VECTOR FIELDS WHOSE LINEARISATION IS HURWITZ ALMOST EVERYWHERE

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 Jacobian matrix DX(p) is Hurwitz for Lebesgue almost all p is an element of R-2. Then the singularity set of X is either an empty set, a one-point set or a non-discrete set. Moreover, if X has a hyperbolic singularity, then X is topologically equivalent to the radial vector field (x, y) bar right arrow (-x, -y). This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.