Dynamical systems on the Liouville plane and the related strictly contact systems

TL;DR

This paper classifies vector fields on the Liouville plane that preserve the Liouville form, analyzes their local bifurcations, and explores their relation to strictly contact vector fields in three dimensions, providing a comprehensive understanding of their local dynamics.

Contribution

It introduces a new classification of univariate functions related to Liouville-preserving vector fields and describes their local bifurcations and connections to contact systems, extending existing theories.

Findings

Classification of Liouville-preserving vector fields

Description of local bifurcations of low codimension

Relations between planar vector fields and 3D contact systems

Abstract

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. We discuss, in addition, their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.