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Complicated behaviors of the compound system with periodic switches between two nonlinear systems are investigated in detail. Through 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 the 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. With the variation of the parameters, cascading of doubling increase of the switching points can be obtained, leading to chaos via period-doubling bifurcations. Furthermore, 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.
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Keywords:
- Rssler system /
- Chua's circuits /
- periodic switching /
- periodic 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, May 26-29, 2002 II-544
[3] Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[4] Zhang W, Yu P 2000 J. Sound Vib. 231 145
[5] Kim S C, Kim Y C, Yoon B Y, Kang M 2007 Computer Netw. 51 606
[6] Jing Z J, Yang Z Y, Jiang T 2006 Chaos Soliton. Fract. 27 722
[7] Bhattacharyya R, Mukhopadhyay B 2010 Nonlinear Anal-Real. 11 3824
[8] Branicky M S 1998 IEEE Trans. Automat. Contr. 43 475
[9] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[10] Cheng D, Guo L, Lin Y, Wang Y 2005 IEEE Trans. Automat. Contr. 50 661
[11] Huang H, Qu Y Z, A, Li H X 2005 Phys. Lett. A 345 345
[12] Pettersson S, Lennartson B 1996 Proc. of the 35th IEEE Conference on Control and Decision Kobe, Japan, December 11-13, 1996 1202
[13] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 70502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 61 70502]
[14] Nishiuchi Y, Ueta T, Kawakami H 2006 Chaos Soliton. Fract. 27 941
[15] Leine R I 2006 Physica D 223 121
[16] Li S L, Zhang Z D, Wu T Y 2012 Acta Phys. Sin. 61 60504 (in Chinese) [李绍龙, 吴天一, 张正娣, 毕勤胜 2012 61 60504]
[17] Rössler O E 1976 Phys. Lett. A 57 397
[18] Yang T 1997 Phys. Lett. A 232 356
[19] Madan R N 1993 Chua's Circuit: A Paradigm for Chaos (Singapore: World Scientific Press) p122
[20] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326 (in Chinese) [陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
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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, May 26-29, 2002 II-544
[3] Putyrski M, Schultz C 2011 Chem. Biol. 18 1126
[4] Zhang W, Yu P 2000 J. Sound Vib. 231 145
[5] Kim S C, Kim Y C, Yoon B Y, Kang M 2007 Computer Netw. 51 606
[6] Jing Z J, Yang Z Y, Jiang T 2006 Chaos Soliton. Fract. 27 722
[7] Bhattacharyya R, Mukhopadhyay B 2010 Nonlinear Anal-Real. 11 3824
[8] Branicky M S 1998 IEEE Trans. Automat. Contr. 43 475
[9] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[10] Cheng D, Guo L, Lin Y, Wang Y 2005 IEEE Trans. Automat. Contr. 50 661
[11] Huang H, Qu Y Z, A, Li H X 2005 Phys. Lett. A 345 345
[12] Pettersson S, Lennartson B 1996 Proc. of the 35th IEEE Conference on Control and Decision Kobe, Japan, December 11-13, 1996 1202
[13] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 70502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 61 70502]
[14] Nishiuchi Y, Ueta T, Kawakami H 2006 Chaos Soliton. Fract. 27 941
[15] Leine R I 2006 Physica D 223 121
[16] Li S L, Zhang Z D, Wu T Y 2012 Acta Phys. Sin. 61 60504 (in Chinese) [李绍龙, 吴天一, 张正娣, 毕勤胜 2012 61 60504]
[17] Rössler O E 1976 Phys. Lett. A 57 397
[18] Yang T 1997 Phys. Lett. A 232 356
[19] Madan R N 1993 Chua's Circuit: A Paradigm for Chaos (Singapore: World Scientific Press) p122
[20] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326 (in Chinese) [陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
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