Vanishing twist in the Hamiltonian Hopf bifurcation

PHYSICA D-NONLINEAR PHENOMENA

卷:201

Article

Dullin, HR ; Ivanov, A

关键词: Hamiltonian Hopf bifurcation;    KAM;    isoenergetic non-degeneracy;    vanishing twist;    elliptic integrals;   

DOI  :  10.1016/j.physd.2004.12.004

来源: Elsevier

【 摘 要 】

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 Poincare 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. (C) 2004 Elsevier B.V. All rights reserved.

【 授权许可】

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