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

Abstract

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.

1. 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.

2. 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 𝑁₁ and 𝑁₂ nodes which are 𝑛₁ and 𝑛₂ 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 𝑥̇_𝑖=𝑓𝑥_𝑖+𝑐₁^𝑁1_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑥_𝑘−𝑥_𝑖+𝑑₁^𝑁2_𝑘=1𝑏_𝑖𝑘Γ₁𝑦_𝑘−𝑥_𝑖,𝑖=1,2,…,𝑁₁,𝑦̇_𝑗=𝑔𝑦_𝑗+𝑐₂^𝑁2_{𝑘=1,𝑘≠𝑗}𝑎_𝑗𝑘𝑦_𝑘−𝑦_𝑗+𝑑₂^𝑁1_𝑘=1𝑏_𝑗𝑘Γ₂𝑥_𝑘−𝑦_𝑗,𝑗=1,2,…,𝑁₂. (2.1)

Equivalently, (2.1) can be rewritten as 𝑥̇_𝑖=𝑓𝑥_𝑖+𝑐₁^𝑁1_𝑘=1𝑎_𝑖𝑘𝑥_𝑘+𝑑₁^𝑁2_𝑘=1𝑏_𝑖𝑘Γ₁𝑦_𝑘−𝑑₁^𝑁2_𝑘=1𝑏_𝑖𝑘𝑥_𝑖,𝑖=1,2,…,𝑁₁,𝑦̇_𝑗=𝑔𝑦_𝑗+𝑐₂^𝑁2_𝑘=1𝑎_𝑗𝑘𝑦_𝑘+𝑑₂^𝑁1_𝑘=1𝑏_𝑗𝑘Γ₂𝑥_𝑘−𝑑₂^𝑁1_𝑘=1𝑏_𝑗𝑘𝑦_𝑗,𝑗=1,2,…,𝑁₂, 2.1^′ where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛1)^𝑇∈𝑅^𝑛1 and 𝑦_𝑗=(𝑦_𝑗1,𝑦_𝑗2,…,𝑦_𝑗𝑛2)^𝑇∈𝑅^𝑛1 are state vectors of the nodes. 𝑓∶Ω₁→𝑅^𝑛1(Ω₁⊆𝑅^𝑛1) and 𝑔∶Ω₂→𝑅^𝑛2(Ω₂⊆𝑅^𝑛2) are smooth nonlinear function vectors. 𝑐₁,𝑐₂>0 are the inner coupling strength of each cluster, 𝑑₁,𝑑₂>0 are the external excitation intensity acting on each cluster. Γ₁∈𝑅^𝑛1×𝑛2, Γ₂∈𝑅^𝑛2×𝑛1 are internal coupling matrices when the corresponding oscillators used to output, and have form [𝐼,0] or [𝐼,0]^𝑇(where 𝐼 denotes identity matrix, 0 is a proper dimension zero matrix). Obviously, there has Γ₁=Γ₂^𝑇. The matrix 𝐴₁=(𝑎_𝑖𝑗)∈𝑅^𝑁1×𝑁1 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 𝐴₁ are 𝑎_𝑖𝑖=−∑^𝑁1_𝑗=1𝑎_𝑖𝑗(𝑖=1,2,…,𝑁₁). With these assumptions, the eigenvalues of matrix 𝐴₁ can be given by 0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁1. The matrix 𝐴₂=(𝑎_𝑖𝑗)∈𝑅^𝑁2×𝑁2 which represents the internal connection in 𝑦-cluster is the same as 𝐴₁, and its eigenvalues can be given by 0=𝜇₁>𝜇₂≥⋯≥𝜇_𝑁2. The matrices 𝐵₁=(𝑏_𝑖𝑗)∈𝑅^𝑁1×𝑁2 and 𝐵₂=(𝑏_𝑖𝑗)∈𝑅^𝑁2×𝑁1 represent 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 𝑐_𝑖𝑘=𝑐_𝑗𝑘,𝑖,𝑗=1,…,𝑚,𝑘=1,…,𝑛. (2.2) Moreover, if the external input matrices 𝐵₁ and 𝐵₂ 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 “same-input” condition. From this condition, we have 𝑏_𝑖𝑘=𝑏_1𝑘,∑^𝑁2_𝑘=1𝑏_𝑖𝑘=𝑏(𝑖=1,2,…,𝑁₁), and 𝑏_𝑖𝑘=𝑏_1𝑘, ∑^𝑁2_𝑘=1𝑏_𝑖𝑘=𝑏(𝑖=1,2,…,𝑁₂). 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 𝑠₁=(1/𝑁₁)∑^𝑁1_𝑖=1𝑥_𝑖,𝑠₂=(1/𝑁₂)∑^𝑁2_𝑗=1𝑦_𝑗 and 𝑓(𝑠₁)=(1/𝑁₁)∑^𝑁1_𝑖=1𝑓(𝑥_𝑖), 𝑔(𝑠₂)=(1/𝑁₂)∑^𝑁2_𝑗=1𝑔(𝑦_𝑗).

Then, we have 𝑠₁̇=𝑓𝑠₁+𝑑₁^𝑁2_𝑘=1𝑏_1𝑘Γ₁𝑦_𝑘−𝑑₁𝑏𝑠₁,𝑠₂̇=𝑔𝑠₂+𝑑₂^𝑁1_𝑘=1𝑏_1𝑘Γ₂𝑥_𝑘−𝑑₂𝑏𝑠₂. (2.3)

Definition 2.2. The set 𝑆=𝑥^𝑇₁,…,𝑥^𝑇_𝑁1,𝑦^𝑇₁,…,𝑦^𝑇_𝑁2^𝑇∶𝑥_𝑖=𝑠₁,𝑦_𝑗=𝑠₂,𝑖=1,…,𝑁₁,𝑗=1,…,𝑁₂ (2.4) is called cluster synchronous manifold of the network (2.1).
Therefore, the synchronous errors are denoted as 𝑥_𝑖=𝑥_𝑖−𝑠₁(𝑖=1,…,𝑁₁) for the 𝑥-cluster and 𝑦_𝑗=𝑦_𝑗−𝑠₂(𝑗=1,…,𝑁₂) for the 𝑦-cluster. Denote 𝑋=(𝑥^𝑇₁,…,𝑥^𝑇_𝑁1)^𝑇, 𝑌=(𝑦^𝑇₁,…,𝑦^𝑇_𝑁2)^𝑇. Then the error equations are given by ̇𝑥_𝑖=𝑓𝑥_𝑖−𝑓𝑠₁+𝑐₁^𝑁1_𝑘=1𝑎_𝑖𝑘𝑥_𝑘−𝑑₁𝑏𝑥_𝑖,𝑖=1,2,…,𝑁₁,̇𝑦_𝑖=𝑔𝑦_𝑗−𝑔𝑠₂+𝑐₂^𝑁2_𝑘=1𝑎_𝑗𝑘𝑦_𝑘−𝑑₂𝑏𝑦_𝑗,𝑗=1,2,…,𝑁₂. (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,…,𝑁₁) and lim_𝑡→+∞‖𝑦_𝑗‖=0(𝑗=1,2,…,𝑁₂), 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 𝑓𝐳₁−𝑓𝐳₂≤𝛿_𝑓𝐳₁−𝐳₂,𝑔𝐳₁−𝑔𝐳₂≤𝛿_𝑔𝐳₁−𝐳₂, (2.6) where 𝐳₁,𝐳₂ 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 𝐴₁ and 𝐴₂, one has𝑋^𝑇𝐴₁⊗𝐼_𝑛1𝑋≤𝜆₂^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖,𝑌^𝑇𝐴₂⊗𝐼_𝑛2𝑌≤𝜇₂^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗. (2.7)

Proof. Since 𝐴₁ is a real symmetric matrix, there exists an orthogonal matrix 𝑄 such that 𝐴₁=𝑄Λ₁𝑄^𝑇, where 𝑄𝑄^𝑇=𝐼, Λ₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑁1).
Take transformation 𝜁=(𝑄^𝑇⊗𝐼_𝑛1)𝑋, that is, 𝑋=(𝑄⊗𝐼_𝑛1)𝜁, where 𝜁=(𝜁^𝑇₁,𝜁^𝑇₂,…,𝜁^𝑇_𝑁1), 𝜁^𝑇_𝑖∈𝑅^𝑛1. Denote 𝑄=(𝑞₁,𝑞₂,…,𝑞_𝑁1), where 𝑞₁=(1/𝑁₁)(1,1,…,1)^𝑇 is the eigenvector corresponding to 𝜆₁=0, and we have 𝜁₁=1𝑁₁^𝑁1_𝑖=1𝑥_𝑖=1𝑁₁^𝑁1_𝑖=1𝑥_𝑖−𝑠₁=0. (2.8)
Moreover, 𝜁^𝑇𝜁=𝑋^𝑇𝑄⊗𝐼_𝑛1𝑄^𝑇⊗𝐼_𝑛1𝑋=𝑋^𝑇𝑋, (2.9) that is, ∑^𝑁1_𝑖=1𝜁^𝑇_𝑖𝜁_𝑖=∑^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖. So one has 𝑋^𝑇𝐴₁⊗𝐼_𝑛1𝑋=𝜁^𝑇𝑄^𝑇𝐴₁𝑄⊗𝐼_𝑛1𝜁=𝜁^𝑇Λ₁⊗𝐼_𝑛1𝜁=^𝑁1_𝑖=1𝜁^𝑇_𝑖𝜆_𝑖𝜁_𝑖≤𝜆₂^𝑁1_𝑖=1𝜁^𝑇_𝑖𝜁_𝑖=𝜆₂^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖. (2.10)
Similarly, one can obtain 𝑌^𝑇(𝐴₂⊗𝐼_𝑛2)𝑌≤𝜇₂∑^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗. This completes the proof.

3. 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: 𝛿_𝑓+𝑐₁𝜆₂−𝑑₁𝑏<0,𝛿_𝑔+𝑐₂𝜇₂−𝑑₂𝑏<0. (3.1)

Proof. Consider the function 𝑉(𝑡)=(1/2)∑^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+(1/2)∑^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗.
Its time derivative along the trajectory of (2.5) is 𝑉̇(𝑡)=^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑓𝑥_𝑖−𝑓𝑠₁+^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑓𝑠₁−𝑓𝑠₁+𝑐₁^𝑁1𝑁_𝑖=1¹_𝑘=1𝑥^𝑇_𝑖𝑎_𝑖𝑘𝑥_𝑘−𝑑₁𝑏^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑔𝑦_𝑗−𝑔𝑠₂+^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑔𝑠₂−𝑔𝑠₂+𝑐₂^𝑁2𝑁_𝑗=1²_𝑘=1𝑦^𝑇_𝑗𝑎_𝑗𝑘𝑦_𝑘−𝑑₂𝑏̃^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗, (3.2)
since ^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑓𝑠₁−𝑓𝑠₁=𝑓𝑠₁−𝑓𝑠₁^𝑁1_𝑖=1𝑥_𝑖−𝑠₁^𝑇=0. (3.3)
Similarly, one has ∑^𝑁2_𝑗=1𝑦^𝑇_𝑗(𝑔(𝑠₂)−𝑔(𝑠₂))=0.
One has 𝑉̇(𝑡)≤𝛿_𝑓^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝑐₁^𝑁1𝑁_𝑖=1¹_𝑘=1𝑥^𝑇_𝑖𝑎_𝑖𝑘𝑥_𝑘−𝑑₁𝑏^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝛿_𝑔^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+𝑐₂^𝑁2𝑁_𝑗=1²_𝑘=1𝑦^𝑇_𝑗𝑎_𝑗𝑘𝑦_𝑘−𝑑₂𝑏̃^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗=𝛿_𝑓−𝑑₁𝑏^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝑐₁𝑋^𝑇𝐴₁⊗𝐼_𝑛1𝑋+𝛿_𝑔−𝑑₂𝑏^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+𝑐₂𝑌^𝑇𝐴₂⊗𝐼_𝑛2𝑌. (3.4)
By Lemma 2.4, one has 𝑉̇(𝑡)≤𝛿_𝑓+𝑐₁𝜆₂−𝑑₁𝑏^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝛿_𝑔+𝑐₂𝜇₂−𝑑₂𝑏^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗≤−2𝜗𝑉(𝑡), (3.5) where 𝜗=min{|𝛿_𝑓+𝑐₁𝜆₂−𝑑₁𝑏|,|𝛿_𝑔+𝑐₂𝜇₂−𝑑₂𝑏|}>0. Then one has 𝑉(𝑡)≤𝑉(0)𝑒^−2𝜗𝑡.
Notice that 𝑉(𝑡)≥(1/2)‖𝑥_𝑖‖², so ‖𝑥_𝑖‖≤2𝑉(0)𝑒^−𝜗𝑡→0, that is, lim_𝑡→+∞‖𝑥_𝑖‖=0(𝑖=1,2,…,𝑁₁). Similarly, lim_𝑡→+∞‖𝑦_𝑗‖≤lim_𝑡→+∞2𝑉(0)𝑒^−𝜗𝑡=0(𝑗=1,2,…,𝑁₂), 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, 𝑐₁,𝑐₂ may take any positive number, even zero, if the external excitation intensity is sufficiently large such that 𝑑₁𝑏>𝛿_𝑓 and 𝑑₂𝑏>𝛿_𝑔. Thus, the following corollary is derived.

Corollary 3.2. For the following network: 𝑥̇_𝑖=𝑓𝑥_𝑖+𝑑₁^𝑁2_𝑘=1𝑏_𝑖𝑘Γ₁𝑦_𝑘−𝑥_𝑖,𝑖=1,2,…,𝑁₁,𝑦̇_𝑗=𝑔𝑦_𝑗+𝑑₂^𝑁1_𝑘=1𝑏_𝑗𝑘Γ₂𝑥_𝑘−𝑦_𝑗,𝑗=1,2,…,𝑁₂, (3.6) which is a special case of network (2.1), the cluster synchronous manifold 𝑆 is globally exponentially stable under the conditions 𝑑₁𝑏>𝛿_𝑓  and  𝑑₂𝑏>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.

4. 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 𝑐₁ and 𝑐₂ being larger than their necessary values. To overcome this drawback, we design 𝑐₁ and 𝑐₂ 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 𝑐₁  and  𝑐₂ as adaptive variables, then the cluster synchronous manifold 𝑆 is globally asymptotically stable under the following adaptive laws: 𝑐₁̇(𝑡)=^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖,𝑐₁(0)𝑐̇>0,₂(𝑡)=^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗,𝑐₂(0)>0. (4.1)

Proof. Consider the Lyapunov function 𝑉(𝑡)=12^𝑁1_𝑘=1𝑥^𝑇_𝑘𝑥_𝑘+12^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+12𝛽₁𝛽₁𝑐₁(𝑡)+𝑑₁𝑏−𝛿_𝑓−1²+12𝛽₂𝛽₂𝑐₂(𝑡)+𝑑₂𝑏−𝛿_𝑔−1², (4.2) where 𝛽₁=−𝜆₂>0and𝛽₂=−𝜇₂>0.
Its time derivative along the trajectory of (2.5) is 𝑉̇(𝑡)=^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥̇_𝑖+^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦̇_𝑗+𝛽₁𝑐₁(𝑡)+𝑑₁𝑏−𝛿_𝑓−1𝑐₁̇(𝑡)+𝛽₂𝑐₂(𝑡)+𝑑₂𝑏−𝛿_𝑔−1𝑐₂̇(𝑡)=^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑓𝑥_𝑖−𝑓𝑠₁+𝑐₁(𝑡)^𝑁1𝑁_𝑖=1¹_𝑘=1𝑥^𝑇_𝑖𝑎_𝑖𝑘𝑥_𝑘−𝑑₁𝑏^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝛽₁𝑐₁(𝑡)+𝑑₁𝑏−𝛿_𝑓−1^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑔𝑦_𝑗−𝑔𝑠₂+𝑐₂(𝑡)^𝑁2𝑁_𝑗=1²_𝑘=1𝑦^𝑇_𝑗𝑎_𝑗𝑘𝑦_𝑘−𝑑₂𝑏^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+𝛽₂𝑐₂(𝑡)+𝑑₂𝑏−𝛿_𝑔−1^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗. (4.3) Similar to the proof of Theorem 3.1, one has 𝑉̇(𝑡)≤𝛿_𝑓^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝑐₁(𝑡)𝜆₂^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝛽₁𝑐₁(𝑡)−𝛿_𝑓−1^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖+𝛿_𝑔^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+𝑐₂(𝑡)𝜇₂^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+𝛽₂𝑐₂(𝑡)−𝛿_𝑔−1^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗=−^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖−^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗≤0. (4.4)
By the LaSalle-Yoshizawa theorem [19, 20], we have lim_𝑡→+∞‖𝑥_𝑖‖=0(𝑖=1,2,…,𝑁₁) and lim_𝑡→+∞‖𝑦_𝑗‖=0(𝑗=1,2,…,𝑁₂), which means that the cluster synchronous manifold 𝑆 is globally asymptotically stable. Now the proof is completed.

Similarly, we can further obtain the following theorem.

Theorem 4.2. For the network (2.1), take the external excitation intensities 𝑑₁,𝑑₂ as adaptive variables, then the cluster synchronous manifold 𝑆 is globally asymptotically stable under the following adaptive laws: 𝑑̇₁(𝑡)=^𝑁1_𝑖=1𝑥^𝑇_𝑖𝑥_𝑖,𝑑₁(0)𝑑̇>0,₂(𝑡)=^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗,𝑑₂(0)>0. (4.5)

Proof. Consider the Lyapunov function 𝑉(𝑡)=(1/2)∑^𝑁1_𝑘=1𝑥^𝑇_𝑘𝑥_𝑘+(1/2)∑^𝑁2_𝑗=1𝑦^𝑇_𝑗𝑦_𝑗+(1/2𝑏)(𝑏𝑑₁(𝑡)−𝑐₁𝜆₂−𝛿_𝑓−1)²+(1/2𝑏)(𝑏𝑑₂(𝑡)−𝑐₂𝜇₂−𝛿_𝑔−1)², 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).

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 Lü oscillators [22] as the 𝑦-cluster. That is, the node’s dynamics is 𝑓(𝑥_𝑖)=(10(𝑥_𝑖2−𝑥_𝑖1),28𝑥_𝑖1−𝑥_𝑖2−𝑥_𝑖1𝑥_𝑖3,𝑥_𝑖1𝑥_𝑖2−8/3𝑥_𝑖3)^𝑇(𝑖=1,…,50), or 𝑔(𝑦_𝑗)=(36(𝑦_𝑗2−𝑦_𝑗1)+𝑦_𝑗4,20𝑦_𝑖2−𝑦_𝑗1𝑦_𝑗3,𝑦_𝑗1𝑦_𝑗2−3𝑦_𝑗3,𝑦_𝑗1𝑦_𝑗3+𝑦_𝑗4)^𝑇(𝑗=1,…,30). The coupling matrix 𝐴₁ of the 𝑥-cluster is taken as an adjacency matrix of scale-free network with 50 nodes, The coupling matrix 𝐴₂ of the 𝑦-cluster is taken as an adjacency matrix of small-world network with 30 nodes, and the matrices 𝐵₁, 𝐵₂, Γ₁, and Γ₂ 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 𝑐₁(0)=𝑐₂(0)=𝑑₁(0)=𝑑₂(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.

(a) The synchrony state 𝑥𝑖(1≤𝑖≤50)

(b) The synchrony states 𝑦𝑗(1≤𝑗≤30)

Figure 1 

Cluster synchronization in the linearly coupled network (2.1) with 50 lorenz oscillators in the 𝑥-cluster and 30 hyperchaotic Lü oscillators in the 𝑦-cluster.

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.

(a) The synchrony state 𝑥𝑖(1≤𝑖≤50)

(b) The synchrony states 𝑦𝑗(1≤𝑗≤30)

Figure 2 

Cluster synchronization of the linearly coupled network (3.6) with 50 lorenz oscillators in the 𝑥-cluster and 30 hyperchaotic Lü oscillators in the 𝑦-cluster.

(a)   The synchrony state 𝑥𝑖(1≤𝑖≤50)

(b) The synchrony states 𝑦𝑗(1≤𝑗≤30)

(c) The estimation of coupling strength 𝑐1

(d) The estimation of coupling strength 𝑐2

Figure 3 

Cluster synchronization of the adaptively coupled network (2.1) with 50 lorenz oscillators in the 𝑥-cluster and 30 hyperchaotic Lü oscillators in the 𝑦-cluster.

(a) The synchrony state 𝑥𝑖(1≤𝑖≤50).

(b) The synchrony states 𝑦𝑗(1≤𝑗≤30).

(c) The estimation of coupling strength 𝑑1.

(d) The estimation of coupling strength 𝑑2.

Figure 4 

Cluster synchronization of the adaptively coupled network (3.6) with 50 lorenz oscillators in the 𝑥-cluster and 30 hyperchaotic Lü oscillators in the 𝑦-cluster.

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 𝑑₁ and 𝑑₂ are more conducive to the cluster synchronization.

6. 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.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grants 61164020, 61004101, the Natural Science Foundation of Guangxi under Grant 2011GXNSFA018147, and Guangxi Key Laboratory of Spatial Information and Geomatics under Grant 1103108-24.