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探讨了周期时间开关及控制阈值下在两个Rayleigh型子系统之间切换的电路系统随参数变化的复杂动力学演化过程, 通过对子系统平衡点的分析, 给出了参数空间中Fold分岔和Hopf分岔的条件, 考察了切换面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 得到了切换面处系统可能存在的各种分岔行为, 进而讨论了系统不同行为的产生机理, 指出系统的相轨迹存在分别由周期开关和控制阈值决定的两类不同的分界点, 而系统轨迹与非光滑分界面的多次碰撞将导致系统由周期倍化分岔导致混沌振荡.
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关键词:
- Rayleigh振子 /
- 开关切换 /
- 控制阈值 /
- 非光滑分岔
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.-
Keywords:
- Rayleigh oscillator /
- switch /
- control threshold /
- non-smooth bifurcation
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[29] Zhusubaliyev Z H, Mosekilde E 2003 Bifurcation and Chaos inPiecewise-Smooth Dynamical Systems (Singapore: World Scientific)
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[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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