Hopf bifurcation at infinity and dissipative vector fields of the plane

Alarcón, Begoña; Rabanal, Roland

doi:10.1090/proc/13462

Hopf bifurcation at infinity and dissipative vector fields of the plane
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by Begoña Alarcón and Roland Rabanal

Proc. Amer. Math. Soc. 145 (2017), 3033-3046

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.

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, DOI 10.5802/aif.179
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, DOI 10.1088/0951-7715/1/4/008
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