2003/05/20 by Holger R. Dullin, Alexey V. Ivanov
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Microtubule and mitosis dynamics #Protein Structure and Dynamics #Quantum chaos and dynamical systems #nlin.CD #nlin.SI
paper · pdf · doi:10.1016/j.physd.2004.12.004
published as Physica D, 201:27--44, 2005 · 18 pages, 4 figures
arxiv created 2003/05/20 · openalex publication_date 2005/01/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 near 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 to non-twist maps near the Hamiltonian Hopf bifurcation.