Global Synchronization in Complex Networks with Adaptive Coupling

Abstract

Global synchronization in adaptive coupling networks is studied in this paper. A new simple adaptive controller is proposed based on a concept of asymptotically stable led by partial state variables. Under the proposed adaptive update law, the network can achieve global synchronization without calculating the eigenvalues of the outer coupling matrix. The update law is only dependent on partial state variables of individual oscillators. Numerical simulations are given to show the effectiveness of the proposed method, in which the unified chaotic system is chosen as the nodes of the network with different topologies.

1. Introduction

Synchronization in complex networks of identical chaotic oscillators has been studied extensively and deeply in various fields of science and engineering in the last few years [1–18]. Some effective methods have been proposed to investigate the stability of the synchronous state of linearly coupled networks. In [2], Pecora and Carroll developed a useful approach, called the Master Stability Function, to the local synchronization for any linear coupling networks. Based on the calculation of the eigenvalues of the outer coupling matrix, this method has been widely used in local stability studies of synchronization in linearly coupled complex networks [3–6]. Global synchronization based on the eigenvalues of the outer coupling matrix was also obtained for undirected [1] and directed [7] networks. These studies show that both local and global synchronization mainly depend upon the eigenvalues of the outer coupling matrix. However, the eigenvalues can be calculated only for simple coupling schemes. For more complicated networks, it becomes a difficult task.

In [8–10], an alternate way is developed to achieve network synchronization, which does not depend on explicit knowledge of the eigenvalues of the outer coupling matrix. This approach combines the Lyapunov function approach with graph theoretical reasoning. It guarantees global synchronization, not just local stability, and is also valid for time-varying networks. In [11–18], adaptive control strategy is used to guarantee local and global synchronization in complex networks. Especially in [11], Zhang et al. present a concept of 𝑥_𝑘-leading asymptotically stable, and study the synchronization in complex networks with adaptive coupling. In [12], Huang studies the global synchronization in an adaptive weighted network, but the coupling strengths between two connected nodes are vector functions.

In [11], however, leading asymptotically stable is achieved by only one state variable. In actual fact, some systems are difficult to meet this condition. Those systems are asymptotically stable led by more than one state variable; moreover, the coupling strengths are not vector functions in practice. Inspired by the upper notions, this paper presents the concept of 𝑥_𝑘(𝑘=1,2,…,𝑠)(𝑠≤𝑛) leading asymptotically stable which means that the system achieves asymptotically stable led by partial state variables and studies the global synchronization in an adaptive coupling network whose coupling strengths are not vector function. By using Lyapunov’s direct method, under a simple adaptive update law, such network can achieve global synchronization finally. The synchronization strategy is setting adaptive coupling strength between two connected nodes, and the adaptive update law of 𝑎_𝑖𝑗 is only relation to partial state variables of connected nodes 𝑖 and 𝑗. The regular star coupled network and ring coupled network are simulated, in which the unified chaotic system is chosen as the dynamical node in the network. The simulation results also show the effectiveness of the proposed adaptive method.

Another active research named consensus or agreement considers the information exchange between the connected agents [19–22]. The aim is to set consensus protocols or coupling scheme to reach agreement on graph and the network could have switching topology [20, 21]. Of course, the consensus state is usually time-invariant [20], such as, average-consensus, max-consensus, min-consensus, and so on. To some extent, consensus is the special case of synchronization but quite independently studied.

2. Network Model and Preliminaries

Consider an undirected complex dynamical network consisting of 𝑁 identical coupled nodes. The dynamical behavior of the network can be described by the following ordinary differential equations: 𝑥̇_𝑖=𝐹(𝑥_𝑖)+^𝑁_𝑗=1𝑎_𝑖𝑗(𝑡)Γ𝑥_𝑗,𝑖=1,2,…,𝑁, (2.1) where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 is the state vector representing the state variables of node 𝑖. 𝐹(𝑥_𝑖)=(𝐹₁(𝑥_𝑖),𝐹₂(𝑥_𝑖),…,𝐹_𝑛(𝑥_𝑖))^𝑇 is a smooth nonlinear vector-valued function. Γ=diag(𝛾₁,…,𝛾_𝑘0,0,…,0)∈𝑀_𝑛(𝑅) is the inner coupling matrix, where 𝛾_𝑖>0(𝑖=1,2,…,𝑘₀) are some constants.

The outer coupling matrix 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡))_𝑁×𝑁 represents not only the topological structure of the network, but also the weight strength, in which 𝑎_𝑖𝑗(𝑡) is defined as follows: if there is a connection between node 𝑖 and node 𝑗(𝑗≠𝑖), then 𝑎_𝑖𝑗(𝑡)=𝑎_𝑗𝑖(𝑡)>0; otherwise, 𝑎_𝑖𝑗(𝑡)=𝑎_𝑗𝑖(𝑡)=0(𝑗≠𝑖). Suppose network (2.1) is diffusively coupled, that is, the diagonal elements of matrix 𝐴(𝑡) are defined by

𝑎_𝑖𝑖(𝑡)=−^𝑁_{𝑗=1𝑗≠𝑖}𝑎_𝑖𝑗(𝑡),𝑖=1,2,…,𝑁. (2.2) Here the coupling strengths depend on the corresponding two nodes (i.e., 𝑎_𝑖𝑗(𝑡) depends on subscripts 𝑖 and 𝑗) and time 𝑡. Simply speaking, our network is a weighted network. For convenience, we replace 𝑎_𝑖𝑗(𝑡) with 𝑎_𝑖𝑗 in the following.

Definition 2.1. An equilibrium point 𝑥=0 of a dynamical system 𝑥̇=𝑓(𝑥),𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 is 𝑥_𝑘(𝑘=1,2,…,𝑠)(𝑠≤𝑛) leading asymptotically stable if the asymptotical stability of 𝑥_𝑘=0(𝑘=1,2,…,𝑠) can lead the asymptotical stability of 𝑥_𝑘=0(𝑘=1,2,…,𝑛).

Remark 2.2. The 𝑥_𝑘-leading asymptotically stable in [11] is a special case of 𝑠=1 in our definition.

Letting 𝑒_𝑖𝑗=(𝑒_𝑖𝑗1,𝑒_𝑖𝑗2,…,𝑒_𝑖𝑗𝑛)^𝑇=𝑥_𝑗−𝑥_𝑖, it is obvious that 𝑒_𝑖𝑖=0,𝑒_𝑗𝑖=−𝑒_𝑖𝑗, and the error system is

𝑒̇_𝑖𝑗=𝐹(𝑥_𝑗)−𝐹(𝑥_𝑖)+^𝑁_𝑘=1(𝑎_𝑗𝑘Γ𝑒_𝑗𝑘−𝑎_𝑖𝑘Γ𝑒_𝑖𝑘),𝑖,𝑗=1,2,…,𝑁. (2.3) The coupled network (2.1) is said to achieve global synchronization if

lim_𝑡→∞‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖=0,thatis,𝑒_𝑖𝑗→0,(𝑡→∞),𝑖,𝑗=1,2,…,𝑁, (2.4) from arbitrary initial values.

3. Main Result

Now we can drive the following main result.

Theorem 3.1. Suppose that system (2.3) satisfies the following conditions.(I)Nonlinear function 𝐹∶𝑅^𝑛→𝑅^𝑛 satisfies the uniform Lipschitz condition, that is, for all 𝑥,𝑦, there exists a constant 𝑙>0 satisfying ||𝐹(𝑥)−𝐹(𝑦)||≤𝑙||𝑥−𝑦||.(II)Equilibrium point 𝑒_𝑖𝑗=0 is 𝑒_𝑖𝑗𝑘(𝑘=1,2,…,𝑠;𝑖,𝑗=1,2,…,𝑁)(𝑠≤𝑘₀) leading asymptotically stable.(III)The nonzero weight strength 𝑎_𝑖𝑗(𝑖≠𝑗) satisfy the following adaptive update law 𝑎̇_𝑖𝑗=𝑘_𝑖𝑗𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑗=𝑘_𝑖𝑗(𝑥_𝑖−𝑥_𝑗)^𝑇𝑃Γ(𝑥_𝑖−𝑥_𝑗). (3.1) Then network (2.1) with coupling strength (3.1) will achieve global synchronization as 𝑡→∞, where 𝑘_𝑖𝑗>0 are some arbitrary constants, 𝑃=diag(1,…,1_𝑠,0,…,0)∈𝑀_𝑛(𝑅).

Proof. Let the Lyapunov function be 1𝑉=2^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗+𝑁2^𝑁𝑁_𝑖=1𝑗=11𝑘_𝑖𝑗(𝑎_𝑖𝑗−𝑐)², (3.2) where 𝑐 is a large positive constant. Obviously, it is positive definite about 𝑒_𝑖𝑗𝑘(𝑘=1,2,…,𝑠). The time derivative of 𝑉 along the trajectory of the system (2.1) with coupling strength (3.1) is given by
𝑉̇=^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑗𝑃𝑒̇_𝑖𝑗+𝑁^𝑁𝑁_𝑖=1𝑗=11𝑘_𝑖𝑗(𝑎_𝑖𝑗−𝑐)𝑎̇_𝑖𝑗=^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑗𝑃(𝐹(𝑥_𝑗)−𝐹(𝑥_𝑖))+^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1(𝑎_𝑗𝑘𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑗𝑘−𝑎_𝑖𝑘𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑘)+𝑁^𝑁𝑁_𝑖=1𝑗=11𝑘_𝑖𝑗(𝑎_𝑖𝑗−𝑐)𝑎̇_𝑖𝑗. (3.3) From condition (I), the first sum 𝑆₁=∑^𝑁_𝑖=1∑^𝑁_𝑗=1𝑒^𝑇_𝑖𝑗𝑃(𝐹(𝑥_𝑗)−𝐹(𝑥_𝑖))≤∑^𝑁_𝑖=1∑^𝑁_𝑗=1𝑙𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗 holds. And the second sum can be calculated as follows. Since 𝑒^𝑇_𝑖𝑗=−𝑒^𝑇_𝑗𝑖, we can see that 𝑆₂=^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1(𝑎_𝑗𝑘𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑗𝑘−𝑎_𝑖𝑘𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑘)=−^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1(𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘+𝑎_𝑖𝑘𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑘). (3.4) Renaming the summation index 𝑖 by 𝑗 in the second term and vice versa, the second term becomes identical to the first and we obtain 𝑆₂=−2^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘. (3.5) Using 𝑒_𝑗𝑗=0, we get 𝑆₂=−2(^𝑁𝑁_𝑖=1^𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘+^𝑁𝑁_𝑖=1^𝑁_{𝑗=1𝑘=𝑗}𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘). (3.6) Renaming 𝑗 by 𝑘 in the second term and vice versa, and using the symmetry of 𝐴(𝑡), we obtain 𝑆₂=−2(^𝑁𝑁_𝑖=1^𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘+^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑗=𝑘}𝑎_𝑗𝑘𝑒^𝑇_𝑘𝑖𝑃Γ𝑒_𝑘𝑗)=−2(^𝑁𝑁_𝑖=1^𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑖𝑃Γ𝑒_𝑗𝑘+^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑗=𝑘}𝑎_𝑗𝑘𝑒^𝑇_𝑖𝑘𝑃Γ𝑒_𝑗𝑘)=−2^𝑁𝑁_𝑖=1^𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘(𝑒^𝑇_𝑗𝑖+𝑒^𝑇_𝑖𝑘)𝑃Γ𝑒_𝑗𝑘. (3.7) Since 𝑒^𝑇_𝑗𝑖+𝑒^𝑇_𝑖𝑘=𝑥^𝑇_𝑖−𝑥^𝑇_𝑗+𝑥^𝑇_𝑘−𝑥^𝑇_𝑖=𝑒^𝑇_𝑗𝑘 and the symmetry of 𝐴(𝑡), we get 𝑆₂=−2^𝑁𝑁_𝑖=1^𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑘𝑃Γ𝑒_𝑗𝑘=−2𝑁^𝑁𝑗_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑘𝑃Γ𝑒_𝑗𝑘=−𝑁^𝑁𝑁_𝑗=1𝑘=1𝑎_𝑗𝑘𝑒^𝑇_𝑗𝑘𝑃Γ𝑒_𝑗𝑘. (3.8) So 𝑉̇≤^𝑁𝑁_𝑖=1𝑗=1𝑙𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗−𝑁^𝑁𝑁_𝑖=1𝑗=1𝑎_𝑖𝑗𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑗+𝑁^𝑁𝑁_𝑖=1𝑗=11𝑘_𝑖𝑗(𝑎_𝑖𝑗−𝑐)𝑎̇_𝑖𝑗=^𝑁𝑁_𝑖=1𝑗=1𝑙𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗−𝑁^𝑁𝑁_𝑖=1𝑗=1𝑎_𝑖𝑗𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑗+𝑁^𝑁𝑁_𝑖=1𝑗=1(𝑎_𝑖𝑗−𝑐)𝑒^𝑇_𝑖𝑗𝑃Γ𝑒_𝑖𝑗=^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖𝑗[𝑙𝑃−𝑐𝑁𝑃Γ]𝑒_𝑖𝑗. (3.9) Because of 𝑠≤𝑘₀, for large constant 𝑐, it holds that 𝑙𝑃−𝑐𝑁𝑃Γ=diag(𝑐₁,𝑐₂,…,𝑐_𝑠,0,…,0)∈𝑀_𝑛(𝑅), where 𝑐_𝑖<0(𝑖=1,2,…,𝑠). So 𝑉̇ is negative definite about 𝑒_𝑖𝑗𝑘(𝑘=1,2,…,𝑠). According to Lyapunov’s direct method, starting with arbitrary initial values, 𝑒_𝑖𝑗𝑘→0(𝑡→∞)(𝑘=1,2,…,𝑠). Since equilibrium point 𝑒_𝑖𝑗=0 of system (2.3) is 𝑒_𝑖𝑗𝑘(𝑘=1,2,…,𝑠)(𝑠≤𝑘₀) leading asymptotically stable, so network (2.1) can reach global synchronization as 𝑡→∞. This completes the proof.

Remark 3.2. Condition (I) is easily satisfied if 𝜕𝐹_𝑖/𝜕𝑥_𝑗 (𝑖,𝑗=1,2,…,𝑛) are bounded. And a number of chaotic systems satisfy this condition, such as Rössler system and unified chaotic system.

Remark 3.3. In practice, we can set 𝑘_𝑖𝑗=𝑘, and the adaptive update law is 𝑎̇_𝑖𝑗=𝑘𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗=𝑘(𝑥_𝑖−𝑥_𝑗)^𝑇𝑃(𝑥_𝑖−𝑥_𝑗).

Remark 3.4. By (3.1), the nonzero coupling strength 𝑎_𝑖𝑗(𝑡)→𝐶_𝑖𝑗 when the network achieves synchronization, where 𝐶_𝑖𝑗 are some positive constants.

Remark 3.5. Our synchronization strategy is setting adaptive coupling strengths between two nodes to achieve global synchronization, and our adaptive update law of 𝑎_𝑖𝑗 is only relation to partial state variables of connected nodes 𝑖 and 𝑗.

4. Illustrative Example

To demonstrate the theoretical result in Section 3, two regular networks with the unified chaotic system are chosen as simulation examples in this section.

The unified chaotic [23] system is described by

𝑥̇₁=(25𝑎+10)(𝑥₂−𝑥₁𝑥̇),₂=(28−35𝑎)𝑥₁+(29𝑎−1)𝑥₂−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂−(𝑎+8)𝑥₃3. (4.1) When 𝑎∈[0,1], it has well-known chaotic behavior.

We consider Γ=diag(𝛾₁,𝛾₂,𝛾₃),𝛾_i>0(𝑖=1,2,3), that is,

𝑥̇_𝑖1=(25𝑎+10)(𝑥_𝑖2−𝑥_𝑖1)+𝛾₁^𝑁_𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝑗𝑥_𝑗1,𝑥̇_𝑖2=(28−35𝑎)𝑥_𝑖1+(29𝑎−1)𝑥_𝑖2−𝑥_𝑖1𝑥_𝑖3+𝛾₂^𝑁_𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝑗𝑥_𝑗2𝑥̇,(𝑖=1,2,…,𝑁),_𝑖3=𝑥_𝑖1𝑥_𝑖2−(𝑎+8)𝑥_𝑖33+𝛾₃^𝑁_𝑗=1𝑐_𝑖𝑗𝑎_𝑖𝑗𝑥_𝑗3. (4.2) The error system is

𝑒̇_𝑖𝑗1=(25𝑎+10)(𝑒_𝑖𝑗2−𝑒_𝑖𝑗1)+𝛾₁^𝑁_𝑘=1(𝑐_𝑗𝑘𝑎_𝑗𝑘𝑒_𝑗𝑘1−𝑐_𝑖𝑘𝑎_𝑖𝑘𝑒_𝑖𝑘1𝑒̇),_𝑖𝑗2=(28−35𝑎)𝑒_𝑖𝑗1+(29𝑎−1)𝑒_𝑖𝑗2−𝑥_𝑗1𝑥_𝑗3+𝑥_𝑖1𝑥_𝑖3+𝛾₂^𝑁_𝑘=1(𝑐_𝑗𝑘𝑎_𝑗𝑘𝑒_𝑗𝑘2−𝑐_𝑖𝑘𝑎_𝑖𝑘𝑒_𝑖𝑘2𝑒̇)(𝑖,𝑗=1,2,…,𝑁),_𝑖𝑗3=𝑥_𝑗1𝑥_𝑗2−𝑥_𝑖1𝑥_𝑖2−(𝑎+8)𝑒_𝑖𝑗33+𝛾₃^𝑁_𝑘=1(𝑐_𝑗𝑘𝑎_𝑗𝑘𝑒_𝑗𝑘3−𝑐_𝑖𝑘𝑎_𝑖𝑘𝑒_𝑖𝑘3). (4.3) In order to use our Theorem, we need the following Lemma.

Lemma 4.1. For 𝑎∈[0,1], system (4.3) is 𝑒_𝑖𝑗𝑘(𝑘=1,2) leading asymptotically stable.

Proof. If 𝑒_𝑖𝑗𝑘=0(𝑘=1,2) for all 𝑖 and 𝑗, choose the following Lyapunov function: 1𝑉=2^𝑁𝑁_𝑖=1𝑗=1𝑒²_𝑖𝑗3. (4.4) The derivative of 𝑉along with the solution of (4.2) is 𝑉̇=^𝑁𝑁_𝑖=1𝑗=1𝑒_𝑖𝑗3𝑒̇_𝑖𝑗3=^𝑁𝑁_𝑖=1𝑗=1[−(𝑎+8)𝑒²_𝑖𝑗33]+^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1(𝑎_𝑗𝑘𝑒_𝑖𝑗3𝑒_𝑗𝑘3−𝑎_𝑖𝑘𝑒_𝑖𝑗3𝑒_𝑖𝑘3). (4.5) Similar to the calculation of 𝑆₂, we can get ∑^𝑁_𝑖=1∑^𝑁_𝑗=1∑^𝑁_𝑘=1(𝑎_𝑗𝑘𝑒_𝑖𝑗3𝑒_𝑗𝑘3−𝑎_𝑖𝑘𝑒_𝑖𝑗3𝑒_𝑖𝑘3)<0 and for all 𝑎∈[0,1], 𝑉̇<0. So it is said that 𝑒_𝑖𝑗3→0(𝑡→∞)(𝑖,𝑗=1,2,…,𝑁).
If we set 𝑎̇_𝑖𝑗=𝑘_𝑖𝑗𝛾₁(𝑥_𝑖1−𝑥_𝑗1)²+𝑘_𝑖𝑗𝛾₂(𝑥_𝑖2−𝑥_𝑗2)², the network (4.3) satisfies the conditions of the Theorem and will achieve global synchronization finally.

Remark 4.2. According to Remark 3.3, we choose 𝑎̇_𝑖𝑗=𝑘𝑒^𝑇_𝑖𝑗𝑃𝑒_𝑖𝑗=𝑘(𝑥_𝑖1−𝑥_𝑗1)²+𝑘(𝑥_𝑖2−𝑥_𝑗2)², (4.6) in our simulation, and Γ=diag(1,2,3),𝑘=0.01,𝑁=7.

We consider two types of regular network: star coupled network and ring coupled network. Note that our adaptive strategy is also valid for other type of complex networks.

(1)Star coupled network

Figure 1 shows the synchronization errors ||𝑒_𝑖||=||𝑥_𝑖−𝑥₁|| (𝑖=2,3,…,7) of the star coupled network when 𝑎=0 and 𝑎=1, respectively. From Figure 1, we can see that the synchronization errors evolve and converge to zero. In Figure 2, we plot the curve of the weight strength 𝑎_𝑖𝑗 of 𝑎=0 and 𝑎=1. We can see that 𝑎_𝑖𝑗 converges to some constants, respectively.

(a)

(b)

Figure 1 

Synchronization errors $||e_i||$ ($i=\mathrm{2,3},\dots ,7$) of the star coupled network.

(a)

(b)

Figure 2 

Weight strength $a_{ij}$ evolving for the star coupled network.

(2)Ring coupled network

Figure 3 shows the synchronization errors ||𝑒_𝑖||=||𝑥_𝑖−𝑥₁|| (𝑖=2,3,…,7) of the ring coupled network when 𝑎=0 and 𝑎=1, respectively. It is said that the network achieves global synchronization finally. Figure 4 shows that the weight strength 𝑎_𝑖𝑗 of 𝑎=0 and 𝑎=1 converges to some constants, respectively.

(a)

(b)

Figure 3 

Synchronization errors $\parallel e_i\parallel \mathrm{\hspace{0.17em}\hspace{0.17em}}(i=\mathrm{2,3},\dots ,7)$ of the ring coupled network.

(a)

(b)

Figure 4 

Weight strength $a_{ij}$ evolving for the ring coupled network.

5. Conclusion

In this paper, we presented a new concept of asymptotically stable led by partial state variables and designed a simple adaptive controller for global synchronization in weight complex dynamical networks. For this network, we proved by using Lyapunov’s direct method that the states of such complex network can achieve global synchronization finally under our adaptive update law. Our synchronization strategy is to set adaptive coupling strength, and our adaptive update law of 𝑎_𝑖𝑗 is only relation to partial state variables of connected nodes 𝑖 and 𝑗. The complex networks with star coupled and ring coupled are simulated, in which the unified chaotic system is chosen as the dynamical node of the network. The simulation results also show the effectiveness of our method.

Acknowledgments

The authors are grateful to the anonymous referees for constructive comments. This work is supported by the Foundation of Zhangzhou Normal University under Grants no. SK09017 and the Foundation for Supporting Universities in Fujian Province of China under Grant no. JK2009020.