Pinning Group Synchronization in Complex Dynamical Networks with Different Groups of Oscillators

Abstract

This paper investigates the pinning control schemes and corresponding criteria for group synchronization in a complex dynamical network with different types of chaotic oscillators. We present the linear pinning and adaptive pinning control schemes to make different groups of oscillators synchronize to their own synchronization states, respectively. The globally asymptotically stable criteria for group synchronization are derived, which indicate that the group synchronization can be realized only by pinning a part of nodes in a general network. Finally, some numerical simulations are provided to verify the theoretical results.

1. Introduction

Nowadays, complex networks widely exist in nature and society, such as the World Wide Web, the Internet, traffic network, communication network, power network, and social network. Over the past three decades, complex networks have received increasing attention from various fields such as biology, physics, chemistry, engineering, and information [1–3].

Synchronization is a kind of typical collective behaviors and basic motion in nature. As a typical network dynamics, the synchronization of complex networks has been intensively studied recently. Different synchronization phenomena have been revealed, such as global synchronization, generalized synchronization, partial synchronization, and cluster synchronization [4–11]. In particular, the group (cluster) synchronization, which means that nodes in the same group achieve the same synchronization state while nodes in different groups achieve different synchronization states, is considered to be significant in biological science, laser technology, and communication engineering [1, 2, 12]. In [13], the authors investigated the group synchronization in complex networks with two clusters of coupled dynamical systems and revealed the group synchronization motions whenever there are connections or not within a cluster. In [14], the authors investigated the group synchronization in delay-coupled dynamical networks with groups of different and identical local dynamics, respectively. In [15], we presented a complex dynamical network model with 𝑀 groups of different chaotic oscillators and derived some criteria to guarantee the group synchronization for a linearly or adaptively coupled network, respectively, under the “same input” condition. However, in many cases, the “same input” condition is too conservative to satisfy. If the “same input” condition cannot be satisfied, does the network achieve group synchronization? In this paper, we will propose control schemes to solve this problem.

As we known now, the real-world complex networks normally have a large number of nodes. Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes. To reduce the number of the controllers, a usual approach, which is called the pinning control, is used to control part of nodes. Wang and Chen [16] and Li et al. [17] proposed some effective methods to pin a complex dynamical network to its equilibrium. In [18], Chen et al. proved that a single controller can pin a coupled complex network to a homogenous solution. Zhou et al. presented a simple approximate formula for estimating the detailed number of pinning nodes and the magitude of the coupling strength for a given general complex dynamical network in [19]. However, the researchers mentioned above focused on the global synchronization of complex dynamical networks by pinning control. In this work, we will use the pinning control schemes to realize the group synchronization in a general complex dynamical network.

In this paper, based on the model in [15], we further investigate the group synchronization in a complex network containing 𝑀 groups of different oscillators. Different from [15], this work mainly focuses on the pinning control schemes and the corresponding criteria to realize the group synchronization in a complex dynamical network. In particular, the network has a general structure rather than the “same input” condition.

The rest of the paper is organized as follows. In Section 2, the network model of our research and mathematical preliminaries are introduced. In Section 3, we present the linear feedback and adaptive feedback pinning schemes, respectively. The corresponding criteria for the globally asymptotically stable of the group synchronization manifold are derived. In Section 4, some numerical simulations are provided to verify the theoretical results. Finally, a brief summary is given in Section 5.

2. Model Description and Mathematical Preliminaries

Let us consider a complex dynamical network with 𝑀 groups; each group contains a number of identical dynamical systems, but the individual dynamic system in different groups can be different. Suppose that there are 𝑁_𝐼 nodes in 𝐼th group, and the individual dynamics satisfies 𝑥̇^𝐼_𝑖=𝐹_𝐼(𝑥^𝐼_𝑖). The 𝑀-groups network is described by 𝑥̇^𝐼_𝑖=𝐹_𝐼𝑥^𝐼_𝑖+𝑐^𝑁𝐼_𝑘≠𝑖𝑎^𝐼𝐼_𝑖𝑘𝑥^𝐼_𝑘−𝑥^𝐼_𝑖+𝑐^𝑀𝑁_𝐽≠𝐼^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑥^𝐽_𝑗−𝑥^𝐼_𝑖,𝑖=1,2,…,𝑁_𝐼,𝐽=1,2,…,𝑀, (1) where 𝑥^𝐼_𝑖∈𝑅^𝑛𝐼 is state vector of the 𝑖th node in the 𝐼th group. 𝐹_𝐼 is a smooth nonlinear function vector which describes the node’s dynamics of the 𝐼th group. 𝑐>0 is the coupling strength among nodes. The matrix 𝐴_𝐼𝐽=(𝑎^𝐼𝐽_𝑖𝑗)∈𝑅^{𝑁𝐼×𝑁𝐽} represents the connection between 𝐼th and 𝐽th groups, where if node 𝑖 in 𝐼th cluster and node 𝑗 in 𝐽th cluster is connected, 𝑎^𝐼𝐽_𝑖𝑗=1; otherwise, 𝑎^𝐼𝐽_𝑖𝑗=0. In particular, matrix 𝐴_𝐼𝐼 represents the internal connection in 𝐼th group, while 𝐴_𝐼𝐽(𝐼≠𝐽) represents the external connection between the 𝐼th and 𝐽th group.

Furthermore, denote the block matrix 𝐴=(𝐴_𝐼𝐽), which represents the structure of network (1). The diagonal elements of matrix 𝐴 satisfy 𝑎^𝐼𝐼_𝑖𝑖=−∑^𝑀_𝐼=1∑^𝑁𝐼_𝑗=1𝑎^𝐼𝐽_𝑖𝑗,𝑖=1,2,…,𝑁_𝐼. Then 𝐴 is a 𝑁×𝑁 symmetric irreducible matrix with zero-row-sum, and Rank(𝐴)=𝑁−1.

Symbol Description. The constant 𝐼 represents 𝐼th group or the set of nodes in 𝐼th group. 𝐼̃ represents the subset of 𝐼 containing the nodes with external connections; that is, 𝐼̃={𝑖∶𝑖∈𝐼,∑^𝑀_𝐽≠𝐼∑^𝑁𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗≠0}, and 𝐼⧵𝐼̃={𝑖∶𝑖∈𝐼,𝑎^𝐼𝐽_𝑖𝑗=0 for 𝐼≠𝐽}. 𝑁_𝐼 and 𝑙_𝐼 denote the number of nodes of set 𝐼 (𝐼th group) and 𝐼̃, respectively. 𝑁 is the total number of nodes in network (1). ‖⋅‖ is the Euclidian 2-norm; that is, ‖𝑥‖=𝑥^𝑇𝑥 for vector 𝑥.

Assumption 1 (A1). Suppose that there exists positive constant 𝜀_𝑘 such that 𝐹_𝑘𝐳_𝟏−𝐹_𝑘𝐳_𝟐≤𝜀_𝑘𝐳_𝟏−𝐳_𝟐, (2) where 𝐳_𝟏,𝐳_𝟐 are time-varying vectors.

Definition 2. The set 𝑆=𝑥¹₁,…,𝑥¹_𝑁1,…,𝑥^𝑀₁,…,𝑥^𝑀_𝑁𝑀∶𝑥^𝐼_𝑖=𝑠_𝐼,𝑖=1,…,𝑁_𝐼,𝐼=1,…,𝑀 (3) is called group synchronous manifold of network (1).

Here, 𝑠_𝐼(𝑡) is the solution of the individual dynamic system of the 𝐼th group satisfying 𝑠_𝐼̇(𝑡)=𝐹_𝐼(𝑠_𝐼(𝑡)). Our objective is to stable the homogenous trajectory of 𝑥^𝐼_𝑖(𝑖=1,2,…,𝑁_𝐼) to 𝑠_𝐼(𝑡),𝐼=1,2,…,𝑀. 𝑠_𝐼(𝑡) may be an equilibrium point, a periodic orbit, an aperiodic orbit, or a chaotic orbit in the phase space.

The controlled network (1) can be rewritten as 𝑥̇^𝐼_𝑖=𝐹_𝐼𝑥^𝐼_𝑖+𝑐^𝑁𝐼_𝑘≠𝑖𝑎^𝐼𝐼_𝑖𝑘𝑥^𝐼_𝑘−𝑥^𝐼_𝑖+𝑐^𝑀𝑁_𝐽≠𝐼^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑥^𝐽_𝑗−𝑥^𝐼_𝑖+𝑢^𝐼_𝑖,𝑖=1,2,…,𝑁_𝐼,𝐽=1,2,…,𝑀. (4)

Define the synchronous error vectors of the 𝐼th group as 𝑒^𝐼_𝑖=𝑥^𝐼_𝑖−𝑠_𝐼(𝑖=1,…,𝑁_𝐼) and 𝐸_𝐼=((𝑒^𝐼₁)^𝑇,…,(𝑒^𝐼_𝑁𝐼)^𝑇)^𝑇. Then the error equations are given by 𝑒^𝐼_𝑖̇=𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑁𝐼_𝑘≠𝑖𝑎^𝐼𝐼_𝑖𝑘𝑒^𝐼_𝑘+𝑐^𝑀𝑁_𝐽≠𝐼^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗+𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽−𝑥^𝐼_𝑖+𝑢^𝐼_𝑖,𝑖=1,2,…,𝑁_𝐼,𝐽=1,2,…,𝑀. (5)

For 𝑖∈𝐼̃, there is ^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽−𝑥^𝐼_𝑖=^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽−^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑥^𝐼_𝑖=^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽. (6)

For 𝑖∈I⧵𝐼̃, then 𝑎^𝐼𝐽_𝑖𝑗=0 for 𝐼≠𝐽; there is ^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽−𝑥^𝐼_𝑖=^𝑁𝐼_𝑘=1𝑎^𝐼𝐼_𝑖𝑘𝑠_𝐼−𝑥^𝐼_𝑖+^𝑀𝑁_𝐽≠𝐼^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽−𝑥^𝐼_𝑖=^𝑁𝐼_𝑘=1𝑎^𝐼𝐼_𝑖𝑘𝑠_𝐼−𝑥^𝐼_𝑖=^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐼−𝑥^𝐼_𝑖=0. (7)

Then the controlled error system (5) can be rewritten as 𝑒^𝐼_𝑖̇=𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑁𝐼_𝑘≠𝑖𝑎^𝐼𝐼_𝑖𝑘𝑒^𝐼_𝑘+𝑐^𝑀𝑁_𝐽≠𝐼^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗+𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽+𝑢^𝐼_𝑖̃𝑒̇,𝑖∈𝐼,_𝐼𝑖=𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼s_𝐼+𝑐^𝑁𝐼_𝑘≠𝑖𝑎^𝐼𝐼_𝑖𝑘𝑒^𝐼_𝑘+𝑢^𝐼_𝑖,𝑖∈𝐼⧵𝐼,̃𝐼=1,2,…,𝑀. (8)

Obviously, the stability problem of group synchronous manifold 𝑆 of the controlled network (1) is equivalent to the stability of system (8) at zero, that is, lim_𝑡→+∞‖𝑒^𝐼_𝑖‖=0(𝑖=1,…,𝑁_𝐼,𝐼=1,…,𝑀), which implies that group synchronization is achieved.

In order to analyze the group synchronization, the following lemmas are needed.

Lemma 3 (see [20]). Suppose that 𝐹, 𝐻 are the 𝑁×𝑁 Hermitian matrices, and the eigenvalues of 𝐹, 𝐻,𝐹+𝐻 are 𝛼₁≥𝛼₁≥⋯≥𝛼_𝑁, 𝛽₁≥𝛽₁≥⋯≥𝛽_𝑁, and 𝛾₁≥𝛾₁≥⋯≥𝛾_𝑁, respectively. Then there is 𝛼_𝑖+𝛽_𝑁≤𝛾_𝑖≤𝛼_𝑖+𝛽₁,1≤𝑖≤𝑁. (9)

Lemma 4 (see [18]). If 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 is an irreducible matrix with Rank(𝐺)=𝑁−1, satisfying 𝑔_𝑖𝑗=𝑔_𝑗𝑖≥0 for 𝑖≠𝑗, and ∑^𝑁_𝑗=1𝑔_𝑖𝑗=0(𝑖=1,…,𝑁), then, for any constant 𝜀>0, all eigenvalues of the matrix 𝑔𝐺=¹¹−𝜀𝑔₁₂⋯𝑔_1𝑁𝑔₂₁𝑔₂₂⋯𝑎_2𝑁𝑔⋮⋮⋱⋮_𝑁1𝑔_𝑁2⋯𝑔_𝑁𝑁 (10) are negative.

Corollary 5. Suppose that 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 satisfies the conditions of Lemma 4. Matrix Δ=diag{𝜀₁,…,𝜀_𝑙,0,…,0} for 𝜀_𝑙>0,1≤𝑙<𝑁. Then all eigenvalues of 𝐺=𝐺−Δ are negative.

Proof. If 𝑙=1, obviously, the conclusion is established by Lemma 4. If 𝑙>1, then 𝐺 can be rewritten as 𝐺=𝐺−Δ, where 𝐺 is in the form of (10), Δ=diag{0,𝜀₂,…,𝜀_𝑙,0,…,0}.
By Lemma 4, all eigenvalues of 𝐺 are negative. Denote the eigenvalues of 𝐺 and 𝐺 as 𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁 and 0>𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁, respectively.
According to Lemma 3, one has 𝜆_𝑖−max_{1≤𝑖≤𝑙}𝜀_𝑖≤𝜆_𝑖≤𝜆_𝑖𝜆+0<0,1≤𝑖≤𝑁−𝑙+1,_𝑁−𝜀_𝑖≤𝜆_𝑖≤𝜆₁−𝜀_𝑖<0,𝑁−𝑙+1<𝑖≤𝑁. (11)
That is, all eigenvalues of 𝐺 are negative. Thus the proof is completed.

Remark 6. From the derivation process, the conclusion of Corollary 5 is also valid when 𝜀₁,…,𝜀_𝑙 are in any positions of the diagonal of Δ; that is to say, the result holds without any dependence on the sequence of the diagonal elements of Δ.

3. Schemes and Criteria

In this section, we present the linear feedback and adaptive feedback pinning schemes, respectively, and derive the corresponding criteria to achieve the group synchronization in network (1).

3.1. Linear Feedback Pinning Control for Group Synchronization

For network (1), we add the following linear feedback pinning controllers: 𝑢^𝐼_𝑖=−𝑐𝑑^𝐼_𝑖𝑒^𝐼_𝑖−𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽̃̃,𝑖∈𝐼,0,𝑖∈𝐼⧵𝐼. (12)

Then the error equation (8) can be rewritten as 𝑒^𝐼_𝑖̇=𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐𝑑^𝐼_𝑖𝑒^𝐼_𝑖,𝑑^𝐼_𝑖>0for𝑖∈𝐼;̃𝑑^𝐼_𝑖=0for𝑖∈𝐼⧵𝐼,̃𝑖=1,2,…,𝑁_𝐼,𝐼=1,2,…,𝑀. (13)

Theorem 7. Suppose that 𝐴1 holds. The group synchronous manifold 𝑆 of the controlled network (1) under the linear feedback pinning controllers (12) is globally asymptotically stable if the following inequality holds: min_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼+𝜆_max(𝐴−𝐷)<0, (14) where 𝐴=(𝐴_𝐼𝐽)_𝑁×𝑁, 𝐷=diag{𝑑¹₁,…,𝑑¹_𝑙1,0,…,0_𝑁1,…,𝑑^𝑀₁,…,𝑑^𝑀_𝑙𝑀,0,…,0_𝑁𝑀}.

Proof. Consider the Lyapunov function 𝑉(𝑡)=(1/2)∑^𝑀_𝐼=1∑^𝑁𝐼_𝑖=1(𝑒^𝐼_𝑖)^𝑇𝑒^𝐼_𝑖.
Its time derivative along the trajectory of (13) is 𝑉̇(𝑡)=^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑑^𝐼_𝑖𝑒^𝐼_𝑖≤^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇‖⋅‖𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑀𝑁_𝐼=1^𝐼𝑀_𝑖=1^𝑁_𝐽=1^𝐽_𝑗=1𝑒^𝐼_𝑖^𝑇𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑑^𝐼_𝑖𝑒^𝐼_𝑖≤^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝜀_𝐼𝑥^𝐼_𝑖−𝑠_𝐼+𝑐^𝑀𝑁_𝐼=1^𝐼𝑀_𝑖=1^𝑁_𝐽=1^𝐽_𝑗=1𝑒^𝐼_𝑖^𝑇𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑑^𝐼_𝑖𝑒^𝐼_𝑖=^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝜀_𝐼𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖+𝑐^𝑀𝑁_𝐼=1^𝐼𝑀_𝑖=1^𝑁_𝐽=1^𝐽_𝑗=1𝑒^𝐼_𝑖^𝑇𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑑^𝐼_𝑖𝑒^𝐼_𝑖=𝐸^𝑇𝜀_𝐼𝐼_𝑁𝐼⊗𝐼_𝑀𝐸+𝑐𝐸^𝑇𝐴𝐸−𝑐𝐸^𝑇𝐷𝐸=𝐸^𝑇𝜀_𝐼𝐼_𝑁𝐼⊗𝐼_𝑀𝐸+𝑐𝐸^𝑇𝐴𝐸−𝑐𝐸^𝑇𝐷𝐸=𝑐𝐸^𝑇(𝑃+(𝐴−𝐷))𝐸, (15) where 𝑃=diag{𝑐⁻¹𝜀₁,…,𝑐⁻¹𝜀_1𝑁1,…,𝑐⁻¹𝜀_𝑀,…,𝑐⁻¹𝜀_𝑀𝑁𝑀}, 𝐸=((𝐸₁)^𝑇,…,(𝐸_𝑀)^𝑇)^𝑇.
According to the Corollary 5, 𝐴−𝐷 is negative definite.
Denote the eigenvalues of 𝐴−𝐷 and 𝑃+(𝐴−𝐷) as 0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁 and 𝛾₁≥𝛾₂≥⋯≥𝛾_𝑁, respectively. The eigenvalues of 𝑃 are 𝛽_𝑗,𝑗=1,2,…,𝑁; there is min_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼≤𝛽_𝑗≤max_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼. (16)
According to Lemma 3, one has 𝜆_𝑁+𝛽_𝑗≤𝛾_𝑗≤𝜆₁+𝛽_𝑗≤𝜆₁+max_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼=𝜆_max(𝐴−𝐷)+max_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼<0,𝑗=1,2,…,𝑁. (17)
That is, 𝑃+(𝐴−𝐷) is negative definite. Then one can obtain lim_𝑡→+∞‖𝑒^𝐼_𝑖‖=0(𝑖=1,…,𝑁_𝐼,𝐼=1,…,𝑀), which means that the group synchronous manifold 𝑆 is globally asymptotically stable. Thus the proof is completed.

Remark 8. From Theorem 7, one can obtain the sufficient condition (14) by means of adjusting the coupling strength 𝑐. Therefore, the group synchronization can be guaranteed.

Remark 9. In this linear feedback pinning scheme, one only needs to pin a few nodes which have external connections; then the network achieves the group synchronization with large enough coupling strength. In particular, it is only relevant to network connectivity, without any limitation to the network structure. For a cluster network, in general, the connection within a cluster is more than the external connection between clusters. When the network has a cluster property, the higher the clustering, the better the control effect. Since the network with higher clustering means that the number of nodes with external connection is less, thus the less nodes are needed to be pinned. Therefore, the higher the degree of clustering, the more obvious the advantages.

3.2. Adaptive Feedback Pinning Control for Group Synchronization

In order to avoid estimating the feedback control gain, in this section, we present an adaptive feedback pining scheme to realize the group synchronization in network (1).

For network (1), we add the adaptive feedback pinning controllers as follows: 𝑢^𝐼_𝑖=−𝑐𝑑^𝐼_𝑖(𝑡)𝑒^𝐼_𝑖−𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑠_𝐽,𝑑̇^𝐼_𝑖(𝑡)=𝑐𝑑𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖̃̃,𝑑>0,for𝑖∈𝐼;0,for𝑖∈𝐼⧵𝐼. (18)

Then the error equation (8) can be rewritten as 𝑒^𝐼_𝑖̇=𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐𝑑^𝐼_𝑖𝑒^𝐼_𝑖,𝑑̇^𝐼_𝑖=𝑐𝑑𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖for𝑖∈𝐼;̃𝑑^𝐼_𝑖=0for𝑖∈𝐼⧵𝐼,̃𝑖=1,2,…,𝑁_𝐼,𝐼=1,2,…,𝑀. (19)

Theorem 10. Suppose that 𝐴1 holds. The group synchronous manifold 𝑆 of the controlled network (1) under the adaptive feedback pinning controllers (18) is globally asymptotically stable if the following inequality holds: min_{1≤𝐼≤𝑀}𝑐⁻¹𝜀_𝐼+𝜆_max𝐴−𝐷<0, (20) where 𝐴=(𝐴_𝐼𝐽)_𝑁×𝑁,𝐷=diag{𝑑,…,𝑑_𝑙1,0,…,0_{𝑁1−𝑙1},…,𝑑,…,𝑑_𝑙𝑀,0,…,0_{𝑁𝑀−𝑙𝑀}}.

Proof. Consider the Lyapunov function 𝑊(𝑡)=12^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖+12𝑑^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑑^𝐼_𝑖−𝑑². (21)
Its time derivative along the trajectory of (19) is 𝑊̇=^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖̇+1𝑑^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑑^𝐼_𝑖−𝑑𝑑̇^𝐼_𝑖=^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝐹_𝐼𝑥^𝐼_𝑖−𝐹_𝐼𝑠_𝐼+𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑀𝑁_𝐽=1^𝐽_𝑗=1𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝑑^𝐼_𝑖e^𝐼_𝑖+𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑑^𝐼_𝑖−𝑑𝑒^𝐼_𝑖^𝑇𝑒^𝐼_𝑖≤^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇𝜀_𝑖𝑒^𝐼_𝑖+𝑐^𝑀𝑁_𝐼=1^𝐼𝑀_𝑖=1^𝑁_𝐽=1^𝐽_𝑗=1𝑒^𝐼_𝑖^𝑇𝑎^𝐼𝐽_𝑖𝑗𝑒^𝐽_𝑗−𝑐^𝑀𝑁_𝐼=1^𝐼_𝑖=1𝑒^𝐼_𝑖^𝑇d𝑒^𝐼_𝑖=𝑐𝐸^𝑇𝑃+𝐴−𝐷𝐸. (22)
The rest is similar to the proof of Theorem 7; one has lim_𝑡→+∞‖𝑒^𝐼_𝑖‖=0(𝑖=1,…,𝑁_𝐼,𝐼=1,…,𝑀), which means that the group synchronous manifold 𝑆 is globally asymptotically stable. Thus the proof is completed.

Remark 11. In Theorem 10, the adaptive constant 𝑑 can be large enough such that inequality (20) holds. Therefore, the group synchronization can be easily guaranteed.

4. Numerical Simulations

In this section, an illustrative example is provided to verify the above theoretical analysis.

We consider a network (see Figure 1) which consists of three groups with 4 Lorenz chaotic oscillators [20] as the 𝑥-group, 5 Chua oscillators [21] as the 𝑦-group, and 6 Chen oscillators [22] as the 𝑧-group, respectively. The Lorenz, Chua, and Chen chaotic attractors are displayed in Figure 2.

Figure 1 

The topology connection of network (23).

(a)

(b)

(c)

Figure 2 

The chaotic attractors of Lorenz (a), Chua (b), and Chen (c).

The network as an example of the controlled network (1) is described by 𝑥̇_𝑖=𝐹₁𝑥_𝑖+𝑐⁴_𝑗=1𝑎_𝑖𝑗𝑥_𝑗+𝑐⁵_𝑗=1𝑏_𝑖𝑗𝑦_𝑗−𝑥_𝑖+𝑐⁶_𝑗=1ℎ_𝑖𝑗𝑧_𝑗−𝑥_𝑖+𝑢¹_𝑖𝑦̇,1≤𝑖≤4,_𝑖=𝐹₂𝑦_𝑖+𝑐⁵_𝑗=1𝑎_𝑖𝑗𝑦_𝑗+𝑐⁴_𝑗=1𝑏_𝑖𝑗𝑥_𝑗−𝑦_𝑖+𝑐⁶_𝑗=1ℎ_𝑖𝑗𝑧_𝑗−𝑦_𝑖+𝑢²_𝑖𝑧̇,1≤𝑖≤5,_𝑖=𝐹₃𝑧_𝑖+𝑐⁶_𝑗=1𝑎_𝑖𝑗𝑧_𝑗+𝑐⁴_𝑗=1𝑏̃_𝑖𝑗𝑥_𝑗−𝑧_𝑖+𝑐⁵_𝑗=1ℎ_𝑖𝑗𝑦_𝑗−𝑧_𝑖+𝑢³_𝑖,1≤𝑖≤6, (23) where 𝐹₁(𝑥_𝑖)=(10(𝑥_𝑖2−𝑥_𝑖1),28𝑥_𝑖1−𝑥_𝑖2−𝑥_𝑖1𝑥_𝑖3,𝑥_𝑖1𝑥_𝑖2−8/3𝑥_𝑖3)^𝑇; 𝐹₂(𝑦_𝑖)=(10(𝑦_𝑖2−𝑦_𝑖1)−𝑓(𝑦_𝑖1),𝑦_𝑖1−𝑦_𝑖2+𝑦_𝑖3,−14.7𝑦_𝑖2)^𝑇, 𝑓(𝑣)=𝑚₀𝑣+0.5(𝑚₁−𝑚₀)(|𝑥+1|−|𝑥−1|),𝑚₀=−0.68,𝑚₁=−1.27; 𝐹₃(𝑧_𝑖)=(35(𝑧_𝑖2−𝑧_𝑖1),−7𝑧_𝑖1+28𝑧_𝑖2−𝑧_𝑖1𝑧_𝑖3,𝑧_𝑖1𝑧_𝑖2−8/3𝑧_𝑖3)^𝑇. The controllers 𝑢^𝐼_𝑖 are taken as (12) and (18), respectively.

In the simulation, we take the initial values 𝑥_𝑖(0)=(−5+0.5𝑖,0.5𝑖,5+0.5𝑖)^𝑇(1≤𝑖≤4), 𝑦_𝑖(0)=(−5+0.5𝑖,0.5𝑖,5+0.5𝑖,10+0.5𝑖)^𝑇(1≤𝑖≤5), 𝑧_𝑖(0)=(−5+0.5𝑖,0.5𝑖,5+0.5𝑖)^𝑇(1≤𝑖≤6), and 𝑑(0)=1.

Figures 3 and 4 display the numerical simulation results of network (23) under the linear feedback and adaptive feedback pinning control, respectively. They both show that a set of nodes belonging to each group synchronizes to the same behavior; that is, the group synchronization has been achieved.

(a)

(b)

(c)

Figure 3 

Group synchronization of network (23) with linear feedback pinning controllers (12). (a) The synchrony states 𝑥𝑖(1≤𝑖≤4) of Lorenz oscillators in 𝑥-group; (b) the synchrony states 𝑦𝑖(1≤𝑖≤5) of Chua oscillators in 𝑦-group; (c) the synchrony states 𝑧𝑖(1≤𝑖≤6) of Chen oscillators in 𝑧-group.

(a)

(b)

(c)

Figure 4 

Group synchronization of network (23) with adaptive feedback pinning controllers (18). (a) The synchrony states 𝑥𝑖(1≤𝑖≤4) of Lorenz oscillators in 𝑥-group; (b) the synchrony states 𝑦𝑖(1≤𝑖≤5) of Chua oscillators in 𝑦-group; (c) the synchrony states 𝑧𝑖(1≤𝑖≤6) of Chen oscillators in 𝑧-group.

Remark 12. By the above theoretical analysis and instance, it is concluded that the pinning control strategy can be applied to large networks with tens of thousands of nodes based on the accurate network topology. In fact, this control strategy depends on the structure information rather than the size of the network.

5. Conclusions

In this paper, we have investigated the group synchronization in a complex dynamical network consisting of different groups of coupled chaotic oscillators. The linear feedback and adaptive feedback pinning schemes are presented, respectively. The corresponding sufficient conditions are derived to ensure each group synchronizes to the same states. Our study shows that the global asymptotic stability of the group synchronous manifold can be guaranteed by controlling a few nodes which have external connections. For the large networks with obvious cluster property, our pinning control schemes can save much cost in practical.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation (no. 61164020), the National Social Science Foundation (no. 13BTJ009), the Natural Science Foundation of Guangxi (0991244, 2011 GXNSFA018147), Guangxi Key Laboratory of Spatial Information and Geomatics (nos. 1103108-24, 1207115-27).