Singular points in generic two-parameter families of vector fields on 2-manifold

TL;DR

This work provides a rigorous classification of singularities in generic two-parameter families of vector fields on compact 2-manifolds. It leverages multijet transversality to establish finiteness of singular points and to bound non-hyperbolic degeneracies to a small, well-understood set (including $AH_0$, $SN_0$, $SN_1$, $AH_1$, and $BT$ types). The authors introduce explicit degeneracy classes ($AH_k$, $SN_k$, $BT_k$) and perform a codimension analysis via centralizers to justify the allowed singularities and their multiplicities, with extensions to analytic settings. The results lay a rigorous foundation for two-parameter bifurcation theory on $M^2$ and set the stage for a full global bifurcation analysis on the two-sphere.

Abstract

In this paper, we give a full description of all possible singular points that occur in generic 2-parameter families of vector fields on compact 2-manifolds. This is a part of a large project aimed to a complete study of global bifurcations in two-parameter families of vector fields on the two-sphere.

Theorems & Definitions (30)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Conjecture 1
• Remark 3
• Definition 1
• Definition 2
• Definition 3