cluster synchronization in a complex dynamical ... - Semantic Scholar
Jrl Syst Sci & Complexity (2008) 21: 20–33
CLUSTER SYNCHRONIZATION IN A COMPLEX DYNAMICAL NETWORK WITH TWO NONIDENTICAL CLUSTERS∗ Liang CHEN · Jun’an LU
Received: 6 April 2007 / Revised: 18 October 2007 c 2008 Springer Science + Business Media, LLC Abstract This paper further investigates cluster synchronization in a complex dynamical network with two-cluster. Each cluster contains a number of identical dynamical systems, however, the subsystems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. Complete synchronization within each cluster is possible only if each node from one cluster receives the same input from nodes in other cluster. In this case, the stability condition of one-cluster synchronization is known to contain two terms: the first accounts for the contribution of the inner-cluster coupling structure while the second is simply an extra linear term, which can be deduced by the “same-input” condition. Applying the connection graph stability method, the authors obtain an upper bound of input strength for one cluster if the first account is known, by which the synchronizability of cluster can be scaled. For different clusters, there are different upper bound of input strength by virtue of different dynamics and the corresponding cluster structure. Moreover, two illustrative examples are presented and the numerical simulations coincide with the theoretical analysis. Key words Cluster synchronization, complex dynamical network, connection graph, stability.
1 Introduction The properties of complex dynamical network and their physical significance have been further investigated since the the discovery of “small world” networks[1] and “free scale” networks[2] . One common property of network, cluster (community) structure, is an important topic and has been widely applied in biology, physics, computer graphics and sociology[3−4] . A network is composed of many clusters and the connection of nodes in one cluster is more close than that of nodes in the other clusters. For example, WWW can be seen as a network constituted by many web stations, and all web stations are discussing the same interesting topics of their own[5] . Networks of dynamical systems have been used in physics, biology, and other science branches. The simplest and most striking interaction between dynamical systems is their synchronization. Synchronization is a very common phenomenon in real systems and has been an Liang CHEN · Jun’an LU School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email: [email protected]; [email protected]. ∗ The work is supported by the National Natural Science Foundation of China under Grant Nos. 70771084 and 60574045, and the National Basic Research Program of China under Grant No. 2007CB310805.
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
21
active research topic since the times of Huygens[6] . Over the last few decades, synchronization has been intensively studied in various disciplines, such as biological, epidemiologic, social, economic, mathematical, and engineering science[7−9] . In particular, as a special synchronization phenomenon, cluster synchronization in networks, i.e., the synchronization of subnetworks or groups of networks has received much attention in recent years[10−18] . In fact, the notion of cluster (or partial) synchronization is quite recent. The original study of cluster synchronization was from the natural physical phenomenon rising in coupled oscillators and coupled map lattice. It stands for the intermediary states where the oscillators synchronize with one another in distinct groups, while there is no synchronization amongst the groups[10−11] . Beside cluster synchronization, the coupled system can obviously display the well-known states of spatiotemporal chaos and complete (full) synchronization[12−13] . In [14– 15], self-organized cluster synchronization and drive cluster synchronization of a coupled map network (CMN) have been studied in detail. In [16], the persistence of cluster synchronization regimes in lattices of nonidentical chaotic oscillators was systematic analyzed. It is shown that, by numerical study, in lattices of Lorenz and R¨ ossler systems the cluster synchronization regimes are stable and robust against up to 10% to 15% parameter mismatch. Qin and Chen[17] proposed a coupling scheme to stabilize the selected cluster synchronization patterns for coupled Josephson equations. Ma and Liu[18] showed the cluster synchronization patterns could be achieved by constructing a special coupled matrix for a dynamical network with identical oscillators. With the development of modern science and technology, it is significant that the cluster synchronization can be applied to the laser technology[12] , biological engineering[19] , and secure communications[12−13] . In above, most of the current works on cluster synchronization were based on the networks of identical oscillators. However, many known facts tell us that the real-world networks are composed of various oscillators with different dynamics. Inspired by this, we are stimulated to investigate cluster synchronization in dynamical networks consisting with nonidentical clusters. The dynamics of nodes between nonidentical clusters can be totally different including the dimensions which are different from that of [16]. In this article, we introduce a dynamical network with two nonidentical clusters. Each cluster contains a number of identical dynamical systems, however, the sub-systems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. The global stability of cluster synchronization can be achieved with the “same-input” condition (all nodes in the same cluster can receive the same input as the nodes of outside clusters do). The synchronization criteria for two nonidentical clusters are obtained by using the connection graph stability method which is a very useful tool for dealing with the synchronization of network with continuous dynamical oscillators[20−22] . For different clusters, there are different upper bounds of the input strength by the different dynamics, the coupling strength and the corresponding inner-cluster structure. The stable synchrony can be achieved in a cluster as long as the input strength is strong enough. These synchronization criterions can provide new insights into the key factors leading to self-organization phenomenon between nonidentical clusters in nature. The remaining part is organized as follows. A complex dynamical network consisting of two nonidentical clusters is briefly introduced in Section 2. The synchronization conditions of three cases in the inner clusters and two examples of numerical simulation are given in Sections 3 and 4, respectively. Conclusions are finally drawn in Section 5.
2 Network Modeling Consider a dynamical network with two clusters, which are composed of n1 and n2 linearly and diffusively coupled identical nodes, with each node being a d and d-dimensional dynamical
LIANG CHEN · JUN’AN LU
22 oscillator as
x˙ = F (x),
y˙ = H(y),
(1)
where the above differential equations are defined in a basic state space X and Y , respectively. The proposed two-cluster network is described by ⎧ n1 n1 +n2 ⎪ ⎨ x˙ = F (x ) + g Γ x + c [I i i ik k ik d×d · P yk · I d×d − Γ xi ], (2) k=1 k=n1 +1 ⎪ ⎩ i = 1, 2, · · · , n1 ; ⎧ n1 +n2 n1 ⎪ ⎨ y˙ = H(y ) + g P y + c [I i i k ik d×d · Γ xk · I d×d − P yi ], ik (3) k=n1 +1 k=1 ⎪ ⎩ i = n1 + 1, n1 + 2, · · · , n1 + n2 . Here, the matrices Γ and P determine which variables of the oscillators used to couple. For s clarity, we shall consider the diagonal matrices Γ = diag(1, 1, · · · , 1, 0, · · · , 0)d×d and P = s
diag(1, 1, · · · , 1, 0, · · · , 0)d×d . Note that the dimension of vectors Γ xi is not equal to that of P yj , to keep the coherence of the vectors’ dimensions in (2) and (3) such that the coupling with two different dynamical systems is reasonable, we introduce the generalized matrices I d×d and I d×d whose left diagonal elements are 1, otherwise 0, for example, ⎛ ⎞
1 0 1 0 0 , I 3×2 = ⎝ 0 1 ⎠ . I 2×3 = 0 1 0 0 0 Equivalently, (2) and (3) can be rewritten as ⎧ n1 n1 +n2 ⎪ ⎨ x˙ = F (x ) + Γ [g x − a x ] + c (I i i ik k i i ik d×d · P yk · I d×d ), k=1 k=n +1 1 ⎪ ⎩ i = 1, 2, · · · , n1 ; ⎧ n1 +n2 n1 ⎪ ⎨ y˙ = H(y ) + P [g y − a y ] + c (I · Γx · I ), ⎪ ⎩
i
i
ik k
k=n1 +1
i i
ik
k=1
d×d
k
d×d
(4)
(5)
i = n1 + 1, n1 + 2, · · · , n1 + n2 , n1 +n2 n1 where ai = k=n1 +1 cik for i = 1, 2, · · · , n1 and ai = k=1 cik for i = n1 +1, n1 +2, · · · , n1 +n2 . Then, the connectivity matrix of network has the form
Gn1 ×n1 C n1 ×n2 An1 ×n1 O n1 ×n2 D= + , (6) C n2 ×n1 Gn2 ×n2 O n2 ×n1 An2 ×n2 where G = (gij )n1 ×n1 and G = (g ij )n2 ×n2 are respectively the inner connectivity matrices of clusters (2) and (3) representing the coupling strength and topological structure of the corresponding clusters, and assumed to be symmetric and satisfy zero row-sums: ⎧ n1 ⎪ ⎪ ⎪ g = g ≥ 0, i = j, g = − gij , i = 1, 2, · · · , n1 , ij ji ii ⎪ ⎪ ⎨ j=1,j=i (7) n 1 +n2 ⎪ ⎪ ⎪ ⎪ i = j, g ii = − g ij , i = n1 + 1, · · · , n1 + n2 . ⎪ ⎩ g ij = g ji ≥ 0, j=n1 +1,j=i
23
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
Correspondingly, C = (cij )n1 ×n2 and C = (cij )n2 ×n1 are nonnegative matrices, called the outer connectivity matrices of clusters (2) and (3) being the connecting status between two clusters, and the diagonal matrices A = diag(−a1 , −a2 , · · · , −an1 ) and A = diag(−an1 +1 , −an1 +2 , · · · , − an1 +n2 ) show the input strengths of clusters (2) and (3), separately. Up to now, we have established the model of a complex dynamical network with two nonidentical clusters. Here, we do not assume irreducibility of the inner-cluster matrices G and G in models (2) and (3). Therefore, there are many cases of graph connectivity in the inner clusters (2) and (3). We mainly consider cluster synchronization in three cases as shown in Figure 1. First, in the inner cluster, there are not any connection between nodes, that is, all nodes in the cluster are isolated if without connection with outside clusters’ nodes. Second, the graph of cluster is connectivity, that is, the inner-cluster matrix of cluster is irreducible. Third, there exist some isolated nodes in the cluster, and the other nodes are arbitrarily connected.
[X
Y\
D (a) Figure 1
[X
\Y
(b) E
Y \
[X
(c) F
Three cases of connectivity in the network with two clusters. (a) No connections between nodes in inner clusters. (b) Arbitrary Connections of the nodes in inner clusters. (c) The coexistence of isolated nodes and nodes with arbitrary connections in inner cluster.
3 Cluster Synchronization of Complex Dynamical Networks In this section, we use the connection graph stability method to prove some cluster synchronization theorems of a network with two-cluster. Firstly, we present some necessary definitions. Definition 3.1 A network with clusters is said to realize cluster synchronization, if the nodes in the same cluster synchronize with each other. In systems (2) and (3), it is said that the network realizes cluster synchronization, if for xi (t, ( x0 , y 0 )) and xj (t, ( x0 , y 0 )) of arbitrary nodes i and j in cluster (2), ||xi (t, ( x0 , y 0 )) − xj (t, ( x0 , y 0 ))|| → 0 as t → 0; and for yi (t, ( x0 , y 0 )) and yj (t, ( x0 , y 0 )) of arbitrary nodes i and j in cluster (3), ||yi (t, ( x0 , y 0 )) − yj (t, ( x0 , y 0 ))|| → 0 as t → 0, where ( x0 , y 0 ) denotes arbitrary T T T T initial states (xT (0), · · · , x (0), y (0), · · · , y (0)) . n1 n1 +n2 1 n1 +1 T T T n1 Definition 3.2 S = { x = (xT 1 , x2 , · · · , xn1 ) ∈ X |xi = xj = s ∈ X, i, j = 1, 2, · · · , n1 } and S = {y = (ynT1 +1 , · · · , ynT1 +n2 )T ∈ Y n2 |yi = yj = s ∈ Y, i, j = n1 + 1, n1 + 2, · · · , n1 + n2 } are called the synchronization manifold of clusters (2) and (3). Definition 3.3 A matrix B = (bij )m×n is said to satisfy condition A1, if its elements satisfy bik = bjk , i, j = 1, 2, · · · , m; k = 1, 2, · · · , n.
In systems (4) and (5), if the outer connectivity matrices of clusters (2) and (3), C = (cij )n1 ×n2 and C = (cij )n2 ×n1 satisfy condition A1, then the input strengths of nodes in the same cluster are equal, that is, ai = aj = a, i, j = 1, 2, · · · , n1 ; and ai = aj = a, i, j = n1 + 1, · · · , n1 + n2 . In this case, condition A1 is also called the “same-input” condition.
24
LIANG CHEN · JUN’AN LU
In the following, the global stability of synchronization manifolds under three types of innercluster structure will be investigated inventively. 3.1
Case 1: Cluster Synchronization in the Case of No Connections Between Nodes in the Inner Clusters
Consider a special case of network composed of cluster (2) and cluster (3) as shown in Figure 1(a). In the inner cluster (2) or (3), all nodes are unattached, that is, there are not any connection between the nodes of clusters (2) or (3). In this case, we have the following assumption. Assumption 3.1 There are not any connection between nodes in the inner clusters (2) and (3), that is, G = (gij )n1 ×n1 = On1 ×n1 and G = (g ij )n2 ×n2 = On2 ×n2 . For clarity, we only consider cluster (2) to study cluster synchronization of the network, and the corresponding result to cluster (3) will be obtained obviously. For cluster (2), we introduce the notation for the differences Xij = xj − xi ,
i, j = 1, 2, · · · , n1 ,
(8)
and by Assumption 3.1, we obtain the difference equations as the form X˙ ij = F (xj ) − F (xi ) − aΓ Xij .
(9)
Using a compact vector notation for the function difference 1 d F (xj ) − F (xi ) = F (βxj + (1 − β)xi )dβ 0 dβ 1 = DF (βxj + (1 − β)xi )dβ Xij ,
(10)
where DF is a d × d Jacobi matrix of F , we obtain 1 X˙ ij = DF (βxj + (1 − β)xi )dβ − aΓ Xij .
(11)
0
0
It is obvious that the global stability of the origin O = {Xij = 0, i, j = 1, 2, · · · , n1 } amounts to the global stability of synchronization manifold S. Consider Lyapunov functions of the form · Xij , i, j = 1, 2, · · · , n1 , ij = 1 X T · M W (12) 2 ij = diag m 1 , m where M 1, m 2, · · · , m s, M 2 > 0, · · · , m s > 0, and the (d − s) × (d − s) 1 > 0, m matrix M1 is positive definite. The derivatives of the Lyapunov functions with respect to the system (11) are 1 ˙ T W ij = Xij M DF (βxj + (1 − β)xi )dβ − aΓ Xij . (13) 0
˙ If W ij < 0, then the origin O = {Xij = 0, i, j = 1, 2, · · · , n1 } is global stability. for some dynamical systems such that However, there not always exists an appropriate M ˙ W ij < 0 holds. We should make an assumption except the special cases. Assumption 3.2 There exists a critical value a∗ such that for arbitrary x the quadratic T form Xij M (DF (x) − aΓ )Xij is negative definite if a > a∗ .
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
25
How to find the critical value a∗ for the coupled systems (Lorenz system, Chen system and so on) satisfying Assumption 3.2? Considering two coupled identical systems x˙ 1 = F (x1 ) + εΓ (x2 − x1 ) and x˙ 2 = F (x2 ) + εΓ (x1 − x2 ), where ε is the coupled strength of the systems. The above two systems can come to synchronization due to the stability of the zero solutions of their difference equations 1 ˙ X12 = DF (βx2 + (1 − β)x1 )dβ − 2εΓ X12 0
comparing with (11). If there exists a critical value ε∗ for the two coupled systems, then for any ε > ε∗ , the zero solution of the above equation is globally asymptotically stable, i.e., the above two systems come to synchronization. In turn, for system (11), a∗ = 2ε∗ is just the critical number sufficient for global synchronization of any two oscillators in cluster (2). Then ˙ ∗ ∗ W ij < 0 (Xij = 0) as long as a > a , that is, there exists sufficient large drive strength a to make the equilibrium state O of system (11) globally stable. The following theorem gives explicit criterion for stability of synchronization of system (2). For cluster (3), we set that there are corresponding a, ε∗ , and Assumption 3.2 with that for cluster (2), which are needed throughout the paper. Hereafter, we have the following theorem. Theorem 3.1 Under Assumption 3.1 and Assumption 3.2 (or 3.2 ), suppose that F : Ω → X(H : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ}(Ω = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (G = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ (or a > 2ε∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators that can realize global synchronization, then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.1 Here, we can get the information from the above theorem that the synchronizability of a cluster completely depend on the dynamics of an individual node. With the same drive strength, the larger the critical ε∗ or ε¯∗ is, the harder the cluster comes to synchronization. 3.2 Case 2: Cluster Synchronization in the Case of Arbitrary Connections between Nodes in the Inner Clusters In this section, we consider the case that all nodes are arbitrary connections in the inner clusters as shown in Figure 1(b), namely, the inner-coupled matrices G and G are irreducible and their graphs are connected. Assumption 3.3 The graphs of clusters (2) and (3) are connected, that is, the symmetric inner-coupled matrices G and G are irreducible. Let the outer-coupled matrices C and C of clusters (2) and (3) satisfy the condition A1. By introducing the differences (8), we get the difference equations X˙ ij = F (xj ) − F (xi ) +
n1
{gjk Γ Xjk − gik Γ Xik } − aΓ Xij .
(14)
k=1
Add and subtract an additional term rΓ Xij from system (14) to get 1 ˙ Xij = DF (βxj + (1 − β)xi )dβ − (a + r)Γ Xij 0
+rΓ Xij +
n1 k=1
{gjk Γ Xjk − gik Γ Xik },
(15)
LIANG CHEN · JUN’AN LU
26
where r > 0 is an auxiliary scalar parameter. The auxiliary terms ±rΓ Xij bridge the coupling terms with the individual system. The positive term +rΓ Xij can be damped by the coupling terms. In turn, the negative term −rΓ Xij can be seen as the contribution of the coupling terms on damp instabilities caused by eigenvalues with positive real parts of the Jacobian DF . Introducing an auxiliary system 1 X˙ ij = DF (βxj + (1 − β)xi )dβ − (a + r)Γ Xij , (16) 0
we consider Lyapunov functions of the form Wij =
1 T X · M · Xij , 2 ij
i, j = 1, 2, · · · , n1 ,
(17)
where M = diag(m1 , m2 , · · · , ms , M1 ), m1 > 0, · · · , ms > 0, and the (d − s) × (d − s) matrix M1 is positive definite. Similar to the discussion of (13), under the Assumption 3.2, if a + r > 2ε∗ holds, the ˙ ij < derivatives of Lyapunov function (17) with respect to system (16) is negative, that is, W 0 (Xij = 0). Construct the Lyapunov function for the globally stable system (15) 1 1 1 X T · M · Xij . 4 i=1 j=1 ij
n
V =
n
(18)
The corresponding time derivative has the form 1 1 1 1 1 ˙ ij + 1 W V˙ = X T rM Γ Xij 2 i=1 j=1 2 i=1 j=1 ij
n
n
n
1 1 1 1 T T M Γ Xik }. {gjk Xij M Γ Xjk − gik Xij 2 i=1 j=1
n
n
+
n
n
(19)
k=1
n1 n1 ˙ W is negative definite if the sufficient condition a + r ≥ 2ε∗ The first sum S1 = 12 i=1 j=1 n1 ijn1 1 T holds. The second sum S2 = 2 i=1 j=1 Xij rM Γ Xij is always positive definite and measures the contribution degree of the coupling terms. And the third term S3 associated with the symmetrized matrix G = (gij ) is always negative definite such that S2 + S3 ≤ 0. Our goal is to obtain the condition of input strength a under which a solution to coupled system (15) converges to 0 as t → ∞. In other words, we need to find a sufficient large a such that V˙ < 0. Using the symmetry of G, we can simplify the third term S3 in the following form[20,22] 1 1 1 1 T T {gjk Xij M Γ Xjk − gik Xij M Γ Xik } 2 i=1 j=1
n
S3 =
n
n
k=1
=−
n1 n1 n1
T gjk Xji M Γ Xjk =
i=1 j=1 k=1
n1 n 1 −1
T n1 gjk Xjk M Γ Xjk .
(20)
k=1 j>k
We require that the sum of S2 + S3 is negative definite, that is, S2 + S 3 =
n n1 1 −1 i=1 j>i
T Xij M (r − n1 gij )Γ Xij ≤ 0.
(21)
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
27
By inequality (21), we obtain a lower bound of r by the connection graph method[20] r∗ ≤ r ≤
n1 gk , bk (n1 , m)
k = 1, 2, · · · , m,
(22)
n1 |(P ij )| is the sum of the lengths of all chosen paths where the quantity bk (n1 , m) = j>i;k∈P ij P ij which pass through a given edge k that belongs to the undirected graph G and gk is the coupling strength of edge k, m is the number of edges of the undirected graph G. Here, we also set there exists a corresponding r∗ in cluster (3) to the quantity r∗ in cluster (2). The lower bound r∗ = min{n1 gk /bk (n1 , m)} meaning the least contribution of coupling terms k
is completely determined by the coupled strength and the inner-structure of cluster. Clearly, given the coupling strengths gk , the bound r∗ depends on the choice of paths P ij . However, the lower bound r∗ is rather different from the “coupling threshold” in [20], which is the coupling strength scaling the synchronizability of a network, while r∗ is a quantity representing the contribution on synchronization in a cluster. Assume that the coupling strength gk = 1 for all k = 1, 2, · · · , m, then one usually takes for P ij the shortest path from node i to node j. For the calculation of quantity r∗ in some regular graphs with all coupling strength gk = 1, k = 1, 2, · · · , m, one can refer to [21]. At present, we have applied the connection graph method to get the useful precondition for the synchronization theorem. Theorem 3.2 Under Assumption 3.2 (or 3.2 ) and Assumption 3.3, suppose that F : Ω → X (orH : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ} (orΩ = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (orG = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ − r∗ (or a > 2ε∗ − r∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators that can realize the global synchronization, r∗ (or r∗ ) is the lower bound of the contribution of the coupled matrix G (or G) on cluster synchronization defined in (22), then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.2 Note that there is a special case that if 2ε∗ − r∗ < 0 (or 2ε∗ − r∗ < 0), the inequality a > 2ε∗ − r∗ (or a > 2ε∗ − r∗ ) always holds because a(or a) selected is always positive. Such special case means that when the contributions on cluster synchronization made by the cluster structure G (or G) is sufficiently large, the systems (2)(or (3)) can achieve stable synchronization even without any input from outside. Theorem 3.2 shows that the cluster synchronization in such case depends on the cluster structures, the dynamics of systems and the input strength from outer cluster and the cluster structure makes the positive contribution for the cluster synchronization. The input strength can be optimal if the cluster structure is changed. 3.3 Case 3: Cluster Synchronization in Case of the Coexistence of Isolated Nodes with Connective Nodes in the Inner Cluster In this part, we consider the cluster with the property that there are some isolated nodes in the cluster, and the other nodes are arbitrarily connected such as that shown in Figure 1(c) which including the case 1 and case 2. By introducing the differences (8), we obtain three kinds of difference equations because there are three kinds of the relations of nodes in this case. The first is the relation of any two isolated nodes in the inner cluster discussed in Subsection 3.1; the second is the relation between the nodes which are arbitrary connections in the inner cluster considered in Subsection
LIANG CHEN · JUN’AN LU
28
3.2; and the last one is the relation between the nodes that one belongs to the set of isolated nodes, and the other are the connective nodes in the inner cluster. Since the first and second kinds of difference equations have been discussed in the above, in this subsection, we only need to focus on the last kind of difference equations denoting those generated by the isolated nodes and the connective nodes. The third kind of difference equations has the form X˙ ij = F (xj ) − F (xi ) +
1
= 0
n
gjk Γ Xjk − aΓ Xij
k=1
n DF (βxj + (1 − β)xi )dβ − aΓ Xij + gjk Γ Xjk ,
(23)
k=1
. Here, we assume that the node j is a connective node where j = 1, 2, · · · , n , i = 1, 2, · · · , n1 − n in the connectivity graph with n nodes (in Figure 1(c), n = 3), and node i is an isolated node with i = 1, 2, · · · , n1 − n . By comparing the conditions of Theorem 3.1 with those of Theorem 3.2, we can easily find that if we use the same condition a > 2ε∗ in the two theorems, the conclusions that synchronization manifolds of system (2) in the first two cases are globally asymptotically stable also hold at the same time. And we will show that the sufficient condition a > 2ε∗ can be applied to the third case such that the trivial equilibriums of system (23), corresponding to the synchronization manifold with the third case, are globally asymptotically stable, too. Similar to the discussion before, we construct the Lyapunov function as follows n
1 T V = X · M · Xij , 2 j=1 ij
(24)
= diag(m 1 ), m where M 1, m 2, · · · , m s, M 1 > 0, · · · , m s > 0, and the (d − s) × (d − s) matrix M1 is positive definite. The derivatives of the Lyapunov functions with respect to system (23) are n
˙ T V = M Xij j=1
1
0
DF (βxj + (1 − β)xi )dβ − aΓ Xij +
n n
T M Γ Xjk . gjk Xij
(25)
j=1 k=1
We denote the first term and the second term in the right side of (25) by S1 and S2 , respectively. Then S1 is negative definite if the sufficient condition a > 2ε∗ holds. We will prove the second sum S2 is also negative definite. Since Xjj = 0 and gjk = gkj , the second sum becomes S2 =
n n
T M Γ Xjk gjk Xij
j=1 k=1
=−
n n −1 k=1 j>k
=−
n n −1 k=1 j>k
T M Γ Xjk − gjk Xji
n n −1
T M Γ Xjk gjk Xji
j=1 jk
=−
n −1 n
T M Γ Xjk < 0. gjk Xjk
(26)
k=1 j>k
Thus, we have proved that the derivatives of the Lyapunov functions with respect to system ˙ (23) are negative definite. Therefore, V < 0 under the condition a > 2ε∗ . From the above discussion, we have the following general theorem. Theorem 3.3 Under Assumption 3.2 (or 3.2 ), suppose that F : Ω → X (or H : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ} (or Ω = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (or G = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ (or a > 2ε∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators sufficient for global synchronization, then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.3 The case discussed in Theorem 3.3 includes cases 1 and 2. The estimation of critical input value for cluster synchronization is conservative. Actually, we think that the critical input value for cluster synchronization in case 3 is between the value of case 1 and the value os case 2, and the value in case 1 is the largest and the value in case 2 is the smallest. Now, we have obtained 3 theorems for 3 different cases of cluster structure. In fact, the results tell us that on the basis of some conditions, global cluster synchronization can be achieved by adjusting the input strength a from the outside of the cluster, which can be regarded as another means to control a cluster to synchronization. When the couple systems and the couple strengths are confirmed to each other, the enough input strength needed to achieve synchronization in a cluster only depends on the inner structure of the cluster.
4 Numerical Simulations To testify the validity of the theorems, we consider two simulation examples as follows. 4.1 Hyperchaotic L¨ u Attractor vs R¨ ossler Chaotic Attractor For applying the theoretical results of case 2, i.e., Theorem 3.2 obtained above, Hyperchaotic L¨ u system[23] and R¨ossler chaotic system[24] are considered as the dynamical nodes of cluster (2) and cluster (3) in the network, respectively. A Hyperchaotic L¨ u system is described by ⎧ x˙ = −a(y − x) + k, ⎪ ⎪ ⎨ y˙ = cy − xz, (27) z˙ = xy − bz, ⎪ ⎪ ⎩˙ k = xz + dk, and a single R¨ossler system is
⎧ ⎨ x˙ = −y − z, y˙ = x + p ∗ y, ⎩˙ z = q + z(x − r).
(28)
LIANG CHEN · JUN’AN LU
30
When a = 36.0, b = 3.0, c = 20.0, d = 1.3, and p = 0.15, q = 0.20, r = 10.0, the above systems have chaotic attractors as shown in Figure 2. The cluster of case 2 as shown in Figure 1(b) are discussed below. The corresponding coupled matrices G and G are given as follows ⎞ ⎛ ⎞ ⎛ −2 1 1 0 0 −2 1 1 0 ⎜ 1 −3 1 1 0 ⎟ ⎟ ⎜ 1 −2 1 0 ⎟ ⎜ ⎟ ⎜ ⎜ 1 1 −4 1 1 ⎟ . (29) G=⎝ , G = ⎟ ⎜ 0 1 −2 1 ⎠ ⎝ 0 1 1 −3 1 ⎠ 1 0 1 −2 0 0 1 1 −2 Let Γ = diag(1, 1, 1, 1) and P = diag(0, 1, 0), which set the coupling ways of the variables of systems (27) and (28).
25
40
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15
z
z¯
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0 40 20
40 20
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0
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y
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−40
15 0
10 5
−5
y¯
x
(a) Hyperchaotic L¨ u attractor, with parameters a = 36.0, b = 3.0, c = 20.0, d = 1.3. Figure 2
5 0
−10 −15
−5 −10
x¯
(b) R¨ ossler chaotic attractor, with parameters p = 0.15, q = 0.20, r = 10.0.
Hyperchaotic L¨ u attractor vs R¨ ossler chaotic attractor
According to Theorem 3.2, we can estimate the drive strength a∗ and a∗ by calculating the coupling strength ε∗ and ε∗ of two identical oscillators. We can use the coupling strength for Hyperchaotic L¨ u system ε∗ = 0.5 and the coupling strength for R¨ ossler chaotic system ∗ ε = 0.2 by numerical simulation. Note the definition of r∗ and r∗ in (22), we can calculate the synchronization contribute values of the clusters structure: when the coupling strength of network is equal to 1, then r∗ = 34 and r∗ = 54 from Figure 1 (b). Here, 2ε∗ − r∗ < 0 (2ε∗ − r∗ < 0), that is, there is sufficiently small drive a(a) such that cluster (2)((3)) can come to synchronization (ref. Remark 3.2). The simulations shown in Figure 3 are the graphs of synchronization in Hyperchaotic L¨ u attractor cluster and R¨ ossler chaotic attractor cluster, which are coincident with the conclusion above. 4.2 Lorenz Chaotic System vs Duffing System Here, we consider the case 3 and the chaotic Lorenz system and Duffing system are selected as the dynamical nodes of cluster (2) and cluster (3) in the network, respectively. A single Lorenz system is described by[25] ⎧ ⎨ x˙ = σ(y − x), y˙ = ρx − xz − y, (30) ⎩ z˙ = xy − βz, and a single Duffing system is given by ¨ + γ x˙ + px3 = q cos(ωt). x
(31)
31
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
20
20
100
0
80
ex
ey
0 −10
−20
−20 −30
−40 0
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t
t 30
60 40
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ek
ez
20 10
0 0 −10
Coupl ed R¨ossler St ates
10
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x¯ z¯
20
0
y¯
−20
−20 0
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6
−40
0
2
t
4
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−40
0
5
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15
t
(a) Synchronization errors of Hyperchaotic L¨ u attractors in cluster (2) versus time. Figure 3
20
25
t
(b) The state variables of R¨ ossler chaotic attractor in cluster (3) versus time.
Graph of synchronization in Hyperchaotic L¨ u attractor cluster and R¨ ossler chaotic attractor cluster
˙ we reformulate it as an autonomous system Letting z = t, y = x, ⎧ ⎨ x˙ = y, y˙ = −γy − µx3 + ν cos(ωz), ⎩˙ z = 1.
(32)
When σ = 10, ρ = 28, β = 83 , and γ = 0.3, µ = 1, ν = 39, ω = 1, the above two systems both have chaotic attractors as shown in Figure 4. The coexistence case is shown in Figure 1 (c) and will be discussed below. 15 50 10 40 5
z
30
y¯
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0
10 −5 0 −20
−10 0 20 y
−20
−10
0
10
20
x
(a) Lorenz chaotic attractor, with parameters σ = 10, ρ = 28, β = 83 . Figure 4
−15 −6
−4
−2
0
x¯
2
4
6
(b) Duffing chaotic attractor, with parameters γ = 0.3, µ = 1, ν = 39, ω = 1.
Lorenz attractor vs Duffing chaotic attractor
We consider a Lorenz cluster with 10 nodes, where a simple ring with 7 nodes and 3 isolated nodes are in coexistence, and a Duffing cluster with 20 nodes, where a star shape graph with 15 nodes and 5 isolated nodes are in coexistence. The corresponding coupled matrices G and G are described as
LIANG CHEN · JUN’AN LU
32
⎞ ··· 0 ··· 0⎟ ⎟ .. .. ⎟ . .⎟ ⎟ , ··· 0⎟ ⎟ ⎟ ··· 0⎟ ··· 0⎠ · · · 0 10×10
⎛
−2 1 0 · · · 1 0 ⎜ 1 −2 1 · · · 0 0 ⎜ ⎜ .. .. .. . . .. .. ⎜ . . . . . . ⎜ G = ⎜ 1 0 · · · 1 −2 0 ⎜ ⎜ 0 ··· ··· ··· 0 0 ⎜ ⎝ 0 ··· ··· ··· 0 0 0 ··· ··· ··· 0 0
⎛
−14 ⎜ 1 ⎜ ⎜ .. ⎜ . ⎜ G=⎜ ⎜ 1 ⎜ 0 ⎜ ⎜ . ⎝ ..
1 ··· 1 0 0 ··· 0 0 .. . . .. .. . . . . 0 · · · · · · −1 0 ··· ··· ··· 0 0 . . . · · · . . · · · .. .. 0 ··· ··· ··· 0 0 1 −1 .. .
⎞ ··· 0 ··· 0⎟ ⎟ .. .. ⎟ . .⎟ ⎟ ··· 0⎟ ⎟ ··· 0⎟ ⎟ .. ⎟ . 0⎠ ··· 0
.
(33)
20×20
Letting Γ = diag(1, 0, 0) and P = diag(1, 0, 0), we select the first variable of systems (30) and (32) to couple in the first function of corresponding systems. According to Theorem 3.3, one can determine the drive strength a∗ and a∗ by calculating the coupling strength ε∗ and ε∗ of two identical oscillators. We can use the coupling strength ε∗ = 5.5 for Lorenz system and the coupling strength ε∗ = 0.4 for Duffing system in numerical 2 ∗ − σ2 is more than 150 which is too simulation. Note that the bound[20] ε∗ = a2 = β(β+1)(ρ+σ) 32(β−1) large to apply in simulations. The simulation results are shown in Figure 5 to illuminate the correctness of Theorem 3.3. In Figure 5 (a), we find that the synchronization errors of nodes of the Lorenz cluster are globally asymptotically convergent to zero. And the Figure 5 (b) also shows that the state variables of Duffing cluster in the network indeed synchronize. 20
0
2
4
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8
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20 0 −20
0
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0
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20
t
(a) Synchronization errors of Lorenz chaotic attractors in cluster (2) versus time. Figure 5
Coup led Du
ex 0 −10
g Systemes Sta tes
150
10
(b) The state variables of Duffing chaotic attractor in cluster (3) versus time.
Graph of synchronization in Lorenz chaotic attractor cluster and Duffing chaotic attractor cluster
5 Conclusions In summary, we have studied global stability of cluster synchronization in a complex dynamical network with two nonidentical clusters. Specifically, the sufficient conditions for the stability of synchronization manifolds in the clusters with the same-input condition are obtained by using the connection graph stability method. Furthermore, there exists an upper bound of input strength of outside clusters which can scale the synchronizability of the cluster. Exceeding the upper bound, the nodes in the same cluster can realize synchrony. Two numerical examples of such network with different chaotic nodes are given to illustrate the validity and feasibility of synchronization criteria. Finally, the synchronous theory of complex network consisting of more clusters and the case of directed networks should be further investigated in the near future.
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
33
References [1] D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature, 1998, 393(4): 440–442. [2] A. L. Barab´ asi and R. Albet, Emergence of scaling in random networks, Science, 1999, 286(15): 509–512. [3] M. E. J. Newman, Community structure in social and biological network, Proc. Natl. Acad. Sci., 2002, 99(12): 7821–7826. [4] P. Gleiser and L. Danon, Community structure in jazz, Advances in Complex Systems, 2003, 6(4): 565–573. [5] G. W Flake, S. R. Lawrence, C. L. Giles, and F. M. Coetzee, Self-organization and identification of Web communities, IEEE Computer , 2002, 35(3): 66–71. [6] C. Hugenii, Horoloquim Oscilatorium, Apud F. Muguet, Parisiis, 1673. [7] L. M. Pecora and T. L. Carroll, Master stability function for synchronized coupled systems, Phys. Rev. Lett., 1998, 80(10): 2109–2112. [8] X. F. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. Bifur. Chaos, 2002, 12(1): 187–192. [9] J. L¨ u and G. Chen, A time-varying complex dynamical network models and its controlled synchronization criteria, IEEE Trans. Auto. Contr., 2005, 50(6): 841–846. [10] Y. Zhang, G. Hu, H. A. Cerdeira, S. Chen, T. Braun, and Y. Yao, Partial synchronization and spontaneous spatial ordering in coupled chaotic systems, Phys. Rev. E, 2001, 63(2): 026211. [11] V. N. Belykh, I. V. Belykh, and E. Mosekilde, Cluster synchronization modes in an ensemble of coupled chaotic oscillators, Phys. Rev. E, 2001, 63(3): 036216. [12] Y. C. Kouomou and P. Woafo, Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shift-invariant set of coupled nonlinear oscillators, Phys. Rev. E, 2003, 67(4): 046205. [13] Y. C. Kouomou and P. Woafo, Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks, Optics Communications, 2003, 223(4–6): 273–282. [14] S. Jalan, R. E. Amritkar, and C. K. Hu, Synchronized clusters in coupled map networks, I. Numerical studies, Phys. Rev. E, 2005, 72(1): 016211. [15] R. E. Amritkar, S. Jalan, and C. K. Hu, Synchronized clusters in coupled map networks, II. Stability analysis, Phys. Rev. E, 2005, 72(1): 016212. [16] I. V. Belykh, V. N. Belykh, K. V. Nevidin, and M. Hasler, Persistent clusters in lattices of coupled nonidentical chaotic systems, Chaos, 2003, 13(1): 165–178. [17] W. X. Qin and G. Chen, Coupling schemes for cluster synchronization in coupled Josephson equation, Physica D, 2004, 197(3–4): 375–391. [18] Z. Ma, Z. Liu, and G. Zhan, A new method to realize cluster synchronization in connected chaotic networks, Chaos, 2006, 16(2): 023103. [19] Kunihiko Kaneko, Relevance of dynamic clustering to biological networks, Physica D, 1994, 75(1– 3): 55–73. [20] I. V. Belykh, V. N. Belykh, and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D, 2004, 195(1–2): 159–187. [21] I. V. Belykh, M. Hasler, M. Lauret, and H. Nijmeijer, Synchronization and graph topology, Int. J. Bifurcation and Chaos, 2005, 15(11): 3423–3433. [22] I. V. Belykh, V. N. Belykh, and M. Hasler, Synchronization in asymmetrically coupled networks with node balance, Chaos, 2006, 16(1): 015102. [23] A. M. Chen, J. A. Lu, and J. H. L¨ u, Generating hyperchaotic L¨ u attrctor via state feedback control, Physica A, 2006, 364(1): 103–110. [24] O. E. R¨ ossler, An equation for continuous chaos, Phys. Lett. A, 1976, 57(5): 397–398. [25] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci, 1963, 20(2): 130–141.
CLUSTER SYNCHRONIZATION IN A COMPLEX DYNAMICAL NETWORK WITH TWO NONIDENTICAL CLUSTERS∗ Liang CHEN · Jun’an LU
Received: 6 April 2007 / Revised: 18 October 2007 c 2008 Springer Science + Business Media, LLC Abstract This paper further investigates cluster synchronization in a complex dynamical network with two-cluster. Each cluster contains a number of identical dynamical systems, however, the subsystems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. Complete synchronization within each cluster is possible only if each node from one cluster receives the same input from nodes in other cluster. In this case, the stability condition of one-cluster synchronization is known to contain two terms: the first accounts for the contribution of the inner-cluster coupling structure while the second is simply an extra linear term, which can be deduced by the “same-input” condition. Applying the connection graph stability method, the authors obtain an upper bound of input strength for one cluster if the first account is known, by which the synchronizability of cluster can be scaled. For different clusters, there are different upper bound of input strength by virtue of different dynamics and the corresponding cluster structure. Moreover, two illustrative examples are presented and the numerical simulations coincide with the theoretical analysis. Key words Cluster synchronization, complex dynamical network, connection graph, stability.
1 Introduction The properties of complex dynamical network and their physical significance have been further investigated since the the discovery of “small world” networks[1] and “free scale” networks[2] . One common property of network, cluster (community) structure, is an important topic and has been widely applied in biology, physics, computer graphics and sociology[3−4] . A network is composed of many clusters and the connection of nodes in one cluster is more close than that of nodes in the other clusters. For example, WWW can be seen as a network constituted by many web stations, and all web stations are discussing the same interesting topics of their own[5] . Networks of dynamical systems have been used in physics, biology, and other science branches. The simplest and most striking interaction between dynamical systems is their synchronization. Synchronization is a very common phenomenon in real systems and has been an Liang CHEN · Jun’an LU School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email: [email protected]; [email protected]. ∗ The work is supported by the National Natural Science Foundation of China under Grant Nos. 70771084 and 60574045, and the National Basic Research Program of China under Grant No. 2007CB310805.
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
21
active research topic since the times of Huygens[6] . Over the last few decades, synchronization has been intensively studied in various disciplines, such as biological, epidemiologic, social, economic, mathematical, and engineering science[7−9] . In particular, as a special synchronization phenomenon, cluster synchronization in networks, i.e., the synchronization of subnetworks or groups of networks has received much attention in recent years[10−18] . In fact, the notion of cluster (or partial) synchronization is quite recent. The original study of cluster synchronization was from the natural physical phenomenon rising in coupled oscillators and coupled map lattice. It stands for the intermediary states where the oscillators synchronize with one another in distinct groups, while there is no synchronization amongst the groups[10−11] . Beside cluster synchronization, the coupled system can obviously display the well-known states of spatiotemporal chaos and complete (full) synchronization[12−13] . In [14– 15], self-organized cluster synchronization and drive cluster synchronization of a coupled map network (CMN) have been studied in detail. In [16], the persistence of cluster synchronization regimes in lattices of nonidentical chaotic oscillators was systematic analyzed. It is shown that, by numerical study, in lattices of Lorenz and R¨ ossler systems the cluster synchronization regimes are stable and robust against up to 10% to 15% parameter mismatch. Qin and Chen[17] proposed a coupling scheme to stabilize the selected cluster synchronization patterns for coupled Josephson equations. Ma and Liu[18] showed the cluster synchronization patterns could be achieved by constructing a special coupled matrix for a dynamical network with identical oscillators. With the development of modern science and technology, it is significant that the cluster synchronization can be applied to the laser technology[12] , biological engineering[19] , and secure communications[12−13] . In above, most of the current works on cluster synchronization were based on the networks of identical oscillators. However, many known facts tell us that the real-world networks are composed of various oscillators with different dynamics. Inspired by this, we are stimulated to investigate cluster synchronization in dynamical networks consisting with nonidentical clusters. The dynamics of nodes between nonidentical clusters can be totally different including the dimensions which are different from that of [16]. In this article, we introduce a dynamical network with two nonidentical clusters. Each cluster contains a number of identical dynamical systems, however, the sub-systems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. The global stability of cluster synchronization can be achieved with the “same-input” condition (all nodes in the same cluster can receive the same input as the nodes of outside clusters do). The synchronization criteria for two nonidentical clusters are obtained by using the connection graph stability method which is a very useful tool for dealing with the synchronization of network with continuous dynamical oscillators[20−22] . For different clusters, there are different upper bounds of the input strength by the different dynamics, the coupling strength and the corresponding inner-cluster structure. The stable synchrony can be achieved in a cluster as long as the input strength is strong enough. These synchronization criterions can provide new insights into the key factors leading to self-organization phenomenon between nonidentical clusters in nature. The remaining part is organized as follows. A complex dynamical network consisting of two nonidentical clusters is briefly introduced in Section 2. The synchronization conditions of three cases in the inner clusters and two examples of numerical simulation are given in Sections 3 and 4, respectively. Conclusions are finally drawn in Section 5.
2 Network Modeling Consider a dynamical network with two clusters, which are composed of n1 and n2 linearly and diffusively coupled identical nodes, with each node being a d and d-dimensional dynamical
LIANG CHEN · JUN’AN LU
22 oscillator as
x˙ = F (x),
y˙ = H(y),
(1)
where the above differential equations are defined in a basic state space X and Y , respectively. The proposed two-cluster network is described by ⎧ n1 n1 +n2 ⎪ ⎨ x˙ = F (x ) + g Γ x + c [I i i ik k ik d×d · P yk · I d×d − Γ xi ], (2) k=1 k=n1 +1 ⎪ ⎩ i = 1, 2, · · · , n1 ; ⎧ n1 +n2 n1 ⎪ ⎨ y˙ = H(y ) + g P y + c [I i i k ik d×d · Γ xk · I d×d − P yi ], ik (3) k=n1 +1 k=1 ⎪ ⎩ i = n1 + 1, n1 + 2, · · · , n1 + n2 . Here, the matrices Γ and P determine which variables of the oscillators used to couple. For s clarity, we shall consider the diagonal matrices Γ = diag(1, 1, · · · , 1, 0, · · · , 0)d×d and P = s
diag(1, 1, · · · , 1, 0, · · · , 0)d×d . Note that the dimension of vectors Γ xi is not equal to that of P yj , to keep the coherence of the vectors’ dimensions in (2) and (3) such that the coupling with two different dynamical systems is reasonable, we introduce the generalized matrices I d×d and I d×d whose left diagonal elements are 1, otherwise 0, for example, ⎛ ⎞
1 0 1 0 0 , I 3×2 = ⎝ 0 1 ⎠ . I 2×3 = 0 1 0 0 0 Equivalently, (2) and (3) can be rewritten as ⎧ n1 n1 +n2 ⎪ ⎨ x˙ = F (x ) + Γ [g x − a x ] + c (I i i ik k i i ik d×d · P yk · I d×d ), k=1 k=n +1 1 ⎪ ⎩ i = 1, 2, · · · , n1 ; ⎧ n1 +n2 n1 ⎪ ⎨ y˙ = H(y ) + P [g y − a y ] + c (I · Γx · I ), ⎪ ⎩
i
i
ik k
k=n1 +1
i i
ik
k=1
d×d
k
d×d
(4)
(5)
i = n1 + 1, n1 + 2, · · · , n1 + n2 , n1 +n2 n1 where ai = k=n1 +1 cik for i = 1, 2, · · · , n1 and ai = k=1 cik for i = n1 +1, n1 +2, · · · , n1 +n2 . Then, the connectivity matrix of network has the form
Gn1 ×n1 C n1 ×n2 An1 ×n1 O n1 ×n2 D= + , (6) C n2 ×n1 Gn2 ×n2 O n2 ×n1 An2 ×n2 where G = (gij )n1 ×n1 and G = (g ij )n2 ×n2 are respectively the inner connectivity matrices of clusters (2) and (3) representing the coupling strength and topological structure of the corresponding clusters, and assumed to be symmetric and satisfy zero row-sums: ⎧ n1 ⎪ ⎪ ⎪ g = g ≥ 0, i = j, g = − gij , i = 1, 2, · · · , n1 , ij ji ii ⎪ ⎪ ⎨ j=1,j=i (7) n 1 +n2 ⎪ ⎪ ⎪ ⎪ i = j, g ii = − g ij , i = n1 + 1, · · · , n1 + n2 . ⎪ ⎩ g ij = g ji ≥ 0, j=n1 +1,j=i
23
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
Correspondingly, C = (cij )n1 ×n2 and C = (cij )n2 ×n1 are nonnegative matrices, called the outer connectivity matrices of clusters (2) and (3) being the connecting status between two clusters, and the diagonal matrices A = diag(−a1 , −a2 , · · · , −an1 ) and A = diag(−an1 +1 , −an1 +2 , · · · , − an1 +n2 ) show the input strengths of clusters (2) and (3), separately. Up to now, we have established the model of a complex dynamical network with two nonidentical clusters. Here, we do not assume irreducibility of the inner-cluster matrices G and G in models (2) and (3). Therefore, there are many cases of graph connectivity in the inner clusters (2) and (3). We mainly consider cluster synchronization in three cases as shown in Figure 1. First, in the inner cluster, there are not any connection between nodes, that is, all nodes in the cluster are isolated if without connection with outside clusters’ nodes. Second, the graph of cluster is connectivity, that is, the inner-cluster matrix of cluster is irreducible. Third, there exist some isolated nodes in the cluster, and the other nodes are arbitrarily connected.
[X
Y\
D (a) Figure 1
[X
\Y
(b) E
Y \
[X
(c) F
Three cases of connectivity in the network with two clusters. (a) No connections between nodes in inner clusters. (b) Arbitrary Connections of the nodes in inner clusters. (c) The coexistence of isolated nodes and nodes with arbitrary connections in inner cluster.
3 Cluster Synchronization of Complex Dynamical Networks In this section, we use the connection graph stability method to prove some cluster synchronization theorems of a network with two-cluster. Firstly, we present some necessary definitions. Definition 3.1 A network with clusters is said to realize cluster synchronization, if the nodes in the same cluster synchronize with each other. In systems (2) and (3), it is said that the network realizes cluster synchronization, if for xi (t, ( x0 , y 0 )) and xj (t, ( x0 , y 0 )) of arbitrary nodes i and j in cluster (2), ||xi (t, ( x0 , y 0 )) − xj (t, ( x0 , y 0 ))|| → 0 as t → 0; and for yi (t, ( x0 , y 0 )) and yj (t, ( x0 , y 0 )) of arbitrary nodes i and j in cluster (3), ||yi (t, ( x0 , y 0 )) − yj (t, ( x0 , y 0 ))|| → 0 as t → 0, where ( x0 , y 0 ) denotes arbitrary T T T T initial states (xT (0), · · · , x (0), y (0), · · · , y (0)) . n1 n1 +n2 1 n1 +1 T T T n1 Definition 3.2 S = { x = (xT 1 , x2 , · · · , xn1 ) ∈ X |xi = xj = s ∈ X, i, j = 1, 2, · · · , n1 } and S = {y = (ynT1 +1 , · · · , ynT1 +n2 )T ∈ Y n2 |yi = yj = s ∈ Y, i, j = n1 + 1, n1 + 2, · · · , n1 + n2 } are called the synchronization manifold of clusters (2) and (3). Definition 3.3 A matrix B = (bij )m×n is said to satisfy condition A1, if its elements satisfy bik = bjk , i, j = 1, 2, · · · , m; k = 1, 2, · · · , n.
In systems (4) and (5), if the outer connectivity matrices of clusters (2) and (3), C = (cij )n1 ×n2 and C = (cij )n2 ×n1 satisfy condition A1, then the input strengths of nodes in the same cluster are equal, that is, ai = aj = a, i, j = 1, 2, · · · , n1 ; and ai = aj = a, i, j = n1 + 1, · · · , n1 + n2 . In this case, condition A1 is also called the “same-input” condition.
24
LIANG CHEN · JUN’AN LU
In the following, the global stability of synchronization manifolds under three types of innercluster structure will be investigated inventively. 3.1
Case 1: Cluster Synchronization in the Case of No Connections Between Nodes in the Inner Clusters
Consider a special case of network composed of cluster (2) and cluster (3) as shown in Figure 1(a). In the inner cluster (2) or (3), all nodes are unattached, that is, there are not any connection between the nodes of clusters (2) or (3). In this case, we have the following assumption. Assumption 3.1 There are not any connection between nodes in the inner clusters (2) and (3), that is, G = (gij )n1 ×n1 = On1 ×n1 and G = (g ij )n2 ×n2 = On2 ×n2 . For clarity, we only consider cluster (2) to study cluster synchronization of the network, and the corresponding result to cluster (3) will be obtained obviously. For cluster (2), we introduce the notation for the differences Xij = xj − xi ,
i, j = 1, 2, · · · , n1 ,
(8)
and by Assumption 3.1, we obtain the difference equations as the form X˙ ij = F (xj ) − F (xi ) − aΓ Xij .
(9)
Using a compact vector notation for the function difference 1 d F (xj ) − F (xi ) = F (βxj + (1 − β)xi )dβ 0 dβ 1 = DF (βxj + (1 − β)xi )dβ Xij ,
(10)
where DF is a d × d Jacobi matrix of F , we obtain 1 X˙ ij = DF (βxj + (1 − β)xi )dβ − aΓ Xij .
(11)
0
0
It is obvious that the global stability of the origin O = {Xij = 0, i, j = 1, 2, · · · , n1 } amounts to the global stability of synchronization manifold S. Consider Lyapunov functions of the form · Xij , i, j = 1, 2, · · · , n1 , ij = 1 X T · M W (12) 2 ij = diag m 1 , m where M 1, m 2, · · · , m s, M 2 > 0, · · · , m s > 0, and the (d − s) × (d − s) 1 > 0, m matrix M1 is positive definite. The derivatives of the Lyapunov functions with respect to the system (11) are 1 ˙ T W ij = Xij M DF (βxj + (1 − β)xi )dβ − aΓ Xij . (13) 0
˙ If W ij < 0, then the origin O = {Xij = 0, i, j = 1, 2, · · · , n1 } is global stability. for some dynamical systems such that However, there not always exists an appropriate M ˙ W ij < 0 holds. We should make an assumption except the special cases. Assumption 3.2 There exists a critical value a∗ such that for arbitrary x the quadratic T form Xij M (DF (x) − aΓ )Xij is negative definite if a > a∗ .
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
25
How to find the critical value a∗ for the coupled systems (Lorenz system, Chen system and so on) satisfying Assumption 3.2? Considering two coupled identical systems x˙ 1 = F (x1 ) + εΓ (x2 − x1 ) and x˙ 2 = F (x2 ) + εΓ (x1 − x2 ), where ε is the coupled strength of the systems. The above two systems can come to synchronization due to the stability of the zero solutions of their difference equations 1 ˙ X12 = DF (βx2 + (1 − β)x1 )dβ − 2εΓ X12 0
comparing with (11). If there exists a critical value ε∗ for the two coupled systems, then for any ε > ε∗ , the zero solution of the above equation is globally asymptotically stable, i.e., the above two systems come to synchronization. In turn, for system (11), a∗ = 2ε∗ is just the critical number sufficient for global synchronization of any two oscillators in cluster (2). Then ˙ ∗ ∗ W ij < 0 (Xij = 0) as long as a > a , that is, there exists sufficient large drive strength a to make the equilibrium state O of system (11) globally stable. The following theorem gives explicit criterion for stability of synchronization of system (2). For cluster (3), we set that there are corresponding a, ε∗ , and Assumption 3.2 with that for cluster (2), which are needed throughout the paper. Hereafter, we have the following theorem. Theorem 3.1 Under Assumption 3.1 and Assumption 3.2 (or 3.2 ), suppose that F : Ω → X(H : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ}(Ω = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (G = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ (or a > 2ε∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators that can realize global synchronization, then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.1 Here, we can get the information from the above theorem that the synchronizability of a cluster completely depend on the dynamics of an individual node. With the same drive strength, the larger the critical ε∗ or ε¯∗ is, the harder the cluster comes to synchronization. 3.2 Case 2: Cluster Synchronization in the Case of Arbitrary Connections between Nodes in the Inner Clusters In this section, we consider the case that all nodes are arbitrary connections in the inner clusters as shown in Figure 1(b), namely, the inner-coupled matrices G and G are irreducible and their graphs are connected. Assumption 3.3 The graphs of clusters (2) and (3) are connected, that is, the symmetric inner-coupled matrices G and G are irreducible. Let the outer-coupled matrices C and C of clusters (2) and (3) satisfy the condition A1. By introducing the differences (8), we get the difference equations X˙ ij = F (xj ) − F (xi ) +
n1
{gjk Γ Xjk − gik Γ Xik } − aΓ Xij .
(14)
k=1
Add and subtract an additional term rΓ Xij from system (14) to get 1 ˙ Xij = DF (βxj + (1 − β)xi )dβ − (a + r)Γ Xij 0
+rΓ Xij +
n1 k=1
{gjk Γ Xjk − gik Γ Xik },
(15)
LIANG CHEN · JUN’AN LU
26
where r > 0 is an auxiliary scalar parameter. The auxiliary terms ±rΓ Xij bridge the coupling terms with the individual system. The positive term +rΓ Xij can be damped by the coupling terms. In turn, the negative term −rΓ Xij can be seen as the contribution of the coupling terms on damp instabilities caused by eigenvalues with positive real parts of the Jacobian DF . Introducing an auxiliary system 1 X˙ ij = DF (βxj + (1 − β)xi )dβ − (a + r)Γ Xij , (16) 0
we consider Lyapunov functions of the form Wij =
1 T X · M · Xij , 2 ij
i, j = 1, 2, · · · , n1 ,
(17)
where M = diag(m1 , m2 , · · · , ms , M1 ), m1 > 0, · · · , ms > 0, and the (d − s) × (d − s) matrix M1 is positive definite. Similar to the discussion of (13), under the Assumption 3.2, if a + r > 2ε∗ holds, the ˙ ij < derivatives of Lyapunov function (17) with respect to system (16) is negative, that is, W 0 (Xij = 0). Construct the Lyapunov function for the globally stable system (15) 1 1 1 X T · M · Xij . 4 i=1 j=1 ij
n
V =
n
(18)
The corresponding time derivative has the form 1 1 1 1 1 ˙ ij + 1 W V˙ = X T rM Γ Xij 2 i=1 j=1 2 i=1 j=1 ij
n
n
n
1 1 1 1 T T M Γ Xik }. {gjk Xij M Γ Xjk − gik Xij 2 i=1 j=1
n
n
+
n
n
(19)
k=1
n1 n1 ˙ W is negative definite if the sufficient condition a + r ≥ 2ε∗ The first sum S1 = 12 i=1 j=1 n1 ijn1 1 T holds. The second sum S2 = 2 i=1 j=1 Xij rM Γ Xij is always positive definite and measures the contribution degree of the coupling terms. And the third term S3 associated with the symmetrized matrix G = (gij ) is always negative definite such that S2 + S3 ≤ 0. Our goal is to obtain the condition of input strength a under which a solution to coupled system (15) converges to 0 as t → ∞. In other words, we need to find a sufficient large a such that V˙ < 0. Using the symmetry of G, we can simplify the third term S3 in the following form[20,22] 1 1 1 1 T T {gjk Xij M Γ Xjk − gik Xij M Γ Xik } 2 i=1 j=1
n
S3 =
n
n
k=1
=−
n1 n1 n1
T gjk Xji M Γ Xjk =
i=1 j=1 k=1
n1 n 1 −1
T n1 gjk Xjk M Γ Xjk .
(20)
k=1 j>k
We require that the sum of S2 + S3 is negative definite, that is, S2 + S 3 =
n n1 1 −1 i=1 j>i
T Xij M (r − n1 gij )Γ Xij ≤ 0.
(21)
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
27
By inequality (21), we obtain a lower bound of r by the connection graph method[20] r∗ ≤ r ≤
n1 gk , bk (n1 , m)
k = 1, 2, · · · , m,
(22)
n1 |(P ij )| is the sum of the lengths of all chosen paths where the quantity bk (n1 , m) = j>i;k∈P ij P ij which pass through a given edge k that belongs to the undirected graph G and gk is the coupling strength of edge k, m is the number of edges of the undirected graph G. Here, we also set there exists a corresponding r∗ in cluster (3) to the quantity r∗ in cluster (2). The lower bound r∗ = min{n1 gk /bk (n1 , m)} meaning the least contribution of coupling terms k
is completely determined by the coupled strength and the inner-structure of cluster. Clearly, given the coupling strengths gk , the bound r∗ depends on the choice of paths P ij . However, the lower bound r∗ is rather different from the “coupling threshold” in [20], which is the coupling strength scaling the synchronizability of a network, while r∗ is a quantity representing the contribution on synchronization in a cluster. Assume that the coupling strength gk = 1 for all k = 1, 2, · · · , m, then one usually takes for P ij the shortest path from node i to node j. For the calculation of quantity r∗ in some regular graphs with all coupling strength gk = 1, k = 1, 2, · · · , m, one can refer to [21]. At present, we have applied the connection graph method to get the useful precondition for the synchronization theorem. Theorem 3.2 Under Assumption 3.2 (or 3.2 ) and Assumption 3.3, suppose that F : Ω → X (orH : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ} (orΩ = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (orG = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ − r∗ (or a > 2ε∗ − r∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators that can realize the global synchronization, r∗ (or r∗ ) is the lower bound of the contribution of the coupled matrix G (or G) on cluster synchronization defined in (22), then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.2 Note that there is a special case that if 2ε∗ − r∗ < 0 (or 2ε∗ − r∗ < 0), the inequality a > 2ε∗ − r∗ (or a > 2ε∗ − r∗ ) always holds because a(or a) selected is always positive. Such special case means that when the contributions on cluster synchronization made by the cluster structure G (or G) is sufficiently large, the systems (2)(or (3)) can achieve stable synchronization even without any input from outside. Theorem 3.2 shows that the cluster synchronization in such case depends on the cluster structures, the dynamics of systems and the input strength from outer cluster and the cluster structure makes the positive contribution for the cluster synchronization. The input strength can be optimal if the cluster structure is changed. 3.3 Case 3: Cluster Synchronization in Case of the Coexistence of Isolated Nodes with Connective Nodes in the Inner Cluster In this part, we consider the cluster with the property that there are some isolated nodes in the cluster, and the other nodes are arbitrarily connected such as that shown in Figure 1(c) which including the case 1 and case 2. By introducing the differences (8), we obtain three kinds of difference equations because there are three kinds of the relations of nodes in this case. The first is the relation of any two isolated nodes in the inner cluster discussed in Subsection 3.1; the second is the relation between the nodes which are arbitrary connections in the inner cluster considered in Subsection
LIANG CHEN · JUN’AN LU
28
3.2; and the last one is the relation between the nodes that one belongs to the set of isolated nodes, and the other are the connective nodes in the inner cluster. Since the first and second kinds of difference equations have been discussed in the above, in this subsection, we only need to focus on the last kind of difference equations denoting those generated by the isolated nodes and the connective nodes. The third kind of difference equations has the form X˙ ij = F (xj ) − F (xi ) +
1
= 0
n
gjk Γ Xjk − aΓ Xij
k=1
n DF (βxj + (1 − β)xi )dβ − aΓ Xij + gjk Γ Xjk ,
(23)
k=1
. Here, we assume that the node j is a connective node where j = 1, 2, · · · , n , i = 1, 2, · · · , n1 − n in the connectivity graph with n nodes (in Figure 1(c), n = 3), and node i is an isolated node with i = 1, 2, · · · , n1 − n . By comparing the conditions of Theorem 3.1 with those of Theorem 3.2, we can easily find that if we use the same condition a > 2ε∗ in the two theorems, the conclusions that synchronization manifolds of system (2) in the first two cases are globally asymptotically stable also hold at the same time. And we will show that the sufficient condition a > 2ε∗ can be applied to the third case such that the trivial equilibriums of system (23), corresponding to the synchronization manifold with the third case, are globally asymptotically stable, too. Similar to the discussion before, we construct the Lyapunov function as follows n
1 T V = X · M · Xij , 2 j=1 ij
(24)
= diag(m 1 ), m where M 1, m 2, · · · , m s, M 1 > 0, · · · , m s > 0, and the (d − s) × (d − s) matrix M1 is positive definite. The derivatives of the Lyapunov functions with respect to system (23) are n
˙ T V = M Xij j=1
1
0
DF (βxj + (1 − β)xi )dβ − aΓ Xij +
n n
T M Γ Xjk . gjk Xij
(25)
j=1 k=1
We denote the first term and the second term in the right side of (25) by S1 and S2 , respectively. Then S1 is negative definite if the sufficient condition a > 2ε∗ holds. We will prove the second sum S2 is also negative definite. Since Xjj = 0 and gjk = gkj , the second sum becomes S2 =
n n
T M Γ Xjk gjk Xij
j=1 k=1
=−
n n −1 k=1 j>k
=−
n n −1 k=1 j>k
T M Γ Xjk − gjk Xji
n n −1
T M Γ Xjk gjk Xji
j=1 jk
=−
n −1 n
T M Γ Xjk < 0. gjk Xjk
(26)
k=1 j>k
Thus, we have proved that the derivatives of the Lyapunov functions with respect to system ˙ (23) are negative definite. Therefore, V < 0 under the condition a > 2ε∗ . From the above discussion, we have the following general theorem. Theorem 3.3 Under Assumption 3.2 (or 3.2 ), suppose that F : Ω → X (or H : Ω → Y ) is continuously differentiable on the positive invariant set Ω = {x ∈ X|||x|| < γ} (or Ω = {y ∈ Y |||y|| < γ}), and the inner connectivity matrix G = (gij )n1 ×n1 (or G = (g ij )n2 ×n2 ) satisfies condition A1. If the input strength a > 2ε∗ (or a > 2ε∗ ), where ε∗ (or ε∗ ) is the coupling strength of two oscillators sufficient for global synchronization, then, the synchronization manifolds S(or S) are globally asymptotically stable for the cluster (2)(or (3)). Remark 3.3 The case discussed in Theorem 3.3 includes cases 1 and 2. The estimation of critical input value for cluster synchronization is conservative. Actually, we think that the critical input value for cluster synchronization in case 3 is between the value of case 1 and the value os case 2, and the value in case 1 is the largest and the value in case 2 is the smallest. Now, we have obtained 3 theorems for 3 different cases of cluster structure. In fact, the results tell us that on the basis of some conditions, global cluster synchronization can be achieved by adjusting the input strength a from the outside of the cluster, which can be regarded as another means to control a cluster to synchronization. When the couple systems and the couple strengths are confirmed to each other, the enough input strength needed to achieve synchronization in a cluster only depends on the inner structure of the cluster.
4 Numerical Simulations To testify the validity of the theorems, we consider two simulation examples as follows. 4.1 Hyperchaotic L¨ u Attractor vs R¨ ossler Chaotic Attractor For applying the theoretical results of case 2, i.e., Theorem 3.2 obtained above, Hyperchaotic L¨ u system[23] and R¨ossler chaotic system[24] are considered as the dynamical nodes of cluster (2) and cluster (3) in the network, respectively. A Hyperchaotic L¨ u system is described by ⎧ x˙ = −a(y − x) + k, ⎪ ⎪ ⎨ y˙ = cy − xz, (27) z˙ = xy − bz, ⎪ ⎪ ⎩˙ k = xz + dk, and a single R¨ossler system is
⎧ ⎨ x˙ = −y − z, y˙ = x + p ∗ y, ⎩˙ z = q + z(x − r).
(28)
LIANG CHEN · JUN’AN LU
30
When a = 36.0, b = 3.0, c = 20.0, d = 1.3, and p = 0.15, q = 0.20, r = 10.0, the above systems have chaotic attractors as shown in Figure 2. The cluster of case 2 as shown in Figure 1(b) are discussed below. The corresponding coupled matrices G and G are given as follows ⎞ ⎛ ⎞ ⎛ −2 1 1 0 0 −2 1 1 0 ⎜ 1 −3 1 1 0 ⎟ ⎟ ⎜ 1 −2 1 0 ⎟ ⎜ ⎟ ⎜ ⎜ 1 1 −4 1 1 ⎟ . (29) G=⎝ , G = ⎟ ⎜ 0 1 −2 1 ⎠ ⎝ 0 1 1 −3 1 ⎠ 1 0 1 −2 0 0 1 1 −2 Let Γ = diag(1, 1, 1, 1) and P = diag(0, 1, 0), which set the coupling ways of the variables of systems (27) and (28).
25
40
20
30
15
z
z¯
50
20
10
10
5 0 10
0 40 20
40 20
0
0
−20
y
−20 −40
−40
15 0
10 5
−5
y¯
x
(a) Hyperchaotic L¨ u attractor, with parameters a = 36.0, b = 3.0, c = 20.0, d = 1.3. Figure 2
5 0
−10 −15
−5 −10
x¯
(b) R¨ ossler chaotic attractor, with parameters p = 0.15, q = 0.20, r = 10.0.
Hyperchaotic L¨ u attractor vs R¨ ossler chaotic attractor
According to Theorem 3.2, we can estimate the drive strength a∗ and a∗ by calculating the coupling strength ε∗ and ε∗ of two identical oscillators. We can use the coupling strength for Hyperchaotic L¨ u system ε∗ = 0.5 and the coupling strength for R¨ ossler chaotic system ∗ ε = 0.2 by numerical simulation. Note the definition of r∗ and r∗ in (22), we can calculate the synchronization contribute values of the clusters structure: when the coupling strength of network is equal to 1, then r∗ = 34 and r∗ = 54 from Figure 1 (b). Here, 2ε∗ − r∗ < 0 (2ε∗ − r∗ < 0), that is, there is sufficiently small drive a(a) such that cluster (2)((3)) can come to synchronization (ref. Remark 3.2). The simulations shown in Figure 3 are the graphs of synchronization in Hyperchaotic L¨ u attractor cluster and R¨ ossler chaotic attractor cluster, which are coincident with the conclusion above. 4.2 Lorenz Chaotic System vs Duffing System Here, we consider the case 3 and the chaotic Lorenz system and Duffing system are selected as the dynamical nodes of cluster (2) and cluster (3) in the network, respectively. A single Lorenz system is described by[25] ⎧ ⎨ x˙ = σ(y − x), y˙ = ρx − xz − y, (30) ⎩ z˙ = xy − βz, and a single Duffing system is given by ¨ + γ x˙ + px3 = q cos(ωt). x
(31)
31
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
20
20
100
0
80
ex
ey
0 −10
−20
−20 −30
−40 0
2
4
6
0
2
4
6
t
t 30
60 40
20
ek
ez
20 10
0 0 −10
Coupl ed R¨ossler St ates
10
60
40
x¯ z¯
20
0
y¯
−20
−20 0
2
4
6
−40
0
2
t
4
6
−40
0
5
10
15
t
(a) Synchronization errors of Hyperchaotic L¨ u attractors in cluster (2) versus time. Figure 3
20
25
t
(b) The state variables of R¨ ossler chaotic attractor in cluster (3) versus time.
Graph of synchronization in Hyperchaotic L¨ u attractor cluster and R¨ ossler chaotic attractor cluster
˙ we reformulate it as an autonomous system Letting z = t, y = x, ⎧ ⎨ x˙ = y, y˙ = −γy − µx3 + ν cos(ωz), ⎩˙ z = 1.
(32)
When σ = 10, ρ = 28, β = 83 , and γ = 0.3, µ = 1, ν = 39, ω = 1, the above two systems both have chaotic attractors as shown in Figure 4. The coexistence case is shown in Figure 1 (c) and will be discussed below. 15 50 10 40 5
z
30
y¯
20
0
10 −5 0 −20
−10 0 20 y
−20
−10
0
10
20
x
(a) Lorenz chaotic attractor, with parameters σ = 10, ρ = 28, β = 83 . Figure 4
−15 −6
−4
−2
0
x¯
2
4
6
(b) Duffing chaotic attractor, with parameters γ = 0.3, µ = 1, ν = 39, ω = 1.
Lorenz attractor vs Duffing chaotic attractor
We consider a Lorenz cluster with 10 nodes, where a simple ring with 7 nodes and 3 isolated nodes are in coexistence, and a Duffing cluster with 20 nodes, where a star shape graph with 15 nodes and 5 isolated nodes are in coexistence. The corresponding coupled matrices G and G are described as
LIANG CHEN · JUN’AN LU
32
⎞ ··· 0 ··· 0⎟ ⎟ .. .. ⎟ . .⎟ ⎟ , ··· 0⎟ ⎟ ⎟ ··· 0⎟ ··· 0⎠ · · · 0 10×10
⎛
−2 1 0 · · · 1 0 ⎜ 1 −2 1 · · · 0 0 ⎜ ⎜ .. .. .. . . .. .. ⎜ . . . . . . ⎜ G = ⎜ 1 0 · · · 1 −2 0 ⎜ ⎜ 0 ··· ··· ··· 0 0 ⎜ ⎝ 0 ··· ··· ··· 0 0 0 ··· ··· ··· 0 0
⎛
−14 ⎜ 1 ⎜ ⎜ .. ⎜ . ⎜ G=⎜ ⎜ 1 ⎜ 0 ⎜ ⎜ . ⎝ ..
1 ··· 1 0 0 ··· 0 0 .. . . .. .. . . . . 0 · · · · · · −1 0 ··· ··· ··· 0 0 . . . · · · . . · · · .. .. 0 ··· ··· ··· 0 0 1 −1 .. .
⎞ ··· 0 ··· 0⎟ ⎟ .. .. ⎟ . .⎟ ⎟ ··· 0⎟ ⎟ ··· 0⎟ ⎟ .. ⎟ . 0⎠ ··· 0
.
(33)
20×20
Letting Γ = diag(1, 0, 0) and P = diag(1, 0, 0), we select the first variable of systems (30) and (32) to couple in the first function of corresponding systems. According to Theorem 3.3, one can determine the drive strength a∗ and a∗ by calculating the coupling strength ε∗ and ε∗ of two identical oscillators. We can use the coupling strength ε∗ = 5.5 for Lorenz system and the coupling strength ε∗ = 0.4 for Duffing system in numerical 2 ∗ − σ2 is more than 150 which is too simulation. Note that the bound[20] ε∗ = a2 = β(β+1)(ρ+σ) 32(β−1) large to apply in simulations. The simulation results are shown in Figure 5 to illuminate the correctness of Theorem 3.3. In Figure 5 (a), we find that the synchronization errors of nodes of the Lorenz cluster are globally asymptotically convergent to zero. And the Figure 5 (b) also shows that the state variables of Duffing cluster in the network indeed synchronize. 20
0
2
4
6
8
10
ey
20 0 −20
0
2
4
6
8
10
ez
50 0 −50
0
2
4
6
8
10
t
100
50
x¯
0
y¯
−50
−100
−150
0
5
10
15
20
t
(a) Synchronization errors of Lorenz chaotic attractors in cluster (2) versus time. Figure 5
Coup led Du
ex 0 −10
g Systemes Sta tes
150
10
(b) The state variables of Duffing chaotic attractor in cluster (3) versus time.
Graph of synchronization in Lorenz chaotic attractor cluster and Duffing chaotic attractor cluster
5 Conclusions In summary, we have studied global stability of cluster synchronization in a complex dynamical network with two nonidentical clusters. Specifically, the sufficient conditions for the stability of synchronization manifolds in the clusters with the same-input condition are obtained by using the connection graph stability method. Furthermore, there exists an upper bound of input strength of outside clusters which can scale the synchronizability of the cluster. Exceeding the upper bound, the nodes in the same cluster can realize synchrony. Two numerical examples of such network with different chaotic nodes are given to illustrate the validity and feasibility of synchronization criteria. Finally, the synchronous theory of complex network consisting of more clusters and the case of directed networks should be further investigated in the near future.
SYNCHRONY IN TWO NONIDENTICAL CLUSTERS
33
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