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arXiv · 1911.12803

Separatrices for real analytic vector fields in the plane

Eduardo Cabrera·Rogério Mol
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Abstract

Let $X$ be a germ of real analytic vector field at $({\mathbb R}^{2},0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order $\nu_{0}(X)$ or the Milnor number $\mu_{0}(X)$ is even, then $X$ has a formal separatrix, that is, a formal invariant curve at $0 \in {\mathbb R}^{2}$. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

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Eduardo Cabrera, Rogério Mol. 2019-11-28. Separatrices for real analytic vector fields in the plane. https://arxiv.org/abs/1911.12803

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