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arXiv · 2603.05753

Diophantine "Tears of the Heart"

Yulij Ilyashenko·Stanislav Minkov·Ivan Shilin
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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.

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Yulij Ilyashenko, Stanislav Minkov, Ivan Shilin. 2026-03-05. Diophantine "Tears of the Heart". https://arxiv.org/abs/2603.05753

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