The structure of Morse flows and co-dimension one gradient flows on the sphere with holes

TL;DR

This paper classifies the topological structures of typical bifurcations in gradient flows on a punctured 2-sphere with up to six singular points, using separatrix diagrams to describe saddle-node and saddle connection configurations.

Contribution

It provides a complete topological classification of bifurcations of gradient flows on the punctured sphere with limited singular points, detailing the role of separatrix diagrams.

Findings

Classified all possible bifurcation structures with up to six singular points.

Described how saddle-node and saddle connection bifurcations are represented in separatrix diagrams.

Provided a framework for understanding flow structures on punctured spheres.

Abstract

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix diagrams of flows. The saddle-node singularity is specified by selecting a separatrix in the diagram of the flow befor the bifurcation and the saddle connection is specified by a separatrix, which conect two saddles.