Synchronization Analysis of Two Coupled Complex Networks with Time Delays

Abstract

This paper studies the synchronized motions between two complex networks with time delays, which include individual inner synchronization in each network and outer synchronization between two networks. Based on the Lyapunov stability theory and the linear matrix equality (LMI), a synchronous criterion for inner synchronization inside each network is derived. Numerical examples are given which fit the theoretical analysis. In addition, the involved numerical results show that the delays between two networks have little effect on inner synchronization. It is also shown that synchronous motions within each network or between two networks are not enhanced if individual intranetwork connections are allowed.

1. Introduction

We refer to network synchronization, that is, synchronizing all the nodes inside a network, as “inner synchronization”, which is widely studied in the science world based on the appearance of small-world [1] and scale-free [2] network models. Afterwards, the improved and expanded work in this respect—that is, introducing weighted connections, time varying coupling matrices, nonlinear coupling function, time delays, and so forth—can be found in the literature [3–10] and many references cited therein. The main approaches on studying synchronization inside a network is decoupling network systems, and studying the low-dimensional systems through the master stability function, or using the tool in equality (LMI toolbox).

Recently “outer synchronization” was proposed, which aimed at studying synchronization between coupled networks. In [11], the authors theoretically and numerically demonstrated the possibility of synchronization between two networks. By the open-plus-closed-loop (OPCL) method [12], synchronization between two networks can be realized with the same topological structures. In reality, if network nodes are of similar properties (same node dynamics), we regard it as one network, otherwise, as more networks. Several examples on two networks are presented in the literature [11, 13].

Actions between two networks are colorful, such as through the nonlinear signals, bipartite connection, or by special nodes. Recently, Sorrentino and Ott studied network synchronization of groups and discussed the feasibility of inner synchronization of individual network [14]. The problem of collective behavior in a network or between two networks is of broad interest. As a first example, in subway systems (e.g., Shanghai, China), when the trains arrive at the platform, the outer and inner doors simultaneously open or close, which shows that both inner synchronization and outer synchronization happen. Another example is two unmanned vehicles are assigned to accomplish independent tasks, such as cooperative searches and attacks [14], which focuses on inner synchronization. If the communicated information between two networks is not amount to lead to desynchronization, we should adopt control skills [15–17] to make synchronization happen between them, which has received increasing attention; see [18–21]. It is also found that present studies on synchronization inside a network and between two networks simultaneously are much less.

Time delays commonly exist in natural systems, since the finite speed of signal transmission over a distance gives rise to finite time delay, which plays an important role in the stability of synchronization [22]. For the two coupled networks, the signals from one network to another often create delays due to the distance. Therefore, motivated by the impact of topological structures and the delays on the dynamics of the networks, this paper mainly focuses on the effect of delays on inner synchronization inside each network and outer synchronization between two coupled networks. In Section 2, network models and synchronization analysis are presented, and numerical examples are shown in Section 3, including two examples for the same or different dimension of node dynamics. Finally, the discussions are included in the last section.

2. Model Presentation and Synchronization Analysis

Consider the following network models: the node dynamics in network 𝕏 is 𝑥̇_𝑖=𝑓(𝑥_𝑖(𝑡)), 𝑖=1,…,𝑁_𝑥, and 𝑦̇_𝑗=𝑔(𝑦_𝑗(𝑡)), 𝑗=1,…,𝑁_𝑦, for the nodes in network 𝕐, where 𝑓(𝑥)∶𝑅^𝑛𝑥→𝑅^𝑛𝑥, 𝑔(𝑦)∶𝑅^𝑛𝑦→𝑅^𝑛𝑦 are continuously differential functions and 𝑥_𝑖(𝑦_𝑗) is an 𝑛_𝑥-dimensional (𝑛_𝑦-dimensional) state vector. The dynamical equations of the network systems with time delays are as follows:𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+^𝑁𝑦_𝑗=1𝐴_𝑖𝑗Γ₁𝑦_𝑗𝑡−𝜏_𝑦,𝑖=1,2,…,𝑁_𝑥,𝑦̇_𝑗(𝑡)=𝑔𝑦_𝑗(𝑡)+^𝑁𝑥_𝑖=1𝐵_𝑗𝑖Γ₂𝑥_𝑖𝑡−𝜏_𝑥,𝑗=1,2,…,𝑁_𝑦, (2.1) where 𝐴 is an 𝑁_𝑥×𝑁_𝑦 dimensional coupling matrix, whose entries (𝐴_𝑖𝑗) represent the intensity of the direct interaction from 𝑖 in network 𝕏 to 𝑗 in network 𝕐, analogously the entries of (𝐵_𝑗𝑖). Matrix Γ₁(Γ₂)∈𝑅^{𝑛𝑥×𝑛𝑦}(𝑅^{𝑛𝑦×𝑛𝑥}) is the inner-coupling matrix. 𝜏_𝑥,𝜏_𝑦 are the time delays between networks. The action sketch between two networks with time delays is shown in Figure 1.

Figure 1 

A simple action sketch between two coupled networks with time delays.

For the more general models, we allow connections in each network, the networked system then reads as𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+^𝑁𝑦_𝑗=1𝐴_𝑖𝑗Γ₁𝑦_𝑗𝑡−𝜏_𝑦+^𝑁𝑥_𝑚=1𝐶_𝑖𝑚Γ₃𝑥_𝑚(𝑡),𝑖=1,2,…,𝑁_𝑥,𝑦̇_𝑗(𝑡)=𝑔𝑦_𝑗(𝑡)+^𝑁𝑥_𝑖=1𝐵_𝑗𝑖Γ₂𝑥_𝑖𝑡−𝜏_𝑥+^𝑁𝑦_𝑘=1𝐷_𝑗𝑘Γ₄𝑦_𝑘(𝑡),𝑗=1,2,…,𝑁_𝑦, (2.2) where 𝐶=(𝐶_𝑖𝑚)∈𝑅^{𝑁𝑥×𝑁𝑥}, 𝐷=(𝐷_𝑗𝑘)∈𝑅^{𝑁𝑦×𝑁𝑦} are the coupling matrices, satisfying the sum of every row being zero. Γ₃∈𝑅^{𝑛𝑥×𝑛𝑥}, Γ₄∈𝑅^{𝑛𝑦×𝑛𝑦} are also inner-coupling matrices.

Let us now consider the possibility that the individual networks achieve synchronization; that is, 𝑥₁(𝑡)=⋯=𝑥_𝑁𝑥(𝑡)=𝑥_𝑠(𝑡) and 𝑦₁(𝑡)=⋯=𝑦_𝑁𝑦(𝑡)=𝑦_𝑠(𝑡). If there exist such synchronous states, satisfying ^𝑁𝑦_𝑗=1𝐴_𝑖𝑗=𝜇₁,∀𝑖∈𝕏,^𝑁𝑥_𝑖=1𝐵_𝑗𝑖=𝜇₂,∀𝑗∈𝕐, (2.3) without loss of generality, we assume that 𝜇₁=𝜇₂=1. Thus, the synchronized state equations are𝑥̇_𝑠(𝑡)=𝑓𝑥_𝑠(𝑡)+Γ₁𝑦_𝑠𝑡−𝜏_𝑦,𝑦̇_𝑠(𝑡)=𝑔𝑦_𝑠(𝑡)+Γ₂𝑥_𝑠𝑡−𝜏_𝑥. (2.4)

Linearizing the synchronous states around 𝑥_𝑠 and 𝑦_𝑠, we get𝛿𝑥̇_𝑖(𝑡)=𝐽(𝑡)𝛿𝑥_𝑖(𝑡)+^𝑁𝑦_𝑗=1𝐴_𝑖𝑗Γ₁𝛿𝑦_𝑗𝑡−𝜏_𝑦,𝑖=1,2,…,𝑁_𝑥,𝛿𝑦̇_𝑗(𝑡)=𝑊(𝑡)𝛿𝑦_𝑗(𝑡)+^𝑁𝑥_𝑖=1𝐵_𝑗𝑖Γ₂𝛿𝑥_𝑖𝑡−𝜏_𝑥,𝑗=1,2,…,𝑁_𝑦, (2.5) where 𝐽(𝑡)=𝐷𝑓(𝑥_𝑠(𝑡)) and 𝑊(𝑡)=𝐷𝑔(𝑦_𝑠(𝑡)) are the Jacobians of 𝑓(𝑥(𝑡)) and 𝑔(𝑦(𝑡)) at 𝑥_𝑠 and 𝑦_𝑠, respectively.

We assume that the (𝑁_𝑥+𝑁_𝑦)-independent solutions of (2.5) can be expressed in the form 𝛿𝑥_𝑖=Φ_𝑥𝑖𝛿𝑥, 𝑖=1,2,…,𝑁_𝑥, 𝛿𝑦_𝑗=Φ_𝑦𝑗𝛿𝑦, 𝑗=1,2,…,𝑁_𝑦, where {Φ_𝑥𝑖},{Φ_𝑦𝑗} are the suitable time-independent scalars and 𝛿𝑥,𝛿𝑦 are the appropriate variables. If the dimension of the space vectors given by the values of Φ_𝑥𝑖,Φ_𝑦𝑗  (𝑖=1,2,…,𝑁_𝑥,𝑗=1,2,…,𝑁_𝑦) is 𝑁_𝑥+𝑁_𝑦, we can see that this assumed form includes all possible linear solutions of (2.5). Substituting this assumption into (2.5), they become Φ_𝑥𝑖𝛿𝑥̇(𝑡)=Φ_𝑥𝑖𝐽(𝑡)𝛿𝑥(𝑡)+^𝑁𝑦_𝑗=1𝐴_𝑖𝑗Φ_𝑦𝑗Γ₁𝛿𝑦𝑡−𝜏_𝑦,𝑖=1,2,…,𝑁_𝑥,Φ_𝑦𝑗𝛿𝑦̇(𝑡)=Φ_𝑦𝑗𝑊(𝑡)𝛿𝑦(𝑡)+^𝑁𝑥_𝑖=1𝐵_𝑗𝑖Φ_𝑥𝑖Γ₂𝛿𝑥𝑡−𝜏_𝑥,𝑗=1,2,…,𝑁_𝑦. (2.6)(2.7) Thus, in order that (2.6) (resp., (2.7)) is satisfied for all 𝑖 (resp., 𝑗), we require that Φ_𝑥⁻¹_𝑖∑_𝑗𝐴_𝑖𝑗Φ_𝑦𝑗=𝜂₁, where 𝜂₁ is independent of 𝑖, and Φ_𝑦⁻¹_𝑗∑_𝑖𝐵_𝑗𝑖Φ_𝑥𝑖=𝜂₂, where 𝜂₂ is independent of 𝑗. Set Φ_𝑥=(Φ_𝑥1,…,Φ_𝑥𝑁𝑥) and Φ_𝑦=(Φ_𝑦1,…,Φ_𝑦𝑁𝑦). From this, we get 𝐴Φ_𝑦=𝜂₁Φ_𝑥, 𝐵Φ_𝑥=𝜂₂Φ_𝑦; that is,0𝐴Φ𝐵0Φ^𝑥_𝑦=𝜂₁Φ_𝑥𝜂₂Φ_𝑦. (2.8)

Substituting (2.8) in (2.6) and (2.7), we obtain𝛿𝑥̇(𝑡)=𝐽(𝑡)𝛿𝑥(𝑡)+𝜂₁Γ₁𝛿𝑦𝑡−𝜏_𝑦,𝛿𝑦̇(𝑡)=𝑊(𝑡)𝛿𝑦(𝑡)+𝜂₂Γ₂𝛿𝑥𝑡−𝜏_𝑥. (2.9) One particular solution of (2.8) is derived when 𝜂₁=𝜂₂=𝜆, thenΘΦ^∗_𝑥Φ^∗_𝑦Φ=𝜆^∗𝑥Φ^∗_𝑦,Θ=0𝐴𝐵0, (2.10) where 𝜆 is the (possibly complex) eigenvalues of the matrix Θ.

Set 𝜂₁=𝜆𝜉, 𝜂₂=𝜆/𝜉, Φ_𝑥=Φ^∗_𝑥, Φ_𝑦=𝜉Φ^∗_𝑦, where 𝜉 is a free parameter. Equation (2.10) becomes0𝐴Φ𝐵0𝜉Φ^∗𝑥∗_𝑦=(𝜆𝜉)Φ^∗_𝑥𝜆𝜉𝜉Φ^∗_𝑦, (2.11) which shows that the solution of (2.11) contains all the possible solutions of (2.8). We rewrite (2.9) as𝛿𝑥̇(𝑡)=𝐽(𝑡)𝛿𝑥(𝑡)+𝜆Γ₁𝛿𝑦𝑡−𝜏_𝑦,𝛿𝑦̇(𝑡)=𝑊(𝑡)𝛿𝑦(𝑡)+𝜆Γ₂𝛿𝑥𝑡−𝜏_𝑥, (2.12) where 𝜆=𝜆₁,𝜆₂,…,𝜆_{𝑁𝑥+𝑁𝑦}.

In [14], the authors gave the explicit analysis on the spectrum of Θ, and two forms of the constructed matrix Θ were shown. Here, we adopt one that matrix Θ has real eigenvalues and only study network (2.1). In the sequel, we utilize the LMI to derive a synchronous theorem.

Theorem 2.1. Consider network model (2.1). If there exist two positive matrices 𝑃,𝑄>0, satisfying ΨΞ=¹00𝜆𝑃Γ₁0Ψ₂𝜆𝑄Γ₂00𝜆Γ^𝑇₂𝑄−𝐼_𝑛𝑥0𝜆Γ^𝑇₁𝑃00−𝐼_𝑛𝑦<0, (2.13) where Ψ₁=𝐽(𝑡)^𝑇𝑃+𝑃𝐽(𝑡)+𝐼_𝑛𝑥, Ψ₂=𝑊(𝑡)^𝑇𝑄+𝑄𝑊(𝑡)+𝐼_𝑛𝑦, then the network (2.1) asymptotically synchronizes to 𝑥_𝑠,𝑦_𝑠 defined by (2.4) for the fixed delays 𝜏_𝑥,𝜏_𝑦>0, respectively.

Proof. Consider (2.12). Choose a Lyapunov-Krasovskii functional as 𝑉(𝑡)=𝛿𝑥^𝑇(𝑡)𝑃𝛿𝑥(𝑡)+𝛿𝑦^𝑇(𝑡)𝑄𝛿𝑦(𝑡)+^𝑡_{𝑡−𝜏𝑥}𝛿𝑥^𝑇(𝛼)𝛿𝑥(𝛼)𝑑𝛼+^𝑡_{𝑡−𝜏𝑦}𝛿𝑦^𝑇(𝛼)𝛿𝑦(𝛼)𝑑𝛼. (2.14) Therefore, 𝑉̇(𝑡)=𝛿𝑥̇^𝑇(𝑡)𝑃𝛿𝑥(𝑡)+𝛿𝑥^𝑇(𝑡)𝑃𝛿𝑥̇(𝑡)+𝛿𝑦̇^𝑇(𝑡)𝑄𝛿𝑦(𝑡)+𝛿𝑦^𝑇(𝑡)𝑄𝛿𝑦̇(𝑡)+𝛿𝑥^𝑇(𝑡)𝛿𝑥(𝑡)−𝛿𝑥^𝑇𝑡−𝜏_𝑥𝛿𝑥𝑡−𝜏_𝑥+𝛿𝑦^𝑇(𝑡)𝛿𝑦(𝑡)−𝛿𝑦^𝑇𝑡−𝜏_𝑦𝛿𝑦𝑡−𝜏_𝑦=𝛿𝑥^𝑇(𝑡)Ψ₁𝛿𝑥(𝑡)+𝛿𝑦^𝑇𝑡−𝜏_𝑦𝜆Γ^𝑇₁𝑃𝛿𝑥(𝑡)+𝛿𝑦^𝑇(𝑡)Ψ₂𝛿𝑦(𝑡)+𝛿𝑥^𝑇𝑡−𝜏_𝑥𝜆Γ^𝑇₂𝑄𝛿𝑦(𝑡)+𝛿𝑦^𝑇(𝑡)𝜆𝑄Γ₂𝛿𝑥^𝑇𝑡−𝜏_𝑥+𝛿𝑥^𝑇(𝑡)𝜆𝑃Γ₁𝛿𝑦^𝑇𝑡−𝜏_𝑦−𝛿𝑥^𝑇𝑡−𝜏_𝑥𝛿𝑥𝑡−𝜏_𝑥−𝛿𝑦^𝑇𝑡−𝜏_𝑦𝛿𝑦𝑡−𝜏_𝑦=𝛿𝛿𝑥(𝑡)𝛿𝑦(𝑡)𝑥(𝑡−𝜏_𝑥)𝛿𝑦(𝑡−𝜏_𝑦)^𝑇Ξ𝛿𝑥(𝑡)𝛿𝑦(𝑡)𝛿𝑥𝑡−𝜏_𝑥𝛿𝑦𝑡−𝜏_𝑦. (2.15) According to the known condition Ξ<0 and Lyapunov-Krasovskii stability Theorem [6], the zero solutions of (2.12) are asymptotically stable, which shows that synchronization inside network 𝕏 and network 𝕐 happens when 𝑡 tends to +∞.

Remark 2.2. Matrix Ξ relies upon 𝑡 and 𝜏_𝑥,𝜏_𝑦. The condition can be changed into Ξ<0 when 𝑡>𝑇 for enough big 𝑇>0. The linear matrix equality method is inadequate for assessing the stability of the synchronous solution when both intragroup and extragroup connections are allowed in the network (2.2).

3. Numerical Examples

In this section, we will give two examples to illustrate our obtained results, which includes two cases: 𝑛_𝑥≠𝑛_𝑦 and 𝑛_𝑥=𝑛_𝑦. For the case 𝑛_𝑥≠𝑛_𝑦, we only consider the inner synchronization inside each network. On the other hand, 𝑛_𝑥=𝑛_𝑦, we discuss the inner synchronization within each network and investigate the outer synchronization between two networks.

Example 3.1. Consider the coupled network dynamical equations [14] below, which are in the form (2.1): 𝑥̇_𝑖(𝑡)=−𝑎𝑥_𝑖(𝑡)+𝑠𝑥_𝑖(𝑡)+^𝑁𝑦_𝑗=1𝐴_𝑖𝑗𝑦_𝑗1𝑡−𝜏_𝑦,𝑖=1,…,𝑁_𝑥,𝑦̇_𝑗1(𝑡)=−𝑦_𝑗1(𝑡)+𝑦_𝑗2(𝑡)+^𝑁𝑥_𝑖=1𝐵_𝑗𝑖𝑥_𝑖𝑡−𝜏_𝑥,𝑦̇_𝑗2(𝑡)=−𝑏𝑦_𝑗1(𝑡),𝑗=1,2,…,𝑁_𝑦, (3.1) where 𝑠(𝑥)=𝑚₁𝑥+1/2(𝑚₀−𝑚₁)(|𝑥+1|−|𝑥−1|). Here, we take 𝑎=4.6, 𝑏=6.02, 𝑚₀=−8/7, 𝑚₁=−5/7, for numerical simulation. We introduce two quantities 𝐸_𝑥=max_{𝑖=1,…,𝑁𝑥}‖𝑥_𝑖−𝑥_𝑠‖ and 𝐸_𝑦=max_{𝑗=1,…,𝑁𝑦}‖𝑦_𝑗−𝑦_𝑠‖ to measure the extent to which synchronization is achieved. The initial values are randomly chosen in (0,1).
The intercoupling matrices are 01𝐴=41214034014141414141,𝐵=2012121201201212120. (3.2)
When 𝑎=4.6, 𝑏=6.02, 𝑚₀=−8/7, 𝑚₁=−5/7, 𝜏_𝑥=𝜏_𝑦=0.8 and 𝐴,𝐵; by using the Matlab LMI Toolbox, we solve the LMI (2.13) for 𝑃>0, 𝑄>0, and obtain 𝑃=(2.5874);𝑄=10.8749−0.9036−0.90361.8461, (3.3) which indicates that inner synchronization inside each network appears. The inner synchronization evolution curves are depicted in Figure 2. We plots 𝐸_𝑥+𝐸_𝑦 with different values of 𝜏_𝑥 or 𝜏_𝑦 in Figure 3, which shows that the delays 𝜏_𝑥,𝜏_𝑦 have little influence on inner synchronization with such intercoupling matrices 𝐴,𝐵. Note that the order of magnitude for 𝐸_𝑥+𝐸_𝑦 is 10⁻⁸.
Next, we change the connection topology of 𝐴,𝐵 and consider a special case—globally connected, the elements of 𝐴_𝑖𝑗=1/𝑁_𝑦, 𝐵_𝑗𝑖=1/𝑁_𝑥, 𝑖=1,…,𝑁_𝑥, and 𝑗=1,…,𝑁_𝑦 with 𝑁_𝑥=50, 𝑁_𝑦=100. The other parameters are the same as in Figure 2. The numerical results are given in Figures 4 and 5. It is shown that the node numbers and connection topology influence the inner synchronization less.

Figure 2 

Inner synchronization evolution curves with 𝜏𝑥=𝜏𝑦=0.8.

(a)

(b)

Figure 3 

The plot shows 𝐸𝑥+𝐸𝑦 at 𝑡=40 with regard to 𝜏𝑥 or 𝜏𝑦, (a) 𝜏𝑦=0.8; (b) 𝜏𝑥=0.8.

Figure 4 

Inner synchronization evolution curves with globally connected topology, where 𝑁𝑥=50, 𝑁𝑦=100, and 𝜏𝑥=𝜏𝑦=0.8.

Figure 5 

The curves of 𝐸𝑥+𝐸𝑦 at 𝑡=40 versus 𝜏𝑦, with 𝜏𝑥=0.8, 𝑁𝑥=50, and 𝑁𝑦=100.

Example 3.2. In this example, we let 𝑛_𝑥=𝑛_𝑦 and 𝑁_𝑥=𝑁_𝑦=𝑁. Therefore, we discuss inner synchronization inside network 𝕏 or 𝕐 and study synchronization between two networks, that is, outer synchronization proposed in [11], denoting the quantity 𝐸_outer=max_{𝑖=1,…,𝑁}‖𝑥_𝑖−𝑦_𝑖‖, to demonstrate that outer synchronization happens. Consider a one-order dynamical system [23] as the dynamical nodes of the complex networks which is described by 𝑥̇_𝑖(𝑡)=−𝑥²_𝑖(𝑡)+𝜌₁𝑥_𝑖(𝑡)+𝑟_𝑥^𝑁_𝑗=1𝐴_𝑖𝑗𝑦_𝑗𝑡−𝜏_𝑦𝑦̇,𝑖=1,2,…,𝑁,_𝑗(𝑡)=−𝑦²_𝑗(𝑡)+𝜌₂𝑦_𝑗(𝑡)+𝑟_𝑦^𝑁_𝑖=1𝐵_𝑗𝑖𝑥_𝑖𝑡−𝜏_𝑥,𝑗=1,2,…,𝑁. (3.4)
The elements of 𝐴_𝑖𝑗=1/𝑁, 𝐵_𝑗𝑖=1/𝑁, 𝑖,𝑗=1,…,𝑁, and 𝑁=50, when 𝜌₁=−0.3, 𝜌₂=0.6, 𝜏_𝑥=0.3, and 𝜏_𝑦=0.5, by solving the LMI (2.13) for 𝑃>0, 𝑄>0, we obtain 𝑃=(2.0946),𝑄=(1.9328), (3.5) satisfying the conditions of a synchronous criterion, then the individual network achieves its own inner synchronized state. Figure 6 plots the inner and outer synchronization evolution curves. In addition, we plot the curves of quantities 𝐸_𝑥,𝐸_𝑦,𝐸_outer with regard to 𝑟_𝑥 for 𝑟_𝑦=0.2 in Figure 7. We find that the individual network easily achieves inner synchronization for the arbitrary values of 𝑟_𝑥, while only when 𝑟_𝑥=1.2, outer synchronization is realized.
In the following, we numerically discuss the delay effect on inner and outer synchronization. Fix the values of 𝜌₁=−0.3, 𝜌₂=0.6, 𝑟_𝑦=0.2, 𝑟_𝑥=1.2, and 𝜏_𝑦=0.5, and let the value of 𝜏_𝑥 vary. We observe that the delay does not influence inner synchronization, and when 𝜏_𝑥>1.5, outer synchronization disappears. Figure 8 plots the curves of 𝐸_𝑥,𝐸_𝑦, and 𝐸_outer with regard to 𝜏_𝑥.
Now, we add the individual connections in network 𝕏 and network 𝕐, and the following network systems are written as: 𝑥̇_𝑖(𝑡)=−𝑥²_𝑖(𝑡)+𝜌₁𝑥_𝑖(𝑡)+𝑟_𝑥^𝑁_𝑗=1𝐴_𝑖𝑗𝑦_𝑗𝑡−𝜏_𝑦+^𝑁_𝑚=1𝐶_𝑖𝑚𝑥_𝑚(𝑡)𝑦̇,𝑖=1,2,…,𝑁,_𝑗(𝑡)=−𝑦²_𝑗(𝑡)+𝜌₂𝑦_𝑗(𝑡)+𝑟_𝑦^𝑁_𝑖=1𝐵_𝑗𝑖𝑥_𝑖𝑡−𝜏_𝑥+^𝑁_𝑘=1𝐷_𝑗𝑘𝑦_𝑘(𝑡),𝑗=1,2,…,𝑁, (3.6) where (−𝐶)=−{𝐶_𝑖𝑚}, (−𝐷)=−{𝐷_𝑗𝑘} are the Laplacian matrices: ∑^𝑁_𝑚=1𝐶_𝑖𝑚=0 for all 𝑖 and ∑^𝑁_𝑘=1𝐷_𝑗𝑘=0 for all 𝑗; obviously, the diffusive couplings ∑^𝑁_𝑚=1𝐶_𝑖𝑚, ∑^𝑁_𝑘=1𝐷_𝑗𝑘 are null in the synchronized manifold. The parameters are chosen as the same as those in Figure 8, and individual couplings 𝐶,𝐷 are taken as two random matrices. Numerical results show that the adding individual connections have little effect on inner and outer synchronization.

Figure 6 

Inner and outer synchronization evolution curves with 𝑟𝑥=0.5, 𝑟𝑦=0.2, 𝜏𝑥=0.3, 𝜏𝑦=0.5, 𝜌1=−0.3, 𝜌2=0.6, and 𝑁=50.

Figure 7 

The curves of 𝐸𝑥,𝐸𝑦, and 𝐸outer versus 𝑟𝑥 for 𝑟𝑦=0.2. The other parameters are 𝜏𝑥=0.3, 𝜏𝑦=0.5, 𝜌1=−0.3, and 𝜌2=0.6.

Figure 8 

The plot show 𝐸𝑥,𝐸𝑦, and 𝐸outer versus 𝜏𝑥 with 𝜏𝑦=0.5, 𝑟𝑦=0.3, 𝑟𝑥=1.2, 𝜌1=−0.3, and 𝜌2=0.6.

4. Conclusions

In conclusion, we have studied inner synchronization inside networks 𝕏 and 𝕐 and outer synchronization between them with time delays. A synchronous criterion for the occurrence of inner synchronization has been derived. Numerical examples show that the delays have little effect on inner synchronization, how to estimate the domain of delays 𝜏_𝑥 and 𝜏_𝑦 is an interesting work. It is noted that in Example 3.2, outer synchronization only happens in the coupling strength 𝑟_𝑥=1.2, and perhaps the node dynamics in networks 𝕏 and 𝕐 plays a central role, or the bidirectional delayed coupling is weak. Because of the various kinds of actions between two networks, how to derive the criteria on inner and outer synchronization simultaneously is an technical challenging. The theoretical understanding is helpful to study the synchronization between two or more neural networks [24] with appropriate couplings. We hope that such work will appear elsewhere.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 10872119, 11005026, and 61003207) and Research Foundation of Hangzhou Dianzi University (no. KYF075610032).