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Complicated behaviors of a compound system with periodic switches between different types of Chen systems are investigated in detail. In the local analysis, the critical conditions such as fold bifurcation and Hopf bifurcation are derived to explore the bifurcations of the compound systems with different stable solutions in the two subsystems. Different types of oscillations of this switched system are observed, of which the mechanism is presented to show that the trajectories of the oscillations can be divided into several parts by the switching points, governed by the two subsystems respectively. Because of the non-smooth characteristics at the switching points, different forms of bifurcations may occur in the compound system, which may result in complicated dynamics such as chaotic oscillations, instead of the simple connections between the trajectories of the two subsystems. By the Poincaré mapping, the location of the fixed point and Floquet characteristic multiplier of switching system are discussed.With the variation of the parameter, the system can evolve into chaos via the cascading of period-doubling bifurcation. Besides, the system can evolve into chaos immediately by saddle-node bifurcations from period solutions.The non-smooth bifurcation mechanism of periodic switching system can be revealed by the research.
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Keywords:
- periodic switches /
- Chen system /
- Poincaré mapping /
- non-smooth bifurcation
[1] Daniéle F P,Pascal C, Laura G 2001 Commun. Nonlinear Sci. Numer. Simulat. 16 916
[2] Ueta T, Kawakami H 2002 International Symposium on Circuits and Systems Toskushima Japan, 2002 May 26-29 II-544
[3] Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[4] Victoriano C, Soledad F G, Emilio F 2012 Physica D: Nonlinear: Phenomena 241 5
[5] Kim S C, Kim Y C, Yoon B Y, Kang M 2007 Computer Networks 51 606
[6] Jing Z J, Yang Z Y, Jiang T 2006 Chaos, Solitons and Fractals 27 722
[7] Santis E D 2011 Systems & Control Letters 60 807
[8] Zhusubaliyev Z H, Mosekilde E 2008 Physics Letters A 372 2237
[9] Kousaka T, Ueta T, Ma Y, Kawakami H 2006 Chaos, Solitons & Fractals 27 1019
[10] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 61 070502]
[11] Andrei A, Yuliy B, Daniel L 2012 System & Control Letters 61 2
[12] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[13] Cheng D, Guo L, Lin Y, Wang Y 2005 IEEE Transactions on Automatic Control 50 661
[14] Matthias A M, Pascal M, Frank A 2012 Journal of Process Control 31 5
[15] Chen Y G, Fei S M, Zhang K J, Yu L 2012 Mathematical and Computer Modelling 56 1
[16] Zhusubaliyev Z T, Mosekilde E 2008 Physics Letters A 372 13
[17] Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 59 7612]
[18] Whiston G S 1992 Journal of Sound and Vibration 152 3
[19] Hu H Y 1995 Journal of Sound and Vibration 187 3
[20] Jin L, Lu Q S 2005 Acta Mechanica Sin. 37 40 (in Chinese) [金俐, 陆启韶 2005 固体力学学报 37 40
[21] Leine R I 2006 Physica D: Nonlinear: Phenomena 223 121
[22] Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[23] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2327 (in Chinese) [陈章耀, 张晓芳, 毕勤胜 2010 59 2327]
[24] Jiang H B, Zhang L P, Chen Z Y, Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese) [姜海波, 张丽萍, 陈章耀, 毕勤胜 2012 61 080505]
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[1] Daniéle F P,Pascal C, Laura G 2001 Commun. Nonlinear Sci. Numer. Simulat. 16 916
[2] Ueta T, Kawakami H 2002 International Symposium on Circuits and Systems Toskushima Japan, 2002 May 26-29 II-544
[3] Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[4] Victoriano C, Soledad F G, Emilio F 2012 Physica D: Nonlinear: Phenomena 241 5
[5] Kim S C, Kim Y C, Yoon B Y, Kang M 2007 Computer Networks 51 606
[6] Jing Z J, Yang Z Y, Jiang T 2006 Chaos, Solitons and Fractals 27 722
[7] Santis E D 2011 Systems & Control Letters 60 807
[8] Zhusubaliyev Z H, Mosekilde E 2008 Physics Letters A 372 2237
[9] Kousaka T, Ueta T, Ma Y, Kawakami H 2006 Chaos, Solitons & Fractals 27 1019
[10] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 61 070502]
[11] Andrei A, Yuliy B, Daniel L 2012 System & Control Letters 61 2
[12] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[13] Cheng D, Guo L, Lin Y, Wang Y 2005 IEEE Transactions on Automatic Control 50 661
[14] Matthias A M, Pascal M, Frank A 2012 Journal of Process Control 31 5
[15] Chen Y G, Fei S M, Zhang K J, Yu L 2012 Mathematical and Computer Modelling 56 1
[16] Zhusubaliyev Z T, Mosekilde E 2008 Physics Letters A 372 13
[17] Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 59 7612]
[18] Whiston G S 1992 Journal of Sound and Vibration 152 3
[19] Hu H Y 1995 Journal of Sound and Vibration 187 3
[20] Jin L, Lu Q S 2005 Acta Mechanica Sin. 37 40 (in Chinese) [金俐, 陆启韶 2005 固体力学学报 37 40
[21] Leine R I 2006 Physica D: Nonlinear: Phenomena 223 121
[22] Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[23] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2327 (in Chinese) [陈章耀, 张晓芳, 毕勤胜 2010 59 2327]
[24] Jiang H B, Zhang L P, Chen Z Y, Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese) [姜海波, 张丽萍, 陈章耀, 毕勤胜 2012 61 080505]
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