Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diophantine "Tears of the Heart"

Yulij Ilyashenko, Stanislav Minkov, Ivan Shilin

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.

Diophantine "Tears of the Heart"

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.
Paper Structure (11 sections, 9 theorems, 23 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. LMF graphs
  3. The set of Diophantine families
  4. Proof of the Main Theorem
  5. Sparkling saddle connections and useful lemmas
  6. No Additional Combinatorics Lemma
  7. The Proof of Theorem \ref{['thm:main']}
  8. Proof of Lemma \ref{['lem:3']}
  9. An explicit description of the ordering
  10. Proof of Proposition \ref{['prop:2']}
  11. Conclusion

Key Result

Theorem 1

There exists a set $\mathcal{A}$ of full Lebesgue measure in the space of coefficients $(\lambda ,\mu ,B_{i},C_{i}) \simeq\mathbb R^6_+$, such that, for any two standard one-parameter families $\{v_\varepsilon\}$ and $\{\tilde{v}_{\tilde{\varepsilon}}\}$ with coefficients in $\mathcal{A}$, weak equi and where $A=-\ln\lambda/\ln(\lambda^2\mu)$, $\tau=s/\ln(\lambda^2\mu)$, $s= \ln \left(\frac{\ln C

Figures (3)

  • Figure 1: The polycycle "tears of the heart" with an exterior and interior saddles
  • Figure 2: The gap and the coordinates $x$ and $y$ at the transversal $\Gamma$.
  • Figure 3: Qualitative plots of the points where the $I$- and $E$-separatrices reach the gap. Due to the monotonicity of the difference $d_{n,k}$, the intersection points (which correspond to $EI$ connections) are positioned as described by Proposition \ref{['prop:2']}.

Theorems & Definitions (16)

  • Definition
  • Definition
  • Theorem 1: on Diophantine families
  • Remark
  • Remark
  • Remark
  • Theorem 2: R. Fedorov, F
  • Definition
  • Proposition 3.1
  • Definition
  • ...and 6 more