Stratification of three-dimensional real flows I: Fitting Domains

TL;DR

This paper studies the local dynamics of three-dimensional real flows near isolated singularities, establishing a systematic way to describe neighborhoods that align with the flow's invariant structures under certain conditions.

Contribution

It introduces a method to construct neighborhoods adapted to the flow's invariant manifolds, assuming hyperbolic singularities, no cycles, and Morse-Smale properties.

Findings

Existence of neighborhoods with boundaries tangent to the flow

Neighborhood boundaries are transverse to invariant manifolds near accumulation points

Framework aids in understanding local flow dynamics around singularities

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.