Vanishing twist in the Hamiltonian Hopf bifurcation

Abstract

The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1 : − 1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus–focus point. We explicitly calculate the frequency (ratio) map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy–momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus–focus point in the image of the energy–momentum map. This is equivalent to the vanishing of twist in a Poincaré map for each energy close to that of the focus–focus point. In addition we show that in a family of periodic orbits (the non-linear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated with non-twist maps near the Hamiltonian Hopf bifurcation.