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Bifurcation 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

When two purely imaginary eigenvalues of opposite Krein signature coalesce, in a Hamiltonian system, a small perturbation can drive them off of the imaginary axis resulting in a linear instability. The most celebrated example of this instability occurs in the restricted 3-body problem at Routh's critical mass ratio. In this paper the collision of eigenvalues is treated as a singularity. A variational form of the Lyapunov–Schmidt method and distinguished parameter ࡃ2-equivariant singularity theory, with the frequency as distinguished parameter, are used to determine the effect of the degeneracy on the branches of periodic solutions in a neighbourhood. Previous results of Meyer and Schmidt[13], Sokol'skij [16] and van der Meer [12] are recovered in the formulation as a co-dimension 1 singularity. The results are extended to include the effect of an additional degeneracy (a co-dimension 2 singularity). The theory is applied to a spinning double pendulum.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 575 - 601
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

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