Reversible quadratic systems

Definitions and mathematical statement

For real parameters $D,F$, consider the reversible quadratic system \[ \dot x=-y+xy,\qquad \dot y=x+Dx^2+Fy^2. \] The equilibrium at the origin is a center. Let $\mathcal A_{D,F}$ be its period annulus, the maximal surrounding family of nonconstant periodic orbits. Choose a smooth transversal meeting each such orbit once and use a regular coordinate $s$ that increases outward. Write $T_{D,F}(s)$ for the least positive time needed to traverse that orbit.

A critical period is an isolated zero of $T'_{D,F}(s)$ at an interior point of the annulus. The problem asks whether \[ \sup_{D,F\in\mathbb R} \#\{\text{critical periods in }\mathcal A_{D,F}\}=2. \] Thus one seeks a universal upper bound of two and parameter choices with two distinct stationary periods. Changes of regular transverse coordinate preserve the count. Changes in the speed along trajectories generally do not preserve the period function and are not allowed.

The claim concerns the entire annulus, including stationary periods far from the center. The center itself and any boundary polycycle are not interior critical periods. Parameters for which the period function is constant are treated as isochronous cases; their continuum of stationary orbits is not counted as a collection of isolated critical periods. This is the global critical-period question for the normalized Loud family, rather than solely the local bifurcation count near an isochronous center.

Short English statement

Is two the sharp global maximum number of isolated critical periods in the reversible quadratic Loud family?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700015 (AMR-046-0015), 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 15: Reversible quadratic systems. Original source indexed by AMR-046-0015. https://arxiv.org/abs/2012.02524.