A reversible bifurcation analysis of the inverted pendulum

Physica D: Nonlinear Phenomena

Volumn 112, Issue 1-2, 1998, Pages 50-63

  Broer, H W ^a   Hoveijn, I ^a   Van Noort, M ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Equivariant singularity theory; Hamiltonian system; Normal form theory; Parametrically forced oscillator; Spatio temporal symmetry

Indexed keywords

EID: 0002954939     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00201-7     Document Type: Article

References (29)

1. V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1980.
2. V.I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, Berlin, 1983.
3. A. Back, J. Guckenheimer, M.R. Myers, F.J. Wicklin, P.A. Worfolk, Dstool: Computer assisted exploration of dynamical systems, Notices Amer. Math. Soc. 39 (1992) 303-309.
4. H.W. Broer, Formal Normal Form Theorems for Vector Fields and Some Consequences for Bifurcations in the Volume Preserving Case, Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 1981.
5. H.W. Broer, Notes on perturbation theory, Erasmus ICP Mathematics and Fundamental Applications, Aristotle University Thessaloniki (1993), 1991.
6. H.W. Broer, S.-N. Chow, Y.-I. Kim, G. Vegter, A normally elliptic hamiltonian bifurcation, ZAMP 44 (1993) 389-432.
7. H.W. Broer, S.-N. Chow, Y.-I. Kim, G. Vegter, The hamiltonian double-zero eigenvalue, Fields Institute Commun. 4 (1995).
8. H.W. Broer, I. Hoveijn, M. van Noort, G. Vegter, in preparation.
9. H.W. Broer, M. Levi, Geometrical aspects of stability theory for Hill's equations, Archive Rat. Mech. Ann. 131 (1995).
10. H.W. Broer, G.A. Lunter, G. Vegter, Equivariant singularity theory with distinguished parameters: two case studies of resonant Hamiltonian systems, Physica D 112 (1997) 64-80.
11. H.W. Broer, F. Takens, Formally symmetric normal forms and genericity, Dynamics Reported 2 (1989).
12. H.W. Broer, G. Vegter, Bifurcational aspects of parametric resonance, Dynamics Reported 1 (New Series) (1992).
13. K.T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963).
14. M. Golubitsky, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer, Berlin, 1985.
15. F.G. Gustafson, On constructing formal integrals of a hamiltonian system near an equilibrium point, Astronomical J. 71 (1966).
16. J.K. Hale, Ordinary Differential Equations, 2nd ed., Krieger, Ney York, 1980.
17. I. Hoveijn, Aspects of resonance in dynamical systems, PhD Thesis, Rijksuniversiteit Utrecht, 1992.
18. I. Hoveijn, B. Krauskopf, The symmetry of a poincaré map as an isotropy group, Preprint W-9613, Groningen (1996).
19. M. Levi, Stability of the inverted pendulum - a topological explanation, SIAM Review 30 (4) (1988).
20. J. Martinet, Singularities of Smooth Functions and Maps, London Mathematical Society Lecture Notes Series, Vol. 58, Cambridge University Press, Cambridge, 1982.
21. J. Meixner, F.W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen, Springer, Berlin, 1954.
22. K. Meyer, Generic Bifurcations in Hamiltonian Systems, Lecture Notes in Mathematics, Vol. 468, Springer, Berlin, 1975.
23. R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam, 1968.
24. J.A. Sanders, F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer, Berlin, 1985.
25. J. Stoker, Nonlinear Vibrations, Interscience, New York, 1950.
26. F. Takens, Forced oscillations and bifurcations, Applications of global analysis I, Comm. Math. Inst. University of Utrecht 3 (1974) 1-59.
27. F. Takens, Singularities of vector fields, Publ. Math. I.H.E.S. 43 (1974).
28. J.C. van der Meer, The Hamiltonian Hopf Bifurcation, Springer, Berlin, 1985.
29. B. van der Pol, M.J.O. Strutt, On the stability of the solution of Mathieu's equation, The London, Edinburgh and Dublin Philosophical Magazine (7th series) 5 (1928).