A complex limit cycle not intersecting the real plane

TL;DR

This paper presents a specific example of a polynomial vector field in real two-dimensional space that has a complex limit cycle entirely contained in the complex plane, not intersecting the real plane.

Contribution

It provides a concrete example of a complex limit cycle in a polynomial vector field that does not intersect the real plane, highlighting a novel phenomenon in dynamical systems.

Findings

Existence of a complex limit cycle not intersecting the real plane

Explicit polynomial vector field example demonstrating this phenomenon

Insight into the structure of singular holomorphic foliations in complex dynamics

Abstract

We give a precise example of a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular holomorphic foliation of $\mathbb{C}^2$ possesses a complex limit cycle which does not intersect the real plane $\mathbb{R}^2$.