Vanishing Twist in the Hamiltonian Hopf Bifurcation

Abstract

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.

18 pages, 4 figures

References in corpus (3)

• Generic Twistless Bifurcations
• Vanishing Twist near Focus-Focus Points
• (Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

Cited by in corpus (7)

• Semi-global symplectic invariants of the spherical pendulum
• Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes
• Taylor series and twisting-index invariants of coupled spin-oscillators
• The 1:+2 / 1:-2 resonance
• Vanishing Twist near Focus-Focus Points
• Generating hyperbolic singularities in completely integrable systems
• (Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation