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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.
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Keywords:
- discrete chaotic system /
- symmetry-breaking /
- period-doubling /
- bifurcation control
[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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[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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