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Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

Liu Hao-Ran Wen Yan Liu Shuang Liu Bin

Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

Liu Hao-Ran, Wen Yan, Liu Shuang, Liu Bin
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  • A coupled nonlinear relative-rotation system is studied, and the Hopf bifurcation is analyzed under the condition of primary resonance and 1:1 internal resonance. In order to control the Hopf bifurcation point, the stability and amplitude of limit cycle, a nonlinear feedback controller is proposed, and numerical calculation can confirm the validity of the method.
    • Funds:
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    Carmeli M 1985 Found. Phys. 15 175

    [2]

    Carmeli M 1986 Inter. J. Theor. Phys. 25 89

    [3]

    Luo S K 1996 J. Beijing Inst. Technol. 16 154(in Chinese) [罗绍凯1996北京理工大学学报16 154]

    [4]

    Luo S K 1996 Appl. Math. Mech. 17 683

    [5]

    Luo S K 1998 Appl. Math. Mech. 19 45

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    Luo S K 2002 Acta Phys. Sin. 51 712(in Chinese)[罗绍凯 2002 物理学报 51 712]

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    Luo S K 2002 Acta Phys. Sin. 51 1416(in Chinese) [罗绍凯 2002 物理学报 51 1416]

    [8]

    Jia L Q 2003 Acta Phys. Sin. 52 1043(in Chinese)[贾利群 2003 物理学报 52 1043]

    [9]

    Fu J L, Chen X W, Luo S K 1999 Appl. Math. Mech. 21 549

    [10]

    Luo S K 2002 Chin. Phys. Lett. 19 449

    [11]

    Luo S K 2003 Appl. Math. Mech. 24 468

    [12]

    Luo S K, Jia L Q, Cai J L 2003 Chin. Phys. 12 841

    [13]

    Luo S K, Guo Y X, Mei F X 2004 Acta Phys. Sin. 53 2413(in Chinese)[罗绍凯、郭永新、梅凤翔 2004 物理学报 53 2413]

    [14]

    Luo S K, Guo Y X, Mei F X 2004 Acta Phys. Sin. 53 1270(in Chinese)[罗绍凯、郭永新、梅凤翔 2004 物理学报 53 1270]

    [15]

    Luo S K, Cai J L, Jia L Q 2005 Chin. Phys. 14 656

    [16]

    Fu J L, Chen L Q 2006 J. Phys. A 358 5

    [17]

    Jia L Q, Luo S K Zhang Y Y 2007 Acta Phys. Sin. 56 6188(in Chinese)[贾利群、罗绍凯、张耀宇 2007 物理学报 56 6188]

    [18]

    Luo S K 2007 Acta Phys. Sin. 56 5580(in Chinese)[罗绍凯 2007 物理学报 56 5580]

    [19]

    Jia L Q, Luo S K, Zhang Y Y 2008 Acta Phys. Sin. 57 2006(in Chinese)[贾利群、罗绍凯、张耀宇 2008 物理学报 57 2006]

    [20]

    Jia L Q, Cui J C, Zhang Y Y, Luo S K 2009 Acta Phys. Sin. 58 16(in Chinese)[贾利群、崔金超、张耀宇、罗绍凯2009 物理学报 58 16]

    [21]

    Dong Q L, Liu B 2002 Acta Phys. Sin. 51 2191(in Chinese)[董全林、刘 彬 2002 物理学报 51 2191]

    [22]

    Dong Q L, Wang K, Zhang C X, Liu B 2004 Acta Phys. Sin. 53 337(in Chinese)[董全林、王 坤、张春熹、刘 彬 2004 物理学报 53 337]

    [23]

    Zhao W, Liu B, Shi P M, Jiang J S 2006 Acta Phys. Sin. 55 3852(in Chinese)[赵 武、刘 彬、时培明、蒋金水 2006 物理学报 55 3852]

    [24]

    Zhao W, Liu B 2005 Acta Phys. Sin. 54 4543(in Chinese)[赵 武、刘 彬 2005 物理学报 54 4543]

    [25]

    Meng Z, Liu B 2007 Acta Phys. Sin. 56 6194(in Chinese) [孟 宗、刘 彬 2007 物理学报 56 6194]

    [26]

    Shi P M, Liu B 2007 Acta Phys. Sin. 56 3678(in Chinese) [时培明、刘 彬 2007 物理学报 56 3678]

    [27]

    Shi P M, Liu B, Liu S 2008 Acta Phys. Sin. 57 4675(in Chinese) [时培明、刘 彬、刘 爽 2008 物理学报 57 4675]

    [28]

    Meng Z, Liu B 2008 Acta Phys. Sin. 57 1329(in Chinese) [孟 宗、刘 彬 2008 物理学报 57 1329]

    [29]

    Shi P M, Liu B, Jiang J S 2009 Acta Phys. Sin. 58 2147(in Chinese) [时培明、刘 彬、蒋金水 2009 物理学报 58 1321]

    [30]

    Liu S, Liu B, Shi P M 2009 Acta Phys. Sin. 58 4383(in Chinese) [刘 爽、刘 彬、时培明 2009 物理学报 58 4383]

    [31]

    Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101(in Chinese) [唐驾时、萧寒 2007 物理学报 56 101]

    [32]

    Zhang W Y, Huseyin K, Ye M 1998 J. Sound Vib. 255 741

  • [1]

    Carmeli M 1985 Found. Phys. 15 175

    [2]

    Carmeli M 1986 Inter. J. Theor. Phys. 25 89

    [3]

    Luo S K 1996 J. Beijing Inst. Technol. 16 154(in Chinese) [罗绍凯1996北京理工大学学报16 154]

    [4]

    Luo S K 1996 Appl. Math. Mech. 17 683

    [5]

    Luo S K 1998 Appl. Math. Mech. 19 45

    [6]

    Luo S K 2002 Acta Phys. Sin. 51 712(in Chinese)[罗绍凯 2002 物理学报 51 712]

    [7]

    Luo S K 2002 Acta Phys. Sin. 51 1416(in Chinese) [罗绍凯 2002 物理学报 51 1416]

    [8]

    Jia L Q 2003 Acta Phys. Sin. 52 1043(in Chinese)[贾利群 2003 物理学报 52 1043]

    [9]

    Fu J L, Chen X W, Luo S K 1999 Appl. Math. Mech. 21 549

    [10]

    Luo S K 2002 Chin. Phys. Lett. 19 449

    [11]

    Luo S K 2003 Appl. Math. Mech. 24 468

    [12]

    Luo S K, Jia L Q, Cai J L 2003 Chin. Phys. 12 841

    [13]

    Luo S K, Guo Y X, Mei F X 2004 Acta Phys. Sin. 53 2413(in Chinese)[罗绍凯、郭永新、梅凤翔 2004 物理学报 53 2413]

    [14]

    Luo S K, Guo Y X, Mei F X 2004 Acta Phys. Sin. 53 1270(in Chinese)[罗绍凯、郭永新、梅凤翔 2004 物理学报 53 1270]

    [15]

    Luo S K, Cai J L, Jia L Q 2005 Chin. Phys. 14 656

    [16]

    Fu J L, Chen L Q 2006 J. Phys. A 358 5

    [17]

    Jia L Q, Luo S K Zhang Y Y 2007 Acta Phys. Sin. 56 6188(in Chinese)[贾利群、罗绍凯、张耀宇 2007 物理学报 56 6188]

    [18]

    Luo S K 2007 Acta Phys. Sin. 56 5580(in Chinese)[罗绍凯 2007 物理学报 56 5580]

    [19]

    Jia L Q, Luo S K, Zhang Y Y 2008 Acta Phys. Sin. 57 2006(in Chinese)[贾利群、罗绍凯、张耀宇 2008 物理学报 57 2006]

    [20]

    Jia L Q, Cui J C, Zhang Y Y, Luo S K 2009 Acta Phys. Sin. 58 16(in Chinese)[贾利群、崔金超、张耀宇、罗绍凯2009 物理学报 58 16]

    [21]

    Dong Q L, Liu B 2002 Acta Phys. Sin. 51 2191(in Chinese)[董全林、刘 彬 2002 物理学报 51 2191]

    [22]

    Dong Q L, Wang K, Zhang C X, Liu B 2004 Acta Phys. Sin. 53 337(in Chinese)[董全林、王 坤、张春熹、刘 彬 2004 物理学报 53 337]

    [23]

    Zhao W, Liu B, Shi P M, Jiang J S 2006 Acta Phys. Sin. 55 3852(in Chinese)[赵 武、刘 彬、时培明、蒋金水 2006 物理学报 55 3852]

    [24]

    Zhao W, Liu B 2005 Acta Phys. Sin. 54 4543(in Chinese)[赵 武、刘 彬 2005 物理学报 54 4543]

    [25]

    Meng Z, Liu B 2007 Acta Phys. Sin. 56 6194(in Chinese) [孟 宗、刘 彬 2007 物理学报 56 6194]

    [26]

    Shi P M, Liu B 2007 Acta Phys. Sin. 56 3678(in Chinese) [时培明、刘 彬 2007 物理学报 56 3678]

    [27]

    Shi P M, Liu B, Liu S 2008 Acta Phys. Sin. 57 4675(in Chinese) [时培明、刘 彬、刘 爽 2008 物理学报 57 4675]

    [28]

    Meng Z, Liu B 2008 Acta Phys. Sin. 57 1329(in Chinese) [孟 宗、刘 彬 2008 物理学报 57 1329]

    [29]

    Shi P M, Liu B, Jiang J S 2009 Acta Phys. Sin. 58 2147(in Chinese) [时培明、刘 彬、蒋金水 2009 物理学报 58 1321]

    [30]

    Liu S, Liu B, Shi P M 2009 Acta Phys. Sin. 58 4383(in Chinese) [刘 爽、刘 彬、时培明 2009 物理学报 58 4383]

    [31]

    Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101(in Chinese) [唐驾时、萧寒 2007 物理学报 56 101]

    [32]

    Zhang W Y, Huseyin K, Ye M 1998 J. Sound Vib. 255 741

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  • Received Date:  08 October 2009
  • Accepted Date:  03 November 2009
  • Published Online:  15 August 2010

Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

  • 1. (1)Institute of Information Technology and Engineering, Yanshan University, Qinhuangdao 066004,China; (2)Institute of mechanical engineering, Yanshan University, Qinhuangdao 066004,China; (3)Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao 066004,China

Abstract: A coupled nonlinear relative-rotation system is studied, and the Hopf bifurcation is analyzed under the condition of primary resonance and 1:1 internal resonance. In order to control the Hopf bifurcation point, the stability and amplitude of limit cycle, a nonlinear feedback controller is proposed, and numerical calculation can confirm the validity of the method.

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