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Further Analysis of Global Synchronisation for Networks of Identical Cells with Delayed Coupling

Published online by Cambridge University Press:  07 September 2015

Chun-Hsien Li  and
Ren-Chuen Chen
Chun-Hsien Li*
Affiliation:
Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
Ren-Chuen Chen
Affiliation:
Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
*
*Corresponding author. Email addresses: chli@nknucc.nknu.edu.tw (C.-H. Li), rcchen@nknucc.nknu.edu.tw (R.-C. Chen)
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Abstract

Synchronisation is one of the most interesting collective motions observed in large-scale complex networks of interacting dynamical systems. We consider global synchronisation for networks of nonlinearly coupled identical cells with time delays, using an approach where the synchronisation problem is converted to solving an homogeneous linear system. This approach is extended to fit networks under more general coupling topologies, and we derive four delay-dependent and delay-independent criteria that ensure the coupled dynamical network is globally synchronised. Some examples show that the four criteria are not mutually inclusive, and numerical simulations also demonstrate our theoretical results.

Keywords

05C8234C1534D0692B20Synchronisationcomplex networkscoupled systemtime delay
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 3 , August 2015 , pp. 238 - 255
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Belykh, V., Belykh, I. and Hasler, M., Connection graph stability method for synchronized coupled chaotic systems, Physica D 195, 159–187 (2004).CrossRefGoogle Scholar
[2]Belykh, V., Belykh, I. and Hasler, M., Generalized connection graph method for synchronization in asymmetrical networks, Physica D 224, 42–51 (2006).Google Scholar
[3]Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. and Hwang, D.-U., Complex networks: Structure and dynamics, Phys. Rep. 424, 175–308 (2006).Google Scholar
[4]Cao, J., Li, P. and Wang, W., Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Phys. Lett. A 353, 318–325 (2006).CrossRefGoogle Scholar
[5]Carroll, T., Amplitude-independent chaotic synchronization, Phys. Rev. E 53, 3117–3122 (1996).Google Scholar
[6]Chen, M., Synchronization in time-varying networks: a matrix measure approach, Phys. Rev. E 76, 016104 (2007).Google Scholar
[7]Chen, T. and Zhu, Z., Exponential synchronization of nonlinear coupled dynamical networks, Int. J. Bifurcation and Chaos 17, 999–1005 (2007).Google Scholar
[8]Fan, Y., Wang, Y., Zhang, Y., and Wang, Q., The synchronization of complex dynamical networks with similar nodes and coupling time-delay, App. Math. Comp. 219, 6719–6728 (2013).Google Scholar
[9]Gao, X., Zhong, S. and Gao, F., Exponential synchronization of neural networks with time-varying delays, Nonlinear Analysis: Theory, Methods & Applications, 71, 2003–2011 (2009).Google Scholar
[10]Greengard, P., The neurobiology of slow synaptic transmission, Science 294, 1024–1030 (2001).CrossRefGoogle ScholarPubMed
[11]Huang, T., Chen, G. and Kurths, J., Synchronization of chaotic systems with time-varying coupling delays, Discrete Contin. Dyn. Syst. Ser. B 16, 1071–1082 (2011).Google Scholar
[12]Ji, D., Lee, D., Koo, J., Won, S., Lee, S. and Park, J., Synchronization of neutral complex dynamical networks with coupling time-varying delays, Nonlinear Dyn. 65, 349–358 (2011).Google Scholar
[13]Juang, J., Li, C.-L. and Liang, Y.-H., Global synchronization in lattices of coupled chaotic systems, Chaos 17, 033111 (2007).CrossRefGoogle ScholarPubMed
[14]Kocarev, L. and Parlitz, U., General approach for chaotic synchronization with applications to communication, Phys. Rev. Lett. 74, 5028–5031 (1995).Google Scholar
[15]Li, Z., Exponential stability of synchronization in asymmetrically coupled dynamical networks, Chaos 18, 023124 (2008).Google Scholar
[16]Li, Z. and Lee, J., New eigenvalue based approach to synchronization in asymmetrically coupled networks, Chaos 17, 043117 (2007).Google Scholar
[17]Li, T., Song, A., Fei, S. and Guo, Y., Synchronization control of chaotic neural networks with time-varying and distributed delays, Nonlinear Analysis: Theory, Methods & Applications 71, 2372–2384 (2009).Google Scholar
[18]Li, C.-H. and Yang, S.-Y., Synchronization in linearly coupled dynamical networks with distributed time delays, Int. J. Bifurcation and Chaos 18, 2039–2047 (2008).Google Scholar
[19]Li, C.-H. and Yang, S.-Y., Synchronization in delayed Cohen-Grossberg neural networks with bounded external inputs, IMA J. App. Math. 74, 178–200 (2009).Google Scholar
[20]Liu, X. and Chen, T., Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling, Physica A 381, 82–92 (2007).Google Scholar
[21]Lu, W. and Chen, T., New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D 213, 214–230 (2006).Google Scholar
[22]Lu, W., Chen, T., and Chen, G., Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay, Physica D 221, 118–134 (2006).CrossRefGoogle Scholar
[23]Lu, J., Ho, D., and Liu, M., Globally exponential synchronization in an array of asymmetric coupled neural networks, Phys. Lett. A 369, 444–451 (2007).Google Scholar
[24]Medvedev, G. and Kopell, N., Synchronization and transient dynamics in the chains of electrically coupled FitzHugh-Nagumo oscillators, SIAM J. App. Math. 61, 1762–1801 (2001).CrossRefGoogle Scholar
[25]Mengue, A. and Essimbi, B., Secure communication using chaotic synchronization in mutually coupled semiconductor lasers, Nonlinear Dyn. 70, 1241–1253 (2012).Google Scholar
[26]Mirollo, R. and Strogatz, S., Synchronization of pulse-coupled biological oscillators, SIAM J. App. Math. 50, 1645–1662 (1990).CrossRefGoogle Scholar
[27]Newman, M., Networks: An Introduction, Oxford University Press, New York (2010).Google Scholar
[28]Newman, M. and Watts, D., Renormalization group analysis of the small-world network model, Phys. Lett. A 263, 341–346 (1999).Google Scholar
[29]Park, M., Kwon, O., Park, J., Lee, S. and Cha, E., Synchronization criteria for coupled neural networks with interval time-varying delays and leakage delay, App. Math. Comp. 218, 6762–6775 (2012).Google Scholar
[30]Pecora, L. and Carroll, T., Driving systems with chaotic signals, Phys. Rev. A 44, 2374–2383 (1991).Google Scholar
[31]Shih, C.-W. and Tseng, J.-P., Global synchronization and asymptotic phase for a ring of identical cells with delayed coupling, SIAM J. Math. Anal. 43, 1667–1697 (2011).Google Scholar
[32]Shih, C.-W. and Tseng, J.-P., A General approach to synchronization of coupled cells, SIAM J. App. Dyn, Systems 12, 1354–1393 (2013).Google Scholar
[33]Toral, R., Masoller, C., Mirasso, C., Ciszak, M., and Calvo, O., Characterization of the anticipated synchronization regime in the coupled FitzHugh-Nagumo model for neurons, Physica A 325, 192–198 (2003).CrossRefGoogle Scholar
[34]Wang, W. and Cao, J., Synchronization in an array of linearly coupled networks with time-varying delay, Physica A 366, 197–211 (2006).Google Scholar
[35]Wang, L., Qian, W., and Wang, Q., Exponential synchronization in complex networks with a single coupling delay, J. Frankl. Insts.-Eng. App. Math. 350, 1406–1423 (2013).Google Scholar
[36]Wang, J.-L., Yang, Z.-C., Huang, T., and Xiao, M., Local and global exponential synchronization of complex delayed dynamical networks with general topology, Discrete Contin. Dyn. Syst. Ser. B 16, 393–408 (2011).Google Scholar
[37]Watts, D. and Strogatz, S., Collective dynamics of ’small-world’ networks, Nature 393, 440–442 (1998).CrossRefGoogle ScholarPubMed
[38]Wu, C.-W. and Chua, L.-O., Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst. I 42, 430–447 (1995).Google Scholar
[39]Xiang, L. and Zhu, J., On pinning synchronization of general coupled networks, Nonlinear Dyn. 64, 339–348 (2011).CrossRefGoogle Scholar
[40]Yang, J. and Zhu, F., Synchronization for chaotic systems and chaos-based secure communications via both reduced-order and step-by-step sliding mode observers, Commun. Nonlinear Sci. Num. Sim. 18, 926–937 (2013).Google Scholar
[41]Yu, W., Cao, J., and Lü, J., Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. App. Dyn. Syst. 7, 108–133 (2008).CrossRefGoogle Scholar
[42]Yu, W., Chen, G., Lu, J., and Kurths, J., Synchronization via pinning control on general complex networks, SIAM J. Control Optim. 51, 1395–1416 (2013).Google Scholar