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

Abstract

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