Infinitesimal Poincar\'e-Bendixson Problem in dimension three

TL;DR

This paper investigates the accumulation sets of orbits near a singularity in three-dimensional real analytic vector fields, revealing their structure as cyclic graphs or single polycycles under certain conditions.

Contribution

It characterizes the structure of accumulation sets in 3D real analytic vector fields, extending the Poincaré-Bendixson theory to higher dimensions with specific singularity conditions.

Findings

Accumulation sets form cyclic graphs with isolated singularities.

Under Morse-Smale conditions, the accumulation set is at most a single polycycle.

The results extend classical 2D dynamics to 3D real analytic vector fields.

Abstract

We describe the sets of accumulation of secants for orbits of real analytic vector fields in dimension three having the origin as only {\omega}-limit point. It is a kind of infinitesimal Poincar\'e-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one blow-up. In the case of hyperbolic reduction of singularities with conditions of type Morse-Smale, we prove that the accumulation set is at most a single polycycle isomorphic to {\mathbb S}^1.