# Sparkling saddle loops of vector fields on surfaces

## TL;DR

This paper investigates bifurcations of vector fields on surfaces with handles, revealing complex phenomena like sparkling saddle loops and Cantor set structures that differ significantly from sphere cases.

## Contribution

It introduces the concept of sparkling saddle loops in the bifurcation analysis of vector fields on surfaces with handles, highlighting new topological complexities.

## Key findings

- Sparkling saddle loops emerge when a separatrix loop is broken.
- Bifurcation diagrams contain Cantor sets with endpoints of gaps.
- Countably many non-equivalent bifurcation germs exist in generic families.

## Abstract

We study bifurcations of vector fields on 2-manifolds with handles in generic one-parameter families unfolding vector fields with a separatrix loop of a hyperbolic saddle. These bifurcations can differ drastically from the analogous bifurcations on the sphere. The reason is that, on a surface, a free separatrix of a hyperbolic saddle may wind toward the separatrix loop of the same saddle. When this loop is broken, sparkling saddle loops emerge. In the orientable case, the parameter values corresponding to these loops form the endpoints of the gaps in a Cantor set contained within the bifurcation diagram. Due to the presence of a Cantor set, there is a countable set of topologically non-equivalent germs of bifurcation diagrams even in generic one-parameter families, in contrast to bifurcations on the sphere.