Vector fields whose linearisation is Hurwitz almost everywhere

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.