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BRIEF NOTES

Onset of Degenerate Hopf Bifurcation of a Vibro-Impact Oscillator

[+] Author and Article Information
GuiLin Wen

School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798Department of Applied Mechanics, Southwest Jiaotong University, Chengdu 610031, P.R. China

JianHua Xie

Department of Applied Mechanics, Southwest Jiaotong University, Chengdu 610031, P.R. China

Daolin Xu

School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798

J. Appl. Mech 71(4), 579-581 (Sep 07, 2004) (3 pages) doi:10.1115/1.1767163 History: Received July 04, 2002; Revised January 27, 2004; Online September 07, 2004
Copyright © 2004 by ASME
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References

Luo,  G. W., and Xie,  J. H., 1998, “Hopf Bifurcation of a Two-Degree-of-freedom Vibrato Impact System,” J. Sound Vib., 213(3), pp. 391–408.
Luo,  G. W., and Xie,  J. H., 2002, “Hopf Bifurcation and Chaos of a Two-Degree-of-Freedom Vibro-Impact System in Two Strong Resonance Cases,” Int. J. Non-Linear Mech., 37, pp. 19–34.
Aidanpaa,  J. O., and Gupta,  R. B., 1992, “Periodic and Chaotic Behavior of a Threshold-Limited Two-Degree-of-Freedom System,” J. Sound Vib., 165(2), pp. 305–327.
Luo,  G. W., Xie,  J. H., and Guo,  S. H. L., 2001, “Periodic Motions and Global Bifurcations of a Two-degree-of-Freedom System With Plastic Vibro-Impact,” J. Sound Vib., 240(5), pp. 837–858.
Chenciner,  A., 1985, “Bifurcations de Points Fixes Elliptiques I-Courbes Invariantes,” IHES Pub. Math., 61, pp. 67–127.
Carr, J., 1981, Applications of Center Manifold Theory (Applied Mathematical Sciences 35), Springer-Verlag, New York, pp. 33–36.
Iooss, G., 1979, Bifurcation of Maps and Applications (Mathematics Studies 36), North-Holland, Amsterdam.
Wen,  G. L., and Xu,  D., 2003, “Control of Degenerate Hopf Bifurcations in Three-Dimensional Maps,” Chaos, 13(2), pp. 486–494.

Figures

Grahic Jump Location
A two-degree-of-freedom impact oscillator
Grahic Jump Location
Degenerate Hopf bifurcation diagram against the parameters (α, μ)
Grahic Jump Location
Coexisting Hopf circles and a fixed point resulted from a degenerate Hopf bifurcation at ω=0.7297(μ<0) and α=0.0313, where the unstable circle separates the stable circle from the stable fixed point. The symbol ‘×’ denotes the location of the last iteration of the Poincaré map (5). A view on the projected section (x2,ẋ2).

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