Mathematical Model and Cluster Synchronization for a Complex Dynamical Network with Two Types of Chaotic Oscillators

We propose a mathematical model of a complex dynamical network consisting of two types of chaotic oscillators and investigate the schemes and corresponding criteria for cluster synchronization. The global asymptotically stable criteria for the linearly or adaptively coupled network are derived to ensure that each group of oscillators is synchronized to the same behavior. The cluster synchronization can be guaranteed by increasing the inner coupling strength in each cluster or enhancing the external excitation. Theoretical analysis and numerical simulation results show that the external excitation is more conducive to the cluster synchronization. All of the results are proved rigorously. Finally, a network with a scale-free subnetwork and a small-world subnetwork is illustrated, and the corresponding numerical simulations verify the theoretical analysis.


Introduction
Since the pioneering works by Watts and Strogatz on the small-world network 1 and by Barabási and Albert on the scale-free network 2 , complex networks have been studied extensively in various disciplines, such as social, biological, mathematical, and engineering sciences 3 .Besides the properties of "small-world" and "scale-free," another common property in real-world complex networks is cluster or community, module structure 4 .Many real networks are composed of several clusters within which the connection of nodes is more than that of nodes between different clusters, or the nodes have some common properties in a cluster.This feature can be seen in many networks such as social networks 5 , biological networks 6 , and citation networks 7 .
Synchronization of coupled chaotic oscillators is one of commonly collective coherent behaviors attracting a growing interest in physics, biology, communication, and other fields of science and technology.Synchronization of complex networks has attracted tremendous attention in recent years.Different synchronization phenomena in complex networks have been studied, such as global synchronization 8, 9 , partial synchronization and cluster synchronization 10-15 .In particular, the cluster synchronization, which implies that nodes in the same group achieve the same synchronization state while nodes in different groups achieve different synchronization state, is considered to be significant in biological science 6 , laser technology, and communication engineering 11 .
More recently, some new progress in cluster synchronization of complex dynamical networks have been reported 16-18 .In 16 , Belykh et al. have studied the cluster synchronization for conditional clusters and unconditional clusters in an oscillator network with given configuration based on a graph theoretical approach, and the corresponding existence and stability conditions are proposed.In 17 , authors presented a linear feedback control strategy to achieve the cluster synchronization for a network with identical oscillators.In 18 authors have investigated the cluster synchronization in a dynamical network consisting of two groups of nonidentical oscillators, and upper bounds of input strength for the synchrony of each cluster are derived under the "same-input" condition.
In this paper, based on 18 , we further investigate the cluster synchronization of a complex network containing two groups of different oscillators.But different from 18 , our research mainly focuses on the inner coupling strength and the external excitation intensity to ensure the cluster synchronization.Intuitively, the inner coupling of clusters is beneficial to the cluster synchronization.Yet our study shows that the external excitation intensity is more conducive to the cluster synchronization under the "same-input" condition.Theoretical analysis and numerical simulation results show that, even without any connection within a cluster, the cluster synchronization can be still achieved by enhancing the external excitation.
The rest of the paper is organized as follows.In Section 2, the mathematical model of our research network is proposed and preliminaries are introduced.The linear coupling criteria for the cluster synchronization are derived in Section 3. The adaptive inner coupling and external excitation schemes and corresponding conditions for the cluster synchronization are presented in Section 4. The numerical simulations are provided to verify the effectiveness of the theoretical analysis in Section 5. Finally, a brief summary of the obtained results is given in Section 6.

Mathematical Model and Preliminaries
Let's consider a dynamical network with two clusters, each cluster contains a number of identical dynamical systems, however, the subsystems composing the two clusters can be different, that is, the individual dynamical system in one cluster can differ from that in the other cluster.Suppose that two clusters are composed of N 1 and N 2 nodes which are n 1 and n 2 dimensional dynamical, oscillator as ẋ f x , ẏ g y , respectively.We call each cluster as x-cluster and y-cluster respectively.The two-cluster-network is described by

2.1
Equivalently, 2.1 can be rewritten as With these assumptions, the eigenvalues of matrix A 1 can be given by 0 which represents the internal connection in y-cluster is the same as A 1 , and its eigenvalues can be given by 0 external connection between two clusters which satisfy "same-input" condition see Definition 2.1 , and all the elements b ij and b ij take 0 or 1.

Definition 2.1. A matrix C
c ij m×n is said to satisfy condition SI, if its elements satisfy Moreover, if the external input matrices B 1 and B 2 satisfy the condition SI, then the network 2.1 is said to satisfy the "same-input" condition.
Note that, we suppose that all network models throughout this paper satisfy "sameinput" condition.From this condition, we have The positive constants b and b are regarded as the external excitation acting on each cluster.It implies that the input of nodes in the same cluster is equal.
Denote T .Then the error equations are given by

2.5
Obviously, the stability problem of cluster synchronous manifold S in the network 2.1 is equivalent to the stability of system 2.5 at zero.Our objective is to find the criteria for the coupling strength such that the network 2.1 achieves cluster synchronization, that is, lim t → ∞ x i 0 i 1, 2, . . ., N 1 and lim t → ∞ y j 0 j 1, 2, . . ., N 2 , which implies that the x-cluster and y-cluster achieve synchronization, respectively.
In order to achieve cluster synchronization, a useful assumption and a lemma are introduced as follows.
Assumption 2.3.A1 Suppose that there exists positive constants δ f and δ g such that where z 1 , z 2 are time-varying vectors.
Note that we assume that A1 is satisfied by all models in this paper.
Lemma 2.4.For the above matrices A 1 and A 2 , one has y T j y j .

2.7
Proof.Since A 1 is a real symmetric matrix, there exists an orthogonal matrix Q such that q 2 , . . ., q N 1 , where q 1 1/ N 1 1, 1, . . ., 1 T is the eigenvector corresponding to λ 1 0, and we have Journal of Applied Mathematics 5 Moreover, x T i x i .

2.10
Similarly, one can obtain j 1 y T j y j .This completes the proof.

Linear Coupling Scheme and Criteria for Cluster Synchronization
In this section, we propose linear coupling schemes to achieve the cluster synchronization, and derive the corresponding criteria for the coupling strength in the network 2.1 .
For the linearly coupled network 2.1 , the following criteria can be derived.
Theorem 3.1.For the linearly coupled network 2.1 , the cluster synchronous manifold S is globally exponentially stable under the following condition: Proof.Consider the function V t 1/2 j 1 y T j y j .Its time derivative along the trajectory of 2.5 is Similarly, one has N 2 j 1 y T j g s 2 − g s 2 0. One has

3.4
By Lemma 2.4, one has where 0 j 1, 2, . . ., N 2 , which implies that the cluster synchronization manifold S of dynamical network 2.1 is globally exponentially stable.Now the proof is completed.
According to Theorem 3.1, for the cluster synchronization, c 1 , c 2 may take any positive number, even zero, if the external excitation intensity is sufficiently large such that d 1 b > δ f and d 2 b > δ g .Thus, the following corollary is derived.

3.6
which is a special case of network 2.1 , the cluster synchronous manifold S is globally exponentially stable under the conditions d 1 b > δ f and d 2 b > δ g .Remark 3.3.In model 3.6 , there is no connection inside clusters.Corollary 3.2 shows that the cluster synchronization can be achieved even if without any connection within a cluster.It implies that the external excitation intensity is more conducive to the cluster synchronization than the cluster's interconnection under the "same-input" condition.

Adaptive Coupling Scheme and Criteria for Cluster Synchronization
Note that, the Lipschitz constants δ f and δ g are required to be known in Theorem 3.1.However, it is often difficult to obtain the precise values of δ f and δ g in some practical systems, hence the constants δ f and δ g are often selected to be larger, which leads to the coupling strengths c 1 and c 2 being larger than their necessary values.To overcome this drawback, we design c 1 and c 2 as adaptive variables, and present a local adaptive coupling scheme to realize cluster synchronization as follows.
Theorem 4.1.For the network 2.1 , take the inner coupling strengths c 1 and c 2 as adaptive variables, then the cluster synchronous manifold S is globally asymptotically stable under the following adaptive laws: y T j y j , c 2 0 > 0.

4.1
Proof.Consider the Lyapunov function where Its time derivative along the trajectory of 2.5 is y T j y j .

4.3
Similar to the proof of Theorem 3.1, one has y T j y j ≤ 0.
Similarly, we can further obtain the following theorem.Theorem 4.2.For the network 2.1 , take the external excitation intensities d 1 , d 2 as adaptive variables, then the cluster synchronous manifold S is globally asymptotically stable under the following adaptive laws: , and similar to the proof of Theorem 4.1, one can obtain the conclusion.Corollary 4.3.For the network 3.6 , the cluster synchronous manifold S is globally asymptotically stable under the adaptive law 4.5 .

Numerical Simulations
In this section, illustrative examples are provided to verify the above theoretical analysis.We consider a network which consists of two clusters with a scale-free sub-network with 50 Lorenz chaotic oscillators 21 as the x-cluster and a small-world sub-network with 30 hyperchaotic L ü oscillators 22 as the y-cluster.That is, the node's dynamics is

Journal of Applied Mathematics
T i 1, . . ., 50 , or g y j 36 y j2 − y j1 y j4 , 20y i2 − y j1 y j3 , y j1 y j2 − 3y j3 , y j1 y j3 y j4 T j 1, . . ., 30 .The coupling matrix A 1 of the x-cluster is taken as an adjacency matrix of scale-free network with 50 nodes, The coupling matrix A 2 of the y-cluster is taken as an adjacency matrix of small-world network with 30 nodes, and the matrices B 1 , B 2 , Γ 1 , and Γ 2 satisfy the conditions of models 2.1 and 3.6 .
In the simulation, we take initial values x i 0 −1 0.5i, −2 0.5i, −3 0.5i T , y j 0 1 0.5j, 2 0.5j, 3 0.5j, 4 0.5j T , where 1 ≤ i ≤ 50, 1 ≤ j ≤ 30, and c 1 0 1. Figures 1 and 3 display the numerical simulation results of network 2.1 linear coupling and adaptive coupling, respectively.It shows that a set of nodes belonging to each cluster synchronize to the same behavior, that is, the cluster synchronization has been achieved quite soon.
Figures 2 and 4 show the numerical simulation results of network 3.6 with linear coupling and the adaptive coupling, respectively, where there are no connections inside the cluster.Obviously, synchronization has been reached quite soon.
Remark 5.1.Figures 1 and 2 show that the cluster synchronization x-cluster and y-cluster can be reached while the complete synchronization cannot be achieved in the network.Furthermore, compared with the coupling strength in Figures 3 and 4, one can also see that the external excitation d 1 and d 2 are more conducive to the cluster synchronization.

Conclusions
In this paper, we have further investigated the cluster synchronization of a complex dynamical network with given configuration which is connected by two groups of different oscillators.we present a linear coupling scheme and the corresponding sufficient condition is derived for the cluster synchronization.Moreover, an adaptive coupling scheme to lead    the cluster synchronization is proposed based on adaptive control technique.Our study shows that the global stability of the cluster synchronization can be guaranteed by increasing coupling strength in each cluster or enhancing the external excitation even if there are no connections insider a cluster.Chaos synchronization of delay systems 23 with adaptive impulsive control method 24 is widely discussed and applied, so we will study cluster synchronization in delay-coupling cluster networks with impulsive control in the future work.

Figure 1 :
Figure 1: Cluster synchronization in the linearly coupled network 2.1 with 50 lorenz oscillators in the x-cluster and 30 hyperchaotic L ü oscillators in the y-cluster.

bFigure 2 :
Figure 2: Cluster synchronization of the linearly coupled network 3.6 with 50 lorenz oscillators in the x-cluster and 30 hyperchaotic L ü oscillators in the y-cluster.

5 a 5 b 1 c 1 t 2 d 2 Figure 3 :
Figure 3: Cluster synchronization of the adaptively coupled network 2.1 with 50 lorenz oscillators in the x-cluster and 30 hyperchaotic L ü oscillators in the y-cluster.

ab 1 c 2 d
The synchrony state x i 1 ≤ i ≤ 50 .The synchrony states y j 1 ≤ j ≤ 30 .The estimation of coupling strength d 1 .The estimation of coupling strength d 2 .

Figure 4 :
Figure 4: Cluster synchronization of the adaptively coupled network 3.6 with 50 lorenz oscillators in the x-cluster and 30 hyperchaotic L ü oscillators in the y-cluster.
2are smooth nonlinear function vectors.c 1 , c 2 > 0 are the inner coupling strength of each cluster, d 1 , d 2 > 0 are the external excitation intensity acting on each cluster.Γ 1 ∈ R n 1 ×n 2 , Γ 2 ∈ R n 2 ×n 1 are internal coupling matrices when the corresponding oscillators used to output, and have form I, 0 or I, 0 Twhere I denotes identity matrix, 0 is a proper dimension zero matrix .Obviously, there hasΓ 1 Γ 2 T .The matrix A 1 a ij ∈ R N1 ×N 1 is a diffusive symmetric irreducible matrix which represents the internal connection in x-cluster.If node i and node j are connected in x-cluster, then a ij a ji 1, otherwise a ij a ji 0. The diagonal elements of A 1 are 3 T : x i s 1 , y j s 2 , i 1, . . ., N 1 , j 1, . . .