Generalized projective lag synchronization between delay-coupled networks under circumstance noise

Zhang Li; Yang Xiao-Li; Sun Zhong-Kui

doi:10.7498/aps.62.240502

Abstract

It is well known that time delay and random noise are universal in complex networks. However, the research on the synchronization of coupled networks that are subjected to delay-coupling and noise perturbation is very rare. In this paper, for two delay-coupled complex networks with different node dynamics, different topological structures and different numbers of nodes, under circumstance noise, the generalized projective lag synchronization between two networks is proposed for the first time. First, a more realistic theoretical framework is constructed for the drive-response network synchronization. Second, according to the LaSalle-type theorem for stochastic differential delay equations, we rigorously prove that the generalized projective lag synchronization between the drive-response networks can be achieved almost surely, by introducing an appropriate controller. Furthermore, numerical simulation is employed to verify the theoretical analysis. The results indicate that the drive-response networks can indeed achieve generalized projective lag synchronization, and that the synchronization is independent of time delay and scaling factor. Moreover, the remarkable influences of the update gain and the coupling delay on synchronization speed are revealed through the numerical results.

Keywords:

complex networks /
generalized projective lag synchronization /
random noise /
time delay

Authors and contacts

1.
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China;
2.
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China
Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272258, 11172342) and the Fundamental Research Fund for the Central Universities, China (Grant No. GK201302001).

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[33]	Sun Z K, Yang X L 2011 Chaos 21 033114
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Cited By

[1]	Huygens C 1669 Instructions Concerning the Use of Pendulum-Watches for Finding the Longitude at Sea 4 (London: Philos. Trans. R. Soc.) p937
[2]	Boccaletti S, Kurths J, Osipov G, Valladares D L, Zhou C S 2002 Phys. Rep. 366 1
[3]	Watts D J, Strogatz S H 1998 Nature 393 440
[4]	Barabási A L, Albert R 1999 Science 286 509
[5]	Luo Q, Wu W, Li L X, Yang Y X, Peng H P 2008 Acta Phys. Sin. 57 1529 (in Chinese) [罗群, 吴薇, 李丽香, 杨义先, 彭海朋 2008 57 1529]
[6]	Jing X D, L L 2009 Acta Phys. Sin. 58 7539 (in Chinese) [敬晓丹, 吕翎 2009 58 7539]
[7]	Li Y, L L, Luan L 2009 Acta Phys. Sin. 58 4463 (in Chinese) [李岩, 吕翎, 栾玲 2009 58 4463]
[8]	L L, Zhang C 2009 Acta Phys. Sin. 58 1462 (in Chinese) [吕翎, 张超 2009 58 1462]
[9]	Liu J G 2012 Chin. Phys. B 21 129506
[10]	Arenas A, Guilera A, Kurths J, Moreno Y, Zhou C S 2008 Phys. Rep. 469 93
[11]	Chen G R, Wang X F, Li X, L J H 2009 Some Recent Advances in Complex Networks Synchronization (Berlin Heidelberg: Springer-Verlag) pp3–16
[12]	Li C P, Sun W G, Kurths J 2007 Phys. Rev. E 76 046204
[13]	Wu X J, Lu H T 2010 Chin. Phys. B 19 070511
[14]	Tang H W, Chen L, Lu J A, Tse C K 2008 Physica A 387 5623
[15]	Li Y, Liu Z R, Zhang J B 2008 Chin. Phys. Lett. 25 874
[16]	Sun M, Zeng C Y, Tian L X 2010 Commun. Nonlinear Sci. Numer. Simul. 15 2162
[17]	Wu X Q, Zheng W X, Zhou J 2009 Chaos 19 013109
[18]	Wu Y Q, Li C P, Wu Y J, Kurths J 2012 Commun. Nonlinear Sci. Numer. Simul. 17 349
[19]	Zheng S, Bi Q S, Cai G L 2009 Phys. Lett. A 373 1553
[20]	Sun M, Zeng C Y, Tian L X 2009 Chin. Phys. Lett. 26 010501
[21]	Wu X J, Lu H T 2012 Commun. Nonlinear Sci. Numer. Simul. 17 3005
[22]	Dai H, Jia L X, Zhang Y B 2012 Chin. Phys. B 21 120508
[23]	Yang Z Q, Zhang Q, Chen Z Q 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2628
[24]	Chen J R, Jiao L C, Wu J S, Wang X H 2009 Chin. Phys. Lett. 26 060505
[25]	Wu X J, Lu H T 2010 Phys. Lett. A 374 3932
[26]	Maritan A, Banavar J R 1994 Phys. Rev. Lett. 72 1451
[27]	Zhou C S, Kurths J 2002 Phys. Rev. Lett. 88 230602
[28]	Yang X L, Xu W, Sun Z K 2006 Phys. Lett. A 353 179
[29]	Guan S G, Lai Y C, Lai C H 2006 Phys. Rev. E 73 046210
[30]	Yang X L, Xu W 2008 Chin. Phys. B 17 2004
[31]	Lin W, Chen G R 2006 Chaos 16 013134
[32]	Xiao Y Z, Tang S F, Xu Y Chaos 22 013110
[33]	Sun Z K, Yang X L 2011 Chaos 21 033114
[34]	Wang G J, Cao J D, Lu J Q 2010 Physica A 389 1480
[35]	Cao L, Ma Y 2012 Int. J. Nonlinear Sci. 13 373
[36]	Sun Y Z, Li W, Ruan J 2013 Commun. Nonlinear Sci. Numer. Simul. 18 989
[37]	Arnold L 1972 Stochastic Differential Equation and Applications (New York: Wiley)
[38]	Friedman A 1975 Stochastic Differential Equations and Applications (New York: Academic Press)
[39]	Shen Y, Luo Q, Mao X R 2006 J. Math. Appl. 318 134
[40]	Mao X R 2002 J. Math. Appl. 268 125

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Vol. 62, Issue 24 - 2013-12-20

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