Vector fields whose linearisation is Hurwitz almost everywhere

TL;DR

This paper investigates vector fields on R^2 whose derivatives are Hurwitz matrices almost everywhere, revealing the structure of their singularity sets and conditions under which they are topologically equivalent to a radial vector field.

Contribution

It generalizes the Bidimensional Global Asymptotic Stability Problem to vector fields with derivatives Hurwitz almost everywhere, extending previous results to non-diffeomorphic cases.

Findings

Singularity set is either empty, a point, or non-discrete.

If a hyperbolic singularity exists, the field is topologically radial.

Generalizes BGAS to non-diffeomorphic vector fields.

Abstract

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided: Let $X:R^2-->R^2$ be a C^1 vector field whose derivative DX(p) is Hurwitz for almost all p in $R^2$. Then the singularity set of X, Sing(X), is either an emptyset, a one--point set or a non-discrete set. Moreover, if Sing(X) contains a hyperbolic singularity then X is topologically equivalent to the radial vector field $(x,y)--> (-x,-y)$. This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.