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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.

Network synchronization, as a collective behavior existing inside a network, has been widely studied since the birth of small-world and scale-free networks [

Generally, we refer to synchronization happening between two networks as outer synchronization [

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 [

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

In this paper, we investigate the synchronization between two discrete-time networks with mutual couplings. The dynamical equations are described as follows:

Let us now consider the possibility whether the individual networks achieve inner synchronization; that is,

Thus the synchronized state equations are

Firstly, we study the system of (

Secondly, we study the stability of (

If

Consider network systems (

Note that Theorem

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 (

The panels exhibit

The plots show

Next, we discuss the effect of node dynamics on the inner and outer synchronization and take

The curves of

The trajectories of

Finally, we discuss the effect of network size

The plots show

The curves of

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

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

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.