## Hopf bifurcation at infinity and dissipative vector fields of the plane

HTML articles powered by AMS MathViewer

- by Begoña Alarcón and Roland Rabanal PDF
- Proc. Amer. Math. Soc.
**145**(2017), 3033-3046 Request permission

## 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.## References

- Begoña Alarcón, Víctor Guíñez, and Carlos Gutierrez,
*Hopf bifurcation at infinity for planar vector fields*, Discrete Contin. Dyn. Syst.**17**(2007), no. 2, 247–258. MR**2257430**, DOI 10.3934/dcds.2007.17.247 - A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer,
*Theory of bifurcations of dynamic systems on a plane*, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian. MR**0344606** - J. Auslander and P. Seibert,
*Prolongations and stability in dynamical systems*, Ann. Inst. Fourier (Grenoble)**14**(1964), no. fasc. 2, 237–267. MR**176180** - R. H. Bing,
*The geometric topology of 3-manifolds*, American Mathematical Society Colloquium Publications, vol. 40, American Mathematical Society, Providence, RI, 1983. MR**728227**, DOI 10.1090/coll/040 - T. R. Blows and C. Rousseau,
*Bifurcation at infinity in polynomial vector fields*, J. Differential Equations**104**(1993), no. 2, 215–242. MR**1231467**, DOI 10.1006/jdeq.1993.1070 - P. Diamond, D. Rachinskii, and M. Yumagulov,
*Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity*, Nonlinear Anal.**42**(2000), no. 6, Ser. A: Theory Methods, 1017–1031. MR**1780452**, DOI 10.1016/S0362-546X(99)00162-5 - Alexandre Fernandes, Carlos Gutierrez, and Roland Rabanal,
*Global asymptotic stability for differentiable vector fields of $\Bbb R^2$*, J. Differential Equations**206**(2004), no. 2, 470–482. MR**2096702**, DOI 10.1016/j.jde.2004.04.015 - J. N. Glover,
*Hopf bifurcations at infinity*, Nonlinear Anal.**13**(1989), no. 12, 1393–1398. MR**1028236**, DOI 10.1016/0362-546X(89)90100-4 - Armengol Gasull, Víctor Mañosa, and Jordi Villadelprat,
*On the period of the limit cycles appearing in one-parameter bifurcations*, J. Differential Equations**213**(2005), no. 2, 255–288. MR**2142367**, DOI 10.1016/j.jde.2004.07.013 - Carlos Gutierrez, Benito Pires, and Roland Rabanal,
*Asymptotic stability at infinity for differentiable vector fields of the plane*, J. Differential Equations**231**(2006), no. 1, 165–181. MR**2287882**, DOI 10.1016/j.jde.2006.07.025 - Carlos Gutierrez and Roland Rabanal,
*Injectivity of differentiable maps $\Bbb R^2\to \Bbb R^2$ at infinity*, Bull. Braz. Math. Soc. (N.S.)**37**(2006), no. 2, 217–239. MR**2266382**, DOI 10.1007/s00574-006-0011-4 - A. Gasull and J. Sotomayor,
*On the basin of attraction of dissipative planar vector fields*, Bifurcations of planar vector fields (Luminy, 1989) Lecture Notes in Math., vol. 1455, Springer, Berlin, 1990, pp. 187–195. MR**1094380**, DOI 10.1007/BFb0085393 - Carlos Gutierrez and Alberto Sarmiento,
*Injectivity of $C^1$ maps $\Bbb R^2\to \Bbb R^2$ at infinity and planar vector fields*, Astérisque**287**(2003), xviii, 89–102 (English, with English and French summaries). Geometric methods in dynamics. II. MR**2040002** - V. Guíñez, E. Sáez, and I. Szántó,
*Simultaneous Hopf bifurcations at the origin and infinity for cubic systems*, Dynamical systems (Santiago, 1990) Pitman Res. Notes Math. Ser., vol. 285, Longman Sci. Tech., Harlow, 1993, pp. 40–51. MR**1213941** - Carlos Gutiérrez and Marco Antonio Teixeira,
*Asymptotic stability at infinity of planar vector fields*, Bol. Soc. Brasil. Mat. (N.S.)**26**(1995), no. 1, 57–66. MR**1339178**, DOI 10.1007/BF01234626 - Xiang Jian He,
*Hopf bifurcation at infinity with discontinuous nonlinearities*, J. Austral. Math. Soc. Ser. B**33**(1991), no. 2, 133–148. MR**1124985**, DOI 10.1017/S0334270000006950 - André Haefliger and Georges Reeb,
*Variétés (non séparées) à une dimension et structures feuilletées du plan*, Enseign. Math. (2)**3**(1957), 107–125 (French). MR**89412** - S. B. Jackson,
*Classroom Notes: A Development of the Jordan Curve Theorem and the Schoenflies Theorem for Polygons*, Amer. Math. Monthly**75**(1968), no. 9, 989–998. MR**1535106**, DOI 10.2307/2315540 - M. A. Krasnosel′skiĭ, N. A. Kuznetsov, and M. G. Yumagulov,
*Localization and construction of cycles for the Hopf bifurcation at infinity*, Dokl. Akad. Nauk**344**(1995), no. 4, 446–449 (Russian). MR**1364308** - M. A. Krasnosel′skiĭ, N. A. Kuznetsov, and M. G. Yumagulov,
*Conditions for the stability of cycles under Hopf bifurcation at infinity*, Avtomat. i Telemekh.**1**(1997), 56–62 (Russian, with Russian summary); English transl., Automat. Remote Control**58**(1997), no. 1, 43–48. MR**1446100** - N. G. Lloyd, T. R. Blows, and M. C. Kalenge,
*Some cubic systems with several limit cycles*, Nonlinearity**1**(1988), no. 4, 653–669. MR**967475** - Jaume Llibre and Roland Rabanal,
*Planar real polynomial differential systems of degree $n>3$ having a weak focus of high order*, Rocky Mountain J. Math.**42**(2012), no. 2, 657–693. MR**2915513**, DOI 10.1216/RMJ-2012-42-2-657 - Jaume Llibre and Roland Rabanal,
*Center conditions for a class of planar rigid polynomial differential systems*, Discrete Contin. Dyn. Syst.**35**(2015), no. 3, 1075–1090. MR**3277186**, DOI 10.3934/dcds.2015.35.1075 - Yang Lijun and Zeng Xianwu,
*Stability of singular Hopf bifurcations*, J. Differential Equations**206**(2004), no. 1, 30–54. MR**2093918**, DOI 10.1016/j.jde.2004.08.002 - Benito Pires and Roland Rabanal,
*Vector fields whose linearisation is Hurwitz almost everywhere*, Proc. Amer. Math. Soc.**142**(2014), no. 9, 3117–3128. MR**3223368**, DOI 10.1090/S0002-9939-2014-12035-1 - Roland Rabanal,
*Center type performance of differentiable vector fields in the plane*, Proc. Amer. Math. Soc.**137**(2009), no. 2, 653–662. MR**2448587**, DOI 10.1090/S0002-9939-08-09686-X - Roland Rabanal,
*On differentiable area-preserving maps of the plane*, Bull. Braz. Math. Soc. (N.S.)**41**(2010), no. 1, 73–82. MR**2609212**, DOI 10.1007/s00574-010-0004-1 - Roland Rabanal,
*Asymptotic stability at infinity for bidimensional Hurwitz vector fields*, J. Differential Equations**255**(2013), no. 5, 1050–1066. MR**3062761**, DOI 10.1016/j.jde.2013.04.037 - Roland Rabanal,
*On the limit cycles of a class of Kukles type differential systems*, Nonlinear Anal.**95**(2014), 676–690. MR**3130553**, DOI 10.1016/j.na.2013.10.013 - Marco Sabatini,
*Hopf bifurcation from infinity*, Rend. Sem. Mat. Univ. Padova**78**(1987), 237–253. MR**934515** - J. Sotomayor,
*Inversion of smooth mappings*, Z. Angew. Math. Phys.**41**(1990), no. 2, 306–310. MR**1045818**, DOI 10.1007/BF00945116 - J. Sotomayor and R. Paterlini,
*Bifurcations of polynomial vector fields in the plane*, Oscillations, bifurcation and chaos (Toronto, Ont., 1986) CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 1987, pp. 665–685. MR**909943** - Eduardo Sáez, Renato Vásquez, and Jorge Billeke,
*About the generalized Hopf bifurcations at infinity for planar cubic systems*, Rev. Colombiana Mat.**29**(1995), no. 2, 89–93. MR**1421060** - L. Zibenman,
*The Osgood-Schoenflies theorem revisited*, Uspekhi Mat. Nauk**60**(2005), no. 4(364), 67–96 (Russian, with Russian summary); English transl., Russian Math. Surveys**60**(2005), no. 4, 645–672. MR**2190924**, DOI 10.1070/RM2005v060n04ABEH003672

## 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
- MR Author ID: 745310
- ORCID: 0000-0003-0622-1878
- Email: rrabanal@pucp.edu.pe
- 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 - Communicated by: Yingfei Yi
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 3033-3046 - MSC (2010): Primary 37G10, 34K18, 34C23
- DOI: https://doi.org/10.1090/proc/13462
- MathSciNet review: 3637951

Dedicated: Dedicated to the memory of Carlos Gutiérrez on the occasion of the fifth anniversary of his death