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# Vanishing twist in the Hamiltonian Hopf Bifurcation

preprint

posted on 2005-07-29, 14:22 authored by Holger R. Dullin, A.V. IvanovThe 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.

## History

## School

- Science

## Department

- Mathematical Sciences

## Pages

530219 bytes## Publication date

2003## Notes

This 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.## Language

- en