Impulsive synchronization of time-scales complex networks with time-varying topology

Commun Nonlinear Sci Numer Simulat 80 (2020) 104981

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Research paper

Impulsive synchronization of time-scales complex networks with time-varying topology Yong Pei a,b, Martin Bohner c,∗, Dechang Pi a a

College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China School of Information, Jiangsu Vocational Institute of Commerce, Nanjing 211168, China c Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA b

a r t i c l e

i n f o

Article history: Available online 23 August 2019 MSC: 34N05 39A13 93D20 Keywords: Impulsive synchronization Time scales Complex networks Dynamical networks Time-varying

a b s t r a c t The dynamics of complex networks (CNs) often does not appear as absolutely continuous or discrete, but more likely to be mixed. This paper discusses the asymptotic synchronization problem of a class of complex networks with time-varying topology (TTCNs). Based on time scales theory, it obtains a general scale-type synchronization condition for TTCNs on different time scales. An impulsive control method is taken into account when the condition is not met, and a scale-type criterion is established in such situation. Simulations verify the effectiveness of our conclusions on different time scales, including continuous, discrete with arbitrary step, and mixed cases. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The synchronization of CNs is a key collective behavior and is widely used in different fields such as physics, neuroscience, sociology and information technology [1–3]. Due to the rich dynamics in CNs, synchronous control has attracted great interest of researchers from different fields [4–16]. Owing to the effectiveness and easy operability, researchers have devoted more time investigating the method of impulsive synchronization in practical applications. However, most of the literature concerning impulsive synchronization control considers continuous or discrete time cases, respectively [10–15]. In fact, the dynamics of many real networks is not merely continuous or discrete, but more intermittent. In addition, most of the literature focuses on networks with time-invariant coupling, while many real networks have time-varying topology. Compared to Zhang et al. [10], three improvements are achieved in this paper. Firstly, for the continuous case, [10, Section 2] restricts the outer coupling matrices to be constant, while we allow them to be time dependent. Secondly, for the discrete case, [10, Section 3] restricts the stepsize to 1, while we allow an arbitrary stepsize h > 0, and additionally, we allow in this case time-dependent outer coupling matrices as well. Thirdly, we do not restrict our results to merely the continuous and discrete cases, but present generalized results for arbitrary time scales, hence unifying the results from [10, Section 2] and [10, Section 3] and extending them to other cases “in between”, e.g., allowing for hybrid situations.

∗

Corresponding author. E-mail address: [email protected] (M. Bohner).

https://doi.org/10.1016/j.cnsns.2019.104981 1007-5704/© 2019 Elsevier B.V. All rights reserved.

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Y. Pei, M. Bohner and D. Pi / Commun Nonlinear Sci Numer Simulat 80 (2020) 104981

We organize the remaining parts as follows. In Section 2, a TTCN model is presented. In Section 3, we discuss a general condition to determine whether the network will synchronize. A scale-type synchronization sufficient condition is derived. In Section 4, the impulsive controller is used when the sufficient condition is not satisfied and a scale-type criterion of impulsive synchronization is created. In Section 5, simulation examples are shown using the chaotic Chen oscillator and the chaotic Hénon map on different time scales, respectively, to verify the effectiveness of our theoretical analysis. Finally, we summarize the whole paper in Section 6. All the time scales related concepts mentioned in this paper can be found in [17], e.g., the notions of right-dense, right-scattered, and regressive. 2. Description of the problem Consider a TTCN of N nodes, each of which is an m-dimensional dynamical system on the time scale T. The network is described as

x i (t ) = f (xi (t )) +

N 

ci j (t )g(x j (t )),

i = 1, 2, · · · , N,

(1)

j=1

where t ∈ T, xi : T → Rm , xi = (xi1 , · · · , xim )T is the state vector of node i, and the derivative x can be written as i

x i (t ) =

⎧ ⎨lims→t,s∈T xi (t ) − xi (s )

if t is right-dense,

t −s ⎩ xi (σ (t )) − xi (t ) μ(t )

if t is right-scattered.

The graininess function μ : T → [0, ∞ ) is defined by μ(t ) = σ (t ) − t, where σ : T → T defined by σ (t ) = inf{s ∈ T : s > t } is the forward jump operator. In (1), the dynamical behavior of each node is described by

f ( xi ) = ( f 1 ( xi ), f 2 ( xi ), · · · , f m ( xi ) ) , T

where f : Rm → Rm is a differentiable nonlinear function. Moreover, g : Rm → Rm is a nonlinear inner coupling differentiable function in each node. The outer coupling matrix C (t ) = (ci j (t )) ∈ RN×N is a real symmetric matrix and is defined as cii (t ) =  − Nj=1, j=i ci j (t ), ci j (t ) = c ji (t ) > 0 (i = j) if there is a link between node i and node j, and otherwise, ci j (t ) = c ji (t ) = 0 (i = j). We denote the eigenvalues of C(t) by

λN (t ) ≤ · · · ≤ λ2 (t ) < λ1 (t ) = 0. From the theory of matrices, we conclude the existence of an eigenvector function matrix

(t ) = (φ1 (t ), φ2 (t ), · · · , φN (t )) ∈ RN×N satisfying

C (t ) = (t )(t )T (t ), where T (t )(t ) = I and (t ) = diag(λ1 (t ), · · · , λN (t )). Assumption 1. There exists a diagonal regressive matrix

 := diag(ψ1 , ψ2 , · · · , ψN ) ∈ RN×N ,

(2)

independent of t ∈ T, and satisfying

 = −1 (t ) (t ), such that we have

 (t )

(3)

= (t ) .

Remark 1. Note that, if C(t) does not depend on t ∈ T, then  = 0, simplifying the calculations below significantly (see [10, Section 2] for T = R and [10, Section 3] for T = Z). 3. Scale-type sufficient condition of asymptotic synchronization Definition 1. A dynamical network achieves asymptotic synchronization if

xi (t ) → x¯ as t → ∞,

i = 1, · · · , N, where f (x¯ ) = 0.

Lemma 1. Consider the linear dynamical systems −1 y k (t ) = (1 + μ (t )ψk ) [D f (x¯ ) + λk (t )Dg(x¯ ) − ψk I ]yk (t ),

k = 1, · · · , N,

(4)

where D f, Dg ∈ Rm×m are respectively, the Jacobian of f and g, λk (t), ψ k are the diagonal items of (t) and  , respectively. If they achieve asymptotic stability at their zero solution, then the network (1) achieves asymptotic synchronization.

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Proof. According to [10, Lemma 1], let E T (t ) = (e1 (t ), · · · , eN (t )) ∈ Rm×N with ei (t ) = xi (t ) − x¯, t ∈ T, i = 1, 2, · · · , N. We have

E  (t ) = E (t )(D f (x¯ ))T + C (t )E (t )(Dg(x¯ ))T . Here, we define a matrix Y (t ) = (y1 (t ), · · · , yN

(t ))T

(5) ∈

Rm×N

by

Y (t ) := −1 (t )E (t ), where (t) is as mentioned above. Hence,

E (t ) = (t )Y (t ). By Assumption 1, the left-hand side of (5) is

E  (t ) = (Y ) (t ) =  (t )Y (t ) + (σ (t ))Y  (t )

= (t ) Y (t ) + [μ(t )(t ) + (t )]Y  (t ) = (t ) Y (t ) + (t )[I + μ(t ) ]Y  (t ), while the right-hand side of (5) is

E (t )(D f (x¯ ))T + C (t )E (t )(Dg(x¯ ))T = (t )Y (t )(D f (x¯ ))T + (t )(t )T (t )(t )Y (t )(Dg(x¯ ))T = (t )Y (t )(D f (x¯ ))T + (t )(t )Y (t )(Dg(x¯ ))T . Therefore, we have





Y  (t ) = [I + μ(t ) ]−1 Y (t )(D f (x¯ ))T + (t )Y (t )(Dg(x¯ ))T −  Y (t ) . So

T  Y

T (t ) = Y  (t )   −1 = D f (x¯ )Y T (t ) + Dg(x¯ )Y T (t )(t ) − Y T (t ) T I + μ(t ) T ,

namely −1 y k (t ) = (1 + μ (t )ψk ) [D f (x¯ ) + λk (t )Dg(x¯ ) − ψk I]yk (t ).

Thus, one can determine the asymptotic synchronization of (1) by using the linear systems (4).

(6) 

In the following theorem, we use the time scales exponential function e introduced in [17, Definition 5.18]: For a matrixvalued function P, we say that P ∈ R provided

I + μ(t )P (t ) is invertible for all t ∈ T. Then the unique solution of the initial value problem

Y  = P (t )Y,

Y (s ) = I

is called the time scales exponential function and denoted by eP (t, s). Theorem 1. Let

Pk (t ) = (1 + μ(t )ψk )−1 [D f (x¯ ) + λk (t )Dg(x¯ ) − ψk I],

k = 1, · · · , N.

If

Pk ∈ R

and

ePk (t, t0 ) → O

as t → ∞,

where t, t0 ∈ T, t > t0 , t0 is the initial time, O is the matrix with all 0 elements, then, at the zero solution, the systems (4) achieve asymptotic stability. Proof. Consider the dynamical system of node k. From (4), we have

y k (t ) = Pk (t )yk (t ).

(7)

We assume the initial values of (7) are yk (t0 ) = y0k ∈ Rm . For Pk ∈ R, according to [17, Theorem 2.35], they should be yk (t ) = ePk (t, t0 )y0k . So, if ePk (t, t0 ) → O as t → ∞, we have yk (t ) → 0 as t → ∞. Hence, (7), i.e., (4) is asymptotically stable about the zero solution.  Definition 2. We define the set R+ of positively regressive matrices by





R+ = R+ T, Rm×m = {A ∈ R : 1 + μ(t )λi (t ) > 0 for all t ∈ T},

4

Y. Pei, M. Bohner and D. Pi / Commun Nonlinear Sci Numer Simulat 80 (2020) 104981

where λi (t) are the eigenvalues of A(t), 1 ≤ i ≤ m. Here, we define

 θk (t ) = λmax PkT

 Pk



(t ) ,

where λmax is the maximum eigenvalue and





PkT  Pk (t ) = PkT (t ) + Pk (t ) + μ(t )PkT (t )Pk (t ).

Lemma 2. If Pk ∈ R+ , then PkT ∈ R+ . Proof. For Pk (t ) ∈ Rm×m , according to Definition 2, we have

1 + μ(t )λi (t ) > 0,

1 ≤ i ≤ m,

where λi (t) are the eigenvalues of Pk (t) and PkT (t ). It can be concluded that 1 + μ(t )λi (t ), 1 ≤ i ≤ m, are the eigenvalues of I + μ(t )PkT (t ), which is invertible. By [17, Definition 5.5], PkT ∈ R. So, from Definition 2, we have PkT ∈ R+ .  Lemma 3. If Pk ∈ R+ , then PkT  Pk ∈ R+ . Proof. From Lemma 2, if PkT ∈ R+ , then I + μ(t )PkT (t ) is invertible for all t ∈ Tκ . Since







I + μ(t ) PkT  Pk (t ) = I + μ(t ) PkT (t ) + Pk (t ) + μ(t )PkT (t )Pk (t )



= I + μ(t )PkT (t ) + μ(t )Pk (t ) + μ2 (t )PkT (t )Pk (t )





= I + μ(t )PkT (t ) [I + μ(t )Pk (t )]

(8)

is the product of two invertible matrices, it is also invertible for each t ∈ Tκ . Hence, PkT  Pk ∈ R. By (8),

1 + μ(t )λ (t ) ≥ 0, i where

λ (t ) i

0 ≤ i ≤ m,





are the eigenvalues of PkT  Pk (t ). Notice that

I + μ(t )(PkT







 Pk )(t ) = I + μ(t )PkT (t ) [I + μ(t )Pk (t )]



= I + μ(t )PkT (t ) |I + μ(t )Pk (t )| > 0. Hence,

1 + μ(t )λ (t ) > 0, i Therefore, PkT  Pk ∈ R+ . Lemma 4. If A ∈

R+ ,

0 ≤ i ≤ m.



then λAi ∈ R+ , where λAi (t ) are the eigenvalues of A(t).

Proof. From Definition 2, we have

1 + μ(t )λAi (t ) > 0,

1 ≤ i ≤ m.

We use [17, Definition 2.45] to conclude λAi ∈ R+ .  t Corollary 1. If Pk ∈ R+ and t θk (u )u → −∞ as t → ∞, then 0

ePk (t, t0 ) → O as t → ∞, where t, t0 ∈ T, t0 < t is the initial time, O is the matrix with all 0 elements. Proof. We choose a function Vk (t ) = yTk (t )yk (t ) and use [17, Theorem 1.16(iii)] along (7) to solve



Vk (t ) = (yTk ) (t )yk (t ) + yTk (t ) = yTk (t )PkT (t )yk (t ) +



σ

y k (t )

 μ(t )(yTk ) (t ) + yTk (t ) Pk (t )yk (t )

μ(t )yTk (t )PkT (t )Pk (t )yk (t ) + yTk (t )Pk (t )yk (t ) + yTk (t )PkT (t )yk (t )   = yTk (t ) μ(t )PkT (t )Pk (t ) + PkT (t ) + Pk (t ) yk (t )

= yTk (t ) PkT  Pk (t )yk (t ), =

and hence

Vk (t ) ≤ θk (t )yTk (t )yk (t ) = θk (t )Vk (t ). Since Pk ∈

R+ ,

according to Lemmas 2–4, we have θk ∈

0 < Vk (t ) ≤ eθk (t, t0 )Vk (t0 ),

(9) R+ .

By [17, Theorem 6.1], we have

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where t > t0 and Vk (t0 ) = 0 is the initial value of Vk (t). We also use [18, Lemma 2] and have

0 < eθk (t, t0 ) ≤ exp



t

t0



θk (u )u .

Therefore,

eθk (t, t0 ) → 0 as t → ∞ if Then

Vk (t ) → 0 as t → ∞ if



t

t0



t

t0

θk (u )u → −∞ as t → ∞.

θk (u )u → −∞ as t → ∞.

That means each item of the vector yk (t) tends to 0. By [17, Theorem 5.24], we have yk (t ) = ePk (t, t0 )yk (t0 ), where yk (t0 ) is not the zero vector. So, we have ePk (t, t0 ) → O as t → ∞.  4. Impulsive synchronization In some cases, the sufficient condition is not satisfied, θ k (t) is always or sometimes greater than 0, the systems (4) may not approach stabilization. We add an impulsive controller to each node and describe the impulsive system as



y (t ) = Pk (t )yk (t ), k



t = tl ,



yk tl+ = (I + (tl ) )yk tl− ,





(10)

l ∈ N,

where yk tl+ = limt →t + yk (t ), yk tl− = limt →t − yk (t ), the time sequence of impulses satisfies l

l

t1 < t2 < · · · < tl < · · · with lim tl = ∞, l→∞

(t ) ∈

Rm×m



is the impulsive control gain, Pk ∈ R+ . Here, we assume yk (tl ) = yk tl− .

Theorem 2. Let γ (t ) = λmax (I + (t ))T (I + (t )). If there exists ξ > 1 such that

ξ γ (tl )eθk (tl , tl−1 ) < 1 for all t ∈ Tκ , l ∈ N,

(11)



where θ k (t) is the maximum eigenvalue of PkT  Pk (t ) as mentioned above, then the impulsive system (10) is asymptotically stable.



Proof. Consider a function Vk (t ) = yTk (t )yk (t ). According to the assumption yk (tl ) = yk tl− , we have Vk (tl ) = Vk (tl− ). As proved in Section 3, we have

Vk (t ) ≤ θk (t )Vk (t ), and hence

t ∈ [tl−1 , tl ),



(12)



+ Vk (t ) ≤ eθk (t, tl−1 )Vk tl−1 .

(13)

It can be obtained from the second equation of (10) that







+ − Vk tl−1 = (I + (tl−1 ))yk (tl−1 )

T 

− (I + (tl−1 ))yk (tl−1 )   − − = yTk (tl−1 ) (I + (tl−1 ))T (I + (tl−1 )) yk (tl−1 )

≤



− − γ (tl−1 )yTk (tl−1 )yk (tl−1 ), l ∈ N.

This means that





+ Vk tl−1 ≤ γ (tl−1 )Vk (tl−1 ),

Let l = 1 in (13). Then

l ∈ N.

(14)



Vk (t ) ≤ eθk (t , t0 )Vk t0+ for all t ∈ [t0 , t1 ). Hence, we have



Vk (t1 ) ≤ eθk (t1 , t0 )Vk t0+ and





Vk t1+ ≤ γ (t1 )Vk (t1 ) ≤ γ (t1 )eθk (t1 , t0 )Vk t0+ . Similarly, for t ∈ [t1 , t2 ),





Vk (t ) ≤ eθk (t , t1 )Vk t1+ ≤ eθk (t , t1 )γ (t1 )eθk (t1 , t0 )Vk t0+

6

and

Y. Pei, M. Bohner and D. Pi / Commun Nonlinear Sci Numer Simulat 80 (2020) 104981





Vk t2+ ≤ γ (t2 )Vk (t2 ) ≤ γ (t2 )eθk (t2 , t1 )γ (t1 )eθk (t1 , t0 )Vk t0+ .



More generally, for t ∈ tl , tl+1





γ (t1 )eθk (t1 , t0 )γ (t2 )eθk (t2 , t1 ) · · · γ (tl )eθk (tl , tl−1 )eθk (t, tl ).

Vk (t ) ≤ Vk t0+

It follows from (11) that

1 e (t, tl ). ξ l θk

Vk (t ) ≤ Vk t0+

(15)

Hence, for ξ > 1, we can obtain that

Vk (t ) → 0 as l → ∞, which yields that yk (t ) → 0 as t → ∞. Therefore, the impulsive system (10) is asymptotically stable.



5. Simulation examples In this part, three simulations with different time scales are given to verify our theoretical results. Precisely, the examples  show how dynamical networks achieve impulsive synchronization with T = R, T = hZ and T = ∞ j=0 [ j ( p + q ), j ( p + q ) + p], respectively. All examples are simulated by random initial values. Example 1. In this example, we suppose T = R, such that μ(t) ≡ 0. We consider the same network model which has been shown as continuous example in [10, Section 4], where the chaotic Chen oscillator is taken to simulate each node. The f and g in (1) are

 f ( xi ) =

35(xi2 − xi1 ) −7xi1 − xi1 xi3 + 28xi2 xi1 xi2 − 3xi3



 and g(x j ) =



x2j1 x j2 , −x j3

respectively. x¯ = [0, 0, 0]T is one of the unstable equilibrium points. Then D f (x¯ ) and Dg(x¯ ) are respectively,



D f (x¯ ) =

−35 −7 0

35 28 0



0 0 −3



and Dg(x¯ ) =

0 0 0

0 1 0



0 0 . −1

Fig. 1. Trajectories of the first state variables in Example 1.

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Suppose the outer-coupling matrix is

⎛−0.2

⎜ 0.1 C (t ) = ⎜ 0 ⎝ 0 0.1

0.1 −0.3 0.1 0.1 0

0 0.1 −0.2 0.1 0

0 0.1 0.1 −0.3 0.1

⎞

0.1 0 ⎟ 0 ⎟, ⎠ 0.1 −0.2

which has eigenvalues 0, −0.138197, −0.238197, −0.3618, −0.4618. If we select  = I, then θ k are 59.942, 59.9156, 59.8964, 59.8728, 59.8536. We take the same impulsive control gain (tl ) = diag(−0.58, −0.68, −0.78 ) and ξ = 1.1. Thus, from (11), we have l < 0.0273, where l = tl+1 − tl . Hence, we can choose l = 0.02 as in the continuous example in [10, Section 4]. Figs. 1–3 show the results of our simulation when T = R. As can be seen, each state variable rapidly converges to zero and the network achieves asymptotic synchronization in a short time.

Fig. 2. Trajectories of the second state variables in Example 1.

Fig. 3. Trajectories of the third state variables in Example 1.

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Example 2. In this example, we suppose T = hZ, where h > 0. Specifically, we let h = 3, such that μ(t) ≡ 3. We consider the same network model which has been shown as a discrete example in [10, Section 4], where the chaotic Hénon map is taken to simulate each node. The f and g in (1) are



f ( xi ) =

xi2 + 1 − 1.4x2i1 0.3xi1





and g(x j ) =



x2j1 , −x j2

respectively. x¯ = [0.631354477, 0.189406343]T is one of its fixed points. Then D f (x¯ ) and Dg(x¯ ) are respectively,



D f (x¯ ) =

−1.7678 0.3

1 0





and Dg(x¯ ) =

1.2627 0



0 . −1

We use the same C(t) and  . Then, θ k are 0.292668, 0.371311, 0.436397, 0.526357, 0.606821. We choose (tl ) = diag(−0.63, −0.63 ) and ξ = 1.1. Then, from (11), we have l ≤ [3.11986] = 3. Without loss of generality, we choose l = 3. Figs. 4–5 illustrate the results of our simulation when T = 3Z. We can conclude that the state variables also converge to zero and the network achieves asymptotic synchronization along with discrete points.

Fig. 4. Trajectories of the first state variables in Example 2.

Fig. 5. Trajectories of the second state variables in Example 2.

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9

∞

Example 3. In this example, we suppose T = j=0 [ j ( p + q ), j ( p + q ) + p], where p, q > 0. Specifically, we let p = 0.02, q =  0.02, such that T = ∞ j=0 [0.04 j, 0.04 j + 0.02], and the graininess function of T is given by



μ(t ) =

0, 0.02,

 if t ∈ ∞ [0.04 j, 0.04 j + 0.02 ),  j=0 if t ∈ ∞ { j=0 0.04 j + 0.02}.

We consider the same model and the same outer coupling matrix C(t), , (tl ) and ξ as in Example 1. In this case, θ k only depends on μ(t). So, when μ(t ) = 0, θ k and l are the same as in Example 1. This means that impulsive control occurs at the beginning of each time segment, except for the first one. When μ(t ) = 0.02, θ k are 71.3734, 71.3335, 71.3046, 71.2689, 71.24. From (11), we estimate that l ≤ 0.023. We also choose l = 0.02, which means that impulsive control occurs at each time point of multiples of 0.02. Figs. 6–8 present the results of our simulation in this case. As can be seen, each state variable also rapidly converges to zero and the network achieves asymptotic synchronization along with each time segment.

Fig. 6. Trajectories of the first state variables in Example 3.

Fig. 7. Trajectories of the second state variables in Example 3.

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Fig. 8. Trajectories of the third state variables in Example 3.

6. Conclusions In this paper, a class of TTDNs has been considered. We have investigated their synchronization and impulsive synchronization on time scales. A scale-type sufficient condition has been derived according to the regressive condition in time scales theory. Moreover, a general impulsive criterion has been established to ensure TTDN achieves asymptotic synchronization. Simulations on different time scale cases have been given to verify our theoretical analysis and to illustrate our conclusions. The proposed ideas and methods can be extended to other impulsive synchronization problems of CNs. Acknowledgement The authors would like to thank all three referees for a careful reading of the manuscript and valuable suggestions. References [1] Tang Y, Qian F, Gao H, Kurths J. Synchronization in complex networks and its application — a survey of recent advances and challenges. Annu Rev Control 2014;38(2):184–98. [2] Couzin ID. Synchronization: the key to effective communication in animal collectives. Trends Cogn Sci 2018;22(10):844–6. [3] Li C, Chen G. Synchronization in general complex dynamical networks with coupling delays. Physica A 2004;343:263–78. [4] Lin Y, Zhang Y. Synchronization of stochastic impulsive discrete-time delayed networks via pinning control. Neurocomput 2018;286:31–40. [5] Sun G, Zhang Y. Exponential stability of impulsive discrete-time stochastic BAM neural networks with time-varying delay. Neurocomput 2014;131:323–30. [6] Zhang Y, Sun J. Stability of impulsive neural networks with time delays. Phys Lett A 2005;348:44–50. [7] Lu J, Chen G. A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans Autom Control 2005;50(6): 841–846. [8] Cai S, Zhou J, Xiang L, Liu Z. Robust impulsive synchronization of complex delayed dynamical networks. Phys Lett A 2008;372(30):4990–5. [9] Chen W-H, Lu X, Zheng WX. Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks. IEEE Trans Neural Netw Learn Syst 2015;26(4):734–48. [10] Zhang Q, Lu J-a, Zhao J. Impulsive synchronization of general continuous and discrete-time complex dynamical networks. Commun Nonlinear Sci Numer Simul 2010;15(4):1063–70. [11] Wu Z, Wang H. Impulsive pinning synchronization of discrete-time network. Adv Difference Equs 2016;2016(1):36. [12] Yang X, Cao J. Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations. Appl Math Comput 2014;227:480–93. [13] Zhao H, Li L, Peng H, Xiao J, Yang Y, Zheng M. Impulsive control for synchronization and parameters identification of uncertain multi-links complex network. Nonlinear Dyn 2016;83(3):1437–51. [14] Yang H, Wang X, Zhong S, Shu L. Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control. Appl Math Comput 2018;320:75–85. [15] Yang X, Lu J, Ho DWC, Song Q. Synchronization of uncertain hybrid switching and impulsive complex networks. Appl Math Model 2018;59:379–92. [16] Liu X, Zhang K. Synchronization of linear dynamical networks on time scales: pinning control via delayed impulses. Automatica 2016;72:147–52. [17] Bohner M, Peterson A. Dynamic equations on time scales. Birkhäuser Boston, Inc., Boston, MA; 2001. ISBN 0-8176-4225-0. doi:10.1007/ 978- 1- 4612- 0201- 1. An introduction with applications. [18] Bohner M. Some oscillation criteria for first order delay dynamic equations. Far East J Appl Math 2005;18(3):289–304.