1) = f(x i (t)) + c ij x j (t), i = 1, 2,..., N, (1) j=1 where f : R n R n is a continuously differentiable function, x i = (x i1, x i2,..., x in ) T R n are the state variables of node i, N is the total number of nodes. C = (c ij ) N N is the coupling between nodes of the whole networks (it is often assumed that there is at most one connection between node i and another node j, and that there are no isolated clusters, that is C is an irreducible matrix), whose entries c ij are defined as follows: if there is a connection between node i and node j (j i), then set c ij > 0, otherwise c ij = 0 (j i); the diagonal elements of C are all zero, assume the sum of each row of C equals a nonzero constant u, that is, u = N j=1 c ij, for i = 1, 2,..., N. Definition For a real n n matrix Z, 2-norm of Z is induced by 2-norm of vector, i.e., Z 2 = ρ(z T Z), where ρ(.) is the maximum eigenvalue of the matrix. The project supported by the Key Programm Projects of the National Natural Science Foundation of China under Grant No. 70431002, the SRF for ROCS, SEM and the Graduate Student Innovation Foundation of Shanghai University Corresponding author, E-mail address: leecp@online.sh.cn
872 SUN Wei-Gang, CHEN Yan, LI Chang-Pin, and FANG Jin-Qing Vol. 48 Hereafter, the network (1) is said to achieve synchronization if x i (t) s(t) as t +, i = 1, 2,..., N, (2) in which s(t) R n, a stable limit set, satisfies s(t + 1) = f(s(t)) + us(t). Remark 1 The case with u = 0 was studied. [5 10] Suppose that s 0 (t) satisfies x(t+1) = f(x(t)). In most situations, s(t) defined in Eq. (2) and s 0 (t) are not equivalent. In this paper, we assume that these two states are both stable. In the following, we establish a synchronized theorem. Let e i (t) = x i (t) s(t) and substitute it into Eq. (1), then its linearized system reads N e i (t + 1) = Df(s(t))e i (t) + c ij e j (t), (3) j=1 where Df(s(t)) is the Jacobian of f(x(t)) at s(t). above equation can also be rewritten as The e i (t + 1) = Df(s(t))e i (t) + (e 1 (t), e 2 (t),..., e N (t))(c i1,..., c in ) T. (4) Let e(t) = (e 1 (t), e 2 (t),..., e N (t)) R n N, equation (4) can be rewritten as e(t + 1) = Df(s(t))e(t) + e(t)c T. (5) where T stands for matrix transpose. Decomposing the coupling matrix according to C T = SJS 1, where J is the Jordan canonical form with complex eigenvalues λ C and S contains the corresponding eigenvectors s, multiplying Eq. (5) from the right with S, and denoting η = es, we obtain where J is a block diagonal matrix, η(t + 1) = Df(s(t))η(t) + η(t)j, (6) J 1... J h, and J k is the block corresponding to the m k multiple eigenvalue λ k of A, λ k 1 0 0 0 λ k 1 0....... 0 0 λ k 1 0 0 0 λ k Let η = [η 1, η 2,, η h ] and η k = [η k,1, η k,2,..., η k,mk ]. We can rewrite Eq. (6) in component form, where k = 1, 2,..., h. Next we give a synchronized theorem. η k,1 (t + 1) = (Df(s(t)) + λ k )η k,1 (t), η k,p+1 (t + 1) = (Df(s(t)) + λ k )η k,p+1 (t) + η k,p (t), 1 p m k 1, (7) Theorem 1 Consider network model (1). Let λ k = α k + jβ k, where j is the imaginary unit, k = 1, 2,..., h, m k is the multiplicity of λ k and h k=1 m k = N. If there exist constants 0 < γ 0 < γ < 1 and an integer t 0 > 0 such that Df(s(t)) 2 + u γ 0, for t > t 0, (8) where. 2 denotes 2-norm defined above, then the synchronized states of Eq. (1) are asymptotically stable. Proof Firstly we consider the first equation of system (7). Let η k,1 = ξ + jζ, where ξ and ζ are both real vectors, η k,1 2 2 = ξ T ξ + ζ T ζ, from Ref. [12], the following inequality holds η k,1 (t + 1) 2 ( Df(s(t)) 2 + λ k ) η k,1 (t) 2. (9) We know 0 is an eigenvalue of the matrix ui N + C, associated with eigenvector (1, 1,..., 1) T, then we can conclude that u is the eigenvalue of the coupling matrix C, and from the Gershgorin s circle theorem, we get η k,1 (t + 1) 2 ( Df(s(t)) 2 + u) η k,1 (t) 2 γ 0 η k,1 (t) 2, (10)
No. 5 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems 873 for all t > t 0. Therefore there exists a constant M > 0 such that η k,1 (t) 2 Mγ t 0, t > t 0, k = 1, 2,..., h. Next we consider the second equation of Eqs. (7). Without loss of generality, let p = 1 and v(t) = η k,2 (t) 2 γ t, then v(t + 1) = η k,2 (t + 1) 2 γ t 1 ( Df(s(t)) 2 + λ k ) η k,2 (t) 2 γ t 1 + η k,1 (t) 2 γ t 1 γ 0 γ v(t) + M ( γ0 ) t ( γ0 ) t+1 T M ( γ0 ) t v(t ) + (t + 1 T ) < + for all t > t0, (11) γ γ γ γ γ thus v(t) is bounded which implies that η k,2 (t) = O(γ t ) for all k = 1, 2,..., h, then the synchronized states of Eq. (1) are asymptotically stable. Remark 2 The above theorem shows that the local synchronization depends on the dynamical behavior of identical system at each node, which is described by the norm of Df(s(t)) 2, and nonzero sum of each row of the coupling matrix. The coupling matrix discussed above may induce bifurcation, in details, u defined by the above C can be regarded as a bifurcation parameter. This case can be seen from the example taken in the following section. 3 Illustrative Examples and Bifurcation Analysis The following numerical examples demonstrate the theoretical results analyzed in Sec. 2. We discuss two cases: asymmetrical and symmetrical network connection. 3.1 Asymmetrical Network Structure For this case, we consider the network topology being a simple connection with 6 dynamical nodes and the coupling matrix being directed (c ij c ji ), and take the node state function as f(x) = x 2 + 0.6x. From the stability condition (8), we can adjust the constant u to make the network be synchronous, so we introduce three real parameters w 1, w 2, w 3 into the coupling matrix as 0 w 1 0 w 2 0 w 3 0 0 w 2 w 1 w 3 0 C = w 2 w 1 0 0 0 w 3 w 3 0 w 2 0 w 1 0. (12) 0 w 2 0 w 3 0 w 1 w 3 0 w 1 0 w 2 0 Due to the synchronized state s(t), which has a sink fixed point u 0.4 if 0.4 < u < 2.4, and from Eq. (8), we can take w 1 = 0.2, w 2 = 0.1 and w 3 = 0.3, then the network will reach the synchronous state. In the following, we plot the curves of error between x i (t) and s(t), see Fig. 1. Denote the quantity D(t) = s(t) s 0 (t), which is nonzero. Fig. 1 time. Synchronization errors for the network (1) with connection matrix (12) and the convergence of D(t) through
874 SUN Wei-Gang, CHEN Yan, LI Chang-Pin, and FANG Jin-Qing Vol. 48 For other choices of u values, the network generally cannot reach synchronization. 3.2 Symmetrical Network Structure In the above case, we only consider the case that the number of nodes is six, but here we consider the case that the number of nodes is N = 200. For simplicity, the elements of the coupling matrix are defined as follows: c ij = c ji = u/(n 1), and c ii = 0, for i, j = 1, 2,..., N. Obviously the sum of each row is constant u. In the following, we take two different node functions to do the numerical simulations. (i) We first consider the node state function f(x) is logistic map, f(x) = ρx(1 x). When 1 < ρ < 3, the map has a sink fixed point 1 1/ρ and a repeller point 0. We take the constants u = 0.2 and ρ = 2.5, which satisfy the stable condition (8). From the derived theorem, we know the network is synchronous. In Fig. 2, we plot the curves of error between x i (t) and s(t). From Fig. 2(b), we can see the network synchronizes another sink fixed point 1 + (u 1)/ρ. In fact, the present synchronized state is determined by the equation s(t+1) = f(s(t)) +us(t). Through a transformation, the above equation changes into ŝ(t + 1) = (ρ + u)ŝ(t)(1 ŝ(t)). If 1 < ρ + u < 3, ŝ(t) is also stable if s 0 (t) is stable. Fig. 2 Synchronization errors for the network (1) and the convergence of D(t) through time, ρ = 2.5, u = 0.2. Fig. 3 Synchronization errors for the network (1). (a) ρ = 3.2, u = 0.2; (b) ρ = 3.7, u = 0.2. When ρ > 3, the map evolves periodically and even behaves chaotically, but this condition does not satisfy the stability condition, so the network cannot achieve synchronization, see Fig. 3. Figure 4(a) is the bifurcation diagram for the case u = 0. If we fix the ρ value, say, let ρ = 2.6, the bifurcation
No. 5 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems 875 diagram of the synchronized state is presented in Fig. 4(b). Fig. 4 The bifurcation diagram. (a) u = 0; (b) ρ = 2.6. Fig. 5 Synchronization errors for the network (1) and the convergence of D(t) through time. Fig. 6 The bifurcation diagram. (ii) Next we take the node state function as f(x) = x 2 + 0.6x and u = 0.6. We give the curves of error between x i (t) and s(t), see Fig. 5. At the present synchronous state, the network exhibits bifurcation if we regard the constant u as a bifurcation parameter. When u = 0, the system is x x 2 + 0.6x, we know that the fixed point x = 0 is stable, x = 0.4
876 SUN Wei-Gang, CHEN Yan, LI Chang-Pin, and FANG Jin-Qing Vol. 48 is unstable. When 0 < u < 0.4, the fixed point x = 0 is stable, another fixed point x = u 0.4 is unstable; when 0.4 < u < 1, the fixed point x = 0 is unstable, however another fixed point x = u 0.4 is stable. For example, when u = 0.6, the system changes into x x 2 + 1.2x, and x = 0 becomes unstable, while x = 0.2 is stable. At the synchronous state, (u, x) = (0.4, 0) is a bifurcation point, the bifurcation diagram is in Fig. 6, the solid line corresponds to stable case, while the dashed line corresponds to unstable case. 4 Conclusion In this paper, synchronization and bifurcation analyses in coupled networks of discrete-time systems are considered. It is observed that the network exhibits bifurcation at the synchronous state, if we regard the constant u as a bifurcation parameter. At last, numerical examples are given to show the correctness of our theoretical results. Our results may be meaningful for the networks design and helpful for understanding of dynamics in natural systems. References [1] S.H. Strogatz, Nature (London) 410 (2001) 268. [2] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [3] A.L. Barabási and R. Albert, Science 286 (1999) 509. [4] R. Albert and A.L. Barabási, Rev. Mod. Phys. 74 (2002) 47. [5] X.F. Wang and G. Chen, Int. J. Bifurc. Chaos 12 (2002) 187. [6] C.P. Li, W.G. Sun, and J. Kurths, Phys. A 361 (2006) 24. [7] C.P. Li, W.G. Sun, and D. Xu, Prog. Theor. Phys. 114 (2005) 749. [8] C.P. Li, W.G. Sun, and J. Kurths, Phys. Rev. E, in press. [9] J. Zhou and T.P. Chen, IEEE Trans. CAS-I 53 (2006) 733. [10] G. Rangarajan and M. Ding, Phys. Lett. A 296 (2002) 204. [11] W.G. Sun, C.X. Xu, C.P. Li, and J.Q. Fang, Commun. Theor. Phys. (Beijing, China) 47 (2007) 1073. [12] J.H. Lü, X. Yu, and G. Chen, Phys. A 334 (2004) 281.