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The complicated dynamical evolution of a circuit system composed of two Rayleigh-types subsystems, which are switched by a periodic switch and a threshold controller, is investigated. Through the analysis of the subsystem equilibrium points, the conditions for Fold bifurcation and Hopf bifurcation in the parameter space are given respectively. The distribution of the generalized Jacobian eigenvalues varying with auxiliary parameter at the switching boundary is presented. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanisms of the different behaviors of the system are discussed. It is pointed that the trajectories of the system have two kinds of turning points, which are determined by the periodic switch and the threshold controller respectively. Meanwhile, the multiple collisions between the trajectories and the non-smooth boundary may lead the system to change from chaos to period-adding bifurcation.
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Keywords:
- Rayleigh oscillator /
- switch /
- control threshold /
- non-smooth bifurcation
[1] Rajasekar S, Parthasarathy S, Lakshmanan M 1992 Chaos, Solitons Fractals 2 271
[2] [3] Zhang W, Yu P 2000 J. Sound and Vibration 231 145
[4] [5] Bogarcz R, Ryczek B 1997 Eng. Trans 45 194
[6] [7] [8] Jing Z J, Yang Z Y, Jiang T 2006 Chaos, Solitons and Fractals 27722
[9] [10] Cveticanin L, Abd El-Latif GM, El-Naggar AM, Ismail GM2008Journal of Sound and Vibration 318 580
[11] Nayfeh A H, Mook D T 1979 Nonlinear Oscillators (New York:Wiley)
[12] [13] [14] Bogoliubov N N, Mitropolski Y S 1961 Asymptotic Methods in theTheory of Non-linear Oscillations (New York: Gordon Breach)
[15] [16] Nayfeh A H 1973 Perturbation Method (New York: Wiley)
[17] Margallo J G, Bejarano J D 1992 Journal of Sound and Vibration156 283
[18] [19] [20] Liu B W 2009 Nonlinear Analysis: Real World Applications 102850
[21] Brogliato B 1999 Nonsmooth Mechanics-Models (New York:Springer-Verlag)
[22] [23] Luo G W, Xie J H 2001 Physica D 148 183
[24] [25] Luo G W, Xie J H 2002 International Journal of Nonlinear Mechanics37 19
[26] [27] [28] Contou-Carrere M N, Daoutidis P 2005 Transactions on AutomaticControl 50 1831
[29] Zhusubaliyev Z H, Mosekilde E 2003 Bifurcation and Chaos inPiecewise-Smooth Dynamical Systems (Singapore: World Scientific)
[30] [31] [32] Zhusubaliyev Z H, Mosekilde E 2008 Physics Letters A 372 2237
[33] [34] Leine R I 2006 Physica D 223 121
[35] [36] Chua L O, Lin G N 1990 Transations on Circuits and Systems 37885
[37] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[38] [39] Xu H D 2005 Ph.D Dissertation (Chengdu: Southwest JiaotongUniversity) (in Chinese)[徐慧东 2005 博士学位论文(成都：西南交通大学)]
[40] [41] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2327[陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
[42] [43] Gonzalo M R, Jason A C Gallas 2010 Physics Letters A 375(2)143
[44] [45] Feng C W, Cai L, Zhang L S, Yang X K, Zhao X H 2010 ActaPhys. Sin. 59 8426 [冯朝文, 蔡理, 张立森, 杨晓阔, 赵晓辉 2010 59 8426]
[46] [47] Kousaka T, Ueta T, Ma Y, Hiroshi Kawakami 2006 Chaos, Solitons Fractals 27 1019
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[1] Rajasekar S, Parthasarathy S, Lakshmanan M 1992 Chaos, Solitons Fractals 2 271
[2] [3] Zhang W, Yu P 2000 J. Sound and Vibration 231 145
[4] [5] Bogarcz R, Ryczek B 1997 Eng. Trans 45 194
[6] [7] [8] Jing Z J, Yang Z Y, Jiang T 2006 Chaos, Solitons and Fractals 27722
[9] [10] Cveticanin L, Abd El-Latif GM, El-Naggar AM, Ismail GM2008Journal of Sound and Vibration 318 580
[11] Nayfeh A H, Mook D T 1979 Nonlinear Oscillators (New York:Wiley)
[12] [13] [14] Bogoliubov N N, Mitropolski Y S 1961 Asymptotic Methods in theTheory of Non-linear Oscillations (New York: Gordon Breach)
[15] [16] Nayfeh A H 1973 Perturbation Method (New York: Wiley)
[17] Margallo J G, Bejarano J D 1992 Journal of Sound and Vibration156 283
[18] [19] [20] Liu B W 2009 Nonlinear Analysis: Real World Applications 102850
[21] Brogliato B 1999 Nonsmooth Mechanics-Models (New York:Springer-Verlag)
[22] [23] Luo G W, Xie J H 2001 Physica D 148 183
[24] [25] Luo G W, Xie J H 2002 International Journal of Nonlinear Mechanics37 19
[26] [27] [28] Contou-Carrere M N, Daoutidis P 2005 Transactions on AutomaticControl 50 1831
[29] Zhusubaliyev Z H, Mosekilde E 2003 Bifurcation and Chaos inPiecewise-Smooth Dynamical Systems (Singapore: World Scientific)
[30] [31] [32] Zhusubaliyev Z H, Mosekilde E 2008 Physics Letters A 372 2237
[33] [34] Leine R I 2006 Physica D 223 121
[35] [36] Chua L O, Lin G N 1990 Transations on Circuits and Systems 37885
[37] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[38] [39] Xu H D 2005 Ph.D Dissertation (Chengdu: Southwest JiaotongUniversity) (in Chinese)[徐慧东 2005 博士学位论文(成都：西南交通大学)]
[40] [41] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2327[陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
[42] [43] Gonzalo M R, Jason A C Gallas 2010 Physics Letters A 375(2)143
[44] [45] Feng C W, Cai L, Zhang L S, Yang X K, Zhao X H 2010 ActaPhys. Sin. 59 8426 [冯朝文, 蔡理, 张立森, 杨晓阔, 赵晓辉 2010 59 8426]
[46] [47] Kousaka T, Ueta T, Ma Y, Hiroshi Kawakami 2006 Chaos, Solitons Fractals 27 1019
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