Projective Synchronization Analysis of Drive-Response Coupled Dynamical Network with Multiple Time-Varying Delays via Impulsive Control

and Applied Analysis 3 where the right and upper Dini’s derivative D+u(t) is defined as D+u(t) = lim h→0 +((u(t + h) − u(t))/h), where h → 0+ means that h approaches zero from the right-hand side. Then u(t) ≤ υ(t) for −τ ≤ t ≤ 0 implies that u(t) ≤ υ(t) for t ≥ 0. 3. Projective Synchronization Analysis 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 B k i ∈ R , the drive-response network (1) can be rewritten as the following controlled impulsive differential equation: ?̇? d (t) = M (z) ⋅ u d (t) + γΓ (u d (t − τ (t)) − u d (t)) , ?̇? (t) = f (u d (t) , u d (t − τ (t)) , z (t) , z (t − τ (t))) , ?̇? r i = M(z) ⋅ u r i (t) + γΓ (u r i (t − τ (t)) − u r i (t))

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 [28][29][30][31][32] and complex dynamical networks [33][34][35][36][37][38][39][40].Xu [29] studied the projective synchronization in coupled partially linear systems via adaptive feedback control.Furthermore, Hu et al. [35] introduced a driveresponse dynamical network model and investigated its projective synchronization properties using pinning control to obtain the desired scaling factor.A short time later, they investigated projective cluster synchronization in a driveresponse dynamical network model with coupled partially linear chaotic systems [36].The impulsive projective synchronization between the drive system and response dynamical network without the time delay was investigated in [37].It is noted that in practical cases time delays are often encountered.Ignoring them may lead to design flaws and incorrect analysis conclusions.Consequently, time delay case should be considered.Recently, Sun et al. [38] studied the projective synchronization in drive-response dynamical networks of partially linear systems with time-varying coupling delay.Chen et al. [39] proposed projective (anticipatory, exact, and lag) synchronization criteria for a drive-response complex network with different scale factors.Moreover, in much of the literature, time delays in the couplings are considered; however, time delays in the dynamical nodes [32,40], which are more complex, are still relatively unexplored.Zheng [40] investigated the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling.Cao et al. [32] proposed projective synchronization of a class of delayed chaotic systems via impulsive control, where the drive-response system can be synchronized to within a scaling factor.On the other hand, it is well known that the impulsive control method [16-20, 32,

Model Description and Preliminaries
The projective synchronization in coupled partially linear delayed chaotic systems via impulsive control is studied in [32].Inspired by [32], the drive-response coupled network model with dynamical nodes delay and multiple coupling delays, in which dynamical nodes are partially linear timedelayed chaotic systems, is described as follows: where the drive system and the response network systems are linked through the variable () ∈  1 .  () ∈   is the state variables of the drive system, and    () ∈   denotes the state variables of the th node in the response network systems.The  and  stand for the drive system and response system, respectively, and () ∈  × is a matrix which depends on the variable ().The constant  is a positive constant and Γ ∈  × a matrix.The constants  > 0 and    > 0 ( = 1, 2, . . ., ) are the nondelayed and the delayed coupling strength to be adjusted, respectively, and the time-varying delays () and   () are bounded by a known constant; that is, 0 ≤ () ≤ , 0 ≤  () ≤ . ∈  × and    ∈  × represent the nondelayed and delayed inner-coupling matrices, respectively. = (  ) × and    = (    ) × are the nondelayed and delayed outer-coupling configuration matrices, respectively, in which   ̸ = 0 ( Remark 1.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 timevarying 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.Definition 2. The projective synchronization is said to take place in drive-response coupled network (1), if there exists constant  ( ̸ = 0) such that lim  → ∞ = ‖   () −   ()‖ = 0 for all , where  is the scaling factor.Lemma 3 (see [41]).Let 0 ≤ (), where the right and upper Dini's derivative  + () is defined as  + () = lim ℎ → 0 + ((( + ℎ) − ())/ℎ), where ℎ → 0 + means that ℎ approaches zero from the right-hand side.Then () ≤ () for − ≤  ≤ 0 implies that () ≤ () for  ≥ 0.

Projective Synchronization Analysis
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    ∈  × , the drive-response network (1) can be rewritten as the following controlled impulsive differential equation: where the impulsive time instants Without loss of generality, we assume that lim  →  −     () =    (  ), which means that the solution of (3) is left continuous at time   .
Remark 4. 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   () =    () −   (), the error dynamical network is characterized by Let () = (  1 (),   2 (), . . .,    ())  , and then (4) can be rewritten in the Kronecker product form as Let ([ 0 − τ,  0 ],   ) be the Banach space of continuous vector-valued functions mapping the interval [ 0 − τ,  0 ] into   with a topology of uniform convergence.‖‖ = sup  0 −τ≤≤ 0 ‖()‖ is used to denote the norm of a function  ∈ ([ 0 − τ,  0 ],   ).For functional differential equation ( 5), its initial condition is given by ] ≤ , based on the theory of impulsive functional differential equation and comparison method, we have the following results.Theorem 5.For given synchronization scaling factor , projective synchronization in the drive-response coupled dynamical network with multiple time-varying delays model will occur if the following inequalities hold: where () = (  ⊗(()−Γ))+(⊗), and then the error system (5) can converge globally exponentially to a decay rate /2, where  > 0 is the solution of That is to say, the coupled dynamical drive-response network with time-varying delays can realize the projective synchronization via impulsive control.
On the other hand, since  > 0,  > 0,  −  − ∑  =1   > 0, and (1/) > 1, one has In the following, we will prove that the following inequality holds: If (20) is not true; that is, it is assumed that there exists a  * > 0 such that From ( 18) and ( 22), one has which contradicts (21), so (20) holds.Letting  → 0, we get Therefore, we have When  → ∞, the error system ( 5) is global exponential asymptotically stable, which implies that the drive-response coupled networks (1) achieve projective synchronization with a scaling factor via impulsive control.This completes the proof of Theorem 5.
From Theorem 5, it is easy to obtain the following corollaries for the drive-response coupled dynamical network without the nodes delay.Corollary 6. Letting () = 0,  ̸ = 0, and    = 0 for  = 1, 2, . . ., , if the following inequality holds: where 0 <  < 1, then the dynamical drive-response network without coupling delays can achieve the projective synchronization.
Remark 9.The value of scaling factor  has no effect on the error dynamics of the system (see ( 4)) because the values of ė  () are independent of the scaling factor .So we can arbitrarily direct the scaling factor  onto any desired value.
Remark 11.Sun et al. [38] studied the projective synchronization in drive-response dynamical networks with timevarying coupling delay, but the time-varying delay in the paper is differential and its derivative is simultaneously required to be not greater than 1, which is a very strict condition.Obviously, we do not need these limit conditions in theorems and corollaries.

Numerical Simulation
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 [32] as the drive system.The Lorenz system with a time delay is described by where  = 10, () = 5 −  − ,  = 16,  = 40,  = 4, Γ =  2 .The system (28) with the above parameters is chaotic, as shown in Figure 1.
For simplicity, the drive-response network systems with two terms of time-varying delayed coupling are described as follows: where According to Theorem 5, we have the result that, for the given scaling factor , if inequalities ( 6) and ( 7) can be satisfied, then the error dynamical system (4) will be stabilized at zero equilibrium asymptotically; that is, the projective synchronization of drive-response network systems will be realized.
In the numerical simulations, we assume  = 0.1,   then, one has Δ < 0.0197.Taking the impulsive interval Δ =  +1 −   = 0.01, then, it is easy to verify that all conditions in Remark 8 are satisfied.The projective synchronization error is defined by ‖()‖ = √ ( 1 −  1 ) 2 + ( 2 −  2 ) 2 ,  = 1, 2, . . ., 5. When the given scaling factor is  = 2, Figure 2 shows the evolution process of the error and the states of the drive-response network without impulsive control.From Figure 2, it is easy to see that the projective synchronization is not achieved.Figure 3 displays the projective synchronization trajectory of the drive-response dynamical networks.Figure 4 shows the evolution process of the error and the states of the drive-response network with impulsive control.When the given scaling factor is  = −0.5, as shown in Figures 5 and 6, the numerical results show that the impulsive controlling scheme for the drive-response coupled dynamical network with time-varying delays is effective.Especially, if  = 0, the delayed system (28) becomes the Lorenz system.But if the other conditions are chosen to be the same as above, one has Δ < 0.0650.We choose impulsive interval Δ =  +1 −   = 0.05, and the projective synchronization can be obtained with the given scaling factor  = −2, and the simulation results are as shown in Figures 7,  8, and 9. Obviously, the numerical simulations confirm the theoretical analysis.
Remark 12.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 [38], the controller used in impulsive method usually is relatively simple and is easy to implement.In the impulsive synchronization, the response networks receive the information from the drive system only in discrete times, which can reduce the information redundancy in the transmitted signal, increase the robustness, and reduce the control cost.Furthermore, from the simulation results, it is clear that the impulsive control scheme is more effective than the adaptive control scheme.

Conclusion
In this paper, the projective synchronization of driveresponse 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 driveresponse 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.

Figure 3 :Figure 4 :
Figure 3: The phase plot of  and  plane with  = 2.

Figure 7 :Figure 8 :
Figure 7: Evolution of (a) state trajectories of drive and response systems and (b) projective synchronization error without impulsive control when () = 0.

Figure 9 :
Figure 9: Evolution of (a) state trajectories of drive (the dash line) and response systems (the solid line) and (b) projective synchronization error under impulsive control.