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建立了一类具有Mathieu-Duffing振子的两质量相对转动系统的非线性动力学方程. 应用多尺度法求解该系统发生主共振-基本参数共振的分岔响应方程，并通过奇异性分析得到系统稳态响应的转迁集. 利用Melnikov方法讨论系统在外激扰动和参激扰动变化下的全局分岔和系统进入混沌状态的可能途径，得到外激和参激幅值变化下系统可能出现多次通向混沌的道路，获得系统发生混沌的必要条件. 最后采用数值方法验证了理论研究的有效性.
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关键词:
- 相对转动 /
- Mathieu-Duffing振子 /
- 混沌 /
- Melnikov方法
The dynamic equation of relative rotation nonlinear dynamic system with Mathieu-Duffing oscillator is investigated. Firstly, the bifurcation response align of the relative rotation system under primary resonance-basic parameters condition is deduced using the method of multiple scales, and a singularity analysis is employed to obtain the transition set of steady motion. Secondly, a global bifurcation of the system, some probable routes leading to chaos and multiple times leading to chaos with parametric and external excitation amplitude changes have been discussed by using Melnikov method, and the necessary condition for chaotic motion of the system is presented. Finally, a numerical method is employed to further prove the effectiveness of the theoretical research.-
Keywords:
- relatively rotation /
- Mathieu-Duffing /
- chaos /
- Melnikov method
[1] Carmeli M 1985 Found. Phys. 15 175
[2] Carmeli M 1986 Int. J. Theor. Phys. 25 89
[3] Luo S K 1996 J. Beijing Inst. Technol. 16 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16 154]
[4] Luo S K 1998 Appl. Math. Mech. 19 45
[5] Huang J C, Jing Z J 2009 Chaos Solitions and Fractels 40 1449
[6] 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]
[7] Chen S Y, Tang J Y 2008 J. Sound Vib. 318 1109
[8] Wang K, Guan X P, Qiao J M 2010 Acta Phys. Sin. 59 3648 (in Chinese) [王坤, 关新平, 乔杰敏 2010 物理学报 59 3648]
[9] Mo J Q, Cheng R J, Ge H X 2011 Acta Phys. Sin. 60 040203 (in Chinese) [莫嘉琪, 程荣军, 葛红霞 2011 物理学报 60 040203]
[10] Tang R R 2012 Acta Phys. Sin. 61 200201 (in Chinese) [唐荣荣 2012 物理学报 61 200201]
[11] Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 9
[12] Liu H R, Zhu ZH L, Shi P M 2010 Acta Phys. Sin. 59 6770 (in Chinese) [刘浩然, 朱占龙, 时培明 2010 物理学报 59 6770]
[13] Li X J, Chen X Q, Yan J 2013 Acta Phys. Sin. 62 090202 (in Chinese) [李晓静, 陈绚青, 严静 2013 物理学报 62 090202]
[14] Yagasaki K 1996 J. Sound Vib. 190 587
[15] Siewe Siewe M, Hongjun Caob, Miguel A F Sanjuán 2009 Chaos Solitions and Fractels 41 772
[16] Vladimir Aslanov, Vadim Yudintsev 2012 Chaos Solitions and Fractels 45 1100
[17] Siewe Siewe M, Moukam Kakmeni F M, Tchawoua C 2004 Chaos Solitions and Fractels 21 841
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[1] Carmeli M 1985 Found. Phys. 15 175
[2] Carmeli M 1986 Int. J. Theor. Phys. 25 89
[3] Luo S K 1996 J. Beijing Inst. Technol. 16 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16 154]
[4] Luo S K 1998 Appl. Math. Mech. 19 45
[5] Huang J C, Jing Z J 2009 Chaos Solitions and Fractels 40 1449
[6] 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]
[7] Chen S Y, Tang J Y 2008 J. Sound Vib. 318 1109
[8] Wang K, Guan X P, Qiao J M 2010 Acta Phys. Sin. 59 3648 (in Chinese) [王坤, 关新平, 乔杰敏 2010 物理学报 59 3648]
[9] Mo J Q, Cheng R J, Ge H X 2011 Acta Phys. Sin. 60 040203 (in Chinese) [莫嘉琪, 程荣军, 葛红霞 2011 物理学报 60 040203]
[10] Tang R R 2012 Acta Phys. Sin. 61 200201 (in Chinese) [唐荣荣 2012 物理学报 61 200201]
[11] Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 9
[12] Liu H R, Zhu ZH L, Shi P M 2010 Acta Phys. Sin. 59 6770 (in Chinese) [刘浩然, 朱占龙, 时培明 2010 物理学报 59 6770]
[13] Li X J, Chen X Q, Yan J 2013 Acta Phys. Sin. 62 090202 (in Chinese) [李晓静, 陈绚青, 严静 2013 物理学报 62 090202]
[14] Yagasaki K 1996 J. Sound Vib. 190 587
[15] Siewe Siewe M, Hongjun Caob, Miguel A F Sanjuán 2009 Chaos Solitions and Fractels 41 772
[16] Vladimir Aslanov, Vadim Yudintsev 2012 Chaos Solitions and Fractels 45 1100
[17] Siewe Siewe M, Moukam Kakmeni F M, Tchawoua C 2004 Chaos Solitions and Fractels 21 841
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