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Hopf bifurcation at infinity and dissipative vector fields of the plane


Authors: Begoña Alarcón and Roland Rabanal
Journal: Proc. Amer. Math. Soc. 145 (2017), 3033-3046
MSC (2010): Primary 37G10, 34K18, 34C23
DOI: https://doi.org/10.1090/proc/13462
Published electronically: January 25, 2017
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Abstract: This work deals with one-parameter families of differentiable (not necessarily $ C^1$) planar vector fields for which the infinity reverses its stability as the parameter goes through zero. These vector fields are defined on the complement of some compact ball centered at the origin and have isolated singularities. They may be considered as linear perturbations at infinity of a vector field with some spectral property, for instance, dissipativity. We also address the case concerning linear perturbations of planar systems with a global period annulus. It is worth noting that the adopted approach is not restricted to consider vector fields which are strongly dominated by the linear part. Moreover, the Poincaré compactification is not applied in this paper.


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Additional Information

Begoña Alarcón
Affiliation: Departamento de Matemática Aplicada, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, CEP 24020-140 Niterói-RJ, Brazil
Email: balarcon@id.uff.br

Roland Rabanal
Affiliation: Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, Lima 32, Perú
Email: rrabanal@pucp.edu.pe

DOI: https://doi.org/10.1090/proc/13462
Keywords: Hopf bifurcation, planar vector fields
Received by editor(s): July 9, 2014
Received by editor(s) in revised form: July 10, 2014, May 23, 2016, and August 24, 2016
Published electronically: January 25, 2017
Additional Notes: The first author was partially supported by CAPES and grants MINECO-15-MTM2014-56953-P from Spain and CNPq 474406/2013-0 from Brazil
The second author was partially supported by Pontifícia Universidad Católica del Perú (DGI:70242.0056) and by Instituto de Ciências Matemáticas e de Computação (ICMC–USP: 2013/16226-8)
This paper was written while the second author served as an Associate Fellow at the Abdus Salam ICTP in Italy. He also acknowledges the hospitality of ICMC–USP in Brazil during the preparation of part of this work
Dedicated: Dedicated to the memory of Carlos Gutiérrez on the occasion of the fifth anniversary of his death
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society