$C^\infty$ symplectic invariants of parabolic orbits and flaps in integrable Hamiltonian systems

TL;DR

This paper develops a smooth symplectic classification of certain singularities in integrable Hamiltonian systems, establishing invariants and normal forms for parabolic orbits and flaps.

Contribution

It introduces a complete set of $C^$ symplectic invariants for cusp singularities, parabolic orbits, and flaps, along with real-analytic classification results and normal forms.

Findings

Action variables form a complete set of symplectic invariants.

Complete symplectic invariant in the real-analytic case is a germ of a real-analytic function.

Constructed symplectic normal forms in both $C^$ and real-analytic categories.

Abstract

In the present paper, we consider a smooth $C^\infty$ symplectic classification of Lagrangian fibrations near cusp singularities, parabolic orbits and cuspidal tori. We show that for these singularities as well as for an arrangement of singularities known as a flap, which arises in the integrable subcritical Hamiltonian Hopf bifurcation, the action variables form a complete set of $C^\infty$ symplectic invariants. We also give a symplectic classification for parabolic orbits in the real-analytic case. Namely, we prove that a complete symplectic invariant in this case is given by a real-analytic function germ in two variables. Additionally, we construct several symplectic normal forms in the $C^\infty$ and/or real-analytic categories, including real-analytic right and right-left symplectic normal forms for parabolic orbits.