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Vector fields whose linearisation is Hurwitz almost everywhere


Authors: Benito Pires and Roland Rabanal
Journal: Proc. Amer. Math. Soc. 142 (2014), 3117-3128
MSC (2010): Primary 34D23, 37B25; Secondary 37C10
Published electronically: May 21, 2014
MathSciNet review: 3223368
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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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Additional Information

Benito Pires
Affiliation: Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, 14040-901 Ribeirão Preto, SP, Brazil
Email: benito@ffclrp.usp.br

Roland Rabanal
Affiliation: Sección Matemática, Pontificia Universidad Católica del Perú, San Miguel, Lima 32, Perú
Email: rrabanal@pucp.edu.pe

DOI: https://doi.org/10.1090/S0002-9939-2014-12035-1
Received by editor(s): November 29, 2011
Received by editor(s) in revised form: September 21, 2012
Published electronically: May 21, 2014
Additional Notes: The first author was partially supported by FAPESP-Brazil (2009/02380-0 and 2008/02841-4)
The second author was partially supported by PUCP-Peru (DAI 2010-0058) and ICTP-Italy (220 (Maths) rr/ab).
Dedicated: Dedicated to the memory of Professor Carlos Gutierrez
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society