A universal method to approach the Poincaré center problem
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911503600320512 |
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| author | García, Isaac A. Giné, Jaume |
| author_facet | García, Isaac A. Giné, Jaume |
| contents | We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic, and if, in addition, $V$ has an essential singularity, then the singularity of $\mathcal{X}$ is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A universal method to approach the Poincaré center problem García, Isaac A. Giné, Jaume Dynamical Systems We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic, and if, in addition, $V$ has an essential singularity, then the singularity of $\mathcal{X}$ is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided. |
| title | A universal method to approach the Poincaré center problem |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2603.09941 |