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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2510.25795 |
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| _version_ | 1866912677984468992 |
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| author | Vera, J. A. |
| author_facet | Vera, J. A. |
| contents | 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 $\det Df\equiv1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture Vera, J. A. Dynamical Systems 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 $\det Df\equiv1$. |
| title | Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2510.25795 |