Analytic invariants for the $1:-1$ resonance

Details Analytic invariants for the $1:-1$ resonance Analytic invariants for the $1:-1$ resonance

2011-05-06

arxiv.org/abs/1105.1349v1

Mathematics

Dynamical Systems

46 pages, 2 figures

Scientific paper

Given an analytic Hamiltonian vector field in $\mathbb{C}^4$ having an equilibrium point satisfying a $1:-1$ resonance, i.e. the eigenvalues of the linearized vector field form a double pair $\pm i \alpha$ with $\alpha>0$, we construct two universal constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the associated Hamiltonian is integrable and moreover we prove that do not vanish on an open and dense set.

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