Stratification of three-dimensional real flows I: Fitting Domains

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 $\pi:M\to\mathbb{R}^3$. The union of the images by $\pi$ of the local invariant manifolds at those hyperbolic points, denoted by $\Lambda$, is composed of trajectories of $\xi$ accumulating to $0 \in \mathbb{R}^3$. Assuming that there are no cycles nor polycycles on the divisor of $\pi$, together with a Morse-Smale type property and a non-resonance condition on the eigenvalues at these points, in this paper we prove the existence of a fundamental system $\{V_n\}$ of neighborhoods well adapted for the description of the local dynamics of $\xi$: the frontier $Fr(V_n)$ is everywhere tangent to $\xi$ except around $Fr(V_n)\cap\Lambda$, where transvesality is mandatory.