# The Complex Network Synchronization via Chaos Control Nodes

Authors: Yin Li; Chun-long Zheng
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We investigate chaos control nodes of the complex network synchronization. The structure of the coupling functions between the connected nodes is obtained based on the chaos control method and Lyapunov stability theory. Moreover a complex network with nodes of the new unified Loren-Chen-Lü system, Coullet system, Chee-Lee system, and the New system is taken as an example; numerical simulations are used to verify the effectiveness of the method.


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## Body

## 1. Introduction

Since the most famous random graph model was proposed by Erdös and Rényi [1], the complex network has attracted much attention in many fields of research, such as biology, physics, computer networks, the World Wide Web (WWW) [2], and so on. Network synchronization has obvious advantages, it has great application value in practice. Therefore, Atay et al. [3] studied synchronization of complex network when delays exist among the nodes; Motter et al. [4] studied the influence of coupling strength on the synchronizing ability of a complex network; Timme et al. [5] studied the web synchronization law of pulse-coupled dynamical systems; Checco et al. [6] studied the synchronization of random web. Lü et al. [7] constructed general complex dynamical networks and studied the synchronization; Lu and Chen [8] studied synchronization analysis of linearly coupled networks of discrete time systems; Han and Lu [9] studied the changes of synchronization ability of coupled networks from ring networks to chain networks; He and Yang [10] studied adaptive synchronization in nonlinearly coupled dynamical networks; Hung et al. [11] studied globally generalized synchronization in scale-free networks.Recently, network synchronization has been an important part of the dynamic study of complex network. It has aroused great interest of scholars both domestically and abroad to build weighted network models and study the characteristics of them. Lü et al. [7] studied chaos synchronization of general complex dynamic networks. Gao et al. [12] realized the adaptive synchronization of complex network. Pei et al. [13] made statistical analysis of a class of real networks; Barrat et al. [14] presented a weighted network model, and studied its dynamic character; Atay et al. [15] studied the synchronization of a complex network with time delay. Hung et al. [11] realized the generalized synchronization of a scale-free network. Qin and Yu [16] achieved the synchronization of the star-network of hyperchaotic Rossler systems. In addition to the above researches, much other work [17, 18] in the field have been done, and complex network synchronization become a focus of attention, such as a random network synchronization, small-world network synchronization, scale-free network synchronization, and so on. However, universal synchronization methods of weighted network still need further exploration and study.The motivation in this paper lies in the complex network synchronization and chaos control importance. Network synchronization is one of the most practical and valuable issues. A synchronization of network means the situation in which the output of all nodes in the study of the complex network is consistent with any given external input signal under a certain condition. The working mechanism of a single chaotic system to track any given external input signal is relatively interesting and significance. Numerical simulations are used to verify the effectiveness of the proposed techniques.It is organized as follows. Firstly, the theory and the method are presented in Section2. Then, different order chaotic systems are adopted as the nodes of this complex network; the structure of the coupling functions among the connected nodes is obtained based on Lyapunov stability theory. With the help of symbolic computation, the temporal evolution of variables and node interaction of the dynamic equation are discussed and simulated by computer in Section 3. The theoretical analysis and the numerical simulations show that this is a universal method and the number of the nodes does not affect the stability of the whole network. Finally, the conclusions are given in Section 4.


## 2. Theory and Method

We summarized the main steps for the complex network synchronization based on Lü et al. [19–23] and Chen et al. [24–27], as follows.Step 1.
Assume the state of nodei is xi, where xi=(xi1,xi2,…,xin)T and xi1,xi2,…,xinϵRn, then the dynamic function for node i without coupling can be described as
(1)x˙i=fi(xi)=Li(xi)+Si(xi),
where Li(xi)=(Ai-Bi)xi,  Ai is the linear coefficient matrix of the system, and Bi is the coefficient matrix of control gain.Step 2.
Considering the coupling of network, the dynamic function for nodei can be described as
(2)x˙i=fi(xi)+gi(x1,x2,…,xm)=(Ai-Bi)xi+Si(xi)+gi(x1,x2,…,xm),
where gi(x1,x2,…,xn) denotes the coupling function. The proportional scale for node i is αi, then we will have αi=diag(αi1,αi2,…,αin).Step 3.
The errors between the state variables of the network are defined as(3)ei=αixi+αi+1xi+1.
We obtain from (2) and (3)
(4)e˙i=αix˙i+αi+1x˙i+1=αi[(Ai-Bi)xi+Si(xi)+gi]+αi+1[(Ai+1-Bi+1)xi+1+Si+1(xi+1)+gi+1]=(Ai-Bi)ei+Siαi+αigi+αi+1(Ai+1-Bi+1)xi+1+αi+1Si+1+αi+1gi+1-(Ai-Bi)αi+1xi+1=(Ai-Bi)ei+[(Ai+1-Bi+1)-(Ai-Bi)]αi+1xi+1+ΔSi(xi,xi+1)+Δgi(xi,xi+1)=(Ai-Bi-Ki)ei+[(Ai+1-Bi+1)-(Ai-Bi)]αi+1xi+1,
where ΔSi(xi,xi+1)=αiSi+αi+1Si+1,  Δgi(xi,xi+1)=αigi+αi+1gi+1+Kiei.Step 4.
ChooseΔgi(xi,xi+1)=-ΔSi(xi,xi+1)+Kiei, then we will have
(5)αjgj=-α1g1-α1S1(x1)-αjSj(xj)+k(α1x1+αjxj),(j=2,3,…,m).
If we choose node 1 as target node, then the coupling function of node j(j=2,3,m) can be described as
(6)gj=-α1αjg1-α1αjS1(x1)-Sj(xj)+k(α1αjx1+xj).
In order to realize the synchronization of the complex network, if we choose other node as target node, the coupling functions of all nodes can also be obtained.Step 5.
Constructing the Lyapunov function according to the weighted complex network with different nodes,(7)V=12∑i=1m-1‍ei2,
and considering (3) and (4), we can obtain the derivative form of V as
(8)V˙=∑i=1m-1‍eie˙i=∑‍{(Ai-Bi-Ki)ei+[(Ai+1-Bi+1)-(Ai-Bi)]αi+1xi+1}ei.
From (8), we can easily see that if
(9)Ai≤Bi+Ki,Ai+1=Bi+1+Ai-Bi.
Then
(10)V˙≤0.
According to Lyapunov stability theory [20] and the weighted complex network [19, 20, 28], the synchronization of the complex network can be realized.


## 3. Application of Chaos Control Method to the Network Synchronization

The unified chaotic system, the New system, Coullet system, and Chee-Lee system are taken as nodes of the network to show the synchronization mechanism mentioned above. Simulation is made as the number of the nodes ism=5.The dynamic equation of the unified chaotic system [21, 29] is presented as follows:
(11)x˙1=(25α+10)(y1-x1),y˙1=(28-35α)x1-x1z1+(29α-1)y1,z˙1=x1y1-8+α3z1,
where α∈[-0.0016,1.15]. When -0.0016<α<0.8, LCL system (11) belongs to the generalized Lorenz system [30]; when α=0.8, LCL system (11) belongs to the generalized Lü system [23]; when 0.8<α<1.15 LCL system (11) belongs to the generalized Chen system [31]. Figures 1(a)–1(c) display these chaotic attractors of the generalized Lorenz system (α=0.2), the generalized Lü system (α=0.8), and the generalized Chen system (α=1.12), respectively.Generlized chaotic attractors: (a) Lorenz, (b) Lü, (c) Chen.
(a)
(b)
(c)The new chaotic system of three-dimensional quadratic autonomous ordinary differential equations [22], which can display the chaotic attractors:
(12)x˙2=-a2b2a2+b2x2-y2z2+c2,y˙2=a2y2+x2z2,z˙2=b2z2+x2y2,
where a2, b2, and c2 are real constants; and x,y, and z are status variables. This system is found to be chaotic in a wide parameter range and has many interesting complex dynamical characteristics. The system is chaotic for the parameters a2=-10,b2=-4, and ∥c∥=19.2; it displays the chaotic attractor as shown in Figure 2(a). Detailed dynamic properties of this system can be found in [22].The chaos attractor of new chaotic system (a) and the Coullet system (b).
(a)
(b)The dynamic equation of Coullet system [19] is described as follows:
(13)x˙3=y3,y˙3=z3,z˙3=a3z3+b3y3+c3x3+x33,
where a3, b3, and c3 are parameters of the system. When the parameters are given as a3=0.45, b3=1.1, and c3=0.8, the chaos attractor of (13) is shown in Figure 2(b).The dynamic equation of new system [23] is described as follows:
(14)x˙4=d(y4-x4),y˙4=-y4+x4z4,z˙4=e-y4x4-fz4,
where d, e, and f are parameters of the system. When the parameters are given as d=5, e=16, and f=1, the chaos attractor of (14) is shown in Figure 3(a).The chaos attractor of (14) system (a) and Chee-Lee system (b).
(a)
(b)The dynamic equation of the Chen-Lee system [32] is described as follows:
(15)x˙5=-y5z5+a5x5,y˙5=x5z5+b5y5,z˙5=13x5y5+c5z5,
where a=5, b=-10, and c=-0.38, (15) is a chaotic system. Figure 3(b) displays the chaotic attractors in (x1,y1,z1)-space. It is easy to see that it admits a symmetry (x5,y5,z5). That is, the system (15) is symmetrical about the three coordinate axes x5,y5, and z5, respectively.Choose the unified chaotic system (16) (node 1) as a target system, that is, coupling function g1i=0, then the dynamic functions of node 2, node 3, node 4, and node 5 with coupling can be described, respectively, as
(16)x˙1=(25α+10)(y1-x1),y˙1=(28-35α)x1-x1z1+(29α-1)y1,z˙1=x1y1-8+α3z1,(17)x˙2=-a2b2a2+b2x2-y2z2+c2+g21,y˙2=a2y2+x2z2+g22,z˙2=b2z2+x2y2+g23,(18)x˙3=y3+g31,y˙3=z3+g32,z˙3=a3z3+b3y3+c3x3+x33+g33,(19)x˙4=d(y4-x4)+g41,y˙4=-y4+x4z4+g42,z˙4=e-y4x4-fz4+g43,(20)x˙5=-y5z5+a5x5+g51,y˙5=x5z5+b5y5+g52,z˙5=13x5y5+c5z5+g53,
where gj(j=2,3,4,5) fulfills (6):
(21)g2=(g21g22g23)=(-15y1+14x1-277x2-1-21x1-6.8y1+8y2-0.267z1+z2)+(y2z2x1z1-x2z2-x1y1-x2y2),g3=(g31g32g33)=(-7.5y1+7x1+72x2-y3-8x1-10.5x1-3.4y1+2y2-z3+6y3-0.13z1+12z2-3.55z3+1.1y3-0.8x3)+(012x1z1-12x1y1+x33),g4=(g41g42g43)=(15y1-14x1-7x2+14x3-5y4+4x421x1+6.8y1-4y2+11y3+0.5y40.267z1-16-z2+6z3)+(0-x1z1-x4z4x1y1+x4y4),g5=(g51g52g53)=(-7.5y1+7x1+3.5x2-7x3-2.5x4-11x5-10.5x1-3.4y1+2y2-5.5y3-1.75y4+6y5-0.133z1+0.5z2-3z3-0.5z4-1.62z5)+(z5y50.5x1z1-x5z5-0.5x1y1-0.33x5y5).
The relative weight of the coupling strength between the nodes of the network is arbitrarily taken as K=[-1,-2,-3,4,3,2,-2,-1,-2,-3,-2,-1],  αi=[1,1,1,-2,-2,-2,0.5,0.5,0.5,2,2,2], then the network synchronization error between system (16) and system (17) will be e21=x1+x2,  e22=y1+y2,  and e23=z1+z2 and the network synchronization error between system (17) and system (18) will be e31=x2-2x3,  e32=y2-2y3, and e33=z2-2z3, and the network synchronization error between system (18) and system (19) will be e41=x3-0.5x4, e42=y3-0.5y3, and e43=z3-0.5z4, and the network synchronization error between system (19) and system (20) will be e51=x4+2x5, e52=y4+2y5, and e53=z4+2z5. The simulations are shown in Figures 4, 5, 6, and 7.Network synchronization errors between node 1 and node 2.
(a)
(b)
(c)Network synchronization errors between node 2 and node 3.
(a)
(b)
(c)Network synchronization errors between node 3 and node 4.
(a)
(b)
(c)Network synchronization errors between node 4 and node 5.
(a)
(b)
(c)For further details, the dynamic attractors for each node with coupling network synchronization are shown in Figures8 and 9. The network synchronization temporal evolution of variables is shown in Figures 10, 11, 12, and 13. From Figures 8 and 9, we can see that the network synchronization has been realized on the required proportional scale.The chaos attractors between node 1 and node 2 (a) and between node 2 and node 3 (b).
(a)
(b)The chaos attractors between node 3 and node 4 (a) and between node 4 and node 5 (b).
(a)
(b)Figure 10
The temporal evolution of variables between node 1 (“⋯”) and node 2 (“—”).Figure 11
The temporal evolution of variables between node 2 (“⋯”) and node 3 (“—”).Figure 12
The temporal evolution of variables between node 3 (“⋯”) and node 4 (“—”).Figure 13
The temporal evolution of variables between node 4 (“⋯”) and node 5 (“—”).


## 4. Summary and Discussion

In this paper, the complex network synchronization is investigated. With the help of symbolic computation, different order chaotic systems are adopted as the nodes of this complex network; the structure of the coupling functions among the connected nodes is obtained based on Lyapunov stability theory. Being of network and physical interests, the temporal evolution of variables and node interaction of the dynamic equation are discussed and simulated by computer. This method has universal significance for network synchronization, and the weight value of the coupling strength between the nodes and the number of the nodes does not affect synchronization of the whole network.



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