# Synchronization between Two Discrete-Time Networks with Mutual Couplings

Authors: Meng Xiao; Weigang Sun; Fangyue Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate synchronization between two discrete-time networks with mutual
couplings, including inner synchronization inside each network and outer synchronization between two networks.
We then obtain a synchronized criterion for the inner synchronization inside each network by the
method of linear matrix inequality and derive a relationship between the inner and outer synchronization.
Finally, we show numerical examples to verify our theoretical analysis and discuss the effect of coupling
strengths, node dynamics, and topological structures on the inner and outer synchronization. Compared to
the inner synchronization inside each network, the outer synchronization between two networks is difficult to
achieve.


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## Body

## 1. Introduction

Network synchronization, as a collective behavior existing inside a network, has been widely studied since the birth of small-world and scale-free networks [1–3]. The main focus is to investigate the interplay between the complexity in the overall topology and the local dynamics of the coupled nodes [4–6]. The topological structures may be globally connected, random, small-world, and scale-free. There are many applications using the synchronization of networks [7], for instance, secure communication and multirobot coordination control. Apart from the complete synchronization appearing inside a network, there are some other types of synchronization, such as phase synchronization, generalized synchronization, lag synchronization, and cluster synchronization [8–12].Generally, we refer to synchronization happening between two networks as outer synchronization [13], which is distinguished from inner synchronization inside a network. Compared to the inner synchronization, outer synchronization of two networks is more complex, which involves more system parameters. In 2007, Li et al. first proposed the concept of outer synchronization and applied the open-plus-closed-loop method to realize the outer synchronization between two networks with identical topologies [13]. Shortly later, using the adaptive control method, Tang et al. achieved the outer synchronization between two networks with different topological structures [14]. In [15], Wu et al. studied the generalized outer synchronization between two networks with different dimensions of node dynamics. In addition, there are many works on the outer synchronization, that is, introducing the noise, time delay, fractional order node dynamics, and unknown parameters [16–21].In the above-mentioned works on the outer synchronization, the researchers usually applied the control methods to realize the outer synchronization and did not study the inner synchronization inside a network. In reality, the mutual coupling forms between two networks are colorful; for instance, Wu et al. investigated the outer synchronization between two networks with two active forms nonlinear signals and reciprocity [22]; however, these two coupling forms do not make the outer synchronization happen. In addition, the inner synchronization inside each network was not considered. In [23], Sorrentino and Ott provided a method to study the inner synchronization of two groups. The problem of collective behaviors inside a network and between two networks is of broad interest. For example, in subway systems, when the trains reach the platform, the outer and inner doors simultaneously open or close, showing that both inner and outer synchronization happen [24]. It is also found that present studies on the synchronization between two networks with various couplings are much less, then studying the effect of various couplings on the synchronization is interesting and meaningful.Inspired by the above discussions, we study synchronization between two discrete-time networks with mutual couplings, including inner synchronization inside each network and outer synchronization between them. By the Lyapunov stability theory and linear matrix inequality, we obtain a synchronous theorem on the inner synchronization inside each network and a relationship between the inner and outer synchronization. Numerical simulations show that the inner synchronization is easier to achieve than the outer synchronization. In addition, given the mutual coupling matrices and appropriate node dynamics, we can adjust coupling strengths to realize the inner and outer synchronization simultaneously. In Section2, network models and synchronization analysis are presented, and numerical examples are shown in Section 3. Finally, the discussions are included in the last section.


## 2. Model Presentation and Synchronization Analysis

In this paper, we investigate the synchronization between two discrete-time networks with mutual couplings. The dynamical equations are described as follows:(1)xi(t+1)=f(xi(t))+mx∑j=1Naij(yj(t)-xj(t)),hhhhhhhhhhhhhhhhhhi=1,2,…,N,yi(t+1)=g(yi(t))+my∑j=1Nbij(xj(t)-yj(t)),hhhhhhhhhhhhhhhhhhhi=1,2,…,N,
where the node dynamical equations are xi(t+1)=f(xi(t)) and yi(t+1)=g(yi(t)),i=1,…,N. f(·):Rn→Rn and  g(·):Rn→Rn are continuously differential functions. xi(yi) is an n-dimensional state vector. N is the number of network nodes. mxandmy are the coupling strengths. A=(aij)N×N and B=(bij)N×N represent the mutual coupling matrices between these two networks, whose entries aij denote the intensity from i in network  X to j in network Y; analogously, the entries of B are the same defined as A.Let us now consider the possibility whether the individual networks achieve inner synchronization; that is,limt→+∞∥xi(t)-xs(t)∥=0 and limt→+∞∥yi(t)-ys(t)∥=0,i=1…,N, where ∥·∥ denotes the Euclidean norm of a vector. If there exist such synchronous states, satisfying
(2)∑j=1Naij=γx,∀i∈X,∑j=1Nbij=γy,∀i∈Y,
without loss of generality, we set γx=γy=1.Thus the synchronized state equations are(3)xs(t+1)=f(xs(t))+mx(ys(t)-xs(t)),ys(t+1)=g(ys(t))+my(xs(t)-ys(t)).
Linearizing the synchronous states around xs and ys, we obtain
(4)δxi(t+1)=Df(xs(t))δxi(t)+mx∑j=1Naij(δyj(t)-δxj(t)),i=1,2,…,N,δyi(t+1)=Dg(ys(t))δyi(t)+my∑j=1Nbij(δxj(t)-δyj(t)),i=1,2,…,N,
where δxi(t)=xi(t)-xs(t),δyi(t)=yi(t)-ys(t), Df(xs(t)),  and Dg(ys(t)) are the Jacobians of f(x(t)),g(y(t)) at xs and ys, respectively. Assume A=B and let δx(t)=[δx1(t),…,δxN(t)]∈Rn×N  and δy(t)=[δy1(t),…,δyN(t)]∈Rn×N. Then (4) is rewritten as
(5)δx(t+1)=Df(xs(t))δx(t)+mx(δy(t)-δx(t))AT,δy(t+1)=Dg(ys(t))δy(t)+my(δx(t)-δy(t))AT.
Further, let δ(t)=[δx(t)δy(t)]∈R2n×N; then (5) reads
(6)δ(t+1)=𝒟δ(t)+𝒲δ(t)AT,
where 𝒟=[Df(xs(t))00Dg(ys(t))] and 𝒲=[-mxInmxInmyIn-myIn], where In is an identity matrix of size n and AT denotes the transpose of A. Generally, the coupling matrix can be decomposed into AT=ΦJΦ-1, where J is the Jordan canonical form with complex eigenvalues λ∈C and Φ contains the corresponding eigenvectors ϕ. Denote η(t)=δ(t)Φ; we obtain
(7)η(t+1)=𝒟η(t)+𝒲η(t)J,
where J is a block diagonal matrix; that is,
(8)J=[J1⋱Jh],
and Jk is the block corresponding to the Nk multiple eigenvalue λk of AT; that is,
(9)Jk=[λk10⋯00λk1⋯0⋮⋮⋱⋱⋮00⋯λk100⋯0λk].
Let η(t)=[η1(t),η2(t),…,ηh(t)] and ηk(t)=[ηk,1(t),ηk,2(t),…,ηk,Nk(t)]. We can rewrite (7) in a component form as
(10)ηk,1(t+1)=(𝒟+λk𝒲)ηk,1(t),(11)ηk,p+1(t+1)=(𝒟+λk𝒲)ηk,p+1(t)+𝒲ηk,p(t),hhhhhhhhhhhhhhh1≤p≤Nk-1,
where k=1,2,…,h.Firstly, we study the system of (10). Let ηk,1(t)=μk,1(t)+jνk,1(t),λk=αk+jβk, where αk,βk∈R,μk,1,νk,1∈R2n, j is and the imaginary unit. Then (10) reads
(12)μk,1(t+1)=(𝒟+αk𝒲)μk,1(t)-βk𝒲νk,1(t),νk,1(t+1)=(𝒟+αk𝒲)νk,1(t)+βk𝒲μk,1(t).
Construct the Lyapunov function as
(13)V(t)=μk,1T(t)μk,1(t)+νk,1T(t)νk,1(t).
Then,
(14)ΔV(t)=V(t+1)-V(t)=μk,1T(t+1)μk,1(t+1)+νk,1T(t+1)νk,1(t+1)-μk,1T(t)μk,1(t)-νk,1T(t)νk,1(t)=[μk,1(t)νk,1(t)]TMk[μk,1(t)νk,1(t)],
where Mk=PkTPk-diag{I2n,I2n} with Pk=[𝒟+αk𝒲-βk𝒲βk𝒲𝒟+αk𝒲],k=1,2,…,h. If Mk<0,k=1,2,…,h,  that is, these matrices are negative definite, then the zero solution of (10) is asymptotically stable.Secondly, we study the stability of (11). Let ηk,p+1(t)=μk,p+1(t)+jνk,p+1(t); then
(15)μk,p+1(t+1)=(𝒟+αk𝒲)μk,p+1(t)-βk𝒲νk,p+1(t)+𝒲μk,p(t),hhhhhhhhhhhh1≤p≤Nk-1,νk,p+1(t+1)=(𝒟+αk𝒲)νk,p+1(t)+βk𝒲μk,p+1(t)+𝒲νk,p(t),hhhhhhhhhhh1≤p≤Nk-1.
Choose the Lyapunov function as
(16)V¯(t)=μk,p+1T(t)μk,p+1(t)+νk,p+1T(t)νk,p+1(t).
Then we obtain
(17)ΔV¯(t)=V¯(t+1)-V¯(t)=[uk,p(t)vk,p(t)uk,p+1(t)vk,p+1(t)]TLk[uk,p(t)vk,p(t)uk,p+1(t)vk,p+1(t)],
where Lk=QkTQk-diag{0,0,𝒲T𝒲+I2n,𝒲T𝒲+I2n} with
(18)Qk=[00𝒲0000𝒲𝒲0𝒟+αk𝒲-βk𝒲0𝒲βk𝒲𝒟+αk𝒲].IfLk<0,k=1,2,…,h, then the zero solution of (11) is asymptotically stable. Hence we obtain a synchronized theorem for networks (1).Theorem 1.
Consider network systems (1). Assume the mutual coupling matrices A=B. Let λk=αk+jβk be the eigenvalues of A, where αk,βk∈R. If these matrices Mk,Lk<0,k=1,2,…,h, then the networks (1) will achieve inner synchronization inside each network.Remark 2.
Note that Theorem1 only gives a feasibility of the inner synchronization inside each network. When the inner synchronization inside networks X and Y happens, and the synchronized states ∥xs(t)-ys(t)∥→0 for a large time, then the outer synchronization between networks X and Y will be achieved.


## 3. Numerical Examples

In this section, we will give some examples to illustrate our theoretical results obtained in the previous section. We mainly investigate the effect of coupling strengths, node dynamics, and mutual coupling forms on the inner and outer synchronization. We consider the following coupled discrete-time networks, which are in the form of (1):
(19)xi1(t+1)=1+xi2(t)-a1xi1(t)2+mx∑j=1Naij(yj1(t)-xj1(t)),xi2(t+1)=b1xi1(t)+mx∑j=1Naij(yj2(t)-xj2(t)),i=1,2,…,N,yi1(t+1)=1+yi2(t)-a2yi1(t)2+my∑j=1Nbij(xj1(t)-yj1(t)),yi2(t+1)=b2yi1(t)+my∑j=1Nbij(xj2(t)-yj2(t)),hhhhhhhhhhhhhhhi=1,2,…,N,
where the node equations in (19) are both Henön maps, which have colorful dynamical properties, for instance, a1=0.5 and b1=0.3; it has a periodic solution. Since the sum of each row of mutual matrices is one, for simplicity, we take aij=bij=1/N for i,j=1,…,N. To measure the extent to which inner synchronization is achieved, we introduce two quantities, Ex=∥xi(t)-xs(t)∥ and Ey=∥yi(t)-ys(t)∥,i=1,…,N. In addition, we denote another quantity Eouter=∥xi(t)-yi(t)∥ for i=1,…,N to demonstrate whether outer synchronization happens. Given the values of a1=0.3,b1=0.2,a2=0.5,  and b2=0.3, we first study the effect of coupling strengths mx and my on the inner and outer synchronization. Figure 1 shows that the outer synchronization does not happen when the coupling strength is mx<0.5, while the inner synchronization inside network X always appears. In the same way, considering the effect of coupling strength my, the details are shown in Figure 2.Figure 1
The panels exhibitEx,Ey, and Eouter at t=400 with regard to mx for N=10 and my=0.2. The bottom one shows that the inner synchronization inside network X is easily achieved. When mx≥0.5, the inner and outer synchronization simultaneously appear.Figure 2
The plots showEx,Ey, and Eouter at t=400 on my for N=10 and mx=0.3. When my≥0.35, the inner and outer synchronization simultaneously happen.Next, we discuss the effect of node dynamics on the inner and outer synchronization and takeb1=0.2,a2=0.5,  and b2=0.3 and N=10,mx=0.2,  and my=0.3. We then investigate the effect of parameter a1 on the inner and outer synchronization. Similarly, given a1=0.3,b1=0.2,  and a2=0.5 and N=10,mx=0.2,  and my=0.3, we study the influence of b2. The numerical simulations are summarized in Figures 3 and 4, showing that the inner synchronization inside network X always happens, while the inner synchronization inside network Y and the outer synchronization only appear for some values of a1 or b2.Figure 3
The curves ofEx,Ey, and Eouter at t=400 concerning a1 with b1=0.2,a2=0.5, and b2=0.3 and N=10,mx=0.2, and my=0.3.Figure 4
The trajectories ofEx,Ey,  and Eouter at t=400 regarding b2 with a1=0.3,b1=0.2, and a2=0.5 and N=10,mx=0.2, and my=0.3.Finally, we discuss the effect of network sizeN on the inner and outer synchronization with aij=bij=1/N,i,j=1,…,N. Taking the values of a1=0.3,b1=0.2,a2=0.5,  and b2=0.3 and mx=0.2 and my=0.3, we plot the curves of Ex,Ey, and Eouter in Figure 5. In the following, we change the topological structures of mutual coupling matrices and choose A=B as a random matrix; the numerics are shown in Figure 6. It is found that the globally connected and random topological structures have similar effect on the inner and outer synchronization. It is noted that the inner synchronization inside network X always happens. A possible reason is the effect of node dynamics. Furthermore, when the Henön map behaves chaotically, no synchronization appears.Figure 5
The plots showEx,Ey, and Eouter at t=400 on N with a1=0.3,b1=0.2,a2=0.5,  and b2=0.3 and mx=0.2 and my=0.3. The topologies are aij=bij=1/N,i,j=1,…,N. Obviously the inner synchronization inside network Y and outer synchronization do not happen for the network size N<40 except some values of N.Figure 6
The curves ofEx,Ey, and Eouter at t=400 regarding N with a1=0.3,b1=0.2,a2=0.5, and b2=0.3 and mx=0.2,  and my=0.3. The coupling matrix A(B) is a random matrix satisfying the sum of each row being one.


## 4.   Conclusions

The current study investigated the synchronization between two discrete-time networks with mutual couplings and mainly studied inner synchronization inside each network and outer synchronization between them. We then obtained a synchronous theorem on the inner synchronization inside each network in terms of linear matrix inequality, for the lack of a criterion on the outer synchronization. When the inner synchronization is achieved inside each network and the synchronized statesxs and ys  are same for a large time, then the outer synchronization will happen. From the numerical simulations, we see that the inner and outer synchronization simultaneously happen when we adjust the values of coupling strengths and parameters in the node dynamics. The globally connected and random topologies have similar effect on the inner and outer synchronization. In addition, outer synchronization is more difficult to achieve than the inner synchronization, meaning that the outer synchronization needs a strong coupling form.  Because of the diversity of coupling forms between two networks, deriving the criteria on the inner and outer synchronization simultaneously is a technical challenge, which would be discussed in the future.



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