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The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results.

Recently, synchronization of all dynamical nodes in a network is one of the hot topics in the investigation of complex networks. It is well known that there are many useful synchronization phenomena in real life, such as the synchronous transfer of digital or analog signals in communication networks. Adaptive feedback control has witnessed its effectiveness in synchronizing a complex network [

In the process of studying synchronization of complex networks, two main factors should be considered: time delays and stochastic perturbations. Time delays commonly exist in the real world and even vary according to time. In the subsystems, time delay can give rise to chaos, such as delayed neural network and delayed Chua's circuit system in Section

Based on the above analysis, in this paper, we study the synchronization of complex networks with time-varying delays and vector-form stochastic elements. Two different adaptive pinning controllers according to the different properties of inner couplings are considered. To obtain our main results, we first formulate a new complex network with nondelayed and delayed couplings and vector-form Wiener processes. Then, by virtue of an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, we develop several theoretical results guaranteeing the synchronization of the new complex networks. Our adaptive pinning controllers are simple. The coupling matrices can be symmetric or asymmetric. Numerical simulations testify the effectiveness of our theoretical results.

In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions.

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

Consider complex a network consisting of

The initial condition of system (

Our objective of synchronization is to control network (

For convenience of writing, in the sequel, we denote

Subtracting (

Then the objective here is to find some appropriate adaptive pinning controllers

Function class QUAD1(

The function class QUAD1(

From condition (

Before giving our main results, we present the following lemmas, which are needed in the next section.

If

Suppose that

The linear matrix inequality (LMI)

Now we list assumptions as follows.

There exist nonnegative constants

In this section, two different adaptive pinning feedback controllers corresponding to two kinds of properties of the inner coupling

For the the inner coupling matrix

Obviously, case (b) is the special case of case (a). Case (

Suppose that the assumptions

According to

Differentiating

Similarly, from

In view of Lemma

Let

Since

Next, we consider the case (b). Obviously, the adaptive pinning controllers (

Assume that

Suppose that the assumptions

Let

Differentiating both sides of (

Since

If there is no noise perturbation in (

Theorems

In this section, we provide three examples to illustrate the effectiveness of the results obtained above.

Consider the following chaotic delayed neural networks:

Taking

In order to verify our new results, consider the following coupled networks:

Through simple computation, we get that the left eigenvector of

Let

The initial conditions of the numerical simulations are as follows:

Chaotic trajectory of system (

Time evolutions of synchronization errors with the first two nodes being controlled.

Trajectories of

Consider the following coupled neural networks:

The initial conditions of the numerical simulations are as follows:

Trajectories of

Trajectories of

Consider the delayed Chua's circuit:

In the case that the initial condition is chosen as

Take

Now we construct a complex network, which obeys the scale-free distribution of the Barabási-Albert model [

Consider the following complex networks:

Without loss of generality, we take the first 10 nodes to be controlled. The initial conditions of the numerical simulations are as follows:

Chaotic trajectory of system (

BA scale-free network initial graph is complete with

Synchronization errors

Trajectories of control gains

In Example

In this paper, we have studied the adaptive pinning synchronization of the complex networks with delays and vector-form stochastic perturbations, in which dynamical behaviors are more realistic and complicated. Since the new condition QUAD1, our proof is very simple. Via two simple adaptive pinning feedback control schemes, several sufficient conditions guaranteeing the synchronization of the proposed complex networks are obtained. Our results are valid even if only partial states of nodes are coupled. Numerical simulations verified the effectiveness of our results. Our results improve and extend some existing results. Models and results in this paper provide possible new applications for network designers.

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 60874088, and the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2009271, the Scientific Research Fund of Yunnan Provincial Education Department under Grant no. 07Y10085, and the Scientific Research Fund of Yunnan Province under Grant no. 2008CD186.