Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture

TL;DR

This paper characterizes trivial isochronous centers in planar polynomial Hamiltonian systems of odd degrees, revealing a triangular family for all odd degrees and a quadratic-shear family emerging at degrees congruent to 3 mod 4, starting from degree 7.

Contribution

It formalizes the existence of a triangular family of trivial isochronous centers for all odd degrees and identifies a quadratic-shear family appearing specifically at degrees congruent to 3 mod 4.

Findings

Triangular family yields trivial isochronous centers in all odd degrees.

Quadratic-shear family appears at degrees n ≡ 3 mod 4, starting from n=7.

Second branch cannot occur at n=9 due to degree parity constraints.

Abstract

We revisit the characterization of \emph{trivial} isochronous centers for planar polynomial Hamiltonian systems in degrees $5$ and $7$ obtained by Braun--Llibre--Mereu, and we formalize two conclusions suggested by their method. First, a \emph{triangular family} yields trivial (indeed global) isochronous centers in every odd degree $n=2k-1\geq3$. Second, a genuinely different \emph{quadratic--shear} ($Q$) family appears exactly when $n\equiv3\pmod 4$, beginning at $n=7$, explaining the observed \textquotedblleft alternating\textquotedblright\ emergence of a second branch. For $n=9$ this second branch cannot occur by degree parity. Our statements rest on the structure of the degree--7 proof and the general triangular construction in the preprint, together with the standard isochrony characterization $\mathcal{H}=\tfrac{1}{2}(f_{1}^{2}+f_{2}^{2})$ with $\det Df\equiv1$.