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Stability of periodic solutions near a collision of eigenvalues of opposite signature

Published online by Cambridge University Press:  24 October 2008

Thomas J. Bridges
Thomas J. Bridges
Affiliation:
Mathematical Institute, University of Warwick, Coventry CV4 7AL
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Abstract

Some general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 375 - 403
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Benjamin, T. B.. Stable and unstable solitary wave solutions of generalized KdV and BBM equations. (Preprint, Oxford University.)Google Scholar
[2]Benjamin, T. B. and Bridges, T. J.. Reappraisal of the Kelvin–Helmholtz problem. 2. Travelling waves and bifurcations. (Preprint, Oxford University.)Google Scholar
[3]Bridges, T. J.. Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature. Math. Proc. Cambridge Philos. Soc. 108 (1990), 575–601.CrossRefGoogle Scholar
[4]Deimling, K.. Nonlinear Functional Analysis (Springer-Verlag, 1985).CrossRefGoogle Scholar
[5]Deprit, A. and Henrard, J.. A manifold of periodic orbits. Adv. Astronomy and Astrophysics 6 (1968), 1–124.CrossRefGoogle Scholar
[6]Golubitsky, M. and Schaeffer, D.. Singularities and Groups in Bifurcation Theory, vol. 1. Appl. Math. Sci. no. 51 (Springer-Verlag, 1985).CrossRefGoogle Scholar
[7]Mawhin, J. and Willem, M.. Critical Point Theory and Hamillonian Systems. Appl. Math. Sci. no. 74 (Springer-Verlag, 1989).CrossRefGoogle Scholar
[8]van der Meer, J.. Nonsemisimple 1: 1 resonance at an equilibrium. Celest. Mech. 27 (1982), 131–149.CrossRefGoogle Scholar
[9]van der Meer, J.. The Hamiltonian Hopf Bifurcation. Lecture Notes in Math. vol. 1160 (Springer-Verlag, 1985).CrossRefGoogle Scholar
[10]Meyer, K. R. and Schmidt, D. S.. Periodic orbits near L 4 for mass ratios near the critical mass ratio of Routh. Celest. Mech. 4 (1971), 99–109.CrossRefGoogle Scholar
[11]Poincaré, H.. Les Méthodes Nouvelles de la Méchanique Céleste (NASA Translation TT/F–450–452, 1967).Google Scholar
[12]Sokol'skij, A. G.. On the stability of an autonomous Hamiltonian system with two degrees of freedom in the case of equal frequencies. J. Appl. Math. Mech. 38 (1974), 791–799.Google Scholar