Characterization of centers by its complex separatrices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929627587411968 |
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| author | García, Isaac A. Giné, Jaume |
| author_facet | García, Isaac A. Giné, Jaume |
| contents | In this work we deal with analytic families of real planar vector fields $\mathcal{X}_λ$ having a monodromic singularity at the origin for any $λ\in Λ\subset \mathbb{R}^p$ and depending analytically on the parameters $λ$. There naturally appears the so-called center-focus problem which consists in describing the partition of $Λ$ induced by the centers and the foci at the origin. We give a characterization of the centers (degenerated or not) in terms of a specific integral of the cofactor associated to a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also stated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of centers by its complex separatrices García, Isaac A. Giné, Jaume Dynamical Systems 34C05, 34A05, 37G10, 37G15 In this work we deal with analytic families of real planar vector fields $\mathcal{X}_λ$ having a monodromic singularity at the origin for any $λ\in Λ\subset \mathbb{R}^p$ and depending analytically on the parameters $λ$. There naturally appears the so-called center-focus problem which consists in describing the partition of $Λ$ induced by the centers and the foci at the origin. We give a characterization of the centers (degenerated or not) in terms of a specific integral of the cofactor associated to a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also stated. |
| title | Characterization of centers by its complex separatrices |
| topic | Dynamical Systems 34C05, 34A05, 37G10, 37G15 |
| url | https://arxiv.org/abs/2412.09197 |