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New Numeric Invariants of an Unfolding of a Polycycle "Tears of the Heart"

Yulij Ilyashenko, Stanislav Minkov, Ivan Shilin

TL;DR

The paper advances the topological classification of three-parameter unfoldings of a tears-of-the-heart polycycle by introducing new numeric invariants that survive topological equivalence, alongside refined asymptotics for sparkling saddle connections and precise monodromy estimates. Central to the approach are the normalizations of monodromy maps, the description of base parameter dynamics via two interlaced arithmetic progressions, and the emergence of shift-exp-Liouvillian three-parameter families whose sequences yield robust invariants like the irrational relative density A and the relative scale coefficient Xi. The results establish that, under mild genericity and irrationality assumptions, the characteristic numbers λ and μ are themselves invariants, and provide a complete topological/arithmetical framework for comparing unfoldings through the asymptotics of connection sequences and their associated invariants. The work also demonstrates that shift-exp-Liouvillian families form a generic (residual) subset in the moduli space of three-parameter unfoldings, making the invariants both structurally meaningful and widely applicable in the study of planar vector fields with polycycles.

Abstract

In this paper new numeric invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the pairs of the unperturbed arithmetic progressions we give a complete topological classification.

New Numeric Invariants of an Unfolding of a Polycycle "Tears of the Heart"

TL;DR

The paper advances the topological classification of three-parameter unfoldings of a tears-of-the-heart polycycle by introducing new numeric invariants that survive topological equivalence, alongside refined asymptotics for sparkling saddle connections and precise monodromy estimates. Central to the approach are the normalizations of monodromy maps, the description of base parameter dynamics via two interlaced arithmetic progressions, and the emergence of shift-exp-Liouvillian three-parameter families whose sequences yield robust invariants like the irrational relative density A and the relative scale coefficient Xi. The results establish that, under mild genericity and irrationality assumptions, the characteristic numbers λ and μ are themselves invariants, and provide a complete topological/arithmetical framework for comparing unfoldings through the asymptotics of connection sequences and their associated invariants. The work also demonstrates that shift-exp-Liouvillian families form a generic (residual) subset in the moduli space of three-parameter unfoldings, making the invariants both structurally meaningful and widely applicable in the study of planar vector fields with polycycles.

Abstract

In this paper new numeric invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the pairs of the unperturbed arithmetic progressions we give a complete topological classification.

Paper Structure

This paper contains 21 sections, 17 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tears of the heart
  3. New invariants
  4. Improved asymptotics of sparkling saddle connections
  5. Estimates of the monodromy mapping
  6. Definitions
  7. Estimates of the monodromy mapping: proof of Lemma \ref{['lem:est']}
  8. Preliminary remarks
  9. Normalization of the monodromy map
  10. The estimates of the monodromy map
  11. Preliminary description of the base homeomorphism
  12. The connection equation: the general case and the model case
  13. Asymptotics of the sparkling saddle connections: the model case
  14. Asymptotics of the sparkling saddle connections: general case
  15. Topology of two pairs of arithmetic progressions on the line
  16. ...and 6 more sections

Key Result

Theorem 1

The equivalence class of the ratio ${\tau = \frac{\beta_1-\beta_2}{\ln \nu_2}}$, is an invariant of the topological classification of generic three-parameter unfoldings of the vector field $v_0$, in the case when $A$ is irrational.

Figures (2)

  • Figure 1: The polycycle "tears of the heart" with an exterior and interior saddles
  • Figure 2: Breaking of a separatrix loop and sparkling saddle connections

Theorems & Definitions (38)

  • Theorem 1: Goncharuk and Kudryashov, GK:20
  • Remark
  • Theorem 2
  • Theorem 3
  • Remark
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 28 more