Local models for smooth vector fields of the line

TL;DR

This paper classifies singularities of smooth vector fields on the line under various conjugacy relations, providing a comprehensive understanding of their local behavior and unfoldings.

Contribution

It offers the first complete local classification of singularities of smooth vector fields on the line under $C^1$ conjugacy, including normal forms and unfoldings.

Findings

Complete classification of singularities under $C^1$ conjugacy

Normal forms and unfoldings for these singularities

Comparison with $C^0$ and $C^{ abla}$ classifications

Abstract

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of $C^1$--conjugacy. Along the way, we recall the analogous classification, up to $C^0$ and $C^{\infty}$ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the $1$--d case is achieved.