Pinning Synchronization of Nonlinearly Coupled Complex Dynamical Networks on Time Scales

Kong, Fang-Di

doi:https://doi.org/10.1155/2020/8057294

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8057294 | https://doi.org/10.1155/2020/8057294

Pinning Synchronization of Nonlinearly Coupled Complex Dynamical Networks on Time Scales

Fang-Di Kong^1,2

Academic Editor: Soheil Salahshour

Received19 Dec 2019

Accepted21 May 2020

Published08 Jul 2020

Abstract

In this paper, we study the synchronization problem for nonlinearly coupled complex dynamical networks on time scales. To achieve synchronization for nonlinearly coupled complex dynamical networks on time scales, a pinning control strategy is designed. Some pinning synchronization criteria are established for nonlinearly coupled complex dynamical networks on time scales, which guarantee the whole network can be pinned to some desired state. The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Moreover, two numerical examples are given for illustration and verification of the obtained results.

1. Introduction

Complex networks are an important part of our daily life and in nature, due to many systems in the real world which can be modeled by complex dynamical networks, such as the Internet, World Wide Web, and food webs [1]. There have been a lot of researches on complex networks and neural networks (see, for example, [2–8] and references therein). Synchronization of complex dynamical network has been a hot topic in the past decades [9–16]. When considering synchronization problem of complex dynamical network, the controlled synchronization problem of complex dynamical network is significant. In recent years, the pinning synchronization of complex dynamical networks, which means the network to achieve desired synchronization by applying control to a small fraction of network nodes, has become a topic of great interest; see [17–34]. In particular, Liu and Chen [16] investigated the global synchronization for nonlinearly coupled complex networks. Further, they investigated pinning synchronization for continuous-time nonlinearly coupled networks; see [32]. In [33], a pinning control scheme was developed for continuous-time nonlinearly coupled complex dynamical network, while the results were extended to discrete-time case. In [34], the synchronization of continuous-time dynamical networks with nonlinearly coupling function was considered.

In real life, the time domains do not always match the known continuous-time intervals or discrete integer time domains. From practical point of view, it is important to study complex dynamic networks on general time domains. This is the starting point of the present investigation. Recently, synchronization of complex dynamical networks on time scales has attracted considerable attention [35–43], which contains not only synchronization of continuous-time and discrete-time complex dynamical networks but also some continuous-time intervals accompanying some discrete moments.

Motivated by the aforementioned discussions, the synchronization of nonlinearly coupled complex dynamical networks on time scales by applying pinning control scheme will be investigated. The objective in this paper is driving the whole network to some desired state by pinning control strategy. By investigating pinning controlled networks on time scales, some sufficient conditions are presented to guarantee the realization of pinning synchronization for nonlinearly coupled complex dynamical networks on time scales. The main contributions of this paper are listed as follows: (i)The model investigated in this paper generalizes the continuous-time and discrete-time nonlinearly coupled complex dynamical networks to a unique and general framework. Therefore, the obtained results include continuous-time and discrete-time nonlinearly coupled complex dynamical networks as special cases. Moreover, the model investigated in this paper is more general. Our results can be applied to investigate pinning synchronization of nonlinearly coupled complex dynamical networks on a mixed time domain(ii)Linearly coupled complex dynamical networks on time scale are included as a special case of the present work

The rest of this paper is organized as follows. Some foundational knowledge about time scales and some notations and supporting lemmas are simply outlined in Section 2. In Section 3, the pinning synchronization problem of nonlinearly coupled complex dynamical networks on time scales is formulated. In Section 4, the main theorems and some corollaries are established. In Section 5, two numerical examples are given to verify the effectiveness of our results. Finally, conclusions are provided in Section 6.

2. Preliminaries

In this section, we will present some foundational knowledge about time scales and some notations and lemmas which are needed later.

2.1. Foundational Knowledge on Time Scales

Throughout this paper, , , and denote the sets of positive integers, integers, and real numbers, respectively. A time scale is defined as a nonempty closed subset of and denoted by .

Definition 1 (see [44, 45]). Let . Define the forward jump operator by , while the backward jump operator is defined by . In this definition, we put and , where denotes the empty set. If , we say that is right-scattered, while if , we say that is left-scattered. Also, if and , then is called right-dense, and if and , then is called left-dense. The graininess function is defined by . The set is derived from the time scale as follows: if has a left-scattered maximum , then . Otherwise, .

Definition 2 (see [44]). Let . Define the function by , i.e.,

Definition 3 (see [44, 45]). Assume that and , then is called -differentiable at the point if there exists such that for any given , there is an open neighborhood of the point such that In this case, is called the -derivative of at the point and we denote it by . Moreover, we say that is -differentiable (or in short: differentiable) on provided exists for all . The function is called the -derivative of on . If , , then for any , the integral is defined as follows:

Remark 4. If , then , is the usual derivative, and is the usual integral. If , then is the forward shift, is the usual forward difference and .

Lemma 5 (see [44, 45]). If are differentiable at , then

Lemma 6 (see [44, 45]). If is differentiable at , then

Definition 7 (see [44, 45]). A function is called -continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of these functions is denoted by .

Definition 8 (see [44, 45]). We say that a function is regressive (positive regressive) provided holds. The set of all regressive (positive regressive) and -continuous functions is denoted by .

Definition 9 (see [44]). If , then we define the exponential function by where the cylinder transformation is defined by where Log is the principal logarithm function.

Lemma 10 (see [44]). Let and . Then, implies

Lemma 11 (see [43]). For fixed , if and , then as .
Let be an -matrix-valued function on . We say that is differentiable on provided each entry of is differentiable on . In this case, we put , where .

Lemma 12 (see [44]). If a matrix-valued function is differentiable at , then

Lemma 13 (see [44]). Suppose that and are differentiable -matrix-valued functions on . Then, (1)(2), where is a constant(3)

2.2. Notations and Supporting Lemmas

For each interval of , is denoted by . denotes the -dimensional Euclidean space with the Euclidean norm . denotes the set of all real matrices. Let be the -dimensional identity matrix, indicate the diagonal matrix with diagonal entries to , be the transpose of matrix , and and represent the minimum eigenvalue and the maximum eigenvalue of a real symmetric matrix. For a symmetric matrix , write if is positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively). For square matrices and , the notation means that is a positive semidefinite (negative semidefinite) matrix. The symbol denotes the Kronecker product.

Lemma 14 (see [46]). For matrices , , , and with appropriate dimensions, we have the following properties: (1), where is a constant(2)(3)(4)If and are symmetric, then is symmetric(5)For square matrices and , every eigenvalue of arises as a product of eigenvalues of and

Lemma 15 (see [47]). Let , , , where and , where . If , and each row sum of is zero, then

Lemma 16 (see [26]. Suppose is a real symmetric and irreducible matrix, in which and nonzero matrix satisfies . Let . Then, all the eigenvalues of are less than 0.

Lemma 17 (see [48]). If are symmetric, , (-dimensional zero vector) and , then (1) is symmetric for (2) is positive definite is negative definite(3) is positive definite there exists a positive definite matrix such that , that is, is a real square root of (4)(5)(6)if is the eigenvalue of , then is the eigenvalue of ,

3. Problem Formulations

Throughout the rest of the paper, let be a time scale with and . In this section, the nonlinearly coupled complex dynamical network on the time scale will be introduced.

In general, the dynamic for each isolated (uncoupled) node of the dynamical network can be described as where , is the -derivative of on , is continuous and of such a nature that existence and uniqueness of solutions to dynamic equation (10) subject to as well as their dependence on initial values is guaranteed.

Suppose that the dynamical network consists of identical nodes, with each node being an -dimensional dynamical system. Then, the nonlinearly coupled dynamical network can be described by where is the state vector of the th node at time ; the constant represents the coupling strength of network; the nonlinearly coupled function is continuous, which satisfies standard assumptions on existence and uniqueness of solutions to dynamic equation (11) subject to as well as their dependence on initial values; the coupling configuration matrix represents the topological structure of the complex network and is defined as follows: if there exists a connection between the node and the node (), then ; otherwise, , and Suppose that network (11) is connected in the sense of having no isolated clusters, which means that the coupling configuration matrix is irreducible.

Suppose that is a solution of the uncoupled system (10). In order to synchronize the network (11) to the objective state , we will design a pinning control scheme, if the network (11) cannot synchronize to the objective state without control. Without loss of generality, we add the controllers to the first nodes. Hence, we have the pinning-controlled network as follows: with the feedback controllers given by where Define the matrix:

Definition 18 (see [30]). The network (12) is said to be synchronized by pinning control, if We get the following error dynamical network by letting :

It follows from Lemma 16 that the symmetric matrix is negative definite, and so the maximal eigenvalue . From Lemma 17((1)–(3)), it is easy to prove that the matrix is symmetric positive definite.

4. Pinning Synchronization Criteria for Nonlinearly Coupled Complex Dynamical Networks on Time Scales

In order to derive the synchronization criteria for the pinning-controlled network (12), we make the following assumptions:

(A1) (see [18]). The function is assumed to satisfy Lipschitz condition, that is, there exists a constant such that

(A2) (see [9, 21, 49]). There exists a constant , such that

(A3). There exists a constant matrix , such that

(A4) (see [18]). The function is assumed to satisfy Lipschitz condition, that is, there exists a constant such that

We have the following theorems and corollaries, which give some sufficient conditions to guarantee synchronization of the network (12) by pinning control.

Theorem 19. Suppose that assumptions (A1), (A2), (A3), and (A4) hold. If there exists with and such that the following condition is satisfied: then, the network (12) is synchronized by pinning control.

Proof. Consider the Lyapunov function By Lemmas 5, 6, 12, and 13, calculating the -derivative of along the trajectories of the error dynamical network (15), one has

By assumption (A1), the following inequalities can be obtained: where , , , , , , , and

We introduce the notations

By assumption (A2) and Lemma 15, we have

By assumption (A2), we get

By assumption (A3) and Lemma 15, we have

By assumption (A3), we get

By calculating, one has

Hence, by Lemmas 14((1), (2), (4), (5)) and 17 (4) and assumption (A4), we have

In view of (22), (24), (25), (26), (27), (29), and Lemma 14((1), (2)), one can get

From (20), we have

By Lemma 10, it implies that

Since and , as . Thus, the network (12) is synchronized by pinning control.

Theorem 20. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold. Then, the network (12) is synchronized by pinning control, if there exists a constant with such that holds.

Proof. Lemma 11 yields . Choose the Lyapunov function Note that The rest proof of the theorem is analogy with the proof of Theorem 19.

Theorem 21. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and is symmetric in assumption (A3). Then, the network (12) is synchronized by pinning control, if and

Proof. Choose the Lyapunov function Taking into account the inequality (34), by analogy with the proof of Theorem 19, one has Hence, by Lemma 17(4), the following estimate holds: By Lemma 10, we have Since and it follows from Lemma 11 that . Therefore, as . This completes the proof.

Corollary 22. Let for all . Suppose that assumptions (A1), (A2), (A3), and (A4) hold, and is symmetric in assumption (A3). If inequality holds, and , then the network (12) is synchronized by pinning control.

Proof. By Lemmas 14((4), (5)) and 17((5)) and Theorem 21, it is not hard to see the result holds when and .

Next, let us consider the particular case when the function is linear: , where . The network (11) can be written as

We write the network (12) as the following where , and is a solution of system: . Meanwhile, the assumption (A3) can be described as

(A3). There exists a constant matrix , such that

Corollary 23. Let for all and be symmetric positive definite. Suppose that assumptions (A1) and (A3) hold. The network (38) is synchronized by pinning control if there exists a constant with such that the following inequality is satisfied:

Proof. Since is symmetric positive definite, one has . By Lemma 17(3), is also symmetric positive definite. From Lemma 17((4), (6)), it is easy to see that That is, assumptions (A2) and (A4) hold, where and . Now, the result follows from Theorem 20, immediately.

By Theorem 21 and Corollary 23, we have the following corollary.

Corollary 24. Let for all and be symmetric positive definite. Suppose that assumptions (A1) and (A3) hold, and is symmetric in assumption (A3). The network (38) is synchronized by pinning control if and

Obviously, the research of synchronization problem for nonlinearly coupled complex dynamical networks on time scales is more general. It contains continuous-time and discrete-time nonlinearly coupled complex dynamical networks. In addition, it contains linearly coupled complex dynamical networks on time scales.

Remark 25. According to the previous works [12, 26, 31, 35, 39, 40], the synchronization problem of complex dynamical networks with delay on time scales by pinning control strategy can be investigated, but there are some challenges for the effects of time delays and the various forms of time scales.

5. Numerical Examples

In this section, two numerical examples are given to verify the theoretical results established above.

Example 1. Consider 2-dimensional nonlinearly coupled complete network with ten nodes on time scale , which is described by where , in this case the graininess function of is given by with with and the corresponding matrix is Note that each isolated node of network (42) is a 2-dimensional nonlinear system described by It is not difficult to obtain that and satisfy assumptions (A1), (A2), (A3), and (A4) with , , , and constant matrix The objective here is to synchronize the network (42) to the solution (-dimensional zero vector) of (45) by pinning control. From Figure 1, we see the complex dynamical network (42) without control cannot reach synchronization with . Now, we apply pinning control to the network (42) with matrix . By calculations, all conditions of Theorem 21 are fulfilled. Then by Theorem 21, the network (42) can achieve synchronization under the above pinning control strategy. Figure 2 also shows that the synchronization is realized.

Example 2. Consider 2-dimensional nonlinearly star-shaped network with five nodes on time scale , which is described by where ; in this case, the graininess function is , with with and the corresponding matrix is Note that each isolated node of network (47) is a 2-dimensional nonlinear system described by be symmetric positive definite. It is not difficult to obtain that assumptions (A1) and (A3) are satisfied with and constant matrix The objective here is to synchronize the network (47) to the solution (-dimensional zero vector) of (49) by pinning control. From Figure 3, we see the complex dynamical network (47) without control cannot reach synchronization with . Now, we apply pinning control to the network (47) with matrix . By calculations, all conditions of Corollary 24 are fulfilled. By Corollary 24, the network (47) can achieve synchronization under the above pinning control strategy. Figure 4 also shows that the synchronization is realized.

6. Conclusions

In this paper, we have investigated the synchronization problem of nonlinearly coupled complex dynamical networks on time scales by pinning control strategy. Some pinning synchronization criteria have been established which guarantee that the nonlinearly coupled complex dynamical networks on time scales can be pinned to some desired state. Two numerical examples have been given to verify the effectiveness of the obtained results.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This study is supported by the National Natural Science Foundation of China (11761047).

References

S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, no. 6825, pp. 268–276, 2001.
View at: Publisher Site | Google Scholar
X. F. Wang, “Complex networks: topology, dynamics and synchronization,” International Journal of Bifurcation and Chaos, vol. 12, no. 5, pp. 885–916, 2002.
View at: Publisher Site | Google Scholar
G. Chen, X. F. Wang, and X. Li, Introduction to complex network: models, structures and dynamics, Higher Education Press, Beijing, 2012.
M. Syed Ali and P. Balasubramaniam, “Stability analysis of uncertain fuzzy Hopfield neural networks with time delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2776–2783, 2009.
View at: Publisher Site | Google Scholar
M. Syed Ali, R. Vadivel, and R. Saravanakumar, “Event-triggered state estimation for Markovian jumping impulsive neural networks with interval time-varying delays,” International Journal of Control, vol. 92, no. 2, pp. 270–290, 2017.
View at: Publisher Site | Google Scholar
R. Saravanakumar, M. Syed Ali, C. K. Ahn, H. R. Karimi, and P. Shi, “Stability of Markovian jump generalized neural networks with interval time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 8, pp. 1840–1850, 2017.
View at: Publisher Site | Google Scholar
M. Syed Ali, S. Saravanan, and J. Cao, “Finite-time boundedness, L2-gain analysis and control of Markovian jump switched neural networks with additive time-varying delays,” Nonlinear Analysis: Hybrid Systems, vol. 23, pp. 27–43, 2017.
View at: Publisher Site | Google Scholar
M. Syed Ali, M. Usha, Z. Orman, and S. Arik, “Improved result on state estimation for complex dynamical networks with time varying delays and stochastic sampling via sampled-data control,” Neural Networks, vol. 114, pp. 28–37, 2019.
View at: Publisher Site | Google Scholar
C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific, Singapore, 2002.
View at: Publisher Site
C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific, Singapore, 2007.
View at: Publisher Site
W. Lu and T. Chen, “Synchronization analysis of linearly coupled networks of discrete time systems,” Physica D, vol. 198, no. 1-2, pp. 148–168, 2004.
View at: Publisher Site | Google Scholar
C. Li and G. Chen, “Synchronization in general complex dynamical networks with coupling delays,” Physica A, vol. 343, pp. 263–278, 2004.
View at: Publisher Site | Google Scholar
Z. Li and G. Chen, “Global synchronization and asymptotic stability of complex dynamical networks,” IEEE Transactions on Circuits and Systems-II: Express Briefs, vol. 53, no. 1, pp. 28–33, 2006.
View at: Publisher Site | Google Scholar
C. P. Li, W. G. Sun, and J. Kurths, “Synchronization of complex dynamical networks with time delays,” Physica A, vol. 361, no. 1, pp. 24–34, 2006.
View at: Publisher Site | Google Scholar
X. Li and G. Chen, “Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint,” IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 50, no. 11, pp. 1381–1390, 2003.
View at: Publisher Site | Google Scholar
X. Liu and T. Chen, “Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix,” Physica A, vol. 387, no. 16-17, pp. 4429–4439, 2008.
View at: Publisher Site | Google Scholar
J. Zhou, J. Lu, and J. Lü, “Pinning adaptive synchronization of a general complex dynamical network,” Automatica, vol. 44, no. 4, pp. 996–1003, 2008.
View at: Publisher Site | Google Scholar
J. Lu, J. Kurths, J. Cao, N. Mahdavi, and C. Huang, “Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 2, pp. 285–292, 2012.
View at: Publisher Site | Google Scholar
W. Yu, G. Chen, J. Lü, and J. Kurths, “Synchronization via pinning control on general complex networks,” SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 1395–1416, 2013.
View at: Publisher Site | Google Scholar
X. F. Wang and G. Chen, “Pinning control of scale-free dynamical networks,” Physica A, vol. 310, no. 3-4, pp. 521–531, 2002.
View at: Publisher Site | Google Scholar
X. Li, X. Wang, and G. Chen, “Pinning a complex dynamical network to its equilibrium,” IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 51, no. 10, pp. 2074–2087, 2004.
View at: Publisher Site | Google Scholar
T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 54, no. 6, pp. 1317–1326, 2007.
View at: Publisher Site | Google Scholar
W. Yu, G. Chen, and J. Lü, “On pinning synchronization of complex dynamical networks,” Automatica, vol. 45, no. 2, pp. 429–435, 2009.
View at: Publisher Site | Google Scholar
Q. Song and J. Cao, “On pinning synchronization of directed and undirected complex dynamical networks,” IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 57, no. 3, pp. 672–680, 2010.
View at: Publisher Site | Google Scholar
G. Chen, “Pinning control and synchronization on complex dynamical networks,” International Journal of Control, Automation, and Systems, vol. 12, no. 2, pp. 221–230, 2014.
View at: Publisher Site | Google Scholar
Z. X. Liu, Z. Q. Chen, and Z. Z. Yuan, “Pinning control of weighted general complex dynamical networks with time delay,” Physica A, vol. 375, no. 1, pp. 345–354, 2007.
View at: Publisher Site | Google Scholar
H. Su and X. Wang, Pinning Control of Complex Networked Systems: Synchronization, Consensus and Flocking of Networked Systems Via Pinning, Springer, Berlin, Heidelberg, 2013.
View at: Publisher Site
X. Wang and H. Su, “Pinning control of complex networked systems: a decade after and beyond,” Annual Reviews in Control, vol. 38, no. 1, pp. 103–111, 2014.
View at: Publisher Site | Google Scholar
H. Zhang, K. Li, and X. Fu, “On pinning control of some typical discrete-time dynamical networks,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 182–188, 2010.
View at: Publisher Site | Google Scholar
W. Guo, F. Austin, and S. Chen, “Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1631–1639, 2010.
View at: Publisher Site | Google Scholar
R. Cheng, M. Peng, and J. Zuo, “Pinning synchronization of discrete dynamical networks with delay coupling,” Physica A, vol. 450, pp. 444–453, 2016.
View at: Publisher Site | Google Scholar
X. Liu and T. Chen, “Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 1, pp. 113–126, 2015.
View at: Publisher Site | Google Scholar
L. Y. Xiang, Z. X. Liu, Z. Q. Chen, F. Chen, and Z. Z. Yuan, “Pinning control of complex dynamical networks with general topology,” Physica A, vol. 379, no. 1, pp. 298–306, 2007.
View at: Publisher Site | Google Scholar
Y. Li, J. Lou, Z. Wang, and F. E. Alsaadi, “Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers,” Journal of the Franklin Institute, vol. 355, no. 14, pp. 6520–6530, 2018.
View at: Publisher Site | Google Scholar
Q. Cheng and J. Cao, “Synchronization of complex dynamical networks with discrete time delays on time scales,” Neurocomputing, vol. 151, pp. 729–736, 2015.
View at: Publisher Site | Google Scholar
X. Lu, Y. Wang, and Y. Zhao, “Synchronization of complex dynamical networks on time scales via Wirtinger-based inequality,” Neurocomputing, vol. 216, pp. 143–149, 2016.
View at: Publisher Site | Google Scholar
A. Ogulenko, “Asymptotical properties of social network dynamics on time scales,” Journal of Computational and Applied Mathematics, vol. 319, pp. 413–422, 2017.
View at: Publisher Site | Google Scholar
X. Liu and K. Zhang, “Synchronization of linear dynamical networks on time scales: pinning control via delayed impulses,” Automatica, vol. 72, pp. 147–152, 2016.
View at: Publisher Site | Google Scholar
M. Syed Ali and J. Yogambigai, “Synchronization of complex dynamical networks with hybrid coupling delays on time scales by handling multitude Kronecker product terms,” Applied Mathematics and Computation, vol. 291, pp. 244–258, 2016.
View at: Publisher Site | Google Scholar
M. Syed Ali and J. Yogambigai, “Synchronization criterion of complex dynamical networks with both leakage delay and coupling delay on time scales,” Neural Processing Letters, vol. 49, no. 2, pp. 453–466, 2019.
View at: Publisher Site | Google Scholar
Z. Huang, J. Cao, and Y. N. Raffoul, “Hilger-type impulsive differential inequality and its application to impulsive synchronization of delayed complex networks on time scales,” Science China Information Sciences, vol. 61, no. 7, pp. 1–3, 2018.
View at: Publisher Site | Google Scholar
X. Lu, X. Zhang, and Q. Liu, “Finite-time synchronization of nonlinear complex dynamical networks on time scales via pinning impulsive control,” Neurocomputing, vol. 275, pp. 2104–2110, 2018.
View at: Publisher Site | Google Scholar
Q. Xiao, F. L. Lewis, and Z. Zeng, “Event-based time-interval pinning control for complex networks on time scales and applications,” IEEE Transactions on Industrial Electronics, vol. 65, no. 11, pp. 8797–8808, 2018.
View at: Publisher Site | Google Scholar
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
View at: Publisher Site
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
View at: Publisher Site
A. N. Langville and W. J. Stewart, “The Kronecker product and stochastic automata networks,” Journal of Computational and Applied Mathematics, vol. 167, no. 2, pp. 429–447, 2004.
View at: Publisher Site | Google Scholar
Q. Cheng and J. Cao, “Global synchronization of complex networks with discrete time delays and stochastic disturbances,” Neural Computing & Applications, vol. 20, no. 8, pp. 1167–1179, 2011.
View at: Publisher Site | Google Scholar
H. Lütkepohl, Handbook of Matrices, Wiley, New York, NY, USA, 1996.
L. O. Chua and D. N. Green, “A qualitative analysis of the behavior of dynamic nonlinear networks: stability of autonomous networks,” IEEE Transactions on Circuits and Systems, vol. 23, no. 6, pp. 355–379, 1976.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Fang-Di Kong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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