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Abstract <p> In this paper, new numerical 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. </p>

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New Numerical Invariants of an Unfolding of a Polycycle “Tears of the Heart”

  • 期刊论文

  • 发表日期：

    2026年3月27日

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  • Yu.S. Ilyashenko,
  • S. Minkov,
  • I. Shilin

摘要

Abstract

In this paper, new numerical 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.