On the slow transition across instabilities in non-linear dissipative systems

Journal of Sound and Vibration

Volumn 192, Issue 4, 1996, Pages 835-865

  Raman, A ^a   Bajaj, A K ^b   Davies, P ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); INTEGRATION; MATHEMATICAL MODELS; NONLINEAR EQUATIONS; NONLINEAR SYSTEMS; OSCILLATORS (MECHANICAL); STABILITY;

DUFFINGS OSCILLATOR; HOPF BIFURCATION; NONLINEAR DISSIPATIVE SYSTEMS; NONLINEAR MATHIEU EQUATION; PARKINSON SMITH MODEL; PITCHFORD BIFURCATION;

VIBRATIONS (MECHANICAL);

EID: 0030148036     PISSN: 0022460X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsvi.1996.0221     Document Type: Article

References (29)

1. R. HABERMAN 1979 SIAM Journal on Applied Mathematics 37, 69-106. Slowly varying jump and transition phenomena associated with algebraic bifurcation problems.
2. V. A. MITROPOLSKII 1965 Problems of the Asymptotic Theory of Nonstationary Vibrations. Jerusalem: Israel Program for Scientific Translations Ltd.
3. N. N. BOGOLIUBOV and V. A. MITROPOLSKII 1961 Asymptotic Methods in the Theory of Non-linear Oscillations. Delhi: Hindustan Publishing.
4. R. M. EVAN-IWANOWSKI 1976 Resonance Oscillations in Mechanical Systems. New York: Elsevier Scientific.
5. A. H. NAYFEH and K. R. ASFAR 1988 Journal of Sound and Vibration 124, 529-537. Non-stationary parametric oscillations.
6. H. L. NEAL and A. H. NAYFEH 1990 International Journal of Non-Linear Mechanics 25, 275-284. Response of a single-degree-of-freedom system to a non-stationary principal parametric excitation.
7. M. TRAN and R. EVAN-IWANOWSKI 1990 International Journal of Non-Linear Mechanics 25, 285-297. Non-stationary responses of self-excited driven systems.
8. N. R. LEBOVITZ and R. J. SCHAAR 1975 Studies in Applied Mathematics 54, 229-260. Exchange of stabilities in autonomous systems.
9. N. R. LEBOVITZ and R. J. SCHAAR 1975 Studies in Applied Mathematics 56, 1-50. Exchange of stabilities in autonomous systems - II. vertical bifurcation.
10. A. I. NEISHTADT 1988 Differential Equations 23, 1385-1391. Persistance of stability loss for dynamical systems.
11. T. ERNEUX, E. L. REISS, L. J. HOLDEN and M. GIORGIOU 1991 in Dynamic Bifurcations, Lecture Notes in Mathematics. Berlin: Springer-Verlag. Slow passage through bifurcations and limit points.
12. T. ERNEUX and P. MANDEL 1979 Journal of Statistical Physics 5/6, 1059-1070. The slow passage through a steady bifurcation - delay and memory effects.
13. F. DIENER and M. DIENER 1991 in Dynamic Bifurcations, Lecture Notes in Mathematics. Berlin: Springer-Verlag. Maximal delay.
14. R. O'MALLEY 1991 Singular Perturbation Methods for Ordinary Differential Equations. New York: Springer-Verlag.
15. S. WIGGINS 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag.
16. J. CARR 1981 Applications of Center Manifold Theory. New York: Springer-Verlag.
17. I. GEORGIOU 1993 Ph.D. Thesis, Purdue University. Non-linear dynamics and chaotic motions of a singularly perturbed viscoelastic beam.
18. H. G. DAVIES and R. KRISHNA 1993 Preprint. Non-stationary response near generic bifurcations - I and II.
19. E. L. INCE 1956 Ordinary Differential Equations. New York: Dover.
20. M. ABRAMOWITZ and I. A. STEGUN 1970 Handbook of Mathematical Functions. Oxford: Clarendon Press, ninth edition.
21. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: John Wiley and Sons Ltd.
22. G. N. WATSON and E. T. WHITTAKER 1940 A Course of Modern Analysis. Cambridge: Cambridge University Press, fourth edition.
23. V. V. BOLOTIN 1964 The Dynamic Stability of Elastic Systems. San Francisco: Holden-Day.
24. J. M. T. THOMPSON 1982 Instabilites and Catastrophes in Science and Engineering. New York: John Wiley and Sons Ltd.
25. G. V. PARKINSON and N. P. H. BROOKS 1961 Journal of Applied Mechanics 28, 252-259. On the aeroelastic instability of bluff cylinders.
26. G. V. PARKINSON and J. D. SMITH 1965 Quarterly Journal of Mechanics and Applied Mathematics 17, 225. The square prism as an aeroelastic non-linear oscillator.
27. M. NOVAK 1971 Proceedings of the Third International Conference on Wind Effects on Buildings and Structures. Galloping and vortex induced oscillations of structures.
28. C. BAESENS 1991 Physica D 53, 319-375. Slow sweep through a period-doubling cascade: delayed bifurcations and renormalization.
29. T. ERNEUX and P. MANDEL 1986 SIAM Journal on Applied Mathematics 46, 1-15. Imperfect bifurcation with a slowly-varying control parameter.