Synchronization between Two Discrete-Time Networks with Mutual Couplings

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.

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 Section 2, 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: 𝑥_𝑖(𝑡+1)=𝑓𝑥_𝑖(𝑡)+𝑚_𝑥^𝑁_𝑗=1𝑎_𝑖𝑗𝑦_𝑗(𝑡)−𝑥_𝑗(𝑡),𝑦𝑖=1,2,…,𝑁,_𝑖(𝑡+1)=𝑔𝑦_𝑖(𝑡)+𝑚_𝑦^𝑁_𝑗=1𝑏_𝑖𝑗𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡),𝑖=1,2,…,𝑁, (1) where the node dynamical equations are 𝑥_𝑖(𝑡+1)=𝑓(𝑥_𝑖(𝑡)) and 𝑦_𝑖(𝑡+1)=𝑔(𝑦_𝑖(𝑡)),𝑖=1,…,𝑁. 𝑓(⋅)∶𝑅^𝑛→𝑅^𝑛 and 𝑔(⋅)∶𝑅^𝑛→𝑅^𝑛 are continuously differential functions. 𝑥_𝑖(𝑦_𝑖) is an 𝑛-dimensional state vector. 𝑁 is the number of network nodes. 𝑚_𝑥and𝑚_𝑦 are the coupling strengths. 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 and 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁 represent the mutual coupling matrices between these two networks, whose entries 𝑎_𝑖𝑗 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, lim_𝑡→+∞‖𝑥_𝑖(𝑡)−𝑥_𝑠(𝑡)‖=0 and lim_𝑡→+∞‖𝑦_𝑖(𝑡)−𝑦_𝑠(𝑡)‖=0,𝑖=1…,𝑁, where ‖⋅‖ denotes the Euclidean norm of a vector. If there exist such synchronous states, satisfying ^𝑁_𝑗=1𝑎_𝑖𝑗=𝛾_𝑥,∀𝑖∈𝑋,_𝑁𝑗=1𝑏_𝑖𝑗=𝛾_𝑦,∀𝑖∈𝑌, (2) without loss of generality, we set 𝛾_𝑥=𝛾_𝑦=1.

Thus the synchronized state equations are 𝑥_𝑠(𝑡+1)=𝑓𝑥_𝑠(𝑡)+𝑚_𝑥𝑦_𝑠(𝑡)−𝑥_𝑠(𝑡),𝑦_𝑠(𝑡+1)=𝑔𝑦_𝑠(𝑡)+𝑚_𝑦𝑥_𝑠(𝑡)−𝑦_𝑠(𝑡). (3) Linearizing the synchronous states around 𝑥_𝑠 and 𝑦_𝑠, we obtain 𝛿𝑥_𝑖(𝑡+1)=𝐷𝑓𝑥_𝑠(𝑡)𝛿𝑥_𝑖(𝑡)+𝑚_𝑥^𝑁_𝑗=1𝑎_𝑖𝑗𝛿𝑦_𝑗(𝑡)−𝛿𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁,𝛿𝑦_𝑖(𝑡+1)=𝐷𝑔𝑦_𝑠(𝑡)𝛿𝑦_𝑖(𝑡)+𝑚_𝑦^𝑁_𝑗=1𝑏_𝑖𝑗𝛿𝑥_𝑗(𝑡)−𝛿𝑦_𝑗(𝑡),𝑖=1,2,…,𝑁, (4) where 𝛿𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑠(𝑡),𝛿𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑦_𝑠(𝑡), 𝐷𝑓(𝑥_𝑠(𝑡)), and 𝐷𝑔(𝑦_𝑠(𝑡)) are the Jacobians of 𝑓(𝑥(𝑡)),𝑔(𝑦(𝑡)) at 𝑥_𝑠 and 𝑦_𝑠, respectively. Assume 𝐴=𝐵 and let 𝛿𝑥(𝑡)=[𝛿𝑥₁(𝑡),…,𝛿𝑥_𝑁(𝑡)]∈𝑅^𝑛×𝑁 and 𝛿𝑦(𝑡)=[𝛿𝑦₁(𝑡),…,𝛿𝑦_𝑁(𝑡)]∈𝑅^𝑛×𝑁. Then (4) is rewritten as 𝛿𝑥(𝑡+1)=𝐷𝑓𝑥_𝑠(𝑡)𝛿𝑥(𝑡)+𝑚_𝑥𝛿𝑦(𝑡)−𝛿𝑥(𝑡)𝐴^𝑇,𝛿𝑦(𝑡+1)=𝐷𝑔𝑦_𝑠(𝑡)𝛿𝑦(𝑡)+𝑚_𝑦𝛿𝑥(𝑡)−𝛿𝑦(𝑡)𝐴^𝑇. (5) Further, let 𝛿(𝑡)=^{𝛿𝑥(𝑡)}_{𝛿𝑦(𝑡)}∈𝑅^2𝑛×𝑁; then (5) reads 𝛿(𝑡+1)=𝒟𝛿(𝑡)+𝒲𝛿(𝑡)𝐴^𝑇, (6) where 𝒟=^{𝐷𝑓(𝑥𝑠(𝑡))0}_{0𝐷𝑔(𝑦𝑠(𝑡))} and 𝒲=^{−𝑚𝑥𝐼𝑛𝑚𝑥𝐼𝑛}_{𝑚𝑦𝐼𝑛−𝑚𝑦𝐼𝑛}, where 𝐼_𝑛 is an identity matrix of size 𝑛 and 𝐴^𝑇 denotes the transpose of 𝐴. Generally, the coupling matrix can be decomposed into 𝐴^𝑇=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 𝜂(𝑡+1)=𝒟𝜂(𝑡)+𝒲𝜂(𝑡)𝐽, (7) where 𝐽 is a block diagonal matrix; that is, 𝐽𝐽=¹⋱𝐽_ℎ, (8) and 𝐽_𝑘 is the block corresponding to the 𝑁_𝑘 multiple eigenvalue 𝜆_𝑘 of 𝐴^𝑇; that is, 𝐽_𝑘=𝜆_𝑘10⋯00𝜆_𝑘1⋯0⋮⋮⋱⋱⋮00⋯𝜆_𝑘100⋯0𝜆_𝑘. (9) Let 𝜂(𝑡)=[𝜂₁(𝑡),𝜂₂(𝑡),…,𝜂_ℎ(𝑡)] and 𝜂_𝑘(𝑡)=[𝜂_𝑘,1(𝑡),𝜂_𝑘,2(𝑡),…,𝜂_{𝑘,𝑁𝑘}(𝑡)]. We can rewrite (7) in a component form as 𝜂_𝑘,1(𝑡+1)=𝒟+𝜆_𝑘𝒲𝜂_𝑘,1(𝑡),𝜂_𝑘,𝑝+1(𝑡+1)=𝒟+𝜆_𝑘𝒲𝜂_𝑘,𝑝+1(𝑡)+𝒲𝜂_𝑘,𝑝(𝑡),1≤𝑝≤𝑁_𝑘−1, (10)(11) where 𝑘=1,2,…,ℎ.

Firstly, we study the system of (10). Let 𝜂_𝑘,1(𝑡)=𝜇_𝑘,1(𝑡)+𝑗𝜈_𝑘,1(𝑡),𝜆_𝑘=𝛼_𝑘+𝑗𝛽_𝑘, where 𝛼_𝑘,𝛽_𝑘∈𝑅,𝜇_𝑘,1,𝜈_𝑘,1∈𝑅^2𝑛, 𝑗 is and the imaginary unit. Then (10) reads 𝜇_𝑘,1(𝑡+1)=𝒟+𝛼_𝑘𝒲𝜇_𝑘,1(𝑡)−𝛽_𝑘𝒲𝜈_𝑘,1(𝑡),𝜈_𝑘,1(𝑡+1)=𝒟+𝛼_𝑘𝒲𝜈_𝑘,1(𝑡)+𝛽_𝑘𝒲𝜇_𝑘,1(𝑡). (12) Construct the Lyapunov function as 𝑉(𝑡)=𝜇^𝑇_𝑘,1(𝑡)𝜇_𝑘,1(𝑡)+𝜈^𝑇_𝑘,1(𝑡)𝜈_𝑘,1(𝑡). (13) Then, Δ𝑉(𝑡)=𝑉(𝑡+1)−𝑉(𝑡)=𝜇^𝑇_𝑘,1(𝑡+1)𝜇_𝑘,1(𝑡+1)+𝜈^𝑇_𝑘,1(𝑡+1)𝜈_𝑘,1(𝑡+1)−𝜇^𝑇_𝑘,1(𝑡)𝜇_𝑘,1(𝑡)−𝜈^𝑇_𝑘,1(𝑡)𝜈_𝑘,1(𝑡)=𝜇_𝑘,1(𝑡)𝜈_𝑘,1(𝑡)^𝑇𝑀_𝑘𝜇_𝑘,1(𝑡)𝜈_𝑘,1(𝑡), (14) where 𝑀_𝑘=𝑃^𝑇_𝑘𝑃_𝑘−diag{𝐼_2𝑛,𝐼_2𝑛} with 𝑃_𝑘=^{𝒟+𝛼𝑘𝒲−𝛽𝑘𝒲}_{𝛽𝑘𝒲𝒟+𝛼𝑘𝒲},𝑘=1,2,…,ℎ. If 𝑀_𝑘<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 𝜂_𝑘,𝑝+1(𝑡)=𝜇_𝑘,𝑝+1(𝑡)+𝑗𝜈_𝑘,𝑝+1(𝑡); then 𝜇_𝑘,𝑝+1(𝑡+1)=𝒟+𝛼_𝑘𝒲𝜇_𝑘,𝑝+1(𝑡)−𝛽_𝑘𝒲𝜈_𝑘,𝑝+1(𝑡)+𝒲𝜇_𝑘,𝑝(𝑡),1≤𝑝≤𝑁_𝑘𝜈−1,_𝑘,𝑝+1(𝑡+1)=𝒟+𝛼_𝑘𝒲𝜈_𝑘,𝑝+1(𝑡)+𝛽_𝑘𝒲𝜇_𝑘,𝑝+1(𝑡)+𝒲𝜈_𝑘,𝑝(𝑡),1≤𝑝≤𝑁_𝑘−1. (15) Choose the Lyapunov function as 𝑉(𝑡)=𝜇^𝑇_𝑘,𝑝+1(𝑡)𝜇_𝑘,𝑝+1(𝑡)+𝜈^𝑇_𝑘,𝑝+1(𝑡)𝜈_𝑘,𝑝+1(𝑡). (16) Then we obtain Δ𝑉(𝑡)=𝑉(𝑡+1)−𝑉(𝑡)=𝑢_𝑘,𝑝(𝑡)𝑣_𝑘,𝑝(𝑡)𝑢_𝑘,𝑝+1(𝑡)𝑣_𝑘,𝑝+1(𝑡)^𝑇𝐿_𝑘𝑢_𝑘,𝑝(𝑡)𝑣_𝑘,𝑝(𝑡)𝑢_𝑘,𝑝+1(𝑡)𝑣_𝑘,𝑝+1(𝑡), (17) where 𝐿_𝑘=𝑄^𝑇_𝑘𝑄_𝑘−diag{0,0,𝒲^𝑇𝒲+𝐼_2𝑛,𝒲^𝑇𝒲+𝐼_2𝑛} with 𝑄_𝑘=00𝒲0000𝒲𝒲0𝒟+𝛼_𝑘𝒲−𝛽_𝑘𝒲0𝒲𝛽_𝑘𝒲𝒟+𝛼_𝑘𝒲. (18)

If 𝐿_𝑘<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 𝜆_𝑘=𝛼_𝑘+𝑗𝛽_𝑘 be the eigenvalues of 𝐴, where 𝛼_𝑘,𝛽_𝑘∈𝑅. If these matrices 𝑀_𝑘,𝐿_𝑘<0,𝑘=1,2,…,ℎ, then the networks (1) will achieve inner synchronization inside each network.

Remark 2. Note that Theorem 1 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 ‖𝑥_𝑠(𝑡)−𝑦_𝑠(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): 𝑥_𝑖1(𝑡+1)=1+𝑥_𝑖2(𝑡)−𝑎₁𝑥_𝑖1(𝑡)²+𝑚_𝑥^𝑁_𝑗=1𝑎_𝑖𝑗𝑦_𝑗1(𝑡)−𝑥_𝑗1(𝑡),𝑥_𝑖2(𝑡+1)=𝑏₁𝑥_𝑖1(𝑡)+𝑚_𝑥^𝑁_𝑗=1𝑎_𝑖𝑗𝑦_𝑗2(𝑡)−𝑥_𝑗2(𝑡𝑦),𝑖=1,2,…,𝑁,_𝑖1(𝑡+1)=1+𝑦_𝑖2(𝑡)−𝑎₂𝑦_𝑖1(𝑡)²+𝑚_𝑦^𝑁_𝑗=1𝑏_𝑖𝑗𝑥_𝑗1(𝑡)−𝑦_𝑗1(𝑡),𝑦_𝑖2(𝑡+1)=𝑏₂𝑦_𝑖1(𝑡)+𝑚_𝑦^𝑁_𝑗=1𝑏_𝑖𝑗𝑥_𝑗2(𝑡)−𝑦_𝑗2(𝑡),𝑖=1,2,…,𝑁, (19) 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 𝑎_𝑖𝑗=𝑏_𝑖𝑗=1/𝑁 for 𝑖,𝑗=1,…,𝑁. To measure the extent to which inner synchronization is achieved, we introduce two quantities, 𝐸_𝑥=‖𝑥_𝑖(𝑡)−𝑥_𝑠(𝑡)‖ and 𝐸_𝑦=‖𝑦_𝑖(𝑡)−𝑦_𝑠(𝑡)‖,𝑖=1,…,𝑁. In addition, we denote another quantity 𝐸_outer=‖𝑥_𝑖(𝑡)−𝑦_𝑖(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 𝑚_𝑥 and 𝑚_𝑦 on the inner and outer synchronization. Figure 1 shows that the outer synchronization does not happen when the coupling strength is 𝑚_𝑥<0.5, while the inner synchronization inside network 𝑋 always appears. In the same way, considering the effect of coupling strength 𝑚_𝑦, the details are shown in Figure 2.

Figure 1 

The panels exhibit 𝐸𝑥,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 show 𝐸𝑥,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 take 𝑏₁=0.2,𝑎₂=0.5, and 𝑏₂=0.3 and 𝑁=10,𝑚_𝑥=0.2, and 𝑚_𝑦=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,𝑚_𝑥=0.2, and 𝑚_𝑦=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 of 𝐸𝑥,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 size 𝑁 on the inner and outer synchronization with 𝑎_𝑖𝑗=𝑏_𝑖𝑗=1/𝑁,𝑖,𝑗=1,…,𝑁. Taking the values of 𝑎₁=0.3,𝑏₁=0.2,𝑎₂=0.5, and 𝑏₂=0.3 and 𝑚_𝑥=0.2 and 𝑚_𝑦=0.3, we plot the curves of 𝐸_𝑥,𝐸_𝑦, and 𝐸_outer 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 show 𝐸𝑥,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 of 𝐸𝑥,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 states 𝑥_𝑠 and 𝑦_𝑠 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61203155 and 11171084) and Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ12F03003.