Degenerate bifurcation analysis on a parametrically and externally excited mechanical system

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 3, 2001, Pages 689-709

  Zhang, Wei ^a,b   Yu, Pei ^a  

^a UNIVERSITY OF WESTERN ONTARIO   (Canada)

^b BEIJING UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041082846     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812740100233X     Document Type: Article

References (26)

1. Abou-Rayan, A. M., Nayfeh, A. H. & Mook, D. T. [1993] "Nonlinear response of a parametrically excited buckled beam," Nonlin. Dyn. 4, 499-525.
2. Bajaj, A. K. [1987] "Bifurcations in a parametrically excited nonlinear oscillator," Int. J. Nonlin. Mech. 22, 47-59.
3. Blocki, W. [1980] "Ship safety in connection with parametric resonance of the roll," Int. Shipbuild. Progr. 27, 36-53.
4. Chia, C. Y. [1980] Non-linear Analysis of Plate (McGraw-Hill Inc.).
5. Dumortier, F., Roussarie, R. & Sotomayor, J. [1987] "Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case," Ergodic Th. Dyn. Syst. 7, 375-413.
6. Feat, G. & Jones, D. [1984] "Parametric excitation and the stability of a ship subjected to a steady heeling moment," Int. Shipbuild. Progr. 28, 262-267.
7. Joyal, P. [1994] "The cusp of order N," J. Diff. Eqs. 88, 1-14.
8. Li, C. & Rousseau, C. [1989] "A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4," J. Diff. Eqs. 79, 132-167.
9. Li, C. & Rousseau, C. [1990] "Codimension 2 symmetric bifurcations and application to 1:2 resonance," Can. J. Math. XLII, 191-212.
10. Nayfeh, A. H. & Mook, D. T. [1979] Nonlinear Oscillations (Wiley-Interscience, NY).
11. Nayfeh, A. H. & Chin, C. M. [1995] "Nonlinear interactions in a parametrically excited system with wide spaced frequencies," Nonlin. Dyn. 7, 195-216.
12. Petrov, G. S. [1984] "Number of zeroes of complete elliptic integrals," Func. Anal. Appl. 18, 148-150.
13. Petrov, G. S. [1986] "Elliptic integrals and their nonoscillation," Func. Anal. Appl. 20, 37-40.
14. Petrov, G. S. [1987] "Complex zeroes of an elliptic integral," Func. Anal. Appl. 21, 247-248.
15. Rousseau, C. & Zoladak, H. [1991] "Zeroes of complete elliptic integrals for 1:2 resonance," J. Diff. Eqs. 94, 41-54.
16. Sanchez, N. E. & Nayfeh, A. H. [1990] "Nonlinear rolling motions of ships in longitudinal waves," Int. Shipbuild. Progr. 37, 247-272.
17. Sanchez, N. E. & Nayfeh, A. H. [1991] "Rolling of biased ships in quartering seas," 18th Symp. Naval Hydrodynamics, Ann Arbor, pp. 133-139.
18. Sethna, P. R. & Shaw, S. W. [1987] "On codimension-three bifurcations in the motiom of articulated tubes conveying a fluid," Physica D24, 305-327.
19. Shi, S. L. [1981] "A method of constructing cycles without contact around a weak focus," J. Diff. Eqs. 41, 301-312.
20. Yu, P. & Huseyin, K. [1980] "Static and dynamic bifurcation associated with a double zero eigenvalue," Dyn. Stab. Syst. 1, 73-86.
21. Yu, P. & Huseyin, K. [1988] "Bifurcations associated with a three-fold zero eigenvalue," Quart. Appl. Math. 46, 193-216.
22. Zhang, W. & Ye, M. [1994] "Local and global bifurcations of valve mechanism," Nonlin. Dyn. 6, 301-316.
23. 2-symmetry," Acta Mech. Sinica 25, 548-559.
24. Zhang, W. & Chen, Y. S. [1998] "Local and global bifurcations in the rolling motions of the ship," Proc. Third Int. Conf. Nonlinear Mechanics, ed. Chien, W. Z. (Shanghai University Press), pp. 625-630.
25. Zhang, W. & Yu, P. [2000] "A study of the limit cycles associated with a generalized codimension-3 Lienard oscillator," J. Sound Vibr. 231, 145-173.
26. Zhang, Z. F., Ding, T. R., Huang, W. Z. & Dong, Z. X. [1985] Qualitative Theory of Differential Equations (Science Press, Beijing).