A countable number of homoclinic loops as an {\omega}-limit set for a planar vector field

TL;DR

This paper constructs a smooth planar vector field with a countable set of homoclinic loops as its {}-limit set, demonstrating complex dynamical behavior in a non-analytic setting.

Contribution

It provides a novel example of a smooth, non-analytic planar vector field with a countable homoclinic loops {}-limit set, expanding understanding of possible limit set configurations.

Findings

Existence of a smooth, non-analytic vector field with complex limit set

Countable homoclinic loops can form the {}-limit set in planar systems

Demonstrates complex dynamical structures beyond analytic cases

Abstract

We construct an example of a smooth, non analytic planar vector field with an {\omega}-limit set consisting of a countable number of homoclinic loops.