Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Normal forms of Liénard type for analytic unfoldings of nilpotent singularities


Author: Renato Huzak
Journal: Proc. Amer. Math. Soc. 145 (2017), 4325-4336
MSC (2010): Primary 37G05; Secondary 34M45
DOI: https://doi.org/10.1090/proc/13539
Published electronically: March 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using the technique of gluing complex manifolds (equipped with vector fields) developed by Loray and the theory of deformation of complex structures developed by Kodaira and Spencer, we find normal forms of Liénard type for analytic unfoldings of planar singularities with a nonradial linear part. In particular, we improve normal forms of Takens for analytic unfoldings of nilpotent singularities and normal forms of De Maesschalck, Dumortier and Roussarie for analytic unfoldings of nilpotent contact points in planar slow-fast systems.


References [Enhancements On Off] (What's this?)

  • [Bot57] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203-248. MR 0089473
  • [DM14] Peter De Maesschalck, Gevrey normal forms for nilpotent contact points of order two, Discrete Contin. Dyn. Syst. 34 (2014), no. 2, 677-688. MR 3094600, https://doi.org/10.3934/dcds.2014.34.677
  • [DMD11] P. De Maesschalck and F. Dumortier, Classical Liénard equations of degree $ n\geq 6$ can have $ [\frac {n-1}{2}]+2$ limit cycles, J. Differential Equations 250 (2011), no. 4, 2162-2176. MR 2763568, https://doi.org/10.1016/j.jde.2010.12.003
  • [DMD16] Peter De Maesschalck and Thai Son Doan, Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems, preprint (2016).
  • [DMDR11] P. De Maesschalck, F. Dumortier, and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.) 22 (2011), no. 3-4, 165-206. MR 2853605, https://doi.org/10.1016/j.indag.2011.09.008
  • [DMH14] Peter De Maesschalck and Renato Huzak, Slow divergence integrals in classical Liénard equations near centers, J. Dynam. Differential Equations 27 (2015), no. 1, 177-185. MR 3317395, https://doi.org/10.1007/s10884-014-9358-1
  • [DPR07] Freddy Dumortier, Daniel Panazzolo, and Robert Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1895-1904 (electronic). MR 2286102, https://doi.org/10.1090/S0002-9939-07-08688-1
  • [DR96] Freddy Dumortier and Robert Roussarie, Canard cycles and center manifolds, with an appendix by Cheng Zhi Li, Mem. Amer. Math. Soc. 121 (1996), no. 577, x+100. MR 1327208, https://doi.org/10.1090/memo/0577
  • [DR01] Freddy Dumortier and Robert Roussarie, Multiple canard cycles in generalized Liénard equations, J. Differential Equations 174 (2001), no. 1, 1-29. MR 1844521, https://doi.org/10.1006/jdeq.2000.3947
  • [DR09] Freddy Dumortier and Robert Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 4, 723-781. MR 2552119, https://doi.org/10.3934/dcdss.2009.2.723
  • [Dum] Freddy Dumortier, Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations, Bifurcations and periodic orbits of vector fields (Montreal, PQ, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 408, Kluwer Acad. Publ., Dordrecht, 1993, pp. 19-73. MR 1258518
  • [Dum06] Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations 224 (2006), no. 2, 296-313. MR 2223719, https://doi.org/10.1016/j.jde.2005.08.011
  • [FG65] Wolfgang Fischer and Hans Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965 (1965), 89-94 (German). MR 0184258
  • [FG02] Klaus Fritzsche and Hans Grauert, From holomorphic functions to complex manifolds, Graduate Texts in Mathematics, vol. 213, Springer-Verlag, New York, 2002. MR 1893803
  • [Gri65] Ph. A. Griffiths, The extension problem for compact submanifolds of complex manifolds. I. The case of a trivial normal bundle, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 113-142. MR 0190952
  • [HDM14] Renato Huzak and Peter De Maesschalck, Slow divergence integrals in generalized Liénard equations near centers, Electron. J. Qual. Theory Differ. Equ. (2014), No. 66, 10 pp.. MR 3304192
  • [Kod86] Kunihiko Kodaira, Complex manifolds and deformation of complex structures, translated from the Japanese by Kazuo Akao, with an appendix by Daisuke Fujiwara, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. MR 815922
  • [KS58] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328-466. MR 0112154
  • [KS01] M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations 174 (2001), no. 2, 312-368. MR 1846739, https://doi.org/10.1006/jdeq.2000.3929
  • [Lor06] Frank Loray, A preparation theorem for codimension-one foliations, Ann. of Math. (2) 163 (2006), no. 2, 709-722. MR 2199230, https://doi.org/10.4007/annals.2006.163.709
  • [Rou07] Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 441-448. MR 2257444, https://doi.org/10.3934/dcds.2007.17.441
  • [Sav82] V. I. Savelev, Zero-type imbedding of a sphere into complex surfaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1982), 28-32, 85 (Russian, with English summary). MR 671883
  • [Sma00] Steve Smale, Mathematical problems for the next century, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271-294. MR 1754783
  • [SŻa02] Ewa Stróżyna and Henryk Żołądek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations 179 (2002), no. 2, 479-537. MR 1885678, https://doi.org/10.1006/jdeq.2001.4043
  • [Tak74] Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1-59. MR 0478235

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37G05, 34M45

Retrieve articles in all journals with MSC (2010): 37G05, 34M45


Additional Information

Renato Huzak
Affiliation: Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium

DOI: https://doi.org/10.1090/proc/13539
Received by editor(s): June 20, 2016
Received by editor(s) in revised form: October 17, 2016
Published electronically: March 27, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society