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Vanishing twist in the Hamiltonian Hopf Bifurcation
preprintposted on 2005-07-29, 14:22 authored by Holger R. Dullin, A.V. Ivanov
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 Poincar´e map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the nonlinear 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.
- Mathematical Sciences
NotesThis pre-print has been submitted to the journal, Physica D - NonLinear Phenomena [© Elsevier]. The definitive version: DULLIN, H.R. and IVANOV, A.R., 2005. Vanishing twist in the Hamiltonian Hopf Bifurcation. Physica D - NonLinear Phenomena, 201(1-2), pp. 27-44 is available at http://www.sciencedirect.com/science/journal/01672789.