# New structurally unstable families of planar vector fields

## TL;DR

This paper investigates the structural instability of certain families of planar vector fields with specific separatrix configurations, expanding understanding of bifurcations and stability in dynamical systems on the sphere.

## Contribution

It introduces new classes of structurally unstable vector field families with 'ears' and 'glasses' separatrix graphs, extending previous results on 'tears of the heart' bifurcations.

## Key findings

- Families with 'ears' and 'glasses' are structurally unstable.
- Classical saddle loop bifurcation analyzed in detail.
- Provides new examples of structurally unstable vector fields.

## Abstract

We study global bifurcations in generic 3-parameter families of vector fields on $S^2$. In the recent article [arXiv:1506.06797], Ilyashenko, Kudryashov, and Schurov show that 3-parameter unfoldings of vector fields with the polycycle "tears of the heart" are structurally unstable. We consider 3-parameter unfoldings of vector fields with separatrix graphs "ears" and "glasses", and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.