Generic Bifurcation of Hamiltonian Vector Fields with Symmetry

Dellnitz, M., I. Melbourne and J. E. Marsden

Nonlinearity, 5, 979-996

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 (linearized) Hamiltonian system. 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 S¹ symmetry, if the representation on an associated four dmensional symplectic space consists of two complex dual subspaces, then generically the eigenvalues split if the Hamiltonian is indefinite, and they pass if the Hamiltonian is definite. 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. The results are illustrated for the rotating orthogonal double planar pendulum.