一类三次方对称离散混沌系统的分岔控制

张惠; 褚衍东; 丁旺才; 李险峰

doi:10.7498/aps.62.040202

摘要

通过分析对称性破缺分岔机制, 采用了一个直接的、有效的线性控制器, 精确控制了一类三次方对称离散混沌系统发生对称性破缺分岔和倍周期分岔时分岔点的位置. 进而分析了系统对初始值的敏感性和对称性, 选择合适的吸引域, 将对称性破缺分岔进行进一步控制, 从而使得对称性破缺分岔所缺解枝得以恢复. 数值结果表明了该控制器的有效性.

关键词:

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.

Keywords:

作者及机构信息

1.
兰州交通大学机电工程学院, 兰州 730070;

2.
兰州交通大学数学系, 兰州 730070;

3.
香港城市大学土木与建筑工程系, 香港

基金项目: 国家自然科学基金(批准号:11161027,11162007)、甘肃省自然科学重点基金(批准号:1010RJZA067)和兰州交通大学青年科学基金(批准号:2011026)资助的课题.

Authors and contacts

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

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).

参考文献

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[34]	Lai Y C 1996 Phys. Rev. E 53 57
[35]	Szabo K G, Tel T 1989 J. Stat. Phys. 54 925
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[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
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[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

施引文献

[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 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 60 100202]
[9]	Fu W B, Tang J S 2004 Acta Phys. Sin. 53 2889 (in Chinese) [符文斌, 唐驾时 2004 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 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 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 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 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 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 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 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 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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