分段线性电路切换系统的复杂行为及非光滑分岔机理

吴立锋; 关永; 刘勇

doi:10.7498/aps.62.110510

摘要

分析了分段线性电路系统在周期切换下的复杂动力学行为及其产生的机理. 基于平衡点分析, 给出了两子系统Fold分岔和Hopf分岔条件. 考虑了在不同稳定态时两子系统周期切换的分岔特性, 产生了不同的周期振荡, 并揭示了其产生的机理. 在不同的周期振荡中, 切换点的数量随参数变化产生倍化, 导致切换系统由倍周期分岔进入混沌.

关键词:

The complex dynamical and non-smooth bifurcations of a compound system with periodic switches between two piecewise linear chaotic circuits are investigated. Based on the analysis of equilibrium states, the conditions for Fold bifurcation and Hopf bifurcation are derived to explore the bifurcations of the compound system with periodic switches while there are different stable solutions in the two subsystems. Different types of oscillations of the swithing system are observed, and the mechanism is studied and presented. In the difference of periodic oscillations, the number of the swithing points increases doubly with the variation of the parameter, which leads from period-doubling bifurcation to chaos.

Keywords:

作者及机构信息

1.
首都师范大学信息工程学院, 北京 100048;

2.
盐城师范学院数学科学学院, 盐城 224002

基金项目: 国家自然科学基金 (批准号: 61070049, 61202027)、国际科技合作资金 (批准号: 2012DFA11340) 和北京市自然科学基金 (批准号: 4122015)资助的课题.

Authors and contacts

1.
College of Information Engineering, Capital Normal University, Beijing 100048, China;

2.
School of Mathematical Science, Yancheng Teachers University, Yancheng 224002, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61070049, 61202027), the National Key Technology R&D Project of China (No. 2012DFA11340), and the Natural Science Foundation of Beijing, China (Grant No. 4122015).

参考文献

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[7]	Koliopanos C L, Kyprianidis I M, Stouboulos I N, Anagnostopoulos A N, Magafas L 2003 Chaos, Solitons and Fractals 16 173
[8]	Contou-Carrere M N, Daoutidis P 2005 IEEE Trans. Auto. Cont. 50 1831
[9]	Karagiannopoulos C G 2007 J. Electrostatics 65 535
[10]	Zhang X F, Chen X K, Bi Q S 2013 Acta Phys. Sin. 62 010502 (in Chinese) [张晓芳, 陈小可, 毕勤胜 2013 物理学报 62 010502 ]
[11]	Daniele F P, Pascal C, Laura G 2001 Commun. Nonlinear Sci. Numer. Simulat. 16 916
[12]	Ueta T, Kawakami H 2002 Int. Symposium on Circuits and Systems Toskushima Japan, May 26-29, 2002II-544
[13]	Zhusubaliyev Z H, Mosekilde E 2008 Phys. Lett. A 372 2237
[14]	Zhusubaliyev Z H, Mosekilde E 2003 Bifurcation and Chaos in Piecewise-Smooth Dynamical Systems (Singapore: World Scientific )
[15]	Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[16]	Zhang W, Yu P 2000 J. Sound Vib. 231 145
[17]	Sun Z D, Zheng D Z 2001 IEEE Trans. Auto. Cont. 46 291
[18]	Leine R I 2006 Phys. D 223 121
[19]	Guo S Q, Yang S P, Guo J B 2005 J. Vibra. Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[20]	Baglietto M, Battistelli G, Scardovi L 2007 Automatica 43 1442
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[23]	Gao C, Bi Q S, Zhang Z D 2013 Acta Phys. Sin. 62 020504 (in Chinese) [高超, 毕勤胜, 张正娣 2013 物理学报 62 020504 ]
[24]	Sprott J C 2000 Amer. J. Phys. 68 758
[25]	Wang X F, Zhang B 2007 Proceedings of the IEEE International Conference on Automation and Logistics Jinan China August 2462

施引文献

[1]	Bartissol P, Chua L O 1988 IEEE Trans. Circ. Syst. 35 1512
[2]	Bai E W, Lonngren K E 2002 Chaos, Solitons and Fractals 13 1515
[3]	Yu H J, Liu Y Z 2005 Acta Phys. Sin. 54 3029 (in Chinese) [于洪洁, 刘延柱 2005 物理学报 54 3029 ]
[4]	Cveticanin L, Abd El-Latif G M, El-Naggar A M, Ismail G M 2008 J. Sound and Vibration 318 580
[5]	Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 物理学报 59 7612 ]
[6]	Madan R N 1993 Chua's Circuit: A Paradigm for Chaos (Singapore: World Scientific Press) p122
[7]	Koliopanos C L, Kyprianidis I M, Stouboulos I N, Anagnostopoulos A N, Magafas L 2003 Chaos, Solitons and Fractals 16 173
[8]	Contou-Carrere M N, Daoutidis P 2005 IEEE Trans. Auto. Cont. 50 1831
[9]	Karagiannopoulos C G 2007 J. Electrostatics 65 535
[10]	Zhang X F, Chen X K, Bi Q S 2013 Acta Phys. Sin. 62 010502 (in Chinese) [张晓芳, 陈小可, 毕勤胜 2013 物理学报 62 010502 ]
[11]	Daniele F P, Pascal C, Laura G 2001 Commun. Nonlinear Sci. Numer. Simulat. 16 916
[12]	Ueta T, Kawakami H 2002 Int. Symposium on Circuits and Systems Toskushima Japan, May 26-29, 2002II-544
[13]	Zhusubaliyev Z H, Mosekilde E 2008 Phys. Lett. A 372 2237
[14]	Zhusubaliyev Z H, Mosekilde E 2003 Bifurcation and Chaos in Piecewise-Smooth Dynamical Systems (Singapore: World Scientific )
[15]	Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[16]	Zhang W, Yu P 2000 J. Sound Vib. 231 145
[17]	Sun Z D, Zheng D Z 2001 IEEE Trans. Auto. Cont. 46 291
[18]	Leine R I 2006 Phys. D 223 121
[19]	Guo S Q, Yang S P, Guo J B 2005 J. Vibra. Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[20]	Baglietto M, Battistelli G, Scardovi L 2007 Automatica 43 1442
[21]	Branicky M S 1998 IEEE Automat. Contr. 43 475
[22]	Xu X P, Antsaklis P J 2000 Int. J. Contr. 73 1261
[23]	Gao C, Bi Q S, Zhang Z D 2013 Acta Phys. Sin. 62 020504 (in Chinese) [高超, 毕勤胜, 张正娣 2013 物理学报 62 020504 ]
[24]	Sprott J C 2000 Amer. J. Phys. 68 758
[25]	Wang X F, Zhang B 2007 Proceedings of the IEEE International Conference on Automation and Logistics Jinan China August 2462

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