双频信号驱动含分数阶内、外阻尼Duffing振子的振动共振

张路; 谢天婷; 罗懋康

doi:10.7498/aps.63.010506

摘要

本文利用解析和数值的方法研究了由双频周期信号驱动含分数阶内、外阻尼的Duffing振子的振动共振现象，并讨论了分数阶阶数对上述现象的影响. 研究发现：双频周期信号同时驱动的分数阶Duffing振子响应幅值增益Q可随着高频周期激励幅值的改变达到最大值，即出现了和整数阶非线性动力系统相似的振动共振现象，而相应的分数阶导数项则分别为系统提供了内、外两种阻尼力从而导致了系统有效势函数的改变，进而引发了比整数阶动力系统更为丰富的振动共振现象.

关键词:

Abstract

The phenomenon of vibrational resonance (VR) in a Duffing system with both fractional-order external damping and fractional-order intrinsic damping driven by the two-frequency periodic signals is investigated. It is observed that the resonance amplitude Q can be optimized by an appropriate choice of the amplitude of the high-frequency signal. The obtained relationship between VR and the fractional-orders shows that both fractional-order external damping and fractional-order intrinsic damping can induce changes of the shapes of the effective potential function and then lead to more abundant resonance behaviors than in the traditional dynamic systems.

Keywords:

作者及机构信息

1.
四川大学数学学院, 成都 610065;

2.
电子信息控制重点实验室, 成都 610036

基金项目: 国家自然科学基金（批准号：11171238）和国家自然科学基金创新研究群体科学基金（批准号：11221101）资助的课题.

Authors and contacts

1.
Department of Mathematics, Sichuan University, Chengdu 610065, China;

2.
Science and Technology on Electronic Information Control Laboratory, Chengdu 610036, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11171238), and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 11221101).

参考文献

[1]	Landa P S, McClintock P V E 2000 J. Phys. A: Math. Gen. 33 L433
[2]	Gitterman M 2005 Physica A 352 309
[3]	Lin M, Fang L M, Zhu R G 2008 Acta Phys. Sin. 57 2642 (in Chinese) [林敏, 方利民, 朱若谷 2008 物理学报 57 2642]
[4]	Baltans J P 2003 Phys. Rev. E 67 66119
[5]	Wang C J 2011 Chin. Phys. Lett. 28 090504
[6]	He Z Y, Zhou Y R 2011 Chin. Phys. Lett. 28 110505
[7]	Fang C J, Liu X B 2012 Chin. Phys. Lett. 29 050504
[8]	Yang J H, Liu X B 2010 J. Phys. A: Math. Theor. 43 122001
[9]	Guo F, Chen X F 2011 Journal of the Korean Physical Society 56 1567
[10]	Yang J H, Liu H G, Cheng G 2013 Acta Phys. Sin. 62 180503 (in Chinese) [杨建华, 刘后, 广程刚 2013 物理学报 62 180503]
[11]	Yang J H, Liu X.B. 2011 Phys. Scr. 83 065008
[12]	Hilfer R 2003 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[13]	Torvik P J, Bagley R L 1984 J. Appl. Mech. 51 294
[14]	Wang Z H, Hu H Y 2009 Science in China: G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑 39 1495]
[15]	Tofighi A 2003 Physica A 329 29
[16]	Ryabov Y E, Puzenko A 2002 Phys. Rev. B 66 184201
[17]	Narahari Achar B N, Hanneken J W, Clarke T 2002 Physica A 309 275
[18]	Yang J H, Zhu H 2012 Chaos 22 13112
[19]	Petras I 2010 Fractional-order nonlinear system: modeling, analysis and simulation (Beijing: Higher Education Press)

施引文献

[1]	Landa P S, McClintock P V E 2000 J. Phys. A: Math. Gen. 33 L433
[2]	Gitterman M 2005 Physica A 352 309
[3]	Lin M, Fang L M, Zhu R G 2008 Acta Phys. Sin. 57 2642 (in Chinese) [林敏, 方利民, 朱若谷 2008 物理学报 57 2642]
[4]	Baltans J P 2003 Phys. Rev. E 67 66119
[5]	Wang C J 2011 Chin. Phys. Lett. 28 090504
[6]	He Z Y, Zhou Y R 2011 Chin. Phys. Lett. 28 110505
[7]	Fang C J, Liu X B 2012 Chin. Phys. Lett. 29 050504
[8]	Yang J H, Liu X B 2010 J. Phys. A: Math. Theor. 43 122001
[9]	Guo F, Chen X F 2011 Journal of the Korean Physical Society 56 1567
[10]	Yang J H, Liu H G, Cheng G 2013 Acta Phys. Sin. 62 180503 (in Chinese) [杨建华, 刘后, 广程刚 2013 物理学报 62 180503]
[11]	Yang J H, Liu X.B. 2011 Phys. Scr. 83 065008
[12]	Hilfer R 2003 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[13]	Torvik P J, Bagley R L 1984 J. Appl. Mech. 51 294
[14]	Wang Z H, Hu H Y 2009 Science in China: G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑 39 1495]
[15]	Tofighi A 2003 Physica A 329 29
[16]	Ryabov Y E, Puzenko A 2002 Phys. Rev. B 66 184201
[17]	Narahari Achar B N, Hanneken J W, Clarke T 2002 Physica A 309 275
[18]	Yang J H, Zhu H 2012 Chaos 22 13112
[19]	Petras I 2010 Fractional-order nonlinear system: modeling, analysis and simulation (Beijing: Higher Education Press)

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