Synchronization between Two Discrete-Time Networks with Mutual Couplings

and Applied Analysis 3 Let η(t) = [η 1 (t), η 2 (t), . . . , η h (t)] and η k (t) = [η k,1 (t), η k,2 (t), . . . , η k,N k (t)]. We can rewrite (7) in a component form as η k,1 ( t + 1) = (D + λkW) k,1 (t) , (10) η k,p+1 ( t + 1) = (D + λkW) k,p+1 (t) +Wηk,p (t) , 1 ≤ p ≤ N k − 1, (11) where k = 1, 2, . . . , h. Firstly, we study the system of (10). Let η k,1 (t) = μ k,1 (t) + j] k,1 (t), λ k = α k + jβ k , where α k , β k ∈ R, μ k,1 , ] k,1 ∈ R , j is and the imaginary unit. Then (10) reads μ k,1 ( t + 1) = (D + αkW) k,1 (t) − βkW]k,1 (t) , ] k,1 ( t + 1) = (D + αkW) ]k,1 (t) + βkWμk,1 (t) . (12) Construct the Lyapunov function as V (t) = μ T k,1 (t) k,1 ( t) + ]T k,1 (t) ]k,1 (t) . (13) Then, ΔV (t) = V (t + 1) − V (t) = μ T k,1 (t + 1) k,1 ( t + 1) + ]T k,1 (t + 1) ]k,1 (t + 1) − μ T k,1 (t) k,1 ( t) − ]T k,1 (t) ]k,1 (t) = [ μ k,1 ( t) ] k,1 ( t) ] T M k [ μ k,1 ( t) ] k,1 ( t) ] , (14) whereM k = P T k P k −diag{I 2n , I 2n }withP k = [ D+α k W −β k W β k W D+α k W ] , 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) + j] k,p+1 (t); then μ k,p+1 ( t + 1) = (D + αkW) k,p+1 (t) − β k W] k,p+1 ( t) +Wμk,p (t) ,


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][2][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][5][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][9][10][11][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-plusclosed-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][17][18][19][20][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.

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: where the node dynamical equations are   ( + 1) = (  ()) and   ( + 1) = (  ()),  = 1, . . ., . (⋅) :   →   and (⋅) :   →   are continuously differential functions.  (  ) is an -dimensional state vector. is the number of network nodes.  and   are the coupling strengths. = (  ) × and  = (  ) × represent the mutual coupling matrices between these two networks, whose entries   denote the intensity from  in network  to  in network ; analogously, the entries of  are the same defined as .

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 discretetime networks, which are in the form of (1): where the node equations in ( 19) are both Henön maps, which have colorful dynamical properties, for instance,  1 = 0.5 and  1 = 0.3; it has a periodic solution.Since the sum of each row of mutual matrices is one, for simplicity, we take   =   = 1/ for ,  = 1, . . ., .To measure the extent to which inner synchronization is achieved, we introduce two quantities,   = ‖  () −   ()‖ and   = ‖  () −   ()‖,  = 1, . . ., .In addition, we denote another quantity  outer = ‖  () −   ()‖ for  = 1, . . .,  to demonstrate whether outer synchronization happens.Given the values of  1 = 0.3,  1 = 0.2,  2 = 0.5, and  2 = 0.3, we first study the effect of coupling strengths   and   on the inner and outer synchronization.Figure 1 shows that the outer synchronization does not happen when the coupling strength is   < 0.5, while the inner synchronization inside network  always appears.In the same way, considering the effect of coupling strength   , the details are shown in Figure 2. Next, we discuss the effect of node dynamics on the inner and outer synchronization and take  1 = 0.2,  2 = 0.5, and  2 = 0.3 and  = 10,   = 0.2, and   = 0.3.We then investigate the effect of parameter  1 on the inner and outer synchronization.Similarly, given  1 = 0.3,  1 = 0.2, and  2 = 0.5 and  = 10,   = 0.2, and   = 0.3, we study the influence of  2 .The numerical simulations are summarized in Figures 3 and 4, showing that the inner synchronization inside network  always happens, while the inner synchronization inside network  and the outer synchronization only appear for some values of  1 or  2 .
Finally, we discuss the effect of network size  on the inner and outer synchronization with   =   = 1/, ,  = 1, . . ., .Taking the values of  1 = 0.3,  1 = 0.2,  2 = 0.5, and  2 = 0.3 and   = 0.2 and   = 0.3, we plot the curves of   ,   , and  outer in Figure 5.In the following, we change the topological structures of mutual coupling matrices and choose  =  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  always happens.A possible reason is the effect of node dynamics.Furthermore, when the Henön map behaves chaotically, no synchronization appears.

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   and   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.

Figure 1 :
Figure1: The panels exhibit   ,   , and  outer at  = 400 with regard to   for  = 10 and   = 0.2.The bottom one shows that the inner synchronization inside network  is easily achieved.When   ≥ 0.5, the inner and outer synchronization simultaneously appear.