Weak Projective Synchronization in Drive-Response Dynamical Networks with Time-Varying Delay and Parameter Mismatch

This paper investigates the problem of projective synchronization in drive-response dynamical networks (DRDNs) with timevarying delay and parameter mismatch via impulsive control. Owing to projective factor and parameter mismatch, complete projective synchronization cannot be achieved.Therefore, a weak projective synchronization scheme is proposed to ensure that the DRDNs are in a state of synchronization with an error level. Based on the stability analysis of the impulsive functional differential equations, a generalmethod of theweak projective synchronizationwith the error level is derived inDRDNs.Numerical simulations are provided to verify the correctness and effectiveness of the proposed method and results.


Introduction
A complex dynamical network is a set of coupled nodes interconnected by edges, in which each node represents a dynamical system.The structure of many real systems in nature can be described by the complex dynamical networks, such as social relationship networks, metabolic networks, food chain, Internet, the World Wide Web, power grids, and so on [1,2].This has led to much interest in the studies of the complex dynamical networks.In particular, the synchronization of complex networks has received much attention, and many interesting results on synchronization were derived for various complex networks such as time invariant, timevarying, discrete, and impulsive network models [3][4][5][6][7][8][9][10][11].
More recently, projective synchronization on dynamical networks has been reported by Hu et al. in [12], in which the projective synchronization with the desired scaling factor can be realized in drive-response dynamical networks.Projective synchronization has become a hot topic and attracted much attention from authors in many fields, including chaotic systems [13][14][15][16] and complex dynamical networks [17][18][19][20].In these papers, the authors just consider the projective synchronization in DRDNs with coupled partially linear chaotic systems.However, there are always some mismatches between drive system and response network systems in the real world.Indeed, almost all complex dynamical networks have different nodes, such as the nodes in network community and the Internet, are in general different.In this case, the DRDNs cannot synchronize completely.Nevertheless, when parameter mismatch is small enough, the synchronization error can converge to a small region containing the origin.In [21], the authors investigated the effect of parameter mismatch on lag synchronization of chaotic systems.In [22], the synchronization of a class of coupled chaotic delayed systems with parameters mismatch and stochastic perturbation was studied.In [23], the weak synchronization criterion of coupled delayed chaotic systems with parameters mismatches was obtained.In [24], the authors studied the synchronization of two coupled identical chaotic systems with parameter mismatch via using periodically intermittent control.In [25], the weak projective synchronization of neural networks with mixed time-varying delays and parameter mismatch was discussed.Unfortunately, there exist few results of a weak projective synchronization method for DRDNs with time-varying delay and parameter mismatch.Therefore, it is worth proposing a weak projective synchronization method in which the problems mentioned above are considered.

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Journal of Applied Mathematics Motivated by the above discussions, in this paper, we introduce a drive-response dynamical network with timevarying delay and parameter mismatch.It is known that complete synchronization is destroyed by parameter mismatch and projective factor.We propose the weak projective synchronization properties of this model via impulsive control.Based on the obtained results, one can control the projective synchronization error in a predetermined level.Results of numerical example show the effectiveness of the proposed approach.The rest of this paper is organized as follows.In Section 2, the DRDNs model with parameter mismatch and some preliminaries are given.In Section 3, some criteria for the weak projective synchronization are derived.Numerical simulations are shown in Section 4. The conclusion is finally given in Section 5.
The notation throughout the paper is quite standard. and   denote the real number set and -dimensional Euclidean space, respectively.‖ ⋅ ‖ stands for either the Euclidean vector norm or its induced matrix 2-norm. max ()( min ()) represents the maximum (minimum) eigenvalue of the symmetric matrix .() =  max (  + ).sup denotes the upper bound.  is the identity matrix with order .Matrices, if not explicitly stated, are assumed to have compatible dimensions.⊗ is the Kronecker product of two matrices.([−, 0],   ) denotes the set of all functions of bounded variation and right-continuous on any compact subinterval of [−, 0].

Preliminaries.
In order to demonstrate this paper clearly, we give some necessary definitions, assumptions, and lemmas, which are useful in deriving projective synchronization criteria.
Definition 1.The drive system (1) and response dynamical networks (2) are said to be weak projective synchronized with an error level  > 0, if there exists a  ≥ 0 such that ‖   () −   ()‖ ≤  for all  ≥ , where  is a desired scaling factor.
Assumption 2. For any  1 ,  2 ∈   , there exist constants   > 0,   > 0,  = 1, 2, . . ., , such that ‖( Assumption 3. () and  1 () are the time-varying delay satisfying 0 ≤ (),  1 () ≤ , where  is a positive constant.Clearly, this assumption is certainly ensured if the timevarying delay is a constant.Remark 4. It should be pointed out that in Assumption 3 we do not require that the time-varying delay is differential function with a bound of its derivative, which means that the time-varying delay satisfying Assumption 3 includes a wide range of functions.
Assumption 5.It is assumed that the trajectory of the drive system (1) is bounded with Remark 6. Assumption 5 is reasonable due to its chaotic feature.

Main Results
In this section, by combining the stability analysis of impulsive functional differential equations, some sufficient conditions for weak projective synchronization in drive-response dynamical networks with time-varying delay and parameter mismatch under impulsive control are given below.Theorem 9.Under Assumptions 2, 3, and 5, let a nonsingular matrix  ∈  × , 0 <  = sup{  −  −1 } < ∞, and sup ≥0 ‖(,   (), )‖ ≤  < ∞.If the following inequalities hold where Proof.Consider the following Lyapunov functional: where  is a symmetric matrix and  =   .

Corollary
where

Numerical Simulation
In this section, an example is presented to show the effectiveness of the proposed scheme.To show the advantage of the criteria based on matrix measure, a scalar Ikeda oscillator is investigated in the context of weak projective synchronization in the following example.The dynamics of Ikeda oscillator is described by  The corresponding response network systems with parameter mismatch are given by Then, the controlled DRDNs are described as follows: choosing the coupling configuration matrix In the numerical simulations, we assume that  = 1,  =  = 0.1,  0 = −0.9,(1 +  0 ) 2 = 0.01 > 0. Γ =  = 1,  =   .The two coupling delays are () = 2 and  1 () = 2 + 0.02 sin , respectively.After calculations, getting  = 38.092, = 0.3999,  = 1.814, one has Δ < 0.0177.Taking the impulsive interval Δ =  +1 −   = 0.01, then, it is easy to verify that all conditions in Corollary 10 are satisfied.The projective synchronization error is defined by ‖()‖ = √ ( 1 −  1 ) 2 + ( 2 −  2 ) 2 ,  = 1, 2, 3, 4. When  = −0.5, as shown in Figures 2-4. Figure 2 shows attractors of the DRDNs network model.Figure 3 displays time evolutions of state trajectories of the controlled DRDNs (29).The evolution process of the error does not converge to zero as shown in Figure 4; from Figure 4, it is easy to see that the projective synchronization is not achieved.The numerical results show that the impulsive controlling scheme for the drive-response coupled dynamical network model with timevarying delays is effective.

Conclusion
In this paper, the problem of weak projective synchronization in DRDNs with time-varying coupling delay and parameter mismatch has been investigated by employing impulsive control scheme.Some criteria for realizing the weak projective synchronization are established based on the stability analysis of impulsive functional differential equations.Moreover, the DRDNs can be synchronized exponentially within a small error; the error upper bound of weak projective synchronization is estimated easily by the theoretical criteria.Finally, the numerical examples show the effectiveness of the proposed results.However, the results of theoretical analysis in this paper are still conservative.Meanwhile, since the surrounding environment is complex variable, it is desirable to investigate weak projective synchronization problem for complex dynamical networks with noise, stochastic disturbances, and so on, so we will further investigate these problems in the future.