Stratification of three-dimensional real flows II: A generalization of Poincar\'e's planar sectorial decomposition

TL;DR

This paper generalizes Poincaré's classical planar sectorial decomposition to three-dimensional real flows, providing a stratification theorem for analytic vector fields with isolated singularities under certain conditions.

Contribution

It extends the concept of sectorial decomposition from 2D to 3D flows, offering a new stratification theorem for analytic vector fields with hyperbolic singularities.

Findings

Established a stratification theorem for 3D analytic vector fields.

Generalized Poincaré's planar sectorial decomposition to three dimensions.

Abstract

Let $\xi$ be an analytic vector field in $\mathbb{R}^3$ with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities $\pi:M\to\mathbb{R}^3$. Assuming certain conditions to be specified throughout the work at hand, we establish a theorem of stratification of the dynamics of $\xi$ that generalizes to dimension three the classical one, coming from Poincar\'{e}, about the decomposition of the dynamics of an analytic planar vector field into {\em parabolic}, {\em elliptic} or {\em hyperbolic} invariant sectors.