Vector fields whose linearisation is Hurwitz almost everywhere

Roland Rabanal | Pires, Benito

Keywords: Asymptotic stability; injectivity; topological dynamics; planar maps; Markus-Yamabe Conjecture;

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 $X:\R^2\to\R^2$ be a $C^1$ vector field whose derivative $DX(p)$ is Hurwitz for almost all $p\in\R^2$. Then the singularity set of $X$, {\rm Sing}\,(X), is either an emptyset, a one--point set or a non-discrete set. Moreover, if ${\rm Sing}\,(X)$ contains a hyperbolic singularity then $X$ is topologically equivalent to the radial vector field $(x,y)\mapsto (-x,-y)$. This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.

Anexos:

• Hurwitz.pdf