Vector fields whose linearisation is Hurwitz almost everywhere

Pires, Benito; Rabanal, Roland

doi:10.1090/S0002-9939-2014-12035-1

Vector fields whose linearisation is Hurwitz almost everywhere
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by Benito Pires and Roland Rabanal PDF

Proc. Amer. Math. Soc. 142 (2014), 3117-3128 Request permission

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 $X:\mathbb {R}^2\to \mathbb {R}^2$ be a $C^1$ vector field whose Jacobian matrix $DX(p)$ is Hurwitz for Lebesgue almost all $p\in \mathbb {R}^2$. 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 $(x,y)\mapsto (-x,-y)$. This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.

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