Singularity Analysis of Duffing-van der Pol System With Two Bifurcation Parameters Under Multi-Frequency Excitations

QIN Zhao-hong; CHEN Yu-shu

doi:10.3879/j.issn.1000-0887.2010.08.009

Volume 31 Issue 8

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2010 > 31(8): 971-978

QIN Zhao-hong, CHEN Yu-shu. Singularity Analysis of Duffing-van der Pol System With Two Bifurcation Parameters Under Multi-Frequency Excitations[J]. Applied Mathematics and Mechanics, 2010, 31(8): 971-978. doi: 10.3879/j.issn.1000-0887.2010.08.009

Citation:

PDF( 1276 KB)

Singularity Analysis of Duffing-van der Pol System With Two Bifurcation Parameters Under Multi-Frequency Excitations

doi: 10.3879/j.issn.1000-0887.2010.08.009

QIN Zhao-hong¹,
CHEN Yu-shu^1,2

School of Astronautics, Harbin Institute of Technology, P. O. Box 137, Harbin 150001, P. R. China;

Received Date: 1900-01-01
Rev Recd Date: 2010-05-29
Publish Date: 2010-08-15

Abstract

Abstract

Bifurcation properties of Duffing-van der Pol System with two parameters under multi-frequency excitations were studied. It was discussed for three cases 1 λ₁ was considered as bifurcation parameter, 2 λ₂ was considered as bifurcation parameter, 3 λ₁ and λ₂ were both considered as bifurcation parameters. According to the definition of transition sets, the whole parametric space was divided into several different persistent regions by the transition sets for different cases. The bifurcation diagrams in different persistent regions were obtained, which could provide a theoretical basis for optmial design of the system.
- Duffing-van der Pol system,
- two bifurcation parameters,
- singularity analysis

FullText(HTML)

References(18)

References

[1]	Nafyeh A H, Mook D L. Nonlinear Oscillations[M]. New York: Wiley Interscience, 1979: 325-328.
[2]	Lim C W, Wu B S. A new analytical approach to the Duffing-harmonic oscillator [J]. Physics Letters A, 2003, 311(4/5): 365-373. doi: 10.1016/S0375-9601(03)00513-9
[3]	Hu H, Tang J H. Solution of a Duffing-harmonic oscillator by the method of harmonic balance [J]. Journal of Sound and Vibration, 2006, 294(3): 637-639. doi: 10.1016/j.jsv.2005.12.025
[4]	Lim C W, Wu B S, Sun W P. Higher accuracy analytical approximations to the Duffing-harmonic oscillator [J]. Journal of Sound and Vibration, 2006, 296(4/5): 1039-1045. doi: 10.1016/j.jsv.2006.02.020
[5]	Rand R H, Holmes P J. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators [J]. International Journal of Non-Linear Mechanics, 1980, 15(4/5): 387-399. doi: 10.1016/0020-7462(80)90024-4
[6]	Mettin R, Parlitz U, Lauterborn W. Bifurcation structure of the driven van der Pol oscillator [J]. International Journal of Bifurcation and Chaos, 1993, 3(6): 1592-1555.
[7]	Wirkus S, Rand R. The dynamics of two coupled van der Pol oscillators with delay coupling [J]. Nonlinear Dynamics, 2004, 30(3): 205-221.
[8]	Acunto M D. Determination of limit cycles for a modified van der Pol oscillator [J]. Mechanics Research Communications, 2006, 33(1): 93-98. doi: 10.1016/j.mechrescom.2005.06.012
[9]	陈予恕. 非线性振动 [M]. 北京: 高等教育出版社, 2002: 201-208.
[10]	Woafo P, Chedjou J C, Fotsin H B. Dynamics of a system consisting of a van der Pol oscillator coupled to a Duffing oscillator [J]. Physical Review E, 1996, 54: 5929-5934. doi: 10.1103/PhysRevE.54.5929
[11]	Ji J C, Hansen C H. Stability and dynamics of a controlled van der Pol-Duffing oscillator [J]. Chaos, Solutions & Fractals, 2006, 28(2): 555-570.
[12]	Jing Z J, Yang Z Y, Jiang T. Complex dynamics in Duffing-Van der Pol equation [J]. Chaos, Solutions & Fractals, 2006, 27(3): 722-747.
[13]	Holmes P, Rand D. Phase portraits and bifurcation of nonlinear oscillator: x+(α+γx2)+βx+δx3=0[J]. International Journal of Nonlinear Mechanics, 1980, 15(6): 449-458. doi: 10.1016/0020-7462(80)90031-1
[14]	Maccari Attilio. Approximate solution of a class of nonlinear oscillatiors in resonance with a periodic excitation [J]. Nonlinear Dynamics, 1998, 15(4): 329-343. doi: 10.1023/A:1008235820302
[15]	董建宁, 申永军, 杨绍普. 多频激励作用下Duffing-van der Pol系统的分岔分析 [J]. 石家庄铁道学院学报, 2006, 19(1): 62-66.
[16]	Golubistky M, Schaeffer D G. Singularities and Groups in Bifurcation Theory[M]. VolⅠ,Ⅱ. New York: Spring-Verlag, 1985,1988.
[17]	Qin Z H, Chen Y S. Singular analysis of bifurcation systems with two parameters [J]. Acta Mechanica Sinica, 2010, 26(3): 501-507. doi: 10.1007/s10409-010-0334-7
[18]	Chen Y S, Leung A Y T. Bifurcation and Chaos in Engineering[M]. London: Springer, 1998.