一类五次方振子系统的叉形分叉及振动共振研究

杨建华; 刘后广; 程刚

doi:10.7498/aps.62.180503

摘要

研究了一类具有分数阶导数阻尼的五次方振子系统中的叉形分叉及振动共振现象. 基于快慢变量分离法, 消去系统中的高频激励成分, 得到关于慢变量的等效系统, 根据等效系统中稳态平衡点的变化情况研究了系统的叉形分叉现象. 结果表明: 高频信号幅值的递增变化会引起亚临界叉形分叉, 高频信号频率和分数阶导数阻尼阶数的递增变化都会引起超临界叉形分叉; 振动共振和叉形分叉是关联的, 当叉形分叉发生时, 振动共振曲线会出现两个峰值, 否则只会出现一个峰值. 通过解析结果和数值模拟结果的对比, 验证了解析分析的正确性.

关键词:

Abstract

The pitchfork bifurcation and vibrational resonance are investigated in this paper. Based on the method of separating slow motion from fast motion, the equivalent equation to the slow motion is obtained. Then, the pitchfork bifurcation is studied. The results show that the amplitude of the high-frequency signal can induce the subcritical pitchfork bifurcation, while both the frequency of the high-frequency signal and the value of the fractional-order can induce supercritical pitchfork bifurcation. The pattern of the vibrational resonance depends on the pitchfork bifurcation. The vibrational resonance presents double-resonance pattern when the pitchfork bifurcation occurs. Or else, the vibrational resonance presents single-resonance pattern. The analytical predications are in good agreement with the numerical calculation results, which verifies the validity of the theoretical results.

Keywords:

作者及机构信息

1.
中国矿业大学机电工程学院, 徐州 221116

基金项目: 中央高校基本科研业务费专项资金(批准号:2012QNA21)和江苏省高校优势学科建设工程资助的课题.

Authors and contacts

1.
School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China

Funds: Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2012QNA21) and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.

参考文献

[1]	Landa P S, McClintock 2000 J. Phys. A 33 L433
[2]	Gitterman M 2001 J. Phys. A 34 L355
[3]	Blekhman I I, Landa P S 2004 Int. J. Non-Linear Mech. 39 421
[4]	Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Phys. Rev. E 80 046608
[5]	Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Chaos 19 043128
[6]	Baltanas J P, Lopez L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuan M A F 2003 Phys. Rev. E 67 066119
[7]	Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[8]	Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 22103
[9]	Yang J H, Liu X B 2010 J. Phys. A 43 122001
[10]	Yang J H, Liu X B 2010 Chaos 20 033124
[11]	Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[12]	Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[13]	Jeevarathinam C, Rajasekar S, Sanjuan M A F 2011 Phys. Rev. E 83 066205
[14]	Lin M, Huang Y M 2007 Acta Phys. Sin. 56 6173 (in Chinese) [林敏, 黄咏梅 2007 物理学报 56 6173]
[15]	Deng B, Wang J, Wei X, Tsang K M, Chan W L 2010 Chaos 20 013113
[16]	Qin Y M, Wang J, Men C, Deng B, Wei X L 2011 Chaos 21 023133
[17]	Yu H, Wang J, Sun J, Yu H 2012 Chaos 22 033105
[18]	Sun J, Deng B, Liu C, Yu H, Wang J, Wei X, Zhao J 2013 Appl. Math. Model. 37 6311
[19]	Yang J H, Zhu H 2012 Chaos 22 013112
[20]	Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[21]	Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
[22]	Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[23]	Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer-Verlag)

施引文献

[1]	Landa P S, McClintock 2000 J. Phys. A 33 L433
[2]	Gitterman M 2001 J. Phys. A 34 L355
[3]	Blekhman I I, Landa P S 2004 Int. J. Non-Linear Mech. 39 421
[4]	Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Phys. Rev. E 80 046608
[5]	Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan M A F 2009 Chaos 19 043128
[6]	Baltanas J P, Lopez L, Blechman I I, Landa P S, Zaikin A, Kurths J, Sanjuan M A F 2003 Phys. Rev. E 67 066119
[7]	Chizhevsky V N, Smeu E, Giacomelli G 2003 Phys. Rev. Lett. 91 220602
[8]	Chizhevsky V N, Giacomelli G 2006 Phys. Rev. E 73 22103
[9]	Yang J H, Liu X B 2010 J. Phys. A 43 122001
[10]	Yang J H, Liu X B 2010 Chaos 20 033124
[11]	Yang J H, Liu X B 2010 Phys. Scr. 82 025006
[12]	Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[13]	Jeevarathinam C, Rajasekar S, Sanjuan M A F 2011 Phys. Rev. E 83 066205
[14]	Lin M, Huang Y M 2007 Acta Phys. Sin. 56 6173 (in Chinese) [林敏, 黄咏梅 2007 物理学报 56 6173]
[15]	Deng B, Wang J, Wei X, Tsang K M, Chan W L 2010 Chaos 20 013113
[16]	Qin Y M, Wang J, Men C, Deng B, Wei X L 2011 Chaos 21 023133
[17]	Yu H, Wang J, Sun J, Yu H 2012 Chaos 22 033105
[18]	Sun J, Deng B, Liu C, Yu H, Wang J, Wei X, Zhao J 2013 Appl. Math. Model. 37 6311
[19]	Yang J H, Zhu H 2012 Chaos 22 013112
[20]	Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[21]	Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
[22]	Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[23]	Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer-Verlag)

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