English

Separatrices for real analytic vector fields in the plane

Dynamical Systems 2019-12-02 v1 Classical Analysis and ODEs Complex Variables

Abstract

Let XX be a germ of real analytic vector field at (R2,0)({\mathbb R}^{2},0) with an algebracally isolated singularity. We say that XX 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 ν0(X)\nu_{0}(X) or the Milnor number μ0(X)\mu_{0}(X) is even, then XX has a formal separatrix, that is, a formal invariant curve at 0R20 \in {\mathbb R}^{2}. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

Keywords

Cite

@article{arxiv.1911.12803,
  title  = {Separatrices for real analytic vector fields in the plane},
  author = {Eduardo Cabrera and Rogério Mol},
  journal= {arXiv preprint arXiv:1911.12803},
  year   = {2019}
}

Comments

16 pages

R2 v1 2026-06-23T12:30:20.326Z