Bifurcations of the polycycle "tears of the heart": multiple numerical invariants

TL;DR

This paper investigates the complex bifurcation structure of the 'tears of the heart' polycycle in planar vector fields, revealing that such bifurcations can have multiple numerical invariants, indicating rich and intricate dynamics.

Contribution

It extends previous work by demonstrating that bifurcations of 'tears of the heart' can possess arbitrarily many numerical invariants, deepening understanding of their classification.

Findings

Bifurcations can have arbitrarily many numerical invariants

The classification of these bifurcations is more complex than previously thought

Structural instability of 'tears of the heart' bifurcations is confirmed

Abstract

"Tears of the heart" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with "tears of the heart" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. In this article, we study the bifurcations of "tears of the heart" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.