Effects of Scale-Free Topological Properties on Dynamical Synchronization and Control in Coupled Map Lattices

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1 Commun. Theor. Phys. (Beijing, China) 47 (2007) pp c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Effects of Scale-Free Topological Properties on Dynamical Synchronization and Control in Coupled Map Lattices CHEN Wei, 1 FANG Jin-Qing, 2, and KANG Ge-Wen 1 1 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu , China 2 China Institute of Atomic Energy, P.O. Box , Beijing , China (Received March 15, 2006) Abstract In the paper, we study effects of scale-free (SF) topology on dynamical synchronization and control in coupled map lattices (CML). Our strategy is to apply three feedback control methods, including constant feedback and two types of time-delayed feedback, to a small fraction of network nodes to reach desired synchronous state. Two controlled bifurcation diagrams verses feedback strength are obtained respectively. It is found that the value of critical feedback strength γ c for the first time-delayed feedback control is increased linearly as ε is increased linearly. The CML with SF loses synchronization and intermittency occurs if γ > γ c. Numerical examples are presented to demonstrate all results. PACS numbers: k, Da, Ra Key words: scale-free network, coupled map lattice, dynamical synchronization, feedback control, timedelayed feedback 1 Introduction Over the past decades, dynamical synchronization and control in complex dynamical systems have attracted much attention. A typical case is the coupled map lattices (CML), which are often used as a convenient model to study characteristics of some spatiotemporal systems. [1] In the past, however, most of these works have been concentrated on the CML, in which it is assumed that the coupling configuration is completely regular or random. For example, Ding and Yang discussed synchronization in the CML with global coupling and nearest neighbor coupling. [2] Fang and Ali proposed several nonlinear feedback functions to realize control and synchronization of spatiotemporal chaos in the CML with different connection and coupled coefficients. [3 5] Konishi described a decentralized delayed-feedback control for a one-way open CML. [6] Huang proposed an adaptive adjustment method to stabilize a CML system. [7] A great change of the studies above has been taken place in recent years since the small-world and scale-free networks were discovered in complex networks in 1998 and 1999 respectively. [8,11] It is well known that regular networks and random networks are both useful idealizations, but interactions in real world are neither completely regular nor completely random, but lie in somewhere between the extremes of order and randomness. In order to describe the transition from a regular network to a random network, Watts and Strogatz proposed a simple model that can be tuned through this middle ground: [8] a regular lattice where the original links are replaced by random ones with some probability. They found the concept of small-world and conjectured that the same two properties (short paths and high clustering) would hold also for many natural and technological networks. They also conjectured that dynamical systems coupled in this way would display enhanced an signal propagation speed, synchronizability, and computational power, as compared with regular lattices of the same size. Therefore, many empirical examples of small-world networks have been documented recently, in fields ranging from cell biology to society. On the theoretical side, small-world networks are turning out to be a rorschach test in disciplines. Much attention has been paid to the CML with small-world topology, for example, Gade studied the synchronization of the CML with small-world interactions. [9] Synchronization in the CML with small-world delayed interactions was investigated by Li. [10] Now, we will focus on the study of the CML with scale-free, where the degree distribution obeys a power law form. Scale-free networks, like the Internet and the WWW, are characterized by an uneven distribution of degree. Besides the nodes of these networks with a random pattern of connections, some nodes act as very connected hubs, a fact that dramatically influences the way of how the network operates. The scale-free network can be described as follows. [11] Staring with m 0 nodes, at every time step a new node is introduced, which is connected to m already existing nodes, we usually take m = m 0. The probability i that a new node is connected to node i depends on the degree k i of node i : i = k i/ j k j. The project supported by the Key Program of National Natural Science Foundation of China under Grant No and National Natural Science Foundation of China under Grant Nos and Corresponding author, fjq96@126.com

2 362 CHEN Wei, FANG Jin-Qing, and KANG Ge-Wen Vol. 47 In this study, we investigate dynamical bifurcation, intermittency, chaos, synchronization, and control of dynamical behaviors in the CML with scale-free topology using different feedback control methods. We consider the network model with one-dimensional CML as follows: x i (t + 1) = (1 ε)f(x i (t)) + ε N a ij f(x j (t)), (1) where s the total number of chaotic oscillators, t is the discrete time step, i = 1,..., N and j are the lattice sites, ε is the coupling strength satisfying 0 ε < 1, f(x(t)) = 1 ax 2 is the logistic map, parameter a is fixed to 1.9 at which the logistic map shows chaotic state. The number of neighbors of node i is denoted by. The matrix A = {a ij } suggests the scale-free connection topology: if map i sends a signal to j, a ij =1; otherwise, a ij =0. a ii = 0 for all i, and = N a ij. In this paper, we assume the CML is an undirected and un-weighted network with scale-free, thus, a ij = a ji. 2 Dynamical Synchronization in the CML with SF Topology In this section, we analyze some conditions for dynamical synchronization in the CML with scale-free (SF) topology. A synchronized state of network is that the states of all nodes are identical: x i (t) = s(t) for all i. (2) In Ref. [2], Ding and Yang have deduced the explicit conditions for computing the parameter values at which the synchronous state becomes unstable in CML with global coupling. When ε , the CML can reach the synchronous state. Barahona and Pecora investigated the synchronization conditions in a small-world network by master stability functions. [12] But there is something different in the CML with SF topology because the SF topology can affect the synchronizability of the system seriously. In order to observe the behavior of synchronous state, we will work in the variable difference space, so we have x i (t) = x i (t) 1 N x j (t), i = 1,..., N. (3) N The variable plotted at the lattice site i is x i (t), where the line indicates the spatial average of the variable x. The absolute value of this quantity measures the distance between the system state and the synchronization manifold. Figure 1 shows the time series of x i (t) when ε = 0.5, the state of the CML system with SF topology. Figures 2(a) and 2(b) indicate the time series of x i (t) for time steps and lattice sites respectively. It is noticed from Figs. 1 and 2 that fully developed asynchronous chaos occurs, so the synchronization criterion of the original global CML is not held for the CML with SF. Recently, some papers investigated the relation between eigenvalue of the coupling Laplacian matrix and synchronizability of network. [10] The Laplacian matrix is defined to be L ij = 1 if nodes i and j are connected, L ii = k i if node i connected to k i other nodes, and L ij = 0 otherwise. So the synchronization criterion is determined by the second smallest eigenvalue λ 2 of the coupling Laplacian matrix, the larger λ 2 is, the easier it is to synchronize. [10] In the SF networks, the second smallest eigenvalue will become larger with increase of m 0. In other words, the change of m 0 may influence synchronizability in the CML with SF topology. In our numerical simulations, the CML with SF may reach synchronous state when m 0 > 4. With different m 0, we can find a critical value ε c, such that when ε ε c, the CML with SF enters synchronous state. However, the synchronous state may be lost if the coupling strength is too strong. That is to say, there is a value region represented by ε, as shown in Table 1, such that if ε is not in the region, the CML with SF cannot be synchronized. Fig. 1 The time series of x i(t): ε = 0.5, m = m 0 = 8, N = 100. Fig. 2 (a) The time series of x i(t) for time steps; (b) The time series of x i(t) for lattice sites.

3 No. 2 Effects of Scale-Free Topological Properties on Dynamical Synchronization and Control in 363 Table 1 The value region of coupling strength when the CML with SF reaches synchronous state with different m 0 in a realization. N = 100. m 0 coupling strength λ It is clear that in Table 1 the value region of coupling strength expands with increase of the second smallest eigenvalue and m 0. 3 Control of Spatiotemporal Chaos Synchronization in the CML with SF Feedback control can effectively suppress spatiotemporal chaos in regular dynamical system. [13] Recently, Wang and Chen applied pinning feedback control method to an array of Chua s oscillators with SF topology, [14] and showed that the specifically pinning of most highly connected nodes need a smaller number of local controllers as compared with the randomly pinning scheme. In the section, we investigate different feedback control strategies in order to find a better way to suppress and control spatiotemporal chaos in CML with SF topology. First of all, we will consider a simplest control method constant feedback control. [15] This feedback signal does not require any priori knowledge of the dynamics of the system and it does not alter any of the system parameters explicitly. Now, we apply a constant feedback to a small number of the nodes in the CML with SF, and suppose that the nodes i 1, i 2,..., i l are selected, x ik (t + 1) = (1 ε)f(x ik (t)) + ε x ik (t + 1) = (1 ε)f(x ik (t)) + ε N N a ij f(x jk (t)) c, k = 1, 2,..., l, (4a) a ij f(x jk (t)), k = l + 1, l + 2,...,, (4b) where c represents a constant feedback strength, which is always fixed to be a small negative or positive number. [16] Here, we choose c = 0.7 as a numerical example. Figures 3 and 4 show the behavior of the CML with SF using the constant feedback. Figure (3a) shows the application of a constant feedback to every node; Figure (3b) indicates the effect of placing the constant feedback on some most highly connected nodes; Figure (3c) uses the randomly pinning scheme. Both Figures (3b) and (3c) select 20 nodes to be controlled. Similarly, figure 4 indicates the time series of x i (t) for time steps and lattice sites respectively. Fig. 3 The time series of x i(t): ε = 0.5, m 0 = 8, N = 100. (a) Applying the constant feedback controllers on every nodes; (b) On some most highly connected nodes; (c) On some randomly chosen nodes.

4 364 CHEN Wei, FANG Jin-Qing, and KANG Ge-Wen Vol. 47 We can conclude from Figs. 3 and 4 that in order to reach synchronous state by the constant feedback control, we must place the feedback controllers on every nodes. Although the latter two schemes cannot suppress chaos entirely, the first one only can achieve in a sense suppression of chaos. Furthermore, from our numerical simulation, when ε < 0.18, the CML with SF cannot be synchronized no matter what constant feedback used. Figure 5 shows the controlled bifurcation diagram verses the constant feedback strength c for the controlled CML with SF at a chosen site i = 20 (controlled bifurcation diagram of the other nodes is similar too). It can be seen clearly from Fig. 5 that it is a typical anti-bifurcation diagram with increasing of constant feedback strength. In this controlled bifurcation diagram dynamical complex behaviors of a lattice site are changed dramatically from spatiotemporally chaos to intermittency, periodic and fixed-point. Thus it is easy to find corresponding constant feedback strength in Fig. 5 to get any desired states in the CML with SF topology. We can realized chaos control and synchronization via a simplest constant feedback control only based on controlled bifurcation diagram in Fig. 5. Fig. 4 The time series of x i(t) for time steps and lattice sites respectively. (a) Applying the constant feedback controllers on every nodes. (b) On some most highly connected nodes. (c) On some randomly chosen nodes. Fig. 5 The bifurcation diagram for a chosen site i = 20 as a function of feedback strength c. ε = 0.5, m 0 = 8, N = 100. Data for t = 400 consecutive time steps are plotted after eliminating the transients.

5 No. 2 Effects of Scale-Free Topological Properties on Dynamical Synchronization and Control in 365 For comparison with constant feedback, we investigate a time-delayed feedback control to the CML with SF as follows: x ik (t + 1) = (1 ε)f(x ik (t)) + ε x ik (t + 1) = (1 ε)f(x ik (t)) + ε N N a ij f(x jk (t)) γ(x ik (t) x ik (t 1)), k = 1, 2,..., l, (5a) a ij f(x jk (t)), k = l + 1, l + 2,...,, (5b) where controller γ(x ik (t) x ik (t 1)) is called as the first time-delayed feedback. The feature and advantage of this method are that each local controller does not use other site information. γ is a small number and represents the control strength. [17] In order to derive the synchronization condition from the system, we introduce another form of the CML and method represented by Atay and Jost. [18,19] We consider a finite discrete set which carries a neighborhood relationship, denoting as i j when i and j are neighbors. Thus equation (5) can be recast as [ 1 ] x ik (t + 1) = f(x ik (t)) + ε f(x jk (t)) f(x ik (t)) γ(x ik (t) x ik (t 1)), k = 1, 2,..., l, (6a) j j i [ 1 x ik (t + 1) = f(x ik (t)) + ε j j i ] f(x jk (t)) f(x ik (t)), k = l + 1, l + 2,...,. (6b) We consider the synchronized state s(t), which evolves according to s(t+1) = f(s(t)) and has a perturbation of the form x i (t + 1) = s(t + 1) + δα k (t + 1)u k i (t + 1), (7) where u k is orthonormal eigenmodes of the Laplacian matrix L, defined by ( L v) = (1/ ) j i (v j v i ), with corresponding eigenvalues λ k. The eigenvalue of L represents the connection topology as mentioned before. The solution of Eq. (7) is stable with small enough δ, α k (t) 0 for t. Considering the first time-delay feedback, and inserting Eq. (7) into Eq. (6a), and expanding about δ = 0 yields α k (t + 1) + γ(α k (t) α k (t 1)) = f (s(t))α k (t)(1 ελ k ). (8) The sufficient local stability condition is 1 lim T T log α k(t ) α k (0) < 0. (9) If there does not exist any feedback, that is to say, if γ = 0, equation (9) can be rewritten as T 1 1 lim T T log t=0 α k (t + 1) α k (t) Then, from Eq. (9), equation (10) becomes Here, < 0. (10) T 1 1 log 1 ελ k + lim log f (x(t)) < 0. (11) T T t=0 T 1 1 µ = lim log f (x(t)) < 0 (12) T T t=0 is the Lyapunov exponent of f and the stability condition is e µ (1 ελ k ) < 1. (13) More detailed analysis can be found in Refs. [15] and [16]. Fig. 6 Bifurcation diagram for a controlled site as a function of γ. ε = 0.5, m 0 = 8, N = 100. Data for t = 400 consecutive time steps are plotted after eliminating the transients. Comparing the first time-delay feedback with the case without any delay feedback, the first time-delay feedback depends on the values of α k (t) and α k (t 1). Cancellation may be obtained due to γ(α k (t) α k (t 1)), thus the absolute value may be smaller. When colligating these two cases, the synchronization conditions may be easier to achieve if we choose an appropriate feedback strength and apply controllers on some nodes. As shown in Ref. [17], we are able to stabilize the whole system with the value of γ decreased. In other words, if we control the system from chaotic state to synchronization, we cannot choose γ at will. For example, when ε = 0.5, we can only find a critical value of γ denoted by γ c such that we can stabilize the whole system if and only if γ γ c. Figure 6

6 366 CHEN Wei, FANG Jin-Qing, and KANG Ge-Wen Vol. 47 shows the other controlled bifurcation diagram for a controlled node as a function of γ, which demonstrates our simulation results. Moreover, we found that the value of γ c increases almost linearly as ε is increased linearly. Figure 7 gives the relationship between the critical value of γ when the system reach synchronous state and value of ε, where m 0 = 8, N = 100, pinning number is randomly chosen as 10 nodes, and here we use the data from Table 1. The conclusion can be drawn from Fig. 7 that this relation is independent of network size. In addition, similar to the results discussed in constant feedback control strategy, when ε < 0.38, no matter how small γ is, the system cannot be synchronized. Correspondingly, if γ > γ c, the system loses synchronization and intermittency occurs. [20] We also observed such phenomena, as shown in Figs. 8 and 9, where ε = 0.5, m 0 = 8, γ = 0.2 > 0.13, N = 100, and pinning number is randomly chosen as 5 nodes. Obviously, the intermittency occurs in the controlled nodes. Fig. 7 The critical value of γ verses ε. Fig. 8 Intermittency occurs in the five controlled nodes. Fig. 9 Intermittency occurs in the five controlled nodes. (a) The time series of x i(t) for time steps; (b) The time series of x i(t) for lattice sites. Fig. 10 Stabilized period-two for one of the controlled nodes. Furthermore, as shown in Fig. 10, after some time, these controlled nodes exhibit stabilized period-two behavior. It implies that the control signal cannot go to zero and may lead to intermittency if value of γ is chosen too big. In addition, it will be quite different for the synchronous states in the CML system whether SF feedback is used or not. If it is used, the system will converge at a certain point regardless of network size or value of m 0. Otherwise, without feedback control, it is possible for the system to reach synchronous state when the value of ε enters synchronization region (see Table 1), but it cannot converge at a certain point, and obviously such a synchronization is unstable. Therefore, we should use feedback control to drive the system converging to desired state (such as at a fixed point and periodic state), then dynamical synchronization of the system is stabilized. It is also observed that the specifically pinning of the most highly connected nodes has the same control results as the randomly pinning. Figure 11 shows spatiotemporal distribution of x for ε = 0.8 (from Table 1). Figure 11(a) gives the unstable synchronization of the system without any feedback, and figure 11(b) represents that the stabilized synchronization is reached by using feedback control for choosing 10 controlled nodes randomly under the same conditions. Furthermore, we plot distribution of x for time steps in Fig. 12. Figure 13 shows the relationship between the covergence value and

7 No. 2 Effects of Scale-Free Topological Properties on Dynamical Synchronization and Control in 367 ε. It shows that the covergent value is descrased when the coupling strength ε is increased. From Figs. 11 and 12, we conclude that feedback not only suppresses chaos but also can control the CML with SF topology to stabilized desired states. The numerical simulation results demonstrate that the first time-delayed feedback control method is effective for controlling synchronization. Fig. 11 Spatiotemporal distribution of x for ε = 0.8, m 0 = 8, and N = 100. (a) Unstable synchronization without any feedback; (b) Stabilized synchronization using feedback control. Fig. 12 Distribution of x for time steps. (a) Unstable synchronization without any feedback; (b) Stabilized synchronization using feedback control. Fig. 13 The relationship between the convergence value and ε. All dates are averaged over 100 realizations, N = 100. Now, we consider the second time-delayed feedback method, which involves other site information. The controlled system is represented by x ik (t+1) = (1 ε)f(x ik (t)) + ε N ( 1 N a ij f(x jk (t)) γ a ij x ik (t) 1 N ) a ij x ik (t 1), k = 1, 2,..., l, (14a) x ik (t + 1) = (1 ε)f(x ik (t)) + ε N a ij f(x jk (t)), k = l + 1, l + 2,...,. (14b) This type of time-delayed feedback control is plausible in actual experimental system as the feedback required can be acquired from experiments. We introduce K i, which represents the set of labels of the neighbors of node i, k i represents the number of such neighbors, so equation (14) can be recast as [ 1 ] x ik (t + 1) = f(x ik (t)) + ε f(x jk (t)) f(x ik (t)) γ k = 1, 2,..., l, (15a) j j i j K i k j (x ik (t) x ik (t 1)),

8 368 CHEN Wei, FANG Jin-Qing, and KANG Ge-Wen Vol. 47 [ 1 ] x ik (t + 1) = f(x ik (t)) + ε f(x jk (t)) f (x ik (t)), k = l + 1, l + 2,...,. (15b) j j i Inserting Eq. (7) into Eq. (15a) and expanding δ = 0 yields α k (t + 1) + γ k j (α k (t) α k (t 1)) j K i = f (s(t))α k (t)(1 ελ k ). (16) Similarly, we may also get cancellation owing to (γ/ ) j K i k j (α k (t) α k (t 1)). Fig. 14 Comparison of the first time-delayed feedback control with the second one for ε = 0.5, m 0 = 8, γ = 0.1, N = 100, and t = 400, and pinning number is randomly chosen as 10 nodes. (a) The first time-delayed feedback; (b) The second time-delayed feedback. Thus the synchronization conditions may be easily satisfied, as our simulation shows, the control results are almost the same as in the case of the first time-delayed feedback control strategy. But, the second time-delayed feedback control will take more time to reach synchronous state. In Fig. 14, we show the comparison of the first timedelayed feedback control with the second one for ε = 0.5, m 0 = 8, γ = 0.1, N = 100, and t = 400, and pinning number is randomly chosen as 10 nodes. Figure 14(a) represents the value distribution of x for the first time-delayed feedback, and figure 14(b) for the second time-delayed one. Clearly, some nodes in the system exist synchronization error for the second time-delayed feedback under the same conditions, correspondingly, for the first timedelayed feedback synchronization can be realized quickly and a better control result can be obtained. 4 Conclusions In the paper, we have investigated dynamical synchronization and its control in the CML with SF topology. An approximate stability criterion for the synchronization is discussed. To control the CML with SF topology to reach desired synchronous states, three feedback control methods, constant feedback control and two types of time-delayed feedback control, are suggested and investigated and compared. Furthermore, two controlled bifurcation diagrams verse constant feedback strength and coupling feedback strength of the first type time-delayed feedback are obtained respectively. We find that the value of critical feedback strength γ c for the first time-delayed feedback control is increased linearly with increase of ε linearly, but it is independent of network size. In addition, when ε < 0.38, no matter how small γ is, the CML with SF cannot be synchronized. Correspondingly, if γ > γ c, the CML with SF loses synchronization and intermittency occurs. Our numerical results provide some interesting phenomena and rich information for further studying dynamical synchronization and control in complex networks with scale-free properties. Some of issues are still open for complex networks. Acknowledgment The authors thank Drs. Atay and Chun-Guang Li for their useful advices and discussions. References [1] K. Kaneko, Chaos 2 (1992) 279. [2] M. Ding and W. Yang, Phys. Rev. E 56 (1997) [3] J.Q. Fang and M.K. Ali, Disc. Dyn. In Natu. and Soci. 1 (1998) 283. [4] J.Q. Fang and M.K. Ali, Chin. Phys. Lett. 14 (1997) 823. [5] J.Q. Fang and M.K. Ali, Scin. Tech. 8(3) (1997) 129; Nucl. Scin. Tech. 8(4) (1997) 193. [6] K. Konishi, M. Hirai, and H. Kokame, Phys. Rev. E 58 (1997) [7] W.H. Huang, Phys. Rev. E 61 (2000) R1012. [8] D.J. Watts and S.H. Strogats, Nature (London) 393 (1998) 440; D.J. Watts, Small worlds: The Dynamics of Networks Between Order and Randomness, Princeton University Press, Princeton, New Jersey (1999). [9] P.M. Gade and C.K. Hu, Phys. Rev. E 62 (2000) [10] C.G. Li, S.W. Li, X.F. Liao, and J.B. Yu, Physica A 335 (2004) 365. [11] A.L. Barabasi and R.A. Albert, Science 286 (1999) 509. [12] M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89 (2002) [13] G. Hu and Z. Qu, Phys. Rev. Lett. 72 (1994) 68. [14] X.F. Wang and G.R. Chen, Phycica A 301 (2002) 521. [15] Chai Wah Wu, arxiv:nlin.cd/ (2003); Takashi Nishikawa, Adilson E. Motter, Ying-Cheng Lai, and Frank C. Hoppensteadt, Phys. Rev. Lett. 91 (2003) [16] S. Parthasarathy and S. Sinha, Phys. Rev. E 51 (1995) [17] P. Paramananda, M. Hildebrand, and E. Eiswirth, Phys. Rev. E 56 (1997) 239. [18] F. Atay, J. Jost, and A. Wende, Phys. Rev. Lett. 92 (2004) [19] J. Jost and M.P. Joy, Phys. Rev. E 65 (2002) [20] M. Ding and W. Yang, Phys. Rev. E 52 (1995) 207.