噪声环境下时滞耦合网络的广义投影滞后同步

张丽; 杨晓丽; 孙中奎

doi:10.7498/aps.62.240502

摘要

时滞和噪声在复杂网络中普遍存在，而含有耦合时滞和噪声摄动的耦合网络同步的研究工作却极其稀少. 本文针对噪声环境下具有不同节点动力学、不同拓扑结构及不同节点数目的耦合时滞网络，提出了两个网络之间的广义投影滞后同步. 首先，构建了更加贴近现实的驱动-响应网络同步的理论框架；其次，基于随机时滞微分方程LaSalle不变性原理，严格证明了在合理的控制器作用下，驱动网络和响应网络在几乎必然渐近稳定性意义下能够取得广义投影滞后同步；最后，借助于计算机仿真，通过具体的网络模型验证了理论推理的有效性. 数值模拟结果表明，驱动网络与响应网络不但能够达到广义投影滞后同步，而且同步效果不依赖于耦合时滞和比例因子的选取，同时也揭示了更新增益和耦合时滞对同步收敛速度的显著性影响.

关键词:

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

作者及机构信息

1.
陕西师范大学数学与信息科学学院, 西安 710062;

2.
西北工业大学应用数学系, 西安 710072

基金项目: 国家自然科学基金（批准号：11272258，11172342）和中央高校基本科研业务费（批准号：GK201302001）资助的课题.

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).

参考文献

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[9]	Liu J G 2012 Chin. Phys. B 21 129506
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[14]	Tang H W, Chen L, Lu J A, Tse C K 2008 Physica A 387 5623
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[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
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[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

施引文献

[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 物理学报 57 1529]
[6]	Jing X D, L L 2009 Acta Phys. Sin. 58 7539 (in Chinese) [敬晓丹, 吕翎 2009 物理学报 58 7539]
[7]	Li Y, L L, Luan L 2009 Acta Phys. Sin. 58 4463 (in Chinese) [李岩, 吕翎, 栾玲 2009 物理学报 58 4463]
[8]	L L, Zhang C 2009 Acta Phys. Sin. 58 1462 (in Chinese) [吕翎, 张超 2009 物理学报 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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