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In this paper, we find the decreasing law of maximum eigenvalue of the principal sub-matrix for coupling matrix, propose a method of calculating quickly pinning nodes in complex networks, and reveal the relation between the pinning strategy and the number pinning nodes. Numerical simulations show the trends of evolution under the conditions of three pinning strategies in a scale-free network and a small world, and the effectiveness of the pinning synchronization by selecting pining nodes randomly in a scale-free network.
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Keywords:
- complex networks /
- pinning synchronization /
- the principal sub-matrix
[1] Watts D J, Strogatz S H 1998 Nature 393 440
[2] Strogatz S H 2001 Nature 410 268
[3] Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47
[4] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006Phys. Reports 424 175
[5] Newman M E J 2003 SIAM Rev. 45 67
[6] Li C G, Chen G R 2004 Physica A 341 73
[7] Dhamala M, Jirsa V K, Ding M 2004 Phys. Rev. Lett. 92 074104
[8] Yu H J, Zheng N 2008 Acta Phys. Sin. 57 4712(in Chinese) [于洪洁, 郑宁 2008 57 4712]
[9] Bian Q X, Yao H X 2010 Acta Phys. Sin. 59 3027 (in Chinese) [卞秋香, 姚洪兴 2010 59 3027]
[10] Wang X H, Jiao L C, Wu J S 2010 Chin. Phys. B 19 020501
[11] Zeng C Y, Sun M, Tian L X 2010 Acta Phys. Sin. 59 5288 (in Chinese) [曾长燕, 孙梅, 田立新 2010 59 5288]
[12] Lü L, Xia X F 2009 Acta Phys. Sin. 58 814 (in Chinese) [吕翎, 夏晓岚 2009 58 814]
[13] Wu X J, Lu H T 2010 Chin. Phys. B 19 070511
[14] Zhou J, Lu J A, Lü J H 2006 IEEE Trans. on Automatic Control51 652
[15] He G M, Yang J Y 2008 Chaos, Soliton and Fractals 38 1254
[16] Li Z, Chen G R 2004 Phys. Lett. A 324 166
[17] Lellis P D, Bernardo M D, Garofalo F 2008 Chaos 18 37110
[18] Lellis P D, Bernardo M D, Garofalo F 2009 Automatica 45 1312
[19] Zhang H G, Gong DW, Wang Z S 2011 Chin. Phys. B 20 040512
[20] Luo Q, Gao Y, Qi Y N, Gao Y, Wu T, Xu H, Li L X 2009 ActaPhys. Sin. 58 6809 (in Chinese) [罗群, 高雅, 齐雅楠, 高雅, 吴桐, 许欢, 李丽香 2009 58 6809]
[21] Jin X Z, Yang G H 2010 Chin. Phys. B 19 080508
[22] Chen T P, Liu XW, LuWL 2007 IEEE Trans. Circuits Syst.- I 541317
[23] Yu W W, Chen G R, Lü J H 2009 Automatica 45 429
[24] Zhou J, Lu J A, Lü J H 2008 Automatica 44 996
[25] Song Q, Cao J D 2010 IEEE Trans. Circuits Syst.- I 57 672
[26] Wang X F, Chen G R 2002 Physica A 310 521
[27] Porfiri M, Bernardo M D 2008 Automatica 44 3100
[28] Boyd S, Ghaoui L E, Feron E, Balakrishnan V 1994 Linear MatrixInequalities in System and Control Theory (Philadelphia, PA:SIAM) p7
[29] Zheng H R, Chen S L, Mo Z X, Huang X D 2002 Numerical CalculatingMethods (Wuhan: Wuhan University Press) p484 (in Chinese) [郑慧娆, 陈绍林, 莫忠息, 黄象鼎 2002 数值计算方法 (武汉:武汉大学出版社) 第484页]
[30] Lu J H, Chen G R 2002 Inter. J. Bifur. Chaos 12 659
[31] Lorenz E N 1963 J. Atmos. Sci. 20 130
[32] Chen G R, Ueta T 1999 Inter. J. Bifur. Chaos 9 1465
[33] Matsumoto T 1984 IEEE Trans. Circuits Syst. 31 1055
[34] Wang X F, Li X, Chen G R 2006 Theory and Application of ComplexNetworks (Beijing: Tsinghua University Press) p21 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用(北京:清华大学出版社)第21页]
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[1] Watts D J, Strogatz S H 1998 Nature 393 440
[2] Strogatz S H 2001 Nature 410 268
[3] Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47
[4] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006Phys. Reports 424 175
[5] Newman M E J 2003 SIAM Rev. 45 67
[6] Li C G, Chen G R 2004 Physica A 341 73
[7] Dhamala M, Jirsa V K, Ding M 2004 Phys. Rev. Lett. 92 074104
[8] Yu H J, Zheng N 2008 Acta Phys. Sin. 57 4712(in Chinese) [于洪洁, 郑宁 2008 57 4712]
[9] Bian Q X, Yao H X 2010 Acta Phys. Sin. 59 3027 (in Chinese) [卞秋香, 姚洪兴 2010 59 3027]
[10] Wang X H, Jiao L C, Wu J S 2010 Chin. Phys. B 19 020501
[11] Zeng C Y, Sun M, Tian L X 2010 Acta Phys. Sin. 59 5288 (in Chinese) [曾长燕, 孙梅, 田立新 2010 59 5288]
[12] Lü L, Xia X F 2009 Acta Phys. Sin. 58 814 (in Chinese) [吕翎, 夏晓岚 2009 58 814]
[13] Wu X J, Lu H T 2010 Chin. Phys. B 19 070511
[14] Zhou J, Lu J A, Lü J H 2006 IEEE Trans. on Automatic Control51 652
[15] He G M, Yang J Y 2008 Chaos, Soliton and Fractals 38 1254
[16] Li Z, Chen G R 2004 Phys. Lett. A 324 166
[17] Lellis P D, Bernardo M D, Garofalo F 2008 Chaos 18 37110
[18] Lellis P D, Bernardo M D, Garofalo F 2009 Automatica 45 1312
[19] Zhang H G, Gong DW, Wang Z S 2011 Chin. Phys. B 20 040512
[20] Luo Q, Gao Y, Qi Y N, Gao Y, Wu T, Xu H, Li L X 2009 ActaPhys. Sin. 58 6809 (in Chinese) [罗群, 高雅, 齐雅楠, 高雅, 吴桐, 许欢, 李丽香 2009 58 6809]
[21] Jin X Z, Yang G H 2010 Chin. Phys. B 19 080508
[22] Chen T P, Liu XW, LuWL 2007 IEEE Trans. Circuits Syst.- I 541317
[23] Yu W W, Chen G R, Lü J H 2009 Automatica 45 429
[24] Zhou J, Lu J A, Lü J H 2008 Automatica 44 996
[25] Song Q, Cao J D 2010 IEEE Trans. Circuits Syst.- I 57 672
[26] Wang X F, Chen G R 2002 Physica A 310 521
[27] Porfiri M, Bernardo M D 2008 Automatica 44 3100
[28] Boyd S, Ghaoui L E, Feron E, Balakrishnan V 1994 Linear MatrixInequalities in System and Control Theory (Philadelphia, PA:SIAM) p7
[29] Zheng H R, Chen S L, Mo Z X, Huang X D 2002 Numerical CalculatingMethods (Wuhan: Wuhan University Press) p484 (in Chinese) [郑慧娆, 陈绍林, 莫忠息, 黄象鼎 2002 数值计算方法 (武汉:武汉大学出版社) 第484页]
[30] Lu J H, Chen G R 2002 Inter. J. Bifur. Chaos 12 659
[31] Lorenz E N 1963 J. Atmos. Sci. 20 130
[32] Chen G R, Ueta T 1999 Inter. J. Bifur. Chaos 9 1465
[33] Matsumoto T 1984 IEEE Trans. Circuits Syst. 31 1055
[34] Wang X F, Li X, Chen G R 2006 Theory and Application of ComplexNetworks (Beijing: Tsinghua University Press) p21 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用(北京:清华大学出版社)第21页]
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