周期切换下Rayleigh振子的振荡行为及机理

陈章耀; 雪增红; 张春; 季颖; 毕勤胜

doi:10.7498/aps.63.010504

摘要

本文研究了自治与非自治电路系统在周期切换连接下的动力学行为及机理. 基于自治子系统平衡点和极限环的相应稳定性分析和切换系统李雅普诺夫指数的理论推导及数值计算. 讨论了两子系统在不同参数下的稳态解在周期切换连接下的复合系统的各种周期振荡行为，进而给出了切换系统随参数变化下的最大李雅普诺夫指数图及相应的分岔图，得到了切换系统在不同参数下呈现出周期振荡，概周期振荡和混沌振荡相互交替出现的复杂动力学行为并分析了其振荡机理. 给出了切换系统通过倍周期分岔，鞍结分岔以及环面分岔到达混沌的不同动力学演化过程.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
江苏大学土木工程与力学学院, 镇江 212013

基金项目: 国家自然科学基金（批准号：21276115，11302086）、江苏省2013年普通高校研究生科研创新计划项目（批准号：CXZZ13-0653）和镇江市科技攻关项目（GY2012020，GY2013032）资助的课题.

Authors and contacts

1.
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 21276115, 11302086), the Scientific Research Innovation Foundation of Jiangsu Province, China (Grant No. CXZZ13-0653), and the Scientific and Technological Foundation of Zhenjiang, China (Grant Nos. GY2012020, GY2013032).

参考文献

[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]

施引文献

[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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