Examples of infinitesimal non-trivial accumulation of secants in dimension three

TL;DR

This paper constructs specific examples of three-dimensional real analytic vector fields where orbits accumulate in a non-trivial way on a set with algebraic structure, with the origin as the only limit point.

Contribution

It provides explicit examples of secant accumulation sets with algebraic structure in three-dimensional vector fields, expanding understanding of orbit behavior.

Findings

Secants accumulate on algebraic varieties intersected with cones.

The origin is the only omega-limit point for these orbits.

Examples demonstrate complex accumulation phenomena in 3D vector fields.

Abstract

We present non-trivial examples of accumulation of secants for orbits (of real analytic three dimensional vector fields) having the origin as only $\omega$-limit point. These non-trivial sets have the structure of a proper algebraic variety of $\mathbb{S}^2$ intersected with a cone.