Vanishing Twist near Focus-Focus Points

Abstract

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.

13 pages

References in corpus (3)

• Generic Twistless Bifurcations
• A Cantor set of tori with monodromy near a focus-focus singularity
• Vanishing Twist in the Hamiltonian Hopf Bifurcation

Cited by in corpus (9)

• Semi-global symplectic invariants of the spherical pendulum
• Taylor series and twisting-index invariants of coupled spin-oscillators
• Foliations of Isonergy Surfaces and Singularities of Curves
• The 1:+2 / 1:-2 resonance
• Symplectic classification of coupled angular momenta
• Vanishing Twist in the Hamiltonian Hopf Bifurcation
• Rotation Forms and Local Hamiltonian Monodromy
• Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory
• Quantum footprints of Liouville integrable systems