Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture
J. A. Vera
TL;DR
This work refines the classification of trivial isochronous centers for planar polynomial Hamiltonians in odd degrees by establishing a two-branch structure: a universal triangular branch exists for every odd degree $n=2k-1$, and a distinct quadratic-shear branch appears exactly when $n\equiv3\pmod4$, beginning at $n=7$. Building on the standard isochrony criterion $\mathcal{H}=\frac{f_1^2+f_2^2}{2}$ with $\det Df\equiv1$, the authors extend the septic proof framework to higher degrees and show the two branches are inequivalent under linear changes. They also confirm that even degrees provide no trivial centers under the same polynomial hypotheses, while nontrivial isochronous centers exist in degrees $6k+1$, highlighting a broader trivial-vs-nontrivial dichotomy. Overall, the results systematize how degree parity governs the emergence of multiple isochronous-center branches and offer a constructive blueprint for future classifications.
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 $\det Df\equiv1$.