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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.

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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

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