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 Vector fields whose linearisation is Hurwitz almost everywhere


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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 DXp is Hurwitz for almost all p in $R^2$. Then the singularity set of X, SingX, is either an emptyset, a one-point set or a non-discrete set. Moreover, if SingX 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.



Author: Benito Pires; Roland Rabanal

Source: https://archive.org/







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