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A system, which alternates between autonomous and non-autonomous circuit systems observing the time periodic switched rules, is investigated in order to explore its complicated dynamical behaviors. By analyzing the equilibrium point, limiting cycles, and the stability of the autonomous subsystems, as well as deriving the Lyapunov exponents of the switching systems in theory and numerical calculation, we have studied the variation of periodic oscillation behaviors of the compound systems with different stable solutions to the two subsystems. By using the bifurcation diagram of the switched systems and their corresponding largest Lyapunov exponent diagrams, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations and chaotic oscillations with different parameters in the switched systems. Furthermore, dynamical evolutions of the switching system to chaos by period-doubling bifurcations, saddle-node bifurcations and torus bifurcations are observed.
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Keywords:
- switching system /
- bifurcation /
- periodic oscillations /
- chaotic oscillations
[1] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[2] Santis E D 2011 Syst. Control Lett. 60 807
[3] Cheng D Z 2004 Syst. Control Lett. 51 79
[4] Liu F, Song Y D 2011 Syst. Control Lett. 60 787
[5] Hu H, Jiang B, Yang H 2013 Signal Processing 93 1804
[6] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[7] Dalvi A, Guay M 2009 Control Eng. Pract. 17 924
[8] Wyczalek F A 2001 IEEE Aero. El. Sys. Mag. 16 15
[9] Varaiya P P 1993 IEEE T. Automat. Contr. 38 195
[10] Zhang W, Yu P 2000 J. Sound Vib. 231 145
[11] Yu Y, Zhang C, Han X J, Bi Q S 2012 Acta Phys. Sin. 61 200507 (in Chinese) [余跃, 张春, 韩修静, 毕勤胜 2012 物理学报 61 200507]
[12] Zhang C, Han X J, Bi Q S 2013 Nonlinear Dyn. 73 29
[13] Zhang C, Han X J, Bi Q S 2012 Chin. Phys. B 21 100501
[14] Margallo J G, Bejarano J D 1992 J. Sound Vib. 156 283
[15] Ma X D, Bi Q S 2012 Acta Phys. Sin. 61 240506 (in Chinese) [马新东, 毕勤胜 2012 物理学报 61 240506]
[16] Cveticanin L, Abd El-Latif G M, El-Naggar A M, Ismail G M 2008 J. Sound Vib. 318 580
[17] Kousaka T, Ueta T, Ma Y, Kawakami H 2006 Chaos Solitons Fract. 27 1019
[18] 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 物理学报 61 070502]
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[1] Xie G M, Wang L 2005 J. Math. Anal. Appl. 305 277
[2] Santis E D 2011 Syst. Control Lett. 60 807
[3] Cheng D Z 2004 Syst. Control Lett. 51 79
[4] Liu F, Song Y D 2011 Syst. Control Lett. 60 787
[5] Hu H, Jiang B, Yang H 2013 Signal Processing 93 1804
[6] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[7] Dalvi A, Guay M 2009 Control Eng. Pract. 17 924
[8] Wyczalek F A 2001 IEEE Aero. El. Sys. Mag. 16 15
[9] Varaiya P P 1993 IEEE T. Automat. Contr. 38 195
[10] Zhang W, Yu P 2000 J. Sound Vib. 231 145
[11] Yu Y, Zhang C, Han X J, Bi Q S 2012 Acta Phys. Sin. 61 200507 (in Chinese) [余跃, 张春, 韩修静, 毕勤胜 2012 物理学报 61 200507]
[12] Zhang C, Han X J, Bi Q S 2013 Nonlinear Dyn. 73 29
[13] Zhang C, Han X J, Bi Q S 2012 Chin. Phys. B 21 100501
[14] Margallo J G, Bejarano J D 1992 J. Sound Vib. 156 283
[15] Ma X D, Bi Q S 2012 Acta Phys. Sin. 61 240506 (in Chinese) [马新东, 毕勤胜 2012 物理学报 61 240506]
[16] Cveticanin L, Abd El-Latif G M, El-Naggar A M, Ismail G M 2008 J. Sound Vib. 318 580
[17] Kousaka T, Ueta T, Ma Y, Kawakami H 2006 Chaos Solitons Fract. 27 1019
[18] 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 物理学报 61 070502]
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