Diophantine "Tears of the Heart"

TL;DR

This paper investigates the topological invariants of a specific class of vector fields with a 'tears of the heart' polycycle, revealing that a metric approach simplifies the classification for most cases compared to previous topological methods.

Contribution

It shows that, contrary to prior topological results, a metric perspective reduces the number of invariants in generic cases of these vector fields.

Findings

For almost all coefficient values, only two invariants are generated.

The metric approach simplifies the classification of these vector fields.

Previous topological methods identified at least four invariants.

Abstract

Recent studies of topologically generic unfoldings of vector fields featuring a "tears of the heart" polycycle with one internal and one external winding separatrix have shown that, in a special one-parameter subfamily where the "heart" is preserved and the "tear" loop if broken, at least four invariants of weak topological classification appear. In this paper, we demonstrate that the metrical perspective yields a different result: for Lebesgue almost all values of the coefficients related to the original vector field, the special one-parameter family generates only two such invariants.