Structure of optimal gradient flows bifurcations on closed surfaces

TL;DR

This paper classifies the topological structure of typical gradient flow bifurcations on closed surfaces, using chord diagrams to distinguish between saddle-node and saddle connection bifurcations, and lists all such diagrams for low-genus surfaces.

Contribution

It introduces a topological classification method for gradient flow bifurcations on closed surfaces using chord diagrams, providing a complete list for low-genus cases.

Findings

Chord diagrams serve as complete invariants for bifurcation classification.

All bifurcation diagrams are listed for surfaces of genus up to 2 (orientable) and 3 (nonorientable).

Inverse flows are identified for each diagram.

Abstract

We consider structure of typical gradient flows bifurcations on closed surfaces with minimal number of singular points. There are two type of such bifurcations: saddle-node (SN) and saddle connections (SC). The structure of a bifurcation is determinated by codimension one flow in the moment of bifurcation. We use the chord diagrams to specify the flows up to topological trajectory equivivalence. A chord diagram with a marked arc is complete topological invariant of a SN-bifurcations and a chord diagram with T-insert -- of SC-bifurcations. We list all such diagrams for flows on norientable surfaces of genus at most 2 and nonorientable surfaces of genus at most 3. For each of diagram we found inverse one that correspond the inverse flow.