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This paper investigates the pinning control schemes and corresponding criteria for group synchronization in a complex dynamical network with different types of chaotic oscillators. We present the linear pinning and adaptive pinning control schemes to make different groups of oscillators synchronize to their own synchronization states, respectively. The globally asymptotically stable criteria for group synchronization are derived, which indicate that the group synchronization can be realized only by pinning a part of nodes in a general network. Finally, some numerical simulations are provided to verify the theoretical results.

Nowadays, complex networks widely exist in nature and society, such as the World Wide Web, the Internet, traffic network, communication network, power network, and social network. Over the past three decades, complex networks have received increasing attention from various fields such as biology, physics, chemistry, engineering, and information [

Synchronization is a kind of typical collective behaviors and basic motion in nature. As a typical network dynamics, the synchronization of complex networks has been intensively studied recently. Different synchronization phenomena have been revealed, such as global synchronization, generalized synchronization, partial synchronization, and cluster synchronization [

As we known now, the real-world complex networks normally have a large number of nodes. Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes. To reduce the number of the controllers, a usual approach, which is called the pinning control, is used to control part of nodes. Wang and Chen [

In this paper, based on the model in [

The rest of the paper is organized as follows. In Section

Let us consider a complex dynamical network with

Furthermore, denote the block matrix

Suppose that there exists positive constant

The set

Here,

The controlled network (

Define the synchronous error vectors of the

For

For

Then the controlled error system (

Obviously, the stability problem of group synchronous manifold

In order to analyze the group synchronization, the following lemmas are needed.

Suppose that

If

Suppose that

If

By Lemma

According to Lemma

That is, all eigenvalues of

From the derivation process, the conclusion of Corollary

In this section, we present the linear feedback and adaptive feedback pinning schemes, respectively, and derive the corresponding criteria to achieve the group synchronization in network (

For network (

Then the error equation (

Suppose that

Consider the Lyapunov function

Its time derivative along the trajectory of (

According to the Corollary

Denote the eigenvalues of

According to Lemma

That is,

From Theorem

In this linear feedback pinning scheme, one only needs to pin a few nodes which have external connections; then the network achieves the group synchronization with large enough coupling strength. In particular, it is only relevant to network connectivity, without any limitation to the network structure. For a cluster network, in general, the connection within a cluster is more than the external connection between clusters. When the network has a cluster property, the higher the clustering, the better the control effect. Since the network with higher clustering means that the number of nodes with external connection is less, thus the less nodes are needed to be pinned. Therefore, the higher the degree of clustering, the more obvious the advantages.

In order to avoid estimating the feedback control gain, in this section, we present an adaptive feedback pining scheme to realize the group synchronization in network (

For network (

Then the error equation (

Suppose that

Consider the Lyapunov function

Its time derivative along the trajectory of (

The rest is similar to the proof of Theorem

In Theorem

In this section, an illustrative example is provided to verify the above theoretical analysis.

We consider a network (see Figure

The topology connection of network (

The chaotic attractors of Lorenz (a), Chua (b), and Chen (c).

The network as an example of the controlled network (

In the simulation, we take the initial values

Figures

Group synchronization of network (

Group synchronization of network (

By the above theoretical analysis and instance, it is concluded that the pinning control strategy can be applied to large networks with tens of thousands of nodes based on the accurate network topology. In fact, this control strategy depends on the structure information rather than the size of the network.

In this paper, we have investigated the group synchronization in a complex dynamical network consisting of different groups of coupled chaotic oscillators. The linear feedback and adaptive feedback pinning schemes are presented, respectively. The corresponding sufficient conditions are derived to ensure each group synchronizes to the same states. Our study shows that the global asymptotic stability of the group synchronous manifold can be guaranteed by controlling a few nodes which have external connections. For the large networks with obvious cluster property, our pinning control schemes can save much cost in practical.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was jointly supported by the National Natural Science Foundation (no. 61164020), the National Social Science Foundation (no. 13BTJ009), the Natural Science Foundation of Guangxi (0991244, 2011 GXNSFA018147), Guangxi Key Laboratory of Spatial Information and Geomatics (nos. 1103108-24, 1207115-27).