Finite-time generalized projective lag synchronization ...

Chaos, Solitons and Fractals 107 (2018) 195–203

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Finite-time generalized projective lag synchronization criteria for neutral-type neural networks with delay Mingwen Zheng a,b, Zeming Wang c, Lixiang Li d,∗, Haipeng Peng d, Jinghua Xiao a,e, Yixian Yang d, Yanping Zhang b, Cuicui Feng f a

School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China International School, Beijing University of Posts and Telecommunications, Beijing, 100876, China d Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, 100876, China e State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, 100876, China f School of Computer Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China b c

a r t i c l e

i n f o

Article history: Received 3 September 2017 Revised 15 December 2017 Accepted 4 January 2018

Keywords: Generalized projective lag synchronization Neutral-type neural networks Nonlinear feedback controller Gronwall-Bellman inequality

a b s t r a c t In this paper, we mainly investigate the finite-time generalized projective lag synchronization of neutraltype neural network with delay. With the help of the definition of finite-time generalized projective lag synchronization and Gronwall–Bellman inequality, the generalized projective lag synchronization is achieved between drive-response neural networks via a nonlinear feedback controller. Some novel and easily verifiable sufficient conditions are obtained. Many existing works can be regarded as special cases of our conclusions. Finally, we provide two simulation examples to verify the correctness of main results.

1. Introduction Dynamic behavior of complex dynamical systems has always been a hot research topic [1–4]. Among them, the neutral delay system is a class of nonlinear systems that can describe both the current states and the derivative of the past states. It has important applications in population ecosystem, heat exchange system, distributed network system, circuit network, and so on [5–12]. For these complex systems, the simple neural network model sometimes does not reflect the whole process well, but the neural network with neutral-type term is more suitable for describing these phenomena because the influence of the change of the past state on the current state cannot be ignored. At the same time, the existence of neutral term is often one of the important factors to cause the neural network performance to deteriorate. Therefore, the neutral-type neural network has aroused many concerns and extensive research of many scholars [13–18]. In recent years, there have been many researches on the asymptotic or finite-time stability and synchronization control of neutraltype neural networks. In the study of these two types of stability of

∗

Corresponding author. E-mail address: [email protected] (L. Li).

https://doi.org/10.1016/j.chaos.2018.01.009 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

© 2018 Elsevier Ltd. All rights reserved.

neutral-type neural networks, the existing methods are Lyapunov stability theory [19–23], non smooth analysis [24], Lasalle invariance principle [25], ordinary differential equation theory [26,27], and so on. Lyapunov stability theory is the most basic tool to determine the stability of neural networks, and it eventually comes down to the study of constructing and discussing the appropriate energy function (i.e. Lyapunov function) and its properties about the derivative of the system along its trajectory. But the Lyapunov function is not easy to be found, and can only rely on personal experience to establish. The representations of stability criterion may depend on the different proof methods and the different stability theories. For example, the commonly used methods are Linear Matrix Inequality (LMI) method [28–30], M-matrix method [14], algebraic inequality [31], etc. LMI method is widely used to design robust controllers in the field of engineering because it can effectively solve matrix inequality in numerical calculation. It has the disadvantage of the strong stability constraints and the greater conservatism. The synchronization control of neural networks is derived from the synchronization of drive-response chaotic systems proposed by Pecora and Carroll [32]. The synchronization behavior of neural networks is a very important dynamic characteristic. The synchronization of neural networks has made gratifying progress in se-

196

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

cure communication, image processing and so on [33–37]. So far for various kinds of neural network models, the researchers have put forward a variety of synchronization control scheme, such as feedback control [38], adaptive control [25,39,40], impulse control [41], intermittent control [42], sample-data control [43], sliding mode control [44,45] and pinning control [46], etc, and obtained a variety of different synchronization states, such as complete synchronization [47], phase synchronization [48], lag synchronization [37,46,49], projective synchronization [50–52], anti synchronization [16,53], anticipated synchronization [54] and so on, for different applications. From the point of view of synchronization time, synchronization can be divided into asymptotic synchronization and finite-time synchronization. Amongst all kinds of synchronization, (function) projective synchronization can obtain faster communication with its proportional feature [55]. From the practical viewpoint, due to finite signal transmission time and switching speeds, lag synchronization is a reasonable scheme between the driveresponse system [56]. The combination of projective synchronization and lag synchronization can produce better signal transmission and applicable value. There have been a lot of literature on the study of projective lag synchronization. In Ref. [38], Ghada Al-mahbashi et al. investigated projective lag asymptotically synchronization behavior in drive-response coupling delay dynamical network based on Lyapunov stability theory and hybrid feedback control. Using the same method, Wu et al. also investigated generalized projective lag synchronization between different hyperchaotic systems [57]. In Ref. [58], Huang et al. studied projective lag synchronization of coupled delay neural network with parameter mismatch based on adaptive controller. Through the analysis of the existing research, we can sum up the following three aspects: (1) Most of them are based on Lyapunov stability theory; (2) Most of the objects studied are chaotic or neural network systems. (3) There are very few studies directly based on the definition of finite-time projective lag synchronization of the neutral-type neural network. Therefore, considering the practicability of finite-time synchronization, we will mainly study the finite-time generalized projective lag synchronization of drive-response systems. Motivated by the above discussions, we study the finite-time generalized projective lag synchronization of neutral-type neural network with delay on the basis of the definition with the help of a nonlinear feedback controller in this paper. To the best of our knowledge, there is less research on this kind of work. The main contributions of our work are summarized as follows. (1) We investigate the problem of finite-time generalized projective lag synchronization for neutral-type neural network with delay for the first time. Unlike previous studies, we use the definition of finite-time generalized projective lag synchronization to derive the main conclusions without using Lyapunov stability theory. (2) With the help of Grownall-Bellman inequality and a nonlinear feedback controller, some novel sufficient conditions that are easily validated are obtained to ensure the finitetime generalized projective lag synchronization of neutraltype neural network with delay. Meanwhile, our conclusions can be easily extended to the studies of complete lag synchronization, function projective synchronization and lag anti synchronization. Moreover, these conclusions can be applied equally to neural networks without neutral terms. (3) Two simulation examples verify the effectiveness of our conclusions. The rest of this paper is organized as follows. The driveresponse neural network model, two assumptions, a lemma and the definition of finite-time generalized projective lag synchronization are introduced in Section 2. In Section 3, we derive the main results, including two theorems and two corollaries. Two simula-

tion examples are given to verify the correctness of the main results in Section 4. We conclude our work and propose future work In Section 5. Notations: In this paper, for any matrix A, A is positive definite (negative definite) if A > 0 (A < 0). If the dimensions of some matrices have not been explicitly stated, we assume that they have compatible dimensions for matrix operations. R+ denotes the positive real numbers. For ι > 0, define the family of continuous functions ϕ from [−ι, 0] to Rn as C ([−ι, 0]; Rn ). The norm of function f(x) is defined as  f (x ) = max √ x {| f (x )|}. The norm of matrix A is defined as 2-norm, i.e. A = λ, where λ is the maximum eigenvalue of matrix AT A. AT denotes the transpose of matrix A. 2. Network models and preliminaries Consider the following neutral-type neural networks with delay as the drive system

x˙ i (t ) − di x˙ i (t − τ1 ) = −ci xi (t ) +

n 

ai j f j (x j (t ))

j=1

+

n 

(1)

bi j g j (x j (t − τ2 )) + Ii , t ≥ 0, i = 1, 2, . . . , n.

j=1

where xi (t) is the state variable. f( · ), g( · ) are activation functions. aij and bij are the connection weights without delay and with delay, respectively. τ 1 , τ 2 are two constant delays. ci is the selffeedback weight, and I = (I1 , I2 , · · · , In )T is the external input. The initial values of the system (1) are xi (t ) = ϕi (t ) with ϕ (t ) = sup−τ ≤s≤0 ϕ (s ), ϕi (t ) ∈ C([−τ , 0], Rn ), τ= max{τ1 , τ2 }. The vector form of the system (1) can be expressed as

x˙ (t ) − Dx˙ (t − τ1 ) = −Cx(t ) + AF (x(t )) + BG(x(t − τ2 )) + I,

(2)

where x = (x1 (t ), x2 (t ), . . . , xn 1 , d2 , . . . , dn }, C = diag{c1 , c2 , . . . , cn }, A = (ai j )n×n , B = (bi j )n×n , I = (I1 , I2 , . . . , In )T , F (x(t )) = ( f1 (x1 (t )), f2 (x2 (t )), . . . , fn (xn )(x(t )))T , G(x(t )) = (g1 (x1 (t − τ2 )), g2 (x2 (t − τ2 )), . . . , gn (xn )(x(t − τ2 )))T . The corresponding response system is defined as follows

(t ))T

y˙ i (t ) − di y˙ i (t − τ1 ) = −ci yi (t ) + +

n 



∈ Rn , D = diag{d

n 





aij f y j (t )

j=1

 bij g y j (t − τ2 ) + Ii + ui (t ),

(3)

j=1

t ≥ 0, i = 1, 2, . . . , n, where ui (t) is the controller to be designed. The response system (3) can be written into the following vector form

y˙ (t ) − Dy˙ (t − τ1 ) = −Cy(t ) + AF (y(t )) + BG(y(t − τ2 )) + I + U (t ),

(4)

where U (t ) = (u1 (t ), u2 (t ), · · · , un (t ))T . The initial value of the system (3) is yi (t ) = φi (t ) with φ (t ) = sup−τ ≤s≤0 φ (s ), φi (t ) ∈ C([−τ , 0], Rn ). Define the following the error system between drive system (2) and response system (4)

e(t ) = y(t ) − α (t )x(t − κ ), i = 1, 2, . . . , n,

(5)

where α (t ) = diag(α1 (t ), α2 (t ), · · · , αn (t )) is the variable projective scaling factor and x(t − κ ) is the lag state variable, where 0≤κ ≤τ. Remark 1. When t ∈ [−τ , 0], we define the error system (5) as ϒ = φ (t ) − α (t )ϕ (t − κ ) and its norm is ϒ  = sup−τ ≤t≤0 φ (t ) − α (t )ϕ (t − κ ).

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

197

Remark 2. The error system (5) combines generalized projective synchronization and lag synchronization. Generalized projective synchronization can additionally enhance the level of security communication based on the unpredictability of the scaling function α (t). Lag synchronization is mainly due to the existence of propagation delay. So this synchronization is of practical value.

In this paper, we design the simple linear feedback controller as shown below

Remark 3. 0 ≤ κ ≤ τ means that the acquired data must start after the maximum of the delay τ of the drive system (1).

where K = diag{k1 , k2 , · · · , kn }, ki > 0 is the control gain.

Next, some assumptions, lemmas and definitions will be given as the main results. Assumption 1. In this paper, the scaling function α (t) is a differentiable and the derivative of α (t) is uniform bounded, i.e.

a ≤ α (t ) ≤ a, 

a ≤ α˙ (t ) ≤ a , where a, b, a , a ∈ R+ , and a ≤ a, a ≤ a . Assumption 2. The activation functions F( · ), G( · ) are norm bounded and satisfy the Lipschitz condition, i.e.

F (· ) ≤ F˜, G(· ) ≤ G˜ , |F (x1 (t )) − F (x2 (t ))| ≤ L|x1 − x2 |, |G(x1 (t )) − G(x2 (t ))| ≤ M|x1 − x2 |,

(6)

+ α˙ (t )x(t − κ ) − Ke(t ), i = 1, 2, . . . , n,

Theorem 1. Suppose Assumptions 1 ∼ 3 hold, for given projective synchronization scaling function α (t) and the lag synchronization delay κ , the drive system (1) can achieve generalized projective lag synchronization with the response system (3) via the controller (6), if the following conditions are satisfied

D < 1 ,  (1 + 2D + BMt

(1 + a )(AF˜ + BG˜ ) + |1 − a|z  t 1 − D 1 − D  (AL − C − K  + BM ) × exp t ≤ . 1 − D )δ +

(7) Proof. According to the error system (5) of drive-response systems (2) and (4) and the controller (6), we have

= Dy˙ (t − τ1 ) − Cy(t ) + AF (y(t ))

Assumption 3. The internal input I = (I1 , I2 , · · · , In )T is norm bounded, i.e.

+BG(y(t − τ2 )) + I + U (t ) − α˙ (t )x(t − k )



−α (t ) Dx˙ (t − κ − τ1 ) − Cx(t − κ ) +AF (x(t − κ )) + BG(x(t − κ − τ2 )) + I} = Dy˙ (t − τ1 ) − Dα˙ (t − τ1 )x(t − τ1 − k ) −Dα (t − τ1 )x˙ (t − τ1 − k ) − Ce(t )



+A F (y(t )) − F (α (t )x(t − κ )) + F (α (t )x(t − κ ))



I ≤ z,



−α (t )F (x(t − κ )) + B G(y(t − τ2 )) −G(α (t − τ2 )x(t − τ2 − κ ))

where z ∈ R+ . Lemma 1 [59]. Let x(t), a(t), h(t) be continuous functions and satisfy



t

t0

h(s )x(s )ds, t ∈ [t0 , T ),



t

t0

h ( s )a ( s )



t s



h(r )dr , t ∈ [t0 , T ).

Furthermore, if a(t) is nondecreasing, then we have



x(t ) ≤ a(t )exp

t

t0

+G(α (t − τ2 )x(t − τ2 − κ ))



−α (t )G(x(t − τ2 − κ )) + (E − α (t ))I − Ke(t ) = De˙ (t − τ1 ) − (C + K )e(t ) + (E − α (t ))I



+A F (y(t )) − F (α (t )x(t − κ ))





+A F (α (t )x(t − κ )) − α (t )F (x(t − κ ))

where T ≤ +∞, k(t ) ≥ 0, then

x(t ) ≤ a(t ) +

− D(α (t − τ1 ) − α (t ))x˙ (t − τ1 − κ )

e˙ (t ) = y˙ (t ) − α (t )x˙ (t − κ ) − α˙ (t )x(t − κ )

where F˜ , G˜ ∈ R+ , F (x ) = ( f1 (x ), f2 (x ), · · · , fn (x ))T , G(x ) = (g1 (x ), g2 (x ), · · · , gn (x ))T , L = diag{l1 , l2 , · · · , ln }, M = diag{m1 , m2 , · · · , mn }, li , mi ∈ R+ .

x(t ) ≤ a(t ) +

U (t ) = − Dα˙ (t − τ1 )x(t − τ1 − κ )



h(s )ds , t ∈ [t0 , T ].

Definition 1. The drive system (1) is said to achieve finite-time generalized projective lag synchronization with the response system (3) under the controller U(t) if there are positive number {δ,  , T },  > δ, t ∈ [t0 , t0 + T ], and Y < δ implies e(t) <  .

3. Main results In this section, we will design the controller U(t) and present some sufficient conditions for generalized projective lag synchronization between the drive system (1) and response system (3) under the controller U(t).











+B G(y(t − τ2 )) − G(α (t − τ2 )x(t − τ2 − κ )) +B G(y(t − τ2 )) − G(α (t − τ2 )x(t − τ2 − κ ))

where E is the n × n identity matrix. Let T1 (x(t ), α (t ), κ ) = A[F (α (t )x(t − κ )) − α (t )F (x(t − κ ))], T2 (x(t ), α (t ), κ , τ2 ) = B[G(α (t − τ2 )x(t − τ2 − κ )) − α (t )F (x(t − τ2 − κ ))], for the sake of simplicity, we use T1 ( · ), T2 ( · ) instead of them. From Assumption 2, we have

e˙ (t ) ≤ De˙ (t − τ1 ) + (AL − C − K )e(t ) +BMe(t − τ2 ) + T1 (· ) + T2 (· ) + (E − α (t ))I.

(8)

By taking the integral on interval [0, t], we obtain

e(t ) ≤ e(0 ) + De(t − τ1 ) − De(−τ1 )  t  t +(AL − C − K ) e(s )ds + BM e(s − τ2 )ds 0

 +  +

t 0 t 0

T3 (· )ds + T1 (· )ds +

 

0

t 0 t 0

(E − α (s ))Ids T2 (· )ds.

(9)

198

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

Taking the norm on both sides of Eq. (9), we obtain

then Theorem 1 can be also extended to the generalized projective synchronization [51,52].

e(t ) ≤ (1 + 2D )ϒ  + De(t ) +AL − C − K  +BM  +

t 0



t 0



t 0

(e(s ) + ϒ  )ds +

T2 (· )ds +

When α (t ) = α in the error system (5), the finite-time synchronization controller is designed as follows

e(s )ds



t 0

 0

t

U (t ) = −Ke(t )

T1 (· )ds

(E − α (s ))Ids.

Thus, we have the following corollaries.

(10)

According to Assumptions 1 and 2, we have



t 0

T1 (· )ds 

=

t 0

 A F (α (s )x(s − κ )) − α (s )F (x(s − κ )) ds

≤ (1 + a )AF˜t and



t 0

T2 (· )ds =



(11) t

t 0

B G(α (s − τ2 )x(s − τ2 − κ )) −α (s )F (x(s − τ2 − κ )) ds 0

(12)

(E − α (s ))Ids ≤ |1 − a|zt.

(13)

Substituting Eqs. (11)–(13) into Eq. (10) yields

e(t ) ≤ (1 + 2D + BMt )ϒ  +((1 + a )(AF˜ + BG˜ ) + |1 − a|z )t +(AL − C − K  + BM )





t 0

D < 1 ,  (1 + 2D + BMt 1 − D

)δ

+

From Assumptions 1 to 3, we get



Corollary 1. Let ei (t ) = yi (t ) − αi xi (t − κ ). Suppose Assumptions 2 ∼ 3 hold, for given projective synchronization scaling factor α and lag synchronization delay κ , the drive system (1) can achieve finitetime projective lag synchronization with the response system (3) via the controller (14), if the following conditions hold.

E + α(AF˜ + BG˜ ) + E − αz  t 1 − D  (AL − C − K  + BM ) × exp t ≤ . 1 − D



≤ (1 + a )BG˜t.

(14)

e(s )ds

(1+a )(AF˜+BG˜ )+|1−a|z Let a(t ) = (1+21D−+DBMt ) ϒ  + t, h(t ) = 1−D

AL−C−K +BM . Obviously, a(t) is a nondecreasing function accord1−D

ing to the Lemma 1, we obtain

 (1 + 2D + BMt e(t ) ≤ ϒ  1 − D (1 + a )(AF˜ + BG˜ ) + |1 − a|z  + t 1 − D (AL − C − K  + BM ) × t. 1 − D  (1 + 2D + BMt ≤ )δ 1 − D (1 + a )(AF˜ + BG˜ ) + |1 − a|z  + t 1 − D  (AL − C − K  + BM ) × exp t . 1 − D

According to the condition (7), we have e(t) <  . Based on Definition 1, we know that the drive system (1) and the response system (3) can achieve generalized projective lag synchronization under the controller (6). This proof is completed.  Remark 4. Compared with the existing articles on projective or lag synchronization [37,38,40,46,50,51], we do not define the Lyapunov function, but utilize Gronwall–Bellman inequality and the definition of finite-time synchronization. It makes the proof process simple and easy to be understood. Remark 5. The results of Theorem 1 can be extended to the common projective lag synchronization when the scaling function matrix α (t ) = α is a constant [49]. Similarly, if the lag delay κ (t ) = 0,

(15)

Proof. The proof process is similar to that of Theorem 1 and is omitted here.  When κ = 0 in the error system (5), generalized projective lag synchronization is changed to generalized projective synchronization. In this case, we design the following controller.

U (t ) = − Dα˙ (t − τ1 )x(t − τ1 ) − D(α (t − τ1 ) − α (t ))x˙ (t − τ1 ) + α˙ (t )x(t ) − Ke(t ).

(16)

Corollary 2. When the lag delay κ = 0, the conditions (7) also can make the drive system (1) and response system (3) achieve the finitetime generalized projective synchronization under the controller (16). Proof. The proof process is similar to that of Theorem 1 and is omitted here.  Remark 6. In Corollary 2, if αi (t ) = αi is a constant value , then Theorem 1 can be generalized to investigate the complete synchronization (αi = 1) and anti-synchronization (αi = −1). There have been many references to study them [16,47,53]. If D = 0 in the drive system (2), the drive system becomes an ordinary neural network model with delay. The drive-response systems can be rewritten to the following forms

x˙ i (t ) = −ci xi (t ) +

n 

ai j f j (x j (t ))

j=1

+

n 

bi j g j (x j (t − τ2 )) + Ii , t ≥ 0, i = 1, 2, . . . , n,

(17)

j=1

y˙ i (t ) = −ci yi (t ) +

n 

ai j f (y j (t ))

j=1

+

n 

bi j g(y j (t − τ2 )) + Ii + ui (t ),

j=1

t ≥ 0, i = 1, 2, . . . , n.

(18)

Define the following controller

U (t ) = α˙ (t )x(t ) − Ke(t ). In this case, we have the following conclusion.

(19)

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

199

Table 1 The settling time in Example 1 when selecting different scale function αi (t ), i = 1, 2.

α i (t)

1

−1

2

sin(t)

Ts

1.2883

1.6278

1.1609

1.2543

Theorem 2. Suppose Assumption 1 holds. The drive system (17) and response system (18) can achieve generalized projective lag synchronization under the controller (14) if the following condition holds.



(1 + BM )δ + ((1 + a )(AF˜ + BG˜ + |1 − a|z )t )   ×exp (AL − C − K  + BM )t ≤  .



(20)

Proof. The proof process is similar to that of Theorem 1. We omit it here.  Remark 7. According to the conditions (7), (15) and (20), the settling time Ts can be approximately estimated given the parameters  and δ . It is noteworthy that the t estimated by conditions (7), (15) or (20) is not a specific value, but an upper bound of stable time. We take this upper bound as the settling time Ts .

Fig. 1. Limit cycle of system (21) with initial value x0 = (0.4, 0.6 )T .

4. Simulation examples In this section, we will apply two numerical simulations to verify the validity of our conclusions. The first example is used to validate Theorem 1 and its corollaries. The second one is used to test Theorem 2. Example 1. Consider a two-dimensional drive neutral-type neural network with time-varying delays as shown below

x˙ i (t ) − di x˙ i (t − τ1 ) = −ci xi +

2 

ai j f j (x j (t ))

j=1

+

2 

bi j g j (x j (t − τ2 )) + Ii , i = 1, 2, t ≥ 0.

(21)

j=1

The parameters of the model are as follows

0.2 D= 0

B=

−1.0 −0.2



0 1.0 ,C = 0.2 0



 −0.1 ,I = 0 0.5

f ( x ) = g( x ) =



0 1.0 ,A = 1.0 2.0



Fig. 2. The phase diagrams of the drive-response systems (21) and (22) when α (t ) = 1, κ = 0.5.

−0.5 , 1.5

T

0 , τ1 = 0 . 8 , τ2 = 0 . 5 ,



tanh(x1 ) . tanh(x2 )

The corresponding response system is as follows

y˙ i (t ) − di y˙ i (t − τ1 ) = −ci yi +

2 

ai j f j (y j (t ))

j=1

+

2 

bi j g j (x j (t − τ2 )) + Ii + ui (t ), i = 1, 2, t ≥ 0,

(22)

j=1

where ui (t ) = −ki (t )ei (t ) − di e˙ i (t − τ1 ), k1 = 1, k2 = 1, ei (t ) = yi (t ) − α (t )xi (t − k ), i = 1, 2, α (t) is the variable projective scaling factor, and the lag delay κ = 0.5. First, we verify the effectiveness of the selected parameters 1 0 0.2 0 according to Theorem 1. We select L = [ ], M = [ ]. 0 1 0 0.2 The initial values for (21) and (22) are chosen to be x0 = [0.4, 0.6]T , y0 = [0.6, 0.4]T , respectively. We have Y = 0.2 < δ = 1, D = 0.2 < 1. Taking  = 600 > δ, F˜ = G˜ = 1. Table 1 shows the settling time Ts when selecting different scale functions in the

case of satisfying the conditions (7) and (15) of Theorem 1 and Corollary 1. Next, we solve the drive-response systems (21) and (22) with Matlab. The simulation results are as follows. The phase diagram of the drive system (21) is shown in Fig. 1. We can see from Fig. 1 that it has a limit cycle under the above parameters. Figs. 2 and 3 show the phase diagrams and state trajectories of the drive-response systems (21) and (22) when α (t ) = 1, respectively. Figs. 4 and 5 show the phase diagrams and state trajectories of the drive-response systems (21) and (22) when α (t ) = −1, respectively. Figs. 6 and 7 show the phase diagrams and state trajectories of the drive-response systems (21) and (22) when α (t ) = 2, respectively. Figs. 8 and 9 show the phase diagrams and state trajectories of the drive-response systems (21) and (22) when α (t ) = sin(t ), respectively. The error curves are shown in Fig. 10 when α (t ) = sin(t ) in the case of generalized projective synchronization. Example 2. Consider the following three-dimensional driveresponse systems with delay without the neutral item, i.e. D = 0

200

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

Fig. 3. The synchronization trajectories of the drive-response systems (21) and (22) when α (t ) = 1, κ = 0.5.

Fig. 6. The phase diagrams of the drive-response systems (21) and (22) when α (t ) = 2, κ = 0.5.

Fig. 7. The synchronization trajectories of the drive-response systems (21) and (22) when α (t ) = 2, κ = 0.5. Fig. 4. The phase diagrams of the drive-response systems (21) and (22) when α (t ) = −1, κ = 0.5.

Fig. 5. The synchronization trajectories of the drive-response systems (21) and (22) when α (t ) = −1, κ = 0.5.

Fig. 8. The phase diagrams of the drive-response systems (21) and (22) when α (t ) = sin(t ), κ = 0.5.

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

201

Fig. 11. Limit cycle of system (23) with initial value x0 = (0.4, 0.6, 0.8 )T .

Fig. 9. The synchronization trajectories of the drive-response systems (21) and (22) when α (t ) = sin(t ), κ = 0.5.

Fig. 12. The phase diagrams of the drive-response systems (23) and (24) when α (t ) = sin(2t ), κ = 0.5.

+ Fig. 10. The error curves with initial value x0 = (0.4, 0.6 )T , y0 = (0.6, 0.4 )T .

3 

bi j g j (x j (t − τ2 )) + Ii + ui (t ),

j=1

i = 1, 2, 3, t ≥ 0, in Eq. (2).

x˙ i (t ) = −ci xi +

3 

where ui (t ) = α˙ (t )x(t ) − ki (t )ei (t ), k1 = 1, k2 = 1, k3 = 1, ei (t ) = yi (t ) − α (t )xi (t − k ), i = 1, 2, 3, α (t ) = sin(2t ) is the variable projective scaling factor, and the lag delay κ = 0.5.

ai j f j (x j (t ))

j=1

+

3 

bi j g j (x j (t − τ2 )) + Ii , i = 1, 2, 3, t ≥ 0.

(23)

j=1

The parameters of the neural network (23) are as follows



C=

1.0 0 0

 B=

−1.4 −1.1 −1.2

0 1.0 0



0 0 ,A = 1.0

0.1 −0.1 0.3





2.0 −0.5 −1.6

0.2  −0.17 , I = 0 0.53



τ1 = 0 . 5 , τ2 = 0 . 8 , f ( x ) = g ( x ) =



2.5 1.8 −1.2 0

−0.5 −1.05 , −0.6

T

0 ,



tanh(x1 ) tanh(x2 ) . tanh(x3 )

The corresponding response system is as follows

y˙ i (t ) = −ci yi +

3  j=1

ai j f j (y j (t ))

(24)

Similarly, we verify the effectiveness of the selected param1 0 0 eters according to Theorem 1. We select L = [0 1 0], M = 0 0 1 0.2 0 0 [ 0 0.2 0 ], feedback gain k1 = 1, k2 = 1, k3 = 1, Lipschitz 0 0 0.2 constants F˜ = G˜ = 1. The initial values of (23) and (24) are chosen to be x0 = [0.4, 0.6, 0.8]T , y0 = [0.6, 0.8, 0.4]T . We can obtain Y = 0.4. Taking δ = 1 and selecting  = 600 > δ, through the above parameters , we calculate the settling time Ts = 1.0994 when satisfying the condition (20) in Theorem 2. Next, we solve the drive-response systems (23) and (24) with Matlab. The simulation results are as follows. The phase diagram of the drive system (23) is shown in Fig. 11, from which we can see that it has a limit cycle under the above parameters. Figs. 12 and 13 show the phase diagrams and state trajectories of the drive-response systems when α (t ) = sin(2t ), respectively.

202

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203

type neural network with delay because of its important applicable value. Acknowledgments This paper is supported by the National Key Research and Development Program (Grant Nos. 2016YFB0800602), the National Natural Science Foundation of China (Grant Nos. 61472045, 61573067, 61771071). References

Fig. 13. The synchronization trajectories of the drive-response systems (23) and (24) when α (t ) = sin(2t ), κ = 0.5.

Fig. 14. The ( 0.6, 0.8, 0.4 )T .

error

curves

with

initial

values

x 0 = ( 0.4, 0.6, 0.8 ) T , y0 =

The error curves are shown in Fig. 14 when α (t ) = sin(2t ) in the case of generalized projective synchronization. 5. Conclusions The finite-time generalized projective lag synchronization problem of neutral-type neural networks with delay is studied based on the definition of generalized projective lag synchronization. By using Gronwall–Bellman inequality and other inequality techniques, some novel sufficient conditions are derived under a nonlinear feedback controller. Our conclusions can be extended to some other synchronization forms, e.g. complete lag synchronization, lag anti synchronization, projective lag synchronization, function projective lag synchronization, etc. And our conclusions are also applicable to the neural network without neutral terms. Finally, two simulation examples show the effectiveness of our main results. The future work include the following: (1) How to further simplify the synchronization conditions and design a simpler controller? (2) We will explore the quantitative effects of neutral delay, discrete delay or scale function α (t) on synchronization; (3) Inspired by Zhao et al. [60,61], the finite-time generalized projective lag synchronization of multiplex networks seems to be a good research content. In a word, there are still some worthy problems for the finite-time generalized projective lag synchronization of neutral-

[1] Zhao D, Wang L, Xu L, Wang Z. Finding another yourself in multiplex networks. Appl Math Comput 2015;266:599–604. [2] Zhao D, Wang L, Li S, Wang Z, Wang L, Gao B. Immunization of epidemics in multiplex networks. PLoS ONE 2014;9(11):e112018. [3] Wang Z, Jusup M, Wang RW, Shi L, Iwasa Y, Moreno Y, et al. Onymity promotes cooperation in social dilemma experiments. Sci Adv 2017;3(3):e1601444. [4] Li X, Wang Z, Gao C, Shi L. Reasoning human emotional responses from large-scale social and public media. Appl Math Comput 2017;310(C):182–93. [5] Brayton R. Small-signal stability criterion for electrical networks containing lossless transmission lines. IBM J Res Dev 1968;12(6):431–40. [6] Kuang Y. Delay differential equations: with applications in population dynamics, 191. Academic Press; 1993. [7] Bellen A, Guglielmi N, Ruehli AE. Methods for linear systems of circuit delay differential equations of neutral type. IEEE Trans Circuits Syst I 1999;46(1):212–15. [8] Niculescu S-I, Brogliato B. Force measurement time-delays and contact instability phenomenon. Eur J Control 1999;5(2–4):279–89. [9] Niculescu S-I. Delay effects on stability: a robust control approach, 269. Springer Science & Business Media; 2001. [10] Yue D, Han Q-L. A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model. IEEE Trans Circuits Syst II Express Briefs 2004;51(12):685–9. [11] Fridman E, Shaked U. An improved delay-dependent H/sub/spl infin//filtering of linear neutral systems. IEEE Trans Signal Process 2004;52(3):668–73. [12] Zhao D, Wang Z, Xiao G, Gao B, Wang L. The robustness of interdependent networks under the interplay between cascading failures and virus propagation. EPL 2016;115(5). [13] Park JH, Kwon O. Further results on state estimation for neural networks of neutral-type with time-varying delay. Appl Math Comput 2009;208(1):69–75. [14] Zhou W, Zhu Q, Shi P, Su H, Fang J, Zhou L. Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters. IEEE Trans Cybern 2014;44(12):2848–60. [15] Lin X, Zhang X, Wang Y. Robust passive filtering for neutral-type neural networks with time-varying discrete and unbounded distributed delays. J Frankl Inst 2013;350(5):966–89. [16] Wang W, Li L, Peng H, Kurths J, Xiao J, Yang Y. Anti-synchronization control of memristive neural networks with multiple proportional delays. Neural Process Lett 2016;43(1):269–83. [17] Zheng M, Li L, Peng H, Xiao J, Yang Y, Zhao H. Parameters estimation and synchronization of uncertain coupling recurrent dynamical neural networks with time-varying delays based on adaptive control. Neural Comput Appl 2016:1–11. [18] Zheng M, Li L, Peng H, Xiao J, Yang Y, Zhao H. Finite-time stability analysis for neutral-type neural networks with hybrid time-varying delays without using Lyapunov method. Neurocomputing 2017;238:67–75. [19] Ali MS, Saravanakumar R, Arik S. Novel H state estimation of static neural networks with interval time-varying delays via augmented Lyapunov–Krasovskii functional. Neurocomputing 2016;171:949–54. [20] Senan S, Ali MS, Vadivel R, Arik S. Decentralized event-triggered synchronization of uncertain Markovian jumping neutral-type neural networks with mixed delays. Neural Netw 2017;86:32–41. [21] He Y, Liu G-P, Rees D, Wu M. Stability analysis for neural networks with time– varying interval delay. IEEE Trans Neural Networks 2007;18(6):1850–4. [22] Rakkiyappan R, Zhu Q, Chandrasekar A. Stability of stochastic neural networks of neutral type with markovian jumping parameters: a delay-fractioning approach. J Frankl Inst 2014;351(3):1553–70. [23] Liu Y, Wang Z, Liang J, Liu X. Synchronization of coupled neutral-type neural networks with jumping-mode-dependent discrete and unbounded distributed delays. IEEE Trans Cybern 2013;43(1):102–14. [24] Tu Z, Cao J, Alsaedi A, Alsaadi F. Global dissipativity of memristor-based neutral type inertial neural networks. Neural Netw 2017;88:125–33. [25] Zheng C-D, Wei Z, Wang Z. Robustly adaptive synchronization for stochastic Markovian neural networks of neutral type with mixed mode-dependent delays. Neurocomputing 2016;171:1254–64. [26] Kolmanovskii V, Myshkis A. Introduction to the theory and applications of functional differential equations, 463. Springer Science & Business Media; 2013. [27] Zjavka L, Pedrycz W. Constructing general partial differential equations using polynomial and neural networks. Neural Netw 2016;73:58–69. [28] Park JH, Kwon O, Lee S-M. LMI optimization approach on stability for delayed neural networks of neutral-type. Appl Math Comput 2008;196(1):236–44.

M. Zheng et al. / Chaos, Solitons and Fractals 107 (2018) 195–203 [29] Sakthivel R, Anbuvithya R, Mathiyalagan K, Arunkumar A, Prakash P. New LMI-based passivity criteria for neutral-type BAMneural networks with randomly occurring uncertainties. Rep Math Phys 2013;72(3):263–86. [30] Lakshmanan S, Lim C, Prakash M, Nahavandi S, Balasubramaniam P. Neutral– type of delayed inertial neural networks and their stability analysis using the LMI approach. Neurocomputing 2017;230:243–50. [31] Jian J, Wang B. Stability analysis in lagrange sense for a class of bam neural networks of neutral type with multiple time-varying delays. Neurocomputing 2015;149:930–9. [32] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821. [33] Broussard RP, Rogers SK, Oxley ME, Tarr GL. Physiologically motivated image fusion for object detection using a pulse coupled neural network. IEEE Trans Neural Networks 1999;10(3):554–63. [34] Prakash M, Balasubramaniam P, Lakshmanan S. Synchronization of Markovian jumping inertial neural networks and its applications in image encryption. Neural Netw 2016;83:86–93. [35] Kanter I, Kinzel W, Kanter E. Secure exchange of information by synchronization of neural networks. EPL (Europhys Lett) 2002;57(1):141. [36] Sendiña-Nadal I, Leyva I, Navas A, Villacorta-Atienza JA, Almendral JA, Wang Z, et al. Effects of degree correlations on the explosive synchronization of scale-free networks. Phys Rev E 2015;91(3):032811. [37] Wen S, Zeng Z, Huang T, Meng Q, Yao W. Lag synchronization of switched neural networks via neural activation function and applications in image encryption. IEEE Trans Neural Netw Learn Syst 2015;26(7):1493–502. [38] Al-Mahbashi G, Noorani MM, Bakar SA. Projective lag synchronization in drive-response dynamical networks with delay coupling via hybrid feedback control. Nonlinear Dyn 2015;82(3):1569–79. [39] Jiang S, Cai G, Cai S, Tian L, Lu X. Adaptive cluster general projective synchronization of complex dynamic networks in finite time. Commun Nonlinear Sci Numer Simul 2015;28(1):194–200. [40] Al-mahbashi G, Noorani MM, Bakar SA, Al-sawalha MM. Robust projective lag synchronization in drive-response dynamical networks via adaptive control. Eur Phys J Spec Top 2016;225(1):51–64. [41] Li X, Song S. Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method. Commun Nonlinear Sci Numer Simul 2014;19(10):3892–900. [42] Zhang G, Lin X, Zhang X. Exponential stabilization of neutral-type neural networks with mixed interval time-varying delays by intermittent control: a CCL approach. Circuits Syst Signal Process 2014;33(2):371–91. [43] Gan Q, Liang Y. Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. J Frankl Inst 2012;349(6):1955–71.

203

[44] Wai R-J, Muthusamy R. Fuzzy-neural-network inherited sliding-mode control for robot manipulator including actuator dynamics. IEEE Trans Neural Netw Learn Syst 2013;24(2):274–87. [45] Vaidyanathan S, Azar AT. Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan chaotic systems. In: Advances and applications in sliding mode control systems. Springer; 2015. p. 549–69. [46] Sun W, Wang S, Wang G, Wu Y. Lag synchronization via pinning control between two coupled networks. Nonlinear Dyn 2015;79(4):2659–66. [47] Li Y, Li C. Complete synchronization of delayed chaotic neural networks by intermittent control with two switches in a control period. Neurocomputing 2016;173:1341–7. [48] Varela F, Lachaux J-P, Rodriguez E, Martinerie J. The brainweb: phase synchronization and large-scale integration. Nat Rev Neurosci 2001;2(4):229–39. [49] Li N, Cao J. Lag synchronization of memristor-based coupled neural networks via-measure. IEEE Trans Neural Netw Learn Syst 2016;27(3):686–97. [50] Chen S, Cao J. Projective synchronization of neural networks with mixed time– varying delays and parameter mismatch. Nonlinear Dyn 2012;67(2):1397–406. [51] Ghosh D, Banerjee S. Projective synchronization of time-varying delayed neural network with adaptive scaling factors. Chaos, Solitons Fract 2013;53:1–9. [52] Xin B, Wu Z. Projective synchronization of chaotic discrete dynamical systems via linear state error feedback control. Entropy 2015;17(5):2677–87. [53] Cao Y, Wen S, Chen MZ, Huang T, Zeng Z. New results on anti-synchronization of switched neural networks with time-varying delays and lag signals. Neural Netw 2016;81:52–8. [54] Ciszak M, Gutiérrez JM, Cofiño A, Mirasso C, Toral R, Pesquera L, et al. Approach to predictability via anticipated synchronization. Phys Rev E 2005;72(4):046218. [55] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 1999;82(15):3042. [56] Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 1997;78(22):4193. [57] Wu X-J, Lu H-T. Generalized projective lag synchronization between different hyperchaotic systems with uncertain parameters. Nonlinear Dyn 2011;66(1):185–200. [58] Huang J, Li C, Zhang W, Wei P. Weak projective lag synchronization of neural networks with parameter mismatch. Neural Comput Appl 2014;24(1):155–60. [59] Corduneanu C. Principles of differential and integral equations. Chelsea Pub Co; 1991. [60] Zhao D, Li L, Peng H, Luo Q, Yang Y. Multiple routes transmitted epidemics on multiplex networks. Phys Lett A 2014;378(10):770–6. [61] Zhao D, Wang L, Zhi Y, Zhang J, Wang Z. The robustness of multiplex networks under layer node-based attack. Sci Rep 2016;6:24304.