Complex Projective Synchronization in Drive-Response Stochastic Complex Networks by Impulsive Pinning Control

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.

Recently, projective synchronization under various cases of complex dynamical networks has been studied [17][18][19][20][21][22][23][24][25][26][27][28][29].In [19], Du et al. studied the problem of function projective synchronization for general complex dynamical networks with time delay.A hybrid feedback control method is designed to achieve function projective synchronization for complex dynamical networks with constant time or time-varying delay.In [20], Liu investigated the synchronization problem of fractional-order complex networks with nonidentical nodes, and the generalized projective synchronization criterion of fractional-order complex networks with order 0 <  < 1 is obtained.In [21], Yao and Wang explored a new cluster projective synchronization scheme in time-varying delay coupled complex dynamical networks with nonidentical nodes in consideration of the community structure of the networks.
In most existing research, the two complex networks (socalled 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 [17,[19][20][21][23][24][25][26][27][28][29].However, in real world, the systems can often evolve in different directions with a constant intersection angle with respect to complex number; for example,  =   , where  denotes the drive system,  denotes the response system,  > 0 denotes the zoom rate,  ∈ [0,2) denotes the rotation angle, and  = √ −1.This synchronization scheme has a large number of real-life examples.For instance, in distributed computers collaboration, each distributed computer (response system) not only to receive unified command from server (drive system), but also they are mutual to use resources for collaboration [30].Furthermore, in a social network or games in economic activities, and behaviors of individuals (those response systems) will be affected not only by powerful one (the drive system used in the present paper), 2 Discrete Dynamics in Nature and Society but also those with a similar role as themselves [22].Recently, some related works have come out, such as [18,22].In [18,22], Wu et al. introduced the concept of complex projective synchronization based on Lyapunov stability theory, several typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization.
In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [31][32][33].In the past several years, impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical system, such as signal processing system, computer networks, automatic control systems, and telecommunications.In [31], Cai et al. investigated the robust impulsive synchronization of complex delayed dynamical networks.Yang and Cao [32] studied the exponential synchronization of complex dynamical network with a coupling delay and impulsive disturbance.Zhu et al. gave some global impulsive exponential synchronization criteria of time-delayed coupled chaotic systems in [33].Xu et al. studied the synchronization problem of stochastic complex networks with Markovian switching and time-varying delays are investigated by using impulsive pinning control scheme in [34].
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 [19,23,24,35].What is more, uncertainties commonly exist in the real world, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties; the stochastic forms from the same noise of one-dimensional vector to the different noise perturbations of vector were investigated in [9,10,15,17,25].Therefore, it is important to study the effect of time delay and stochastic noise in complex projective synchronization of drive-response networks.
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.
Notation.Throughout this paper, C  and C × denote dimensional complex vectors and the set of  ×  complex matrices, respectively.For Hermite matrix , the notation  > 0 ( < 0) means that the matrix  is positive definite (negative definite).For any complex (real) matrix ,   =   + .For any complex number (or complex vector) , the notations   and   denote its real and imaginary parts, respectively, and  denotes the complex conjugate of . min ()( max ()) represents the smallest(largest) eigenvalue of a symmetric matrix .⊗ is the Kronecker product.The superscript  of   or   denotes the transpose of the vector  ∈ R  or the matrix  ∈ R × .  is identity matrix with  nodes.

Model Description and Preliminaries
Consider a drive-response network coupled with 1 +  identical partially linear stochastic complex network with coupling time delay, which is described as follows: where () = ( 1 ,  2 , . . .,   )  ∈ C  and () ∈ R are the drive system variables and   () = ( 1 ,  2 , . . .,   )  ∈ C  is the state variable of a node  in the response network.(()) ∈ R × is a complex matrix function,  1 > 0,  2 > 0 is the coupling strength, and Γ ∈ R × is the inner coupling matrix. is the coupling time delay.Matrices  = (  ) × and  = (  ) × are the zero-row-sum outer coupling matrix, which denote the network topology and are defined as follows.If there is a connection (information transmission) from node  to node  ( ̸ = ), then   ̸ = 0 and   ̸ = 0; otherwise,   = 0 and   = 0, and   () = ( 1 (),  2 (), . . .,   ())  ∈ R  is a bounded vector-form Weiner process, satisfying Now, two mathematical definitions for the generalized projective synchronization are introduced as follows.
for some  > 0 and some  > 0, then the drive-response networks (1) and (2) are said to achieve complex projective synchronization in mean square, and the parameter  is called a scaling factor.Without loss of generality, let  = (cos + sin ), where  = || is the module of  and  ∈ [0, 2) is the phase of .Therefore, the projective synchronization is achieved when  = 0 or .Furthermore, the complete synchronization is achieved when  = 1 and  = 0, and the antisynchronization is achieved when  = 1 and  =  [22].
The following lemmas and assumption are used throughout the paper.Lemma 3 (see [22]).Let  ×  complex matrix  be Hermitian; then (1)    is real for all  ∈   ; (2) all the eigenvalues of  are real.
Assumption 6 (see [22]).Suppose that there exists a constant  such that the largest eigenvalue of   (()) satisfies Remark 7. All the chaotic systems satisfy Assumption 6 due to the fact that () is bounded [22].

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

Conclusion
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.

Figure 1 :Figure 2 :Figure 3 :
Figure 1: The topology structures of the networks for 9 nodes.