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本文以一类典型的相对转动振动系统为研究对象，研究激励引起的系统混沌运动和安全域侵蚀，并对系统施加时滞位置反馈来抑制这两类复杂动力学行为. 首先，利用Melnikov函数法获得相对转动系统的混沌运动及安全盆侵蚀的激励振幅阈值；其次，通过讨论时滞反馈系统的Hopf分岔条件获得适用于Melnikov函数法的控制参数取值范围，进而利用Melnikov函数法获得时滞受控系统的全局分岔必要条件；最后，利用四阶Rung-Kutta法和点映射法数值模拟了时滞受控系统动力学行为随参数的演变，从而验证解析结果的有效性. 研究发现：在正的增益系数和较短的时滞量下，时滞位置反馈能够有效抑制相对转动系统的混沌运动和安全盆侵蚀现象.A typical relative rotation system is considered whose chaotic motion and basin erosion caused by external excitation is investigated in this paper. And a delayed position feedback control is applied in the system for suppressing the two types of complex dynamical behaviors. Firstly, the excitation amplitude threshold of chaotic motion and the basin erosion of an uncontrolled relative rotation system is obtained by the Melnikov method. Secondly, the condition of Hopf bifurcation of a delay controlled system is discussed so as to obtain the available ranges of control parameters in the Melnikov method. Then the necessary condition for the global bifurcation of a delay controlled system is obtained. Finally, the evolutions of the dynamical behavior of the delay controlled system together with its control parameters are presented numerically using the 4th Runge-Kutta method and the point-to-point mapping method, which confirm the validity of the theoretical prediction. It is found that the chaotic motion and basin erosion can be suppressed effectively by delayed position feedback control when the gain is positive and the time delay is short.
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Keywords:
- relative rotation /
- delayed feedback control /
- safe basin /
- chaos
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[2] Qiao J M, Wnag K, Li X J, Zhang B 2009 Journal of Yanshan University 33 02159 (in Chinese) [乔杰敏, 王坤, 李秀菊, 张波 2009 燕山大学学报 33 02159]
[3] Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1321 (in Chinese) [时培明, 刘彬, 侯东晓 2008 57 1321]
[4] Li H B, Wang B H, Zhang Z Q, Liu S, Li Y N 2012 Acta Phys. Sin. 61 094501 (in Chinese)[李海滨, 王博华, 张志强, 刘爽, 李延树 2012 61 094501]
[5] Meng Z, Fu L Y, Song M H 2013 Acta Phys. Sin. 62 054501 (in Chinese)[孟宗, 付立元, 宋明厚 2013 62 054501]
[6] Wang K 2005 Acta Phys. Sin. 54 5530 (in Chinese) [王坤 2005 54 5530]
[7] 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 59 3648]
[8] Xu J X, Sun Z C 2001 Chin. Phys. 10 599
[9] Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 112
[10] Zhang W M, Li X, Liu S, Li Y Q, Wang B H 2013 Acta Phys. Sin. 62 094502 (in Chinese) [张文明, 李雪, 刘爽, 李雅倩, 王博华 2013 62 094502]
[11] Pyragas K 1993 Physics Letters A 181 203
[12] Wang Z H, Hu H Y 2005 International Journal of Bifurcation and Chaos 15 1787
[13] Xu J, Chung K W, Chan C L 2007 SIAM Journal on Applied Dynamical Systems 6 29
[14] Balanov A G, Janson N B, Schöll E 2005 Physical Review E 71 016222
[15] Shao S, Masri K M, Younis M I 2013 Nonlinear Dynamics 74 247
[16] Thompson J M T, McRobie F A 1993 Proceedings of 1st European Nonlinear Oscillation's Conference Hamburg, German, August 16-20, 1993 p107
[17] Wei D Q, Zhang B, Qiu D Y, Luo X S 2010 Nonlinear Dynamics 61 477
[18] Li S, Li Q, Li J R, Feng J Q 2011 Nonlinear Analysis: Real World Applications 121950
[19] Alsaleem F M Younis M I 2010 Smart Materials and Structures 19 035016
[20] Rong H W, Wang X D, Xu W, Fang T 2008 Journal of Sound and Vibration 313 46
[21] Gan C B 2006 Chaos, Solitions and Fractals 30 920
[22] Shang H L, Xu J 2009 Chaos, Solitons and Fractals 41 1880
[23] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
[24] Shang H L 2012 Acta Phys. Sin. 61 180506 (in Chinese) [尚慧琳 2012 61 180506]
[25] Zhao Y Y, Li C A 2011 Acta Phys. Sin. 60 114305 (in Chinese) [赵艳影, 李昌爱 2011 60 114305]
[26] Thomsen J J, Fidlin A 2003 International Journal of Non-Linear Mechanics 38 389
[27] Tang K T 2007 Mathematical Methods for Engineers and Scientists (New York: Springer-Verlag) pp141-147
[28] Hong L, Xu J X 2000 Acta Phys. Sin. 49 1228 (in Chinese)[洪灵, 徐健学 2000 49 1228]
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[1] Liu H R, Zhou Z L, Shi P M 2010 Acta Phys. Sin. 59 6770 (in Chinese)[刘浩然, 朱占龙, 时培明 2010 59 6770]
[2] Qiao J M, Wnag K, Li X J, Zhang B 2009 Journal of Yanshan University 33 02159 (in Chinese) [乔杰敏, 王坤, 李秀菊, 张波 2009 燕山大学学报 33 02159]
[3] Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1321 (in Chinese) [时培明, 刘彬, 侯东晓 2008 57 1321]
[4] Li H B, Wang B H, Zhang Z Q, Liu S, Li Y N 2012 Acta Phys. Sin. 61 094501 (in Chinese)[李海滨, 王博华, 张志强, 刘爽, 李延树 2012 61 094501]
[5] Meng Z, Fu L Y, Song M H 2013 Acta Phys. Sin. 62 054501 (in Chinese)[孟宗, 付立元, 宋明厚 2013 62 054501]
[6] Wang K 2005 Acta Phys. Sin. 54 5530 (in Chinese) [王坤 2005 54 5530]
[7] 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 59 3648]
[8] Xu J X, Sun Z C 2001 Chin. Phys. 10 599
[9] Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 112
[10] Zhang W M, Li X, Liu S, Li Y Q, Wang B H 2013 Acta Phys. Sin. 62 094502 (in Chinese) [张文明, 李雪, 刘爽, 李雅倩, 王博华 2013 62 094502]
[11] Pyragas K 1993 Physics Letters A 181 203
[12] Wang Z H, Hu H Y 2005 International Journal of Bifurcation and Chaos 15 1787
[13] Xu J, Chung K W, Chan C L 2007 SIAM Journal on Applied Dynamical Systems 6 29
[14] Balanov A G, Janson N B, Schöll E 2005 Physical Review E 71 016222
[15] Shao S, Masri K M, Younis M I 2013 Nonlinear Dynamics 74 247
[16] Thompson J M T, McRobie F A 1993 Proceedings of 1st European Nonlinear Oscillation's Conference Hamburg, German, August 16-20, 1993 p107
[17] Wei D Q, Zhang B, Qiu D Y, Luo X S 2010 Nonlinear Dynamics 61 477
[18] Li S, Li Q, Li J R, Feng J Q 2011 Nonlinear Analysis: Real World Applications 121950
[19] Alsaleem F M Younis M I 2010 Smart Materials and Structures 19 035016
[20] Rong H W, Wang X D, Xu W, Fang T 2008 Journal of Sound and Vibration 313 46
[21] Gan C B 2006 Chaos, Solitions and Fractals 30 920
[22] Shang H L, Xu J 2009 Chaos, Solitons and Fractals 41 1880
[23] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
[24] Shang H L 2012 Acta Phys. Sin. 61 180506 (in Chinese) [尚慧琳 2012 61 180506]
[25] Zhao Y Y, Li C A 2011 Acta Phys. Sin. 60 114305 (in Chinese) [赵艳影, 李昌爱 2011 60 114305]
[26] Thomsen J J, Fidlin A 2003 International Journal of Non-Linear Mechanics 38 389
[27] Tang K T 2007 Mathematical Methods for Engineers and Scientists (New York: Springer-Verlag) pp141-147
[28] Hong L, Xu J X 2000 Acta Phys. Sin. 49 1228 (in Chinese)[洪灵, 徐健学 2000 49 1228]
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