Some aspects of destabilization in reversible dynamical systems with application to follower forces

Nonlinear Dynamics

Volumn 10, Issue 1, 1996, Pages 63-87

  O'Reilly, O M ^a   Malhotra, N K ^b   Namachchivaya, N S ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Dissipation induced destabilization; Local bifurcations; Normal forms; Reversible dynamical systems

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; INVARIANCE; MATHEMATICAL TRANSFORMATIONS; PERTURBATION TECHNIQUES; STABILITY CRITERIA; SYSTEM STABILITY;

DESTABILIZATION; FOLLOWER FORCE PROBLEMS; IRREVERSIBLE PERTURBATION; REVERSIBLE DYNAMICAL SYSTEMS;

CONTROL SYSTEM SYNTHESIS;

EID: 0030151087     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00114799     Document Type: Article

References (27)

1. Arnol'd, V. I., Geometric Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.
2. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.
3. Bibikov, Y. N., Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, Vol. 702, Springer-Verlag, New York, 1979.
4. Birkhoff, G. D., Dynamical Systems, American Mathematical Society, Providence, Rhode Island, 1927.
5. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., and Ratiu, T. S., 'Dissipation induced instabilities', Annales de L'Institut H. Poincaré: Analyse Non Linéaire 11, 1994, 37-90.
6. Devaney, R. L., 'Reversible diffeomorphisms and flows', Transactions of the American Mathematical Society 218, 1976, 89-113.
7. Fu, F. C. L. and Nemat-Nasser, S., 'Stability of solutions of systems of linear differential equations with harmonic coefficients', AIAA Journal 10, 1972, 30-36.
8. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 2nd revised and corrected printing, 1986.
9. Haller, G., 'Gyroscopic stability and its loss in systems with two essential coordinates', International journal of Non-Linear Mechanics, 27, 1992, 113-127.
10. Helmholtz, H., 'Über die physikalische Bedeutung des Princips der kleinsten Wirkung', Crelle Journal fü die reine und angewandte Mathematik 100, 1887, 137-166 and 213-222.
11. Herrmann, G. and Bungay, R. W., 'On the stability of elastic systems subjected to nonconservative forces', ASME Journal of Applied Mechanics 31, 1964, 435-440.
12. Herrmann, G. and Jong, I. Ch., 'On the destabilizing effect of damping in nonconservative elastic systems', ASME Journal of Applied Mechanics 32, 1965, 592-597.
13. Leipholz, H. H., Stability of Elastic Systems, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980.
14. MacKay, R. S., 'Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation', Physics Letters 155A, 1991, 266-268.
15. Moser, J., Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, New Jersey, 1973.
16. Namachchivaya, N. S., Doyle, M. M., Langford, W. F., and Evans, N.W., 'Normal form for generalized Hopf bifurcation with non-semisimple 1:1 resonance', Journal of Applied Mathematics and Physics (ZAMP) 45, 1994, 312-335.
17. Namachchivaya, N. S. and Malhotra, N. K., 'Normal forms and homoclinic chaos, application to structural systems', Fields Institute Communications, 1995, in press.
18. Roberts, J. A. G. and Quispel, G. R. W., 'Chaos and time-reversal symmetry', Physics Reports 216(2/3), 1992, 63-177.
19. Sevryuk, M. B., 'Lower dimensional tori in reversible systems', Chaos 1, 1991, 160-167.
20. Shahruz, S. M. and Ma, F., 'Approximate decoupling of the equations of motion of linear underdamped systems', ASME Journal of Applied Mechanics 110, 1988, 716-720.
21. Takens, F., 'Singularities of vector fields', Publications Mathématique de l'Institut des Hautes Etudes Scientifique 43, 1974, 47-100.
22. Tkhai, V. N., 'On stability of mechanical systems under the action of position forces', Journal of Applied Mathematics and Mechanics (English translation of P.M.M.) 44, 1981, 24-29.
23. Tkhai, V. N., 'The reversibility of mechanical systems', Journal of Applied Mathematics and Mechanics (English translation of P.M.M.) 55, 1991, 461-468.
24. Tompson, W. and Tait, P. G., Treatise on Natural Philosophy, Cambridge University Press, Cambridge, 1912 reprinted edition.
25. van Gils, S. A., Krupa, M., and Langford, W. F., 'Hopf bifurcation with non-semisimple 1:1 resonance', Nonlinearity 13, 1990, 825-850.
26. Wehrli, Ch. and O'Reilly, O., Nonlinear Vibrations, ETH-Zurich, Institut für Mechanik (unpublished lecture notes), 1992.
27. Ziegler, H., 'Die Stabilitatskriterien der Elastomechanik', Ingenieur-Archiv 20, 1952, 49-56.