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Complicated behaviors and non-smooth bifurcation of a switching system with piecewise linearchaotic circuit

Wu Li-Feng Guan Yong Liu Yong

Complicated behaviors and non-smooth bifurcation of a switching system with piecewise linearchaotic circuit

Wu Li-Feng, Guan Yong, Liu Yong
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  • 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.
    • 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).
    [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

  • [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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  • Received Date:  04 March 2013
  • Accepted Date:  26 March 2013
  • Published Online:  05 June 2013

Complicated behaviors and non-smooth bifurcation of a switching system with piecewise linearchaotic circuit

  • 1. College of Information Engineering, Capital Normal University, Beijing 100048, China;
  • 2. School of Mathematical Science, Yancheng Teachers University, Yancheng 224002, China
Fund Project:  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).

Abstract: 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.

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