Period functions for systems with homogeneous components

Definitions and mathematical statement

Fix nonnegative integers $k,\ell$ and nonzero real homogeneous polynomials $P_{2k+1},Q_{2\ell+1}$ of the indicated degrees. Consider \[ \dot x=P_{2k+1}(x,y),\qquad \dot y=Q_{2\ell+1}(x,y). \] The origin is a center if it is an isolated equilibrium surrounded by a punctured neighborhood foliated by periodic orbits. Its period annulus is the maximal family of those surrounding closed orbits. Parameterize the orbits by a smooth transversal, increasing away from the origin, and let $T(s)$ be their least positive periods in the original time variable.

First, characterize the polynomial coefficient pairs for which the origin is a center. Second, for each degree pair, determine the maximum number of oscillations of $T$ on its full period annulus. In the critical-period formulation this means bounding the isolated zeros of $T'(s)$; a change of regular transverse coordinate does not change these zeros. A constant period function is isochronous and is treated separately, not counted as having infinitely many isolated critical periods.

The classification includes degenerate equilibria and is not restricted to perturbations of a linear center. When $k=\ell$, homogeneity makes the center condition and period scaling explicit; the genuinely different-degree case is the main unresolved part. Statements about local extrema arbitrarily close to the origin do not settle the requested global count, since further stationary periods may occur near the outer boundary of the annulus.

Short English statement

Classify centers with unequal homogeneous component degrees and determine how often their periods can change monotonicity.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700013 (AMR-046-0013), 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 13: Period functions for systems with homogeneous components. Original source indexed by AMR-046-0013. https://arxiv.org/abs/2012.02524.