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arXiv · 2412.09197

Characterization of centers by its complex separatrices

Isaac A. García·Jaume Giné
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Abstract

In this work we deal with analytic families of real planar vector fields $\mathcal{X}_\lambda$ having a monodromic singularity at the origin for any $\lambda \in \Lambda \subset \mathbb{R}^p$ and depending analytically on the parameters $\lambda$. There naturally appears the so-called center-focus problem which consists in describing the partition of $\Lambda$ 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.

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Isaac A. García, Jaume Giné. 2024-12-12. Characterization of centers by its complex separatrices. https://arxiv.org/abs/2412.09197

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