^{1}

This paper addresses the hybrid synchronization problem in two nonlinearly coupled complex networks with asymmetrical coupling matrices under pinning control schemes. The hybrid synchronization of two complex networks is the outer antisynchronization between the driving network and the response network while the inner complete synchronization in the driving network and the response network. We will show that only a small number of pinning feedback controllers acting on some nodes are effective for synchronization control of the mentioned dynamical networks. Based on Lyapunov Stability Theory, some simple criteria for hybrid synchronization are derived for such dynamical networks by pinning control strategy. Numerical examples are provided to illustrate the effectiveness of our theoretical results.

Complex networks become more and more important because they abound both in nature and in the artificial networks (easy examples include biological ecosystems, internet connections, the World Wide Web, and various social and neural networks [

Many synchronization patterns have been recorded in the literature such as complete synchronization [

Recently, there is much interest in the coexistence of antisynchronization and complete synchronization (called

Since it is literally impossible to add controllers to all nodes, local feedback injections are applied to only some of the nodes (this is known as

This paper is organized as follows. In Section

In [

In this section, we write down some definitions, notations, and lemmas that will be used throughout this paper.

If

A nonlinear coupling function

Let

Assume that

let

Denote

If

When

In this part, we investigate the hybrid synchronization of a class of nonlinearly coupled complex dynamical networks and obtain some criteria for hybrid synchronization.

Suppose Hypothesis holds,

Denote

Let

Differentiating the function

Because of Hypothesis (

Denote

By Lemma

Combining inequalities (

Evaluating the time derivative of

By Hypothesis and because

Let

Finally, substituting (

The proof of the theorem is complete.

From the proof, one can see that the difficulty in investigating the hybrid synchronization with an asymmetrical coupling matrix

Therefore

Suppose

If

Suppose Hypothesis holds.

The case has been discussed in [

By using the Lyapunov method combined with some other technique, the hybrid synchronization criterion of the pinning-controlled dynamical networks has been obtained. It means that the outer antisynchronization between system (

In this section, we given numerical simulation to verify the theorem given in the previous section. In order to verify our results, we consider the driving complex network [

Let

As for the asymmetrical coupling matrix

The initial values are chosen as

Let

The trajectories evolution of the pinning control gains are shown in Figure

The trajectories evolution of the state variable, (a)

(a) The synchronization error evolution of the driver networks

As for the symmetrical coupling matrix

The initial values are chosen as above. The trajectories evolution of the pinning control gains are shown in Figure

The trajectories evolution of the state variable, (a)

(a) The synchronization error evolution of the driver networks

In this paper, we considered the hybrid synchronization of two nonlinearly coupled complex dynamical networks with asymmetrical coupling matrices under the pinning control scheme. By placing a small number of feedback controllers on some nodes, we obtained some criteria for the hybrid synchronization of such dynamical networks based on Lyapunov Stability Theory. It is shown that under certain conditions, two nonlinearly coupled complex networks can achieve an intriguing hybrid synchronization: the outer antisynchronization between the driving network and the response network and the inner complete synchronization between the driving network and the response network, respectively. Numerical examples were also provided to demonstrate the effectiveness of the theoretical result.

Because the random phenomenon appears frequently in dynamical complex network, in our further efforts, this feature should be taken into account in order to solve problems more practically. More precisely, our future works may be extended to the consensus problem of the multiagent systems with stochastic disturbance, the synchronization issue of the dynamical complex networks with stochastic disturbance, and so on. On the other side, control methods would not be limited to the feedback control any more. For example, in real world, the states of nodes in the networks often suffer from instantaneous perturbations or abrupt changes at certain instants, such as switching phenomena, frequency change, or sudden noise. In order to investigate the synchronization matters for this situation, suitable pinning impulsive controllers could be applied to deal with the impulsive-coupled dynamical networks. Moreover, couplings or communications between nodes in this paper are considered to be continuous. However, there are other modes of the information exchanging among the nodes, such as impulsive communication and intermittent communication. The studies on these problems will be also interesting and meaningful in the further research work.

This work was supported by the National Science Foundation of China under Grant no. 61273220, the Guangdong Education University Industry Cooperation Project (2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JCYJ20120613105730482). The authors are very grateful to the reviewers and editors for their valuable comments on this paper.