The problem of projective synchronization of drive-response coupled dynamical network with delayed system nodes and multiple coupling time-varying delays is investigated. Some sufficient conditions are derived to ensure projective synchronization of drive-response coupled network under the impulsive controller by utilizing the stability analysis of the impulsive functional differential equation and comparison theory. Numerical simulations on coupled time delay Lorenz chaotic systems are exploited finally to illustrate the effectiveness of the obtained results.

In the past few years, it is found that synchronization is one of the most important and interesting collective behaviors of complex networks and has been extensively investigated in different fields of engineering and sociology [

In projective synchronization, the drive-response systems can be synchronized up to a scaling factor. Due to the potential applications in secure communication, the projective synchronization has been extremely investigated including chaotic systems [

Motivated by the above discussions, this paper aims to handle the problem of the impulsive projective synchronization for a drive-response coupled dynamical network with dynamical nodes delay and both nondelayed coupling and multiple delayed couplings. The sufficient conditions for projective synchronization are derived analytically by using the stability analysis of the impulsive functional differential equation, and an impulsive controller is designed. Analytical results show that drive-response coupled dynamical networks with multiple time delays can realize projective synchronization within a scaling factor.

The rest of this paper is organized as follows: in Section

The projective synchronization in coupled partially linear delayed chaotic systems via impulsive control is studied in [

In this paper, it should be pointed out that we do not require that the time-varying delay is a differential function with a bound of its derivative, which means that the time-varying delays include a wide range of functions. Moreover, the coupling configuration matrices are not assumed to be symmetric or irreducible.

In order to derive our main results, some necessary definitions and lemmas are needed.

The projective synchronization is said to take place in drive-response coupled network (

Let

This section addresses the implementation of projective synchronization between the drive and response networks with time delay characteristics. By taking a theoretical approach based on the classic Lyapunov stability theory, we derive the criteria of network projective synchronization and present an impulsive control scheme.

By selecting proper control gain matrix

Compared with continuous control, discontinuous control, including impulsive control and intermittent control, is effective, practical, and applicable in many areas, especially for secure communication. Impulsive controller has a relatively simple structure and is easy to implement. In an impulsive synchronization scheme, the response system receives the information from the drive system only in discrete times and the amount of conveyed information is, therefore, decreased. This is very advantageous in practice due to reduced control cost.

Letting projective synchronization error be

Let

Let

Letting

For given synchronization scaling factor

Consider the following Lyapunov candidate function:

For

It is clear that

From the definition of

When

For any

Since

By the formula for the variation of parameters,one obtains

According to the representation of the Cauchy matrix, we get the following estimation of

For simplicity, let

Defining

On the other hand, since

In the following, we will prove that the following inequality holds:

If (

From (

Therefore, we have

When

From Theorem

Letting

Letting

We consider the equidistant impulsive interval

The value of scaling factor

When

Sun et al. [

In this section, numerical simulations are given to verify and demonstrate the effectiveness of the proposed synchronization scheme for synchronizing the drive-response coupled network with time-delayed dynamical nodes and multiple coupling delays onto a scaling factor. We consider the time delay Lorenz chaotic system [

Chaotic behavior of the time-delayed system (

For simplicity, the drive-response network systems with two terms of time-varying delayed coupling are described as follows:

The coupling configuration matrices

According to Theorem

In the numerical simulations, we assume

Evolution of (a) state trajectories of drive and response systems and (b) projective synchronization error without impulsive control when

The phase plot of

Evolution of (a) state trajectories of drive (the dash line) and response systems (the solid line) and (b) projective synchronization error under impulsive control.

The phase plot of

Evolution of (a) state trajectories of drive (the dash line) and response systems (the solid line) and (b) projective synchronization error under impulsive control.

Especially, if

Evolution of (a) state trajectories of drive and response systems and (b) projective synchronization error without impulsive control when

The phase plot of

Evolution of (a) state trajectories of drive (the dash line) and response systems (the solid line) and (b) projective synchronization error under impulsive control.

In this paper, the projective synchronization problem of drive-response coupled dynamical network with multiple time-varying delays is studied by employing the impulsive control scheme. As well known, compared with the controller used in adaptive control method [

In this paper, the projective synchronization of drive-response coupled dynamical network with time delays dynamical nodes and multiple coupling delays has been studied. Some sufficient conditions for realizing the projective synchronization with a scaling factor are established by using the stability analysis of impulsive delayed systems and comparison method. Numerical simulations have also been given to show the effectiveness and the correctness of the theoretical analysis finally.

In the analysis and simulation study in this paper, we fully considered the impact of the time delay element on the projective synchronization of the drive-response network systems. In order to obtain a generic solution of projective synchronization criteria and control scheme, we neglected the particularities of networks. In fact, the dynamic processes of different oscillators are not always unified; as a result their dynamic characteristics under time delay need to be further investigated. Furthermore, we did not consider the environment factors, for example, noise, on the networks, which often affect the synchronization process of the drive-response network systems. Therefore, with respect to the future work, we will further consider the projective synchronization problem of drive-response network with different dynamics oscillators under different scaling factors. Simultaneously, other environmental factors, for example, the noise, will be taken into account in the study to further improve the robustness of the control solutions.

The authors declare that they have no conflict of interests.

This work was jointly supported by the National Science Foundation of China (Grant nos. 11102076 and 11202085), the Society Science Foundation from Ministry of Education of China (Grant no. 12YJAZH002), the Natural Science Foundation of Zhejiang Province (Grant no. LY13F030016), and the Foundation of Zhejiang Provincial Education Department (Grant no. Y201328316).