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The control of amplitude of limit cycle emerging from the Hopf bifurcation in Langford system under a nonlinear feedback controller is investigate in this paper. Explicit nonlinear control formulae and amplitude approximations in terms of control gains are derived by utilizing the center manifold theory and normal form reduction. Gain-amplitude curves for controlled systems are drawn and verified by numerical simulations. The formulae and expressions for the Langford system present a convenient approach to obtaining an effective analytical control in this system.
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Keywords:
- amplitude control /
- Langford system /
- limit cycle /
- Hopf bifurcation
[1] Berns D W, Moiola J L, Chen G R 1998 Automatica 34 1567
[2] Oueini S S Nayfeh A H 1999 J. Sound and Vibration 224 33
[3] Berns D W, Moiola J L, Chen G R 2000 Int. J. Bifur. Chaos 10 2815
[4] Angulo F, Bernardo M, Fossas E 2005 IEEE Trans. Circ. Sys. 52 366
[5] Tang J S, Chen Z L 2006 Int. J. Bifur. Chaos 16 1
[6] Liu X D, Huang W H 1999 J. Vibrat. Eng. 12 21 (in Chinese) [刘向东, 黄文虎 1999 振动工程学报 12 21]
[7] Qian C Z, Fu W B 2005 J. Dyn. Control 3 7 (in Chinese) [钱长照, 符文彬 2005 动力学与控制学报 3 7]
[8] Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101 (in Chinese) [唐驾时, 萧寒 2005 56 101]
[9] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 57 6162]
[10] Liu S H, Zhao C G 2008 J. Dyn. Control 6 141 (in Chinese) [刘素华, 赵成刚 2008 动力学与控制学报 6 141]
[11] Liu S H, Tang J S 2007 Acta Phys. Sin 56 3145 (in Chinese) [刘素华, 唐驾时 2007 56 3145]
[12] Liu S H, Wang S G 2008 J. Hunan Univ. 35 55 (in Chinese) [刘素华, 王曙光 2008 湖南大学学报 35 55]
[13] Liu S H, Tang J S 2008 Chin. Phys. B 17 1691
[14] Liu S H 2008 Ph. D. Dissertation (Changsha: Hunan University) (in Chinese) [刘素华 2008 博士学位论文 (长沙:湖南大学)]
[15] Hassard B D, Kazarinoff N D, Wan Y 1981 Theory and Applications of Hopf Bifurcation (London: Cambridge Univ.) p35
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[1] Berns D W, Moiola J L, Chen G R 1998 Automatica 34 1567
[2] Oueini S S Nayfeh A H 1999 J. Sound and Vibration 224 33
[3] Berns D W, Moiola J L, Chen G R 2000 Int. J. Bifur. Chaos 10 2815
[4] Angulo F, Bernardo M, Fossas E 2005 IEEE Trans. Circ. Sys. 52 366
[5] Tang J S, Chen Z L 2006 Int. J. Bifur. Chaos 16 1
[6] Liu X D, Huang W H 1999 J. Vibrat. Eng. 12 21 (in Chinese) [刘向东, 黄文虎 1999 振动工程学报 12 21]
[7] Qian C Z, Fu W B 2005 J. Dyn. Control 3 7 (in Chinese) [钱长照, 符文彬 2005 动力学与控制学报 3 7]
[8] Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101 (in Chinese) [唐驾时, 萧寒 2005 56 101]
[9] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 57 6162]
[10] Liu S H, Zhao C G 2008 J. Dyn. Control 6 141 (in Chinese) [刘素华, 赵成刚 2008 动力学与控制学报 6 141]
[11] Liu S H, Tang J S 2007 Acta Phys. Sin 56 3145 (in Chinese) [刘素华, 唐驾时 2007 56 3145]
[12] Liu S H, Wang S G 2008 J. Hunan Univ. 35 55 (in Chinese) [刘素华, 王曙光 2008 湖南大学学报 35 55]
[13] Liu S H, Tang J S 2008 Chin. Phys. B 17 1691
[14] Liu S H 2008 Ph. D. Dissertation (Changsha: Hunan University) (in Chinese) [刘素华 2008 博士学位论文 (长沙:湖南大学)]
[15] Hassard B D, Kazarinoff N D, Wan Y 1981 Theory and Applications of Hopf Bifurcation (London: Cambridge Univ.) p35
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