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Generic bifurcation of Hamiltonian vector fields with symmetry

Dellnitz, Michael and Melbourne, Ian and Marsden, Jerrold E. (1992) Generic bifurcation of Hamiltonian vector fields with symmetry. Nonlinearity, 5 (4). pp. 979-996. ISSN 0951-7715. doi:10.1088/0951-7715/5/4/008.

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One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. We classify this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart) or definiteness properties of the Hamiltonian (Cartan-Oh) to determine whether the eigenvalues split or pass on the imaginary axis.

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Additional Information:©1992 IOP Publishing Ltd. and LMS Publishing Ltd. We would like to thank Marty Golubitsky for very helpful discussions and suggestions. Research supported by the Deutsche Forschungsgemeinschaft and by NSFJDARPA DMS-87W897. Supported in part by NSFJDARPA DMS-8700897. Research partially supported by a Humboldt award.
Issue or Number:4
Record Number:CaltechAUTHORS:DELnonlin92
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:587
Deposited By: Tony Diaz
Deposited On:31 Aug 2005
Last Modified:08 Nov 2021 19:03

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