The complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems is considered. The impulsive pinning control scheme is adopted to achieve complex projective synchronization and several simple and practical sufficient conditions are obtained in a general drive-response network. In addition, the adaptive feedback algorithms are proposed to adjust the control strength. Several numerical simulations are provided to show the effectiveness and feasibility of the proposed methods.

Complex networks become more and more important in modern society. Up to now, the investigation on the synchronization of complex networks has attracted a great deal of attentions due to its potential applications in various fields, such as physics, secure communication, automatic control, biology, and sociology [

Recently, projective synchronization under various cases of complex dynamical networks has been studied [

In most existing research, the two complex networks (so-called driver-response networks) evolve along the same or inverse direction with respect to real number, real matrix, or even real function in a complex plane [

In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [

Besides, due to the finite information transmission and processing speeds among the units, the connection delays in realistic modeling of many large networks with communication must be taken into account, such as [

Based on the above-mentioned content, the complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems and linear coupling time delays by impulsive pinning control scheme are considered in this paper. Several simple and practical criteria for complex projective synchronization are obtained by using Lyapunov functional method, stochastic differential theory, and linear matrix inequality (LMI) approaches.

Consider a drive-response network coupled with

Now, two mathematical definitions for the generalized projective synchronization are introduced as follows.

If there is a complex

Without loss of generality, let

Matrix

The following lemmas and assumption are used throughout the paper.

Let

all the eigenvalues of

If

Consider an

Suppose that there exists a constant

All the chaotic systems satisfy Assumption

Denote

Assumption

Our objective here is to achieve complex projective synchronization in the drive-response networks (

Define the synchronization errors between the drive network (

Supposing that Assumptions

Consider the Lyapunov functional candidate

On the other hand, from the construction of

Letting

Using condition (

When considering the system without delay, that is,

Then, without loss of generality, one has the following corollary.

Supposing that Assumptions

In this section, we conduct some numerical simulations to illustrate the effectiveness of the theorems of the previous section.

Consider a driver-response network coupled with the following complex Lorenz systems:

Consider complex projective synchronization in a drive-response network coupled with

The topology structures of the networks for

Choosing

The evolution of the synchronization trajectory

The evolution of the synchronization errors

The synchronization in drive-response stochastic coupled networks with complex-variable systems is considered in this paper. Because drive-response systems may evolve in different directions with a constant intersection angle in many real situations but may not simultaneously evolve along the same or inverse direction based on real number, real matrix, or even real function in a complex plane, we focus on the projective synchronization of this situation by impulsive pinning control through a theorem and a corollary. Eventually, several numerical simulations verified the validity of those results.

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

The authors thank the referees and the editor for their valuable comments on this paper. This work was supported by Shenzhen Strategic Emerging Industries Projects (ZDSY20120613125016389) and Shenzhen Basic Research Project (JCYJ20130331152625792).