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Bifurcation control of a cubic symmetry discrete chaotic system

Zhang Hui Chu Yan-Dong Ding Wang-Cai Li Xian-Feng

Bifurcation control of a cubic symmetry discrete chaotic system

Zhang Hui, Chu Yan-Dong, Ding Wang-Cai, Li Xian-Feng
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  • A direct and effective linear-controller is employed to exactly control the locations of bifurcation points, both the symmetry-breaking bifurcation and the period-doubling bifurcation, in a cubic symmetry discrete system. Moreover, both the sensibility and the symmetry to the initial values of the system are analyzed. The lack of the solution branches due to the symmetry-breaking bifurcation can be reinstated temporarily by selecting the corresponding basins of attraction. The effectiveness of the controller is verified by numerical simulations.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11161027, 11162007), the Key Foundation of Natural Science of Gansu Province, China (Grant No. 1010RJZA067), and the Young Scholars Science Foundation of Lanzhou Jiaotong University, China (Grant No. 2011026).
    [1]

    Chen G, Morola J L, Wang H O 2000 Int. J. Bif. Chaos 10 511

    [2]

    Chen G, Hill D J, Yu X H 2003 Bifurcation Control: Theory and Applications (Berlin: Springer) pp1-327

    [3]

    Harb A M, Zohdy M A 2002 Nonlin. Anal. 7 37

    [4]

    Wang H O, Abed E G 1995 Automatica 31 1213

    [5]

    Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 物理学报 57 6162]

    [6]

    Chen D, Wang H O, Chen G 2001 IEEE Trans. Circuits Syst. I 48 661

    [7]

    L Z S, Duan L X 2009 Chin. Phys. Lett. 26 050504

    [8]

    Ma W, Wang M Y, Nie H L 2011 Acta Phys. Sin. 60 100202 (in Chinese) [马伟, 王明渝, 聂海龙 2011 物理学报 60 100202]

    [9]

    Fu W B, Tang J S 2004 Acta Phys. Sin. 53 2889 (in Chinese) [符文斌, 唐驾时 2004 物理学报 53 2889]

    [10]

    Abed F H, Wang H O, Chen R C 1994 Physica D 70 154

    [11]

    Tang J S, Ouyang K J 2006 Acta Phys. Sin. 55 4438 (in Chinese) [唐驾时, 欧阳克俭 2006 物理学报 55 4438]

    [12]

    Tang J S, Zhao M H, Han F, Zhang L 2011 Chin. Phys. B 20 020504

    [13]

    Xiao M, Cao J D 2007 J. Math. Anal. Appl. 332 1010

    [14]

    Liang C X, Tang J S 2008 Chin. Phys. B 17 135

    [15]

    Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235

    [16]

    Lu W G, Xu P Y, Zhou L W, Luo Q M 2010 Chin. Phys. Lett. 27 030501

    [17]

    Zong X P, Geng J, Wang P G 2011 Infom. Control 40 343 (in Chinese) [宗晓萍, 耿军, 王培光 2011 信息与控制 40 343]

    [18]

    Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明 2011 物理学报 60 050504]

    [19]

    Yu P, Chen G 2004 Int. J. Bif. Chaos 14 1683

    [20]

    Huang Q W, Tang J S 2011 Commun. Theor. Phys. 55 685

    [21]

    Qian C Z, Tang J S 2006 Acta Phys. Sin. 55 617 (in Chinese) [钱长照, 唐驾时 2006 物理学报 55 617]

    [22]

    Luo X S, Chen G R, Wang B H, Fang J Q, Zou Y L, Quan H J 2003 Acta Phys. Sin. 52 790 (in Chinese) [罗晓曙, 陈关荣, 汪秉宏, 方锦清, 邹艳丽, 全宏俊 2003 物理学报 52 790]

    [23]

    Xiao H, Tang J S, Liang C X 2009 Acta Phys. Sin. 58 2989 (in Chinese) [萧寒, 唐驾时, 梁翠香 2009 物理学报 58 2989]

    [24]

    Leung A Y T, Ji J C, Chen G R 2004 Int. J. Bif. Chaos 14 1423

    [25]

    Ji J C, Leung A Y T 2002 Nonlin. Dyna. 27 411

    [26]

    Ji J C 2001 Nonlin. Dyn. 25 369

    [27]

    Ouyang K J, Tang J S, Liang C X 2009 Chin. Phys. 18 4748

    [28]

    Liu S, Liu H R, Wen Y, Liu B 2010 Acta Phys. Sin. 59 5223 (in Chinese) [刘爽, 刘浩然, 闻岩, 刘彬 2010 物理学报 59 5223]

    [29]

    Yu P, L J H 2011 Int. J. Bif. Chaos 21 2647

    [30]

    Field M, Golubitsky M 1992 Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (2nd Ed.) (Oxford: Oxford University Press) p27

    [31]

    Zou F F 2006 M. S. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [邹芳芳 2006 硕士学位论文 (大连: 大连理工大学)]

    [32]

    Zhang Y, Lei Y M, Fang T 2009 Acta Phys. Sin. 58 3799 (in Chinese) [张莹, 雷佑铭, 方同 2009 物理学报 58 3799]

    [33]

    Chossat P, Golubitsky M 1988 Physica D 32 423

    [34]

    Lai Y C 1996 Phys. Rev. E 53 57

    [35]

    Szabo K G, Tel T 1989 J. Stat. Phys. 54 925

    [36]

    Attili B S 1993 J. Austral. Math. Soc. B 35 103

    [37]

    Werner B, Spence A 1984 SIAM J. Numer. Anal. 21 388

    [38]

    Bishop S R, Sofroniou A, Shi P L 2005 Chaos Soliton. Fract. 25 257

    [39]

    Wang X Y, Meng Q Y 2004 Acta Phys. Sin. 53 388 (in Chinese) [王兴元, 孟庆业 2004 物理学报 53 388]

    [40]

    Wang X Y 2003 Chaos in the Complex Nonlinear System (Beijing: Electronics Industry Press) pp41-42 (in Chinese) [王兴元 2003 复杂非线性系统中的混沌 (北京: 电子工业出版社) 第41–42页]

    [41]

    Liu S, Che X J, Wang Z X 2011 J. Comp. 6 1648

    [42]

    Li X F, Chu Y D, Zhang H 2012 Chin. Phys. B 21 030203

    [43]

    Li X F, Leung A Y T, Chu Y D 2012 Chin. Phys. Lett. 29 010201

  • [1]

    Chen G, Morola J L, Wang H O 2000 Int. J. Bif. Chaos 10 511

    [2]

    Chen G, Hill D J, Yu X H 2003 Bifurcation Control: Theory and Applications (Berlin: Springer) pp1-327

    [3]

    Harb A M, Zohdy M A 2002 Nonlin. Anal. 7 37

    [4]

    Wang H O, Abed E G 1995 Automatica 31 1213

    [5]

    Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 物理学报 57 6162]

    [6]

    Chen D, Wang H O, Chen G 2001 IEEE Trans. Circuits Syst. I 48 661

    [7]

    L Z S, Duan L X 2009 Chin. Phys. Lett. 26 050504

    [8]

    Ma W, Wang M Y, Nie H L 2011 Acta Phys. Sin. 60 100202 (in Chinese) [马伟, 王明渝, 聂海龙 2011 物理学报 60 100202]

    [9]

    Fu W B, Tang J S 2004 Acta Phys. Sin. 53 2889 (in Chinese) [符文斌, 唐驾时 2004 物理学报 53 2889]

    [10]

    Abed F H, Wang H O, Chen R C 1994 Physica D 70 154

    [11]

    Tang J S, Ouyang K J 2006 Acta Phys. Sin. 55 4438 (in Chinese) [唐驾时, 欧阳克俭 2006 物理学报 55 4438]

    [12]

    Tang J S, Zhao M H, Han F, Zhang L 2011 Chin. Phys. B 20 020504

    [13]

    Xiao M, Cao J D 2007 J. Math. Anal. Appl. 332 1010

    [14]

    Liang C X, Tang J S 2008 Chin. Phys. B 17 135

    [15]

    Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235

    [16]

    Lu W G, Xu P Y, Zhou L W, Luo Q M 2010 Chin. Phys. Lett. 27 030501

    [17]

    Zong X P, Geng J, Wang P G 2011 Infom. Control 40 343 (in Chinese) [宗晓萍, 耿军, 王培光 2011 信息与控制 40 343]

    [18]

    Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明 2011 物理学报 60 050504]

    [19]

    Yu P, Chen G 2004 Int. J. Bif. Chaos 14 1683

    [20]

    Huang Q W, Tang J S 2011 Commun. Theor. Phys. 55 685

    [21]

    Qian C Z, Tang J S 2006 Acta Phys. Sin. 55 617 (in Chinese) [钱长照, 唐驾时 2006 物理学报 55 617]

    [22]

    Luo X S, Chen G R, Wang B H, Fang J Q, Zou Y L, Quan H J 2003 Acta Phys. Sin. 52 790 (in Chinese) [罗晓曙, 陈关荣, 汪秉宏, 方锦清, 邹艳丽, 全宏俊 2003 物理学报 52 790]

    [23]

    Xiao H, Tang J S, Liang C X 2009 Acta Phys. Sin. 58 2989 (in Chinese) [萧寒, 唐驾时, 梁翠香 2009 物理学报 58 2989]

    [24]

    Leung A Y T, Ji J C, Chen G R 2004 Int. J. Bif. Chaos 14 1423

    [25]

    Ji J C, Leung A Y T 2002 Nonlin. Dyna. 27 411

    [26]

    Ji J C 2001 Nonlin. Dyn. 25 369

    [27]

    Ouyang K J, Tang J S, Liang C X 2009 Chin. Phys. 18 4748

    [28]

    Liu S, Liu H R, Wen Y, Liu B 2010 Acta Phys. Sin. 59 5223 (in Chinese) [刘爽, 刘浩然, 闻岩, 刘彬 2010 物理学报 59 5223]

    [29]

    Yu P, L J H 2011 Int. J. Bif. Chaos 21 2647

    [30]

    Field M, Golubitsky M 1992 Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (2nd Ed.) (Oxford: Oxford University Press) p27

    [31]

    Zou F F 2006 M. S. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [邹芳芳 2006 硕士学位论文 (大连: 大连理工大学)]

    [32]

    Zhang Y, Lei Y M, Fang T 2009 Acta Phys. Sin. 58 3799 (in Chinese) [张莹, 雷佑铭, 方同 2009 物理学报 58 3799]

    [33]

    Chossat P, Golubitsky M 1988 Physica D 32 423

    [34]

    Lai Y C 1996 Phys. Rev. E 53 57

    [35]

    Szabo K G, Tel T 1989 J. Stat. Phys. 54 925

    [36]

    Attili B S 1993 J. Austral. Math. Soc. B 35 103

    [37]

    Werner B, Spence A 1984 SIAM J. Numer. Anal. 21 388

    [38]

    Bishop S R, Sofroniou A, Shi P L 2005 Chaos Soliton. Fract. 25 257

    [39]

    Wang X Y, Meng Q Y 2004 Acta Phys. Sin. 53 388 (in Chinese) [王兴元, 孟庆业 2004 物理学报 53 388]

    [40]

    Wang X Y 2003 Chaos in the Complex Nonlinear System (Beijing: Electronics Industry Press) pp41-42 (in Chinese) [王兴元 2003 复杂非线性系统中的混沌 (北京: 电子工业出版社) 第41–42页]

    [41]

    Liu S, Che X J, Wang Z X 2011 J. Comp. 6 1648

    [42]

    Li X F, Chu Y D, Zhang H 2012 Chin. Phys. B 21 030203

    [43]

    Li X F, Leung A Y T, Chu Y D 2012 Chin. Phys. Lett. 29 010201

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  • Received Date:  05 August 2012
  • Accepted Date:  26 September 2012
  • Published Online:  05 February 2013

Bifurcation control of a cubic symmetry discrete chaotic system

  • 1. School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China;
  • 2. Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China;
  • 3. Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 11161027, 11162007), the Key Foundation of Natural Science of Gansu Province, China (Grant No. 1010RJZA067), and the Young Scholars Science Foundation of Lanzhou Jiaotong University, China (Grant No. 2011026).

Abstract: A direct and effective linear-controller is employed to exactly control the locations of bifurcation points, both the symmetry-breaking bifurcation and the period-doubling bifurcation, in a cubic symmetry discrete system. Moreover, both the sensibility and the symmetry to the initial values of the system are analyzed. The lack of the solution branches due to the symmetry-breaking bifurcation can be reinstated temporarily by selecting the corresponding basins of attraction. The effectiveness of the controller is verified by numerical simulations.

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