Separatrices for real analytic vector fields in the plane

TL;DR

This paper investigates real analytic vector fields in the plane with isolated singularities, establishing conditions under which such fields possess formal invariant curves, and highlighting the limitations of these conditions for convergent separatrices.

Contribution

It proves that certain algebraic conditions guarantee the existence of formal separatrices in real analytic vector fields with isolated singularities.

Findings

Existence of formal separatrices under even Milnor number or order.

Optimality of conditions, not guaranteeing convergent separatrices.

Characterization of topological generalized curves without saddle-nodes.

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.