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

TL;DR

This paper introduces new numerical invariants for analyzing structurally unstable planar vector fields, utilizing advanced asymptotic analysis and topological invariants of arithmetic progressions to classify their behavior.

Contribution

It develops novel numerical invariants based on asymptotics of saddle connections and topological invariants for arithmetic progressions, enhancing understanding of vector field bifurcations.

Findings

New invariants for unstable vector fields

Complete topological classification of arithmetic progressions

Improved asymptotic analysis of saddle connections

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.