A class of Hamiltonian systems

Definitions and mathematical statement

Let $n\ge1$ and $m>2n$ be integers, and let $H_{2n},H_m\in\mathbb R[x,y]$ be homogeneous polynomials of the indicated degrees. Set \[ H=H_{2n}+H_m,\qquad \dot x=H_y,\qquad \dot y=-H_x, \] and assume that the origin is a center: a punctured neighborhood is filled by periodic orbits surrounding this isolated equilibrium. Its period annulus is the maximal connected family of such surrounding periodic orbits. On each regular level oval $\gamma_h$ in that annulus, let $T(h)>0$ be the least positive period of the Hamiltonian flow.

A critical period means an isolated stationary point of this function: $T'(h)=0$ at an interior energy value. Equivalently one may use a smooth transverse parameter, since stationary points are unchanged by a regular reparametrization. The question is whether there is at most one critical period throughout the full period annulus of the origin.

The count concerns the actual Hamiltonian time, and a nonconstant time rescaling is not allowed. Degenerate centers are included, so their periods may diverge as the energy approaches the center. Endpoints corresponding to the center or the outer boundary are not counted as interior critical periods. The source proves the bound in the range $m\ge4n-2$; the remaining range is $2n<m<4n-2$, beginning with a quartic-plus-quintic Hamiltonian. These degree restrictions do not impose any extra sign assumption beyond the stated existence of a center.

Short English statement

Does the period annulus of a center for a two-term homogeneous Hamiltonian have at most one critical period?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700012 (AMR-046-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 12: A class of Hamiltonian systems. Original source indexed by AMR-046-0012. https://arxiv.org/abs/2012.02524.