Duke Mathematical Journal

On the finite cyclicity of open period annuli

Lubomir Gavrilov and Dmitry Novikov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let Π be an open, relatively compact period annulus of real analytic vector field X0 on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from Π under a given multiparameter analytic deformation Xλ of X0 is finite provided that X0 is either a Hamiltonian or generic Darbouxian vector field

Article information

Source
Duke Math. J., Volume 152, Number 1 (2010), 1-26.

Dates
First available in Project Euclid: 11 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1268317522

Digital Object Identifier
doi:10.1215/00127094-2010-005

Mathematical Reviews number (MathSciNet)
MR2643055

Zentralblatt MATH identifier
1205.34030

Subjects
Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory

Citation

Gavrilov, Lubomir; Novikov, Dmitry. On the finite cyclicity of open period annuli. Duke Math. J. 152 (2010), no. 1, 1--26. doi:10.1215/00127094-2010-005. https://projecteuclid.org/euclid.dmj/1268317522


Export citation

References

  • V. I. Arnold, Arnold's Problems, trans. and rev. ed., Springer, Berlin, 2004.
  • G. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal hilbert sixteenth problem, preprint.
  • M. BobieńSki and P. MardešIć, Pseudo-Abelian integrals along Darboux cycles, Proc. Lond. Math. Soc. (3) 97 (2008), 669--688.
  • M. BobieńSki, P. MardešIć, and D. Novikov, Pseudo-abelian integrals: Unfolding generic exponential, J. Differential Equations 247 (2009), 3357--3376.
  • M. Caubergh, F. Dumortier, and R. Roussarie, Alien limit cycles in rigid unfoldings of a Hamiltonian $2$-saddle cycle. Commun. Pure Appl. Anal. 6 (2007), 1--21.
  • F. Dumortier and R. Roussarie, Abelian integrals and limit cycles. J. Differential Equations 227 (2006), 116--165.
  • L. Gavrilov, Higher order Poincaré-Pontryagin functions and iterated path integrals, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), 663--682.
  • L. Gavrilov, Cyclicity of period annuli and principalization of Bautin ideals, Ergodic Theory Dynam. Systems 28 (2008), 1497--1507.
  • L. Gavrilov and I. D. Iliev, The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields, Amer. J. Math. 127 (2005), 1153--1190.
  • H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460--472.
  • M. Hall, The Theory of Groups, reprint of the 1968 ed., Chelsea, New York, 1976.
  • N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970) 175--232.
  • A. G. Khovanskiǐ, Real analytic manifolds with the property of finiteness, and complex abelian integrals (in Russian), Funktsional. Anal. i Prilozhen. 18 (1984), 40--50.
  • D. Novikov, On limit cycles appearing by polynomial perturbation of Darbouxian integrable systems, Geom. Funct. Anal. 18 (2009), 1750--1773.
  • L. S. Pontryagin, Über Autoschwingungssysteme, die den Hamiltonischen nahe liegen, Phys. Z. Sowjetunion 6 (1934), 25--28.; On dynamics systems close to Hamiltonian systems, Zh. Eksp. Teor. Fiz. 4 (1934), 234--238.
  • R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Brasil. Mat. 17 (1986), 67--101.
  • —, ``A note on finite cyclicity property and Hilbert's 16th problem'' in Dynamical Systems (Valparaiso, Chile, 1986), Lecture Notes in Math. 1331, Springer, Berlin, 1988, 161--168.
  • —, Cyclicité finie des lacets et des points cuspidaux, Nonlinearity 2 (1989), 73--117.
  • —, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, Progr. Math. 164, Birkhäuser, Basel, 1998.
  • —, Melnikov functions and Bautin ideal, Qual. Theory Dyn. Syst. 2 (2001), 67--78.
  • J.-P. Serre, Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University, corrected fifth printing of the 2nd ed., Lecture Notes in Math. 1500, Springer, Berlin, 2006.
  • A. N. Varchenko, Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles (in Russian), Funktsional. Anal. i Prilozhen. 18, no. 2 (1984), 14--25. \endthebibliography