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It is well known that time delay is universal in complex networks. However, in most existing researches outer synchronization is realized between two networks with time delay by adding controllers to all nodes which may bring great economic costs and increase the difficulties in control in practice. In this paper, in order to deal with the problem of outer synchronization between two time-varying coupling networks with node delay and coupling delay, an adaptive pinning control scheme is proposed. First, a more realistic drive-response complex network model is constructed by introducing double delays and asymmetric coupling configuration matrices. Then, we design an adaptive pinning controller which is easy to implement, and choose an effective pinning strategy to control a crucial part of the nodes in the response network. Based on LaSalle' invariance principle and the linear matrix inequality, we may rigorously prove that the outer synchronization between the proposed drive-response networks can be achieved, and meanwhile some sufficient conditions are derived by adopting an appropriate Lyapunov-Krasovskii energy function. Finally, numerical simulation experiments are employed to verify the correctness and the effectiveness of the proposed method. Results indicate that the drive-response networks with double delays can indeed achieve outer synchronization by pinning control. Moreover, the synchronization is independent of coupling delay. And the remarkable influences of coupling delays on the synchronization speed are also revealed.
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Keywords:
- delays /
- time-varying coupling /
- pinning control /
- outer synchronization
[1] Strogatz S H 2001 Nature 410 268
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Han M, Niu Z Q, Han B 2008 Acta Phys. Sin. 57 6824 (in Chinese) [韩敏, 牛志强, 韩冰 2008 物理学报 57 6824]
[4] Li K, Guan S G, Gong X F, Lai C H 2008 Phys. Lett. A 372 7133
[5] L L, Li G, Guo L, Meng L, Zou J R, Yang M 2010 Chin. Phys. B 19 080507
[6] Zhang Q J, Zhao J C 2012 Chin. Phys. B 21 040502
[7] Wang G J, Cao J D, Lu J Q 2010 Physica A 389 1480
[8] Zhang M, L L, L N, Fan X 2012 Acta Phys. Sin. 61 220508 (in Chinese) [张檬, 吕翎, 吕娜, 范鑫 2012 物理学报 61 220508]
[9] Li C P, Sun W G, Kurths J 2007 Phys. Rev. E 76 046204
[10] Tang H W, Chen L, Lu J A, Tse C K 2008 Physica A 387 5623
[11] Wu Z Y, Fu X C 2012 Nonlinear Dyn. 69 685
[12] Sun Y Z, Li W, Ruan J 2013 Commun. Nonlinear Sci. Numer. Simul. 18 989
[13] Lu J Q, Ho D W C, Cao J D, Kurths J 2011 IEEE Trans. neural netw. 22 329
[14] Fang X L, Yang Q, Yan W J 2014 Math. Probl. Eng. 2014 437673
[15] Cai S M, He Q B, Hao J J, Liu Z R 2010 Phys. Lett. A 374 2539
[16] Li K, Guan S G, Gong X F, Lai C H 2008 Phys. Lett. A 372 7133
[17] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese) [梁义, 王兴元 2013 物理学报 62 018901]
[18] Li X, Wang X F, Chen G R 2004 IEEE Trans. Circuits Syst. I-Regul. Pap. 51 2074
[19] Chen T P, Liu X W, Lu W L 2007 IEEE Trans. Circuits Syst. I-Regul. Pap. 54 1317
[20] Guo W L, Austin F, Chen S H, Sun W 2009 Phys. Lett. A 373 1565
[21] Lu J Q, Kurths J, Cao J D, Mahdavi N, Huang C 2012 IEEE Trans. Neural Netw. Learn. Syst. 23 285
[22] Lu J Q, Ho D W C, Cao J D, Kurths J 2013 Nonlinear Anal.-Real World Appl. 14 581
[23] Wang S G, Yao H X 2012 Chin. Phys. B 21 050508
[24] Fan C X, Jiang G P, Jiang F H 2010 IEEE Trans. Circuits Syst. I-Regul. Pap. 57 2991
[25] Zheng S, Bi Q S 2011 Phys. Scr. 84 025008
[26] Hu C, Yu J, Jiang H J, Teng Z D 2011 Phys. Lett. A 375 873
[27] LaSalle J P 1960 Proc Natl. Acad. Sci. U.S.A. 46 363
[28] Yu W W, Chen G R, L J H, Kurths J 2013 SIAM J. Control Optim. 51 1395
[29] Song Q, Cao J D 2010 IEEE Trans. Circuits Syst. I-Regul. Pap. 57 672
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[1] Strogatz S H 2001 Nature 410 268
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Han M, Niu Z Q, Han B 2008 Acta Phys. Sin. 57 6824 (in Chinese) [韩敏, 牛志强, 韩冰 2008 物理学报 57 6824]
[4] Li K, Guan S G, Gong X F, Lai C H 2008 Phys. Lett. A 372 7133
[5] L L, Li G, Guo L, Meng L, Zou J R, Yang M 2010 Chin. Phys. B 19 080507
[6] Zhang Q J, Zhao J C 2012 Chin. Phys. B 21 040502
[7] Wang G J, Cao J D, Lu J Q 2010 Physica A 389 1480
[8] Zhang M, L L, L N, Fan X 2012 Acta Phys. Sin. 61 220508 (in Chinese) [张檬, 吕翎, 吕娜, 范鑫 2012 物理学报 61 220508]
[9] Li C P, Sun W G, Kurths J 2007 Phys. Rev. E 76 046204
[10] Tang H W, Chen L, Lu J A, Tse C K 2008 Physica A 387 5623
[11] Wu Z Y, Fu X C 2012 Nonlinear Dyn. 69 685
[12] Sun Y Z, Li W, Ruan J 2013 Commun. Nonlinear Sci. Numer. Simul. 18 989
[13] Lu J Q, Ho D W C, Cao J D, Kurths J 2011 IEEE Trans. neural netw. 22 329
[14] Fang X L, Yang Q, Yan W J 2014 Math. Probl. Eng. 2014 437673
[15] Cai S M, He Q B, Hao J J, Liu Z R 2010 Phys. Lett. A 374 2539
[16] Li K, Guan S G, Gong X F, Lai C H 2008 Phys. Lett. A 372 7133
[17] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese) [梁义, 王兴元 2013 物理学报 62 018901]
[18] Li X, Wang X F, Chen G R 2004 IEEE Trans. Circuits Syst. I-Regul. Pap. 51 2074
[19] Chen T P, Liu X W, Lu W L 2007 IEEE Trans. Circuits Syst. I-Regul. Pap. 54 1317
[20] Guo W L, Austin F, Chen S H, Sun W 2009 Phys. Lett. A 373 1565
[21] Lu J Q, Kurths J, Cao J D, Mahdavi N, Huang C 2012 IEEE Trans. Neural Netw. Learn. Syst. 23 285
[22] Lu J Q, Ho D W C, Cao J D, Kurths J 2013 Nonlinear Anal.-Real World Appl. 14 581
[23] Wang S G, Yao H X 2012 Chin. Phys. B 21 050508
[24] Fan C X, Jiang G P, Jiang F H 2010 IEEE Trans. Circuits Syst. I-Regul. Pap. 57 2991
[25] Zheng S, Bi Q S 2011 Phys. Scr. 84 025008
[26] Hu C, Yu J, Jiang H J, Teng Z D 2011 Phys. Lett. A 375 873
[27] LaSalle J P 1960 Proc Natl. Acad. Sci. U.S.A. 46 363
[28] Yu W W, Chen G R, L J H, Kurths J 2013 SIAM J. Control Optim. 51 1395
[29] Song Q, Cao J D 2010 IEEE Trans. Circuits Syst. I-Regul. Pap. 57 672
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