Reversible equivariant planar differential systems

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Identify the plane with $\mathbb C$ by $z=x+iy$, and fix integers $n,k\ge1$. Consider the real autonomous differential equation \[\dot z=iz+(z\bar z)^n z^{k+1}.\] Its reversible center at the origin is surrounded by a period annulus, the maximal connected annular region filled with periodic orbits that shrink toward the origin. Parameterize these orbits by their intersection $z=r>0$ with the positive real ray, ordered from the center outward. Write $T(r)$ for the minimal positive period in the original time variable. Monotone decrease means $T(r_2)\le T(r_1)$ for any $r_1<r_2$ in this annulus.

Statement recorded in the supplied source.

For every pair $n,k\ge1$, is the period function $T$ monotonically decreasing throughout the period annulus of the origin? [S2]

Short English statement

Does the oscillation period decrease from the center outward for every member of this reversible complex polynomial family?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4700016 (AMR-046-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Gasull, Some open problems in low dimensional dynamical systems (2020), Section 3 introductory period definitions; Section 3.5, equation (15), Proposition 3.1 and Problem 16, pp.13,15--16. https://arxiv.org/pdf/2012.02524.