##
Generic Bifurcation of Hamiltonian Vector Fields with Symmetry

* Nonlinearity * **5** (1992) 979-996

** Michael Dellnitz, Ian Melbourne and Jerrold E. Marsden **

** Abstract **

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 *S*^{1} 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 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 if the eigenvalues split or pass on the imaginary
axis.

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