Vanishing Twist in the Hamiltonian Hopf Bifurcation

Details Vanishing Twist in the Hamiltonian Hopf Bifurcation Vanishing Twist in the Hamiltonian Hopf Bifurcation

2003-05-20

arxiv.org/abs/nlin/0305037v1

Physica D, 201:27--44, 2005

Nonlinear Sciences

Chaotic Dynamics

18 pages, 4 figures

Scientific paper

10.1016/j.physd.2004.12.004

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 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.

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