Moscow Mathematical Journal

Volume 22, Issue 4, October–December 2022 pp. 595–611.

Separatrices for Real Analytic Vector Fields in the Plane

Authors: Eduardo Cabrera (1) and Rogério Mol (1)
Author institution:(1) Departamento de Matemática - ICEX, Universidade Federal de Minas Gerais, UFMG

Summary:

Let $X$ be a germ of real analytic vector field at $(\mathbb{R}^{2},0)$ with an algebraically 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.

2020 Math. Subj. Class. 32S65, 37F75, 34Cxx, 14P15.

Keywords: Real analytic vector field, formal and analytic separatrix, reduction of singularities, index of vector fields, polar invariants, center-focus vector field.

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