Combination resonance bifurcations and chaos of some nonlinear relative rotation system

Li Hai-Bin; Wang Bo-Hua; Zhang Zhi-Qiang; Liu Shuang; Li Yan-Shu

doi:10.7498/aps.61.094501

Abstract

The bifurcation and chaotic motion of some nonlinear relative rotation system with heteroclinic orbit is studied. By using dissipative system Lagrange equation, the dynamics equation of nonlinear relative rotation system under combination harmonic excitations is established. Firstly, the bifurcation response equation of relative rotation system under combination resonance is deduced with the method of multiple scales. Singularity analysis is employed to obtain the transition set of steady motion. Secondly, the Melnikov function of heteroclinic orbit is solved according to heteroclinic orbit parameter equation of relative rotation system. The critical condition of chaos about Smale commutation is given. Finally, numerical method is employed to analyze the influences of different system parameters on chaotic motion by bifurcation diagram, the maximum Lyapunov, phase trajectory and Poincare map.

Keywords:

Authors and contacts

1.
Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao 066004, China;
2.
M.D. Department of Diagnostic Radiology, The Military General Hospital of Beijing PLA, Beijing 100700, China
Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61104040), and the Natural Science Foundation of Hebei Province, China (Grant No. E2012203090).

References

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[5]	Ma C, Wang X Y 2012 Communications in Nonlinear Science and Numerical Simulation 17 721
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[13]	Chen J S, Zeng Y C, Chen S B, Li J B 2011 Chinese Journal of Scientific Instrument 32 1710 (in Chinese) [陈家胜, 曾以成, 陈仕必, 李家标 2011 仪器仪表学报 32 1710]
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[17]	Li B, Li Y B 2011 Journal of Tianjin University 44 701 (in Chinese) [李彬, 李贻斌 2011 天津大学学报 44 701]
[18]	Wang X Y, Luo C, Meng J 2009 Applied Mathematics and Computation 207 63
[19]	Luo C W, Wang C C, Wei J J 2009 Chaos, Solitons and Fractals 39 1831
[20]	Huang G, Zhang D G, Meng J J, Zhu X Y 2011 Neurocomputing 74 1026
[21]	Luo Z Z, Li Y F, Meng M, Sun Y 2011 Chinese Journal of Scientific Instrument 32 33 (in Chinese) [罗志增, 李亚飞, 孟明, 孙曜 2011 仪器仪表学报 32 33]
[22]	Li C S, Wang H 2009 Chinese Journal of Scientific Instrument 30 477 (in Chinese) [李春胜, 王宏 2009 仪器仪表学报 30 477]
[23]	Wang X Y, Luo C 2006 Applied Mathematics and Computation 183 30
[24]	Liu S, Liu H R, Wen Y, Liu B 2010 Acta Phys. Sin. 59 5223 (in Chinese) [刘爽, 刘浩然, 闻岩, 刘彬 2010 物理学报 59 5223]
[25]	Liu S, Liu B, Zhang Y K, Wen Y 2010 Acta Phys. Sin. 59 38 (in Chinese) [刘爽, 刘彬, 张业宽, 闻岩 2010 物理学报 59 38]
[26]	Liu S, Liu B, Shi P M 2009 Acta Phys. Sin. 58 4383 (in Chinese) [刘爽, 刘彬, 时培明 2009 物理学报 58 4383]
[27]	Ma S J, Xu W, Li W 2006 Acta Phys. Sin. 55 4013 (in Chinese) [马少娟, 徐伟, 李伟 2006 物理学报 55 4013]
[28]	Xue C X, Zhang S Y, Shu X F 2010 Acta Phys. Sin. 59 6599 (in Chinese) [薛春霞, 张善元, 树学锋 2010 物理学报 59 6599]
[29]	Luo S Y, Shao M Z, Luo X H 2010 Acta Phys. Sin. 59 2685 (in Chinese) [罗诗裕, 邵明珠, 罗晓华 2010 物理学报 59 2685]
[30]	Han Z J, Wang J J, Cui Y, Lu G Y, Zhang S Y 2011 Journal of Vibration and Shock 30 135 (in Chinese) [韩志军, 王建军, 崔艳, 路国运, 张善元 2011 振动与冲击 30 135]
[31]	Yong E M, Tang G J 2005 Journal of Astronautics 26 535 (in Chinese) [雍恩米, 唐国金 2005 宇航学报 26 535]
[32]	Chen F Q, Wu Z Q, Chen Y S 2001 Acta Mechanical Sinica 33 661 (in Chinese) [陈芳启, 吴志强, 陈予恕 2001 力学学报 33 661]

Cited By

[1]	Huang Z W, Zhou J Z, He H, Zhang X Y, Wang C Q 2011 Journal of Mechanical Engineering 47 59 (in Chinese) [黄志伟, 周建中, 贺徽, 张孝远, 王常青 2011 机械工程学报 47 59]
[2]	Wang X Y, Wang M J 2008 Physica A 387 3751
[3]	Zhang L S, Cai L, Feng C W 2010 Acta Electronica Sinica 38 1(in Chinese) [张立森, 蔡理, 冯朝文 2010 电子学报 38 1]
[4]	Tang Y, Fang B, Zhang Y W, Li Q F 2011 Journal of Vibration and Shock 30 269 (in Chinese) [唐冶, 方勃, 张业伟, 李庆芬 2011 振动与冲击 30 269]
[5]	Ma C, Wang X Y 2012 Communications in Nonlinear Science and Numerical Simulation 17 721
[6]	Wu G, Cai L 2010 Chinese Journal of Scientific Instrument 31 454 (in Chinese) [吴刚, 蔡理 2010 仪器仪表学报 31 454]
[7]	Huang L L, Yin Q T 2009 Journal of Electronics & Information Technology 31 2402 (in Chinese) [黄丽莲, 尹启天 2009 电子与信息学报 31 2402]
[8]	Bao H M, Zhu Y S 2009 Acta Electronica Sinica 37 1222 (in Chinese) [包浩明, 朱义胜 2009 电子学报 37 1222]
[9]	Wang X Y, Wang L L 2011 Chin. Phys. B 20 050509
[10]	Zheng H Z, Hu J F, Xu W, Liu L D, He Z X 2011 Journal of Electronics & Information Technology 33 844 (in Chinese) [郑皓洲, 胡进峰, 徐威, 刘立东, 何子述 2011 电子与信息学报 33 844]
[11]	Gong M J, Qu S C, Wang X Y 2009 Journal of Electronics & Information Technology 31 1442 (in Chinese) [龚美静, 瞿少成, 王晓燕 2009 电子与信息学报 31 1442]
[12]	Wang X Y, Liang Q Y, Meng J 2008 International Journal of Modern Physics C 19 1389
[13]	Chen J S, Zeng Y C, Chen S B, Li J B 2011 Chinese Journal of Scientific Instrument 32 1710 (in Chinese) [陈家胜, 曾以成, 陈仕必, 李家标 2011 仪器仪表学报 32 1710]
[14]	Fang H, Zhou L X, Yin Z K, Wang J Y 2009 Journal of the China Railway Society 31 56 (in Chinese) [方辉, 周黎霞, 尹忠科, 王建英 2009 铁道学报 31 56]
[15]	Wang X Y, Liang Q Y 2008 Communications in Nonlinear Science and Numerical Simulation 13 913
[16]	Zhang J Z, Li P, Zhang Y Y, Chen S S, Wang Y C 2011 Journal of Shenzhen University Science and Engineering 28 316 (in Chinese) [张建忠, 李璞, 张英英, 陈莎莎, 王云才 2011 深圳大学学报理工版 28 316]
[17]	Li B, Li Y B 2011 Journal of Tianjin University 44 701 (in Chinese) [李彬, 李贻斌 2011 天津大学学报 44 701]
[18]	Wang X Y, Luo C, Meng J 2009 Applied Mathematics and Computation 207 63
[19]	Luo C W, Wang C C, Wei J J 2009 Chaos, Solitons and Fractals 39 1831
[20]	Huang G, Zhang D G, Meng J J, Zhu X Y 2011 Neurocomputing 74 1026
[21]	Luo Z Z, Li Y F, Meng M, Sun Y 2011 Chinese Journal of Scientific Instrument 32 33 (in Chinese) [罗志增, 李亚飞, 孟明, 孙曜 2011 仪器仪表学报 32 33]
[22]	Li C S, Wang H 2009 Chinese Journal of Scientific Instrument 30 477 (in Chinese) [李春胜, 王宏 2009 仪器仪表学报 30 477]
[23]	Wang X Y, Luo C 2006 Applied Mathematics and Computation 183 30
[24]	Liu S, Liu H R, Wen Y, Liu B 2010 Acta Phys. Sin. 59 5223 (in Chinese) [刘爽, 刘浩然, 闻岩, 刘彬 2010 物理学报 59 5223]
[25]	Liu S, Liu B, Zhang Y K, Wen Y 2010 Acta Phys. Sin. 59 38 (in Chinese) [刘爽, 刘彬, 张业宽, 闻岩 2010 物理学报 59 38]
[26]	Liu S, Liu B, Shi P M 2009 Acta Phys. Sin. 58 4383 (in Chinese) [刘爽, 刘彬, 时培明 2009 物理学报 58 4383]
[27]	Ma S J, Xu W, Li W 2006 Acta Phys. Sin. 55 4013 (in Chinese) [马少娟, 徐伟, 李伟 2006 物理学报 55 4013]
[28]	Xue C X, Zhang S Y, Shu X F 2010 Acta Phys. Sin. 59 6599 (in Chinese) [薛春霞, 张善元, 树学锋 2010 物理学报 59 6599]
[29]	Luo S Y, Shao M Z, Luo X H 2010 Acta Phys. Sin. 59 2685 (in Chinese) [罗诗裕, 邵明珠, 罗晓华 2010 物理学报 59 2685]
[30]	Han Z J, Wang J J, Cui Y, Lu G Y, Zhang S Y 2011 Journal of Vibration and Shock 30 135 (in Chinese) [韩志军, 王建军, 崔艳, 路国运, 张善元 2011 振动与冲击 30 135]
[31]	Yong E M, Tang G J 2005 Journal of Astronautics 26 535 (in Chinese) [雍恩米, 唐国金 2005 宇航学报 26 535]
[32]	Chen F Q, Wu Z Q, Chen Y S 2001 Acta Mechanical Sinica 33 661 (in Chinese) [陈芳启, 吴志强, 陈予恕 2001 力学学报 33 661]

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Vol. 61, Issue 9 - 2012-05-05

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