# Global Synchronization in Complex Networks with Adaptive Coupling

Authors: Zhengzhong Yuan; Jianping Cai; Meili Lin
Journal: Mathematical Problems in Engineering
 (2010)

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## 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.


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## Body

## 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 xk-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 xk(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 aij 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 ofN identical coupled nodes. The dynamical behavior of the network can be described by the following ordinary differential equations:
(2.1)ẋi=F(xi)+∑j=1Naij(t)Γxj,i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈Rn is the state vector representing the state variables of node i. F(xi)=(F1(xi),F2(xi),…,Fn(xi))T is a smooth nonlinear vector-valued function. Γ=diag(γ1,…,γk0,0,…,0)∈Mn(R) is the inner coupling matrix, where γi>0(i=1,2,…,k0) are some constants.The outer coupling matrixA(t)=(aij(t))N×N represents not only the topological structure of the network, but also the weight strength, in which aij(t) is defined as follows: if there is a connection between node i and node j(j≠i), then aij(t)=aji(t)>0; otherwise, aij(t)=aji(t)=0(j≠i). Suppose network (2.1) is diffusively coupled, that is, the diagonal elements of matrix A(t) are defined by(2.2)aii(t)=-∑j=1j≠iNaij(t),i=1,2,…,N.
Here the coupling strengths depend on the corresponding two nodes (i.e., aij(t) depends on subscripts i and j) and time t. Simply speaking, our network is a weighted network. For convenience, we replace aij(t) with aij in the following.Definition 2.1.
An equilibrium pointx=0 of a dynamical system ẋ=f(x),x=(x1,x2,…,xn)T∈Rn is xk(k=1,2,…,s)(s≤n) leading asymptotically stable if the asymptotical stability of xk=0(k=1,2,…,s) can lead the asymptotical stability of xk=0(k=1,2,…,n).Remark 2.2.
Thexk-leading asymptotically stable in [11] is a special case of s=1 in our definition.Lettingeij=(eij1,eij2,…,eijn)T=xj-xi, it is obvious that eii=0,eji=-eij, and the error system is(2.3)ėij=F(xj)-F(xi)+∑k=1N(ajkΓejk-aikΓeik),i,j=1,2,…,N.
The coupled network (2.1) is said to achieve global synchronization if(2.4)limt→∞∥xi(t)-xj(t)∥=0,thatis,eij→0,(t→∞),i,j=1,2,…,N,
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 functionF:Rn→Rn 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 pointeij=0 is eijk(k=1,2,…,s;i,j=1,2,…,N)(s≤k0) leading asymptotically stable.(III)
The nonzero weight strengthaij(i≠j) satisfy the following adaptive update law
(3.1)ȧij=kijeijTPΓeij=kij(xi-xj)TPΓ(xi-xj).
Then network (2.1) with coupling strength (3.1) will achieve global synchronization as t→∞, where kij>0 are some arbitrary constants, P=diag(1,…,1︸s,0,…,0)∈Mn(R).Proof.
Let the Lyapunov function be(3.2)V=12∑i=1N∑j=1NeijTPeij+N2∑i=1N∑j=1N1kij(aij-c)2,
where c is a large positive constant. Obviously, it is positive definite about eijk(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
(3.3)V̇=∑i=1N∑j=1NeijTPėij+N∑i=1N∑j=1N1kij(aij-c)ȧij=∑i=1N∑j=1NeijTP(F(xj)-F(xi))+∑i=1N∑j=1N∑k=1N(ajkeijTPΓejk-aikeijTPΓeik)+N∑i=1N∑j=1N1kij(aij-c)ȧij.
From condition (I), the first sum S1=∑i=1N∑j=1NeijTP(F(xj)-F(xi))≤∑i=1N∑j=1NleijTPeij holds. And the second sum can be calculated as follows. Since eijT=-ejiT, we can see that
(3.4)S2=∑i=1N∑j=1N∑k=1N(ajkeijTPΓejk-aikeijTPΓeik)=-∑i=1N∑j=1N∑k=1N(ajkejiTPΓejk+aikeijTPΓeik).
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
(3.5)S2=-2∑i=1N∑j=1N∑k=1NajkejiTPΓejk.
Using ejj=0, we get
(3.6)S2=-2(∑i=1N∑j=1N∑k=1jajkejiTPΓejk+∑i=1N∑j=1N∑k=jNajkejiTPΓejk).
Renaming j by k in the second term and vice versa, and using the symmetry of A(t), we obtain
(3.7)S2=-2(∑i=1N∑j=1N∑k=1jajkejiTPΓejk+∑i=1N∑k=1N∑j=kNajkekiTPΓekj)=-2(∑i=1N∑j=1N∑k=1jajkejiTPΓejk+∑i=1N∑k=1N∑j=kNajkeikTPΓejk)=-2∑i=1N∑j=1N∑k=1jajk(ejiT+eikT)PΓejk.
Since ejiT+eikT=xiT-xjT+xkT-xiT=ejkT and the symmetry of A(t), we get
(3.8)S2=-2∑i=1N∑j=1N∑k=1jajkejkTPΓejk=-2N∑j=1N∑k=1jajkejkTPΓejk=-N∑j=1N∑k=1NajkejkTPΓejk.
So
(3.9)V̇≤∑i=1N∑j=1NleijTPeij-N∑i=1N∑j=1NaijeijTPΓeij+N∑i=1N∑j=1N1kij(aij-c)ȧij=∑i=1N∑j=1NleijTPeij-N∑i=1N∑j=1NaijeijTPΓeij+N∑i=1N∑j=1N(aij-c)eijTPΓeij=∑i=1N∑j=1NeijT[lP-cNPΓ]eij.
Because of s≤k0, for large constant c, it holds that lP-cNPΓ=diag(c1,c2,…,cs,0,…,0)∈Mn(R), where ci<0(i=1,2,…,s). So V̇ is negative definite about eijk(k=1,2,…,s). According to Lyapunov’s direct method, starting with arbitrary initial values, eijk→0(t→∞)(k=1,2,…,s). Since equilibrium point eij=0 of system (2.3) is eijk(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∂Fi/∂xj (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 setkij=k, and the adaptive update law is ȧij=keijTPeij=k(xi-xj)TP(xi-xj).Remark 3.4.
By (3.1), the nonzero coupling strength aij(t)→Cij when the network achieves synchronization, where Cij 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 ofaij is only relation to partial state variables of connected nodes i and j.


## 4. Illustrative Example

To demonstrate the theoretical result in Section3, two regular networks with the unified chaotic system are chosen as simulation examples in this section.The unified chaotic [23] system is described by(4.1)ẋ1=(25a+10)(x2-x1),ẋ2=(28-35a)x1+(29a-1)x2-x1x3,ẋ3=x1x2-(a+8)x33.
When a∈[0,1], it has well-known chaotic behavior.We considerΓ=diag(γ1,γ2,γ3),γi>0(i=1,2,3), that is,(4.2)ẋi1=(25a+10)(xi2-xi1)+γ1∑j=1Ncijaijxj1,ẋi2=(28-35a)xi1+(29a-1)xi2-xi1xi3+γ2∑j=1Ncijaijxj2,(i=1,2,…,N),ẋi3=xi1xi2-(a+8)xi33+γ3∑j=1Ncijaijxj3.
The error system is(4.3)ėij1=(25a+10)(eij2-eij1)+γ1∑k=1N(cjkajkejk1-cikaikeik1),ėij2=(28-35a)eij1+(29a-1)eij2-xj1xj3+xi1xi3+γ2∑k=1N(cjkajkejk2-cikaikeik2)(i,j=1,2,…,N),ėij3=xj1xj2-xi1xi2-(a+8)eij33+γ3∑k=1N(cjkajkejk3-cikaikeik3).
In order to use our Theorem, we need the following Lemma.Lemma 4.1.
Fora∈[0,1], system (4.3) is eijk(k=1,2) leading asymptotically stable.Proof.
Ifeijk=0(k=1,2) for all i and j, choose the following Lyapunov function:
(4.4)V=12∑i=1N∑j=1Neij32.
The derivative of Valong with the solution of  (4.2) is
(4.5)V̇=∑i=1N∑j=1Neij3ėij3=∑i=1N∑j=1N[-(a+8)eij323]+∑i=1N∑j=1N∑k=1N(ajkeij3ejk3-aikeij3eik3).
Similar to the calculation of S2, we can get ∑i=1N∑j=1N∑k=1N(ajkeij3ejk3-aikeij3eik3)<0 and for all a∈[0,1], V̇<0. So it is said that eij3→0(t→∞)(i,j=1,2,…,N).
If we setȧij=kijγ1(xi1-xj1)2+kijγ2(xi2-xj2)2, the network (4.3) satisfies the conditions of the Theorem and will achieve global synchronization finally.Remark 4.2.
According to Remark3.3, we choose
(4.6)ȧij=keijTPeij=k(xi1-xj1)2+k(xi2-xj2)2,
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 networkFigure1 shows the synchronization errors ||ei||=||xi-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 aij of a=0 and a=1. We can see that aij converges to some constants, respectively.Synchronization errors||ei||  (i=2,3,…,7) of the star coupled network.
(a)(b)Weight strengthaij evolving for the star coupled network.
(a)(b)(2)
Ring coupled networkFigure3 shows the synchronization errors ||ei||=||xi-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 aij of a=0 and a=1 converges to some constants, respectively.Synchronization errors∥ei∥(i=2,3,…,7) of the ring coupled network.
(a)(b)Weight strengthaij evolving for the ring coupled network.
(a)(b)


## 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 ofaij 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.



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