On the classification of two dimensional flows

Abstract

Lawrence Markus, a well-known mathematician, published, in 1954, a strong result about classification of planar vector fields. Afterwards, Dean Arnold Neumann generalized such work for continuous flows, and the result became known as the Markus–Neumann Theorem. Recently, in 2018, José Ginés Espín Buendía and Víctor Jiménez López wrote a paper exposing gaps in the two aforementioned articles.

We present here a discussion on these classical results of classification of planar vector fields and continuous flows, going through the three cited works and elaborating on definitions, examples and proofs.