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Mosc. Math. J., 2007, Volume 7, Number 1, Pages 1–20 (Mi mmj268)  

This article is cited in 4 scientific papers (total in 4 papers)

A counterexample to a multidimensional version of the weakened Hilbert's 16th problem

M. Bobieński, H. Żołądek

Institute of Mathematics, Warsaw University

Abstract: In the weakened 16th Hilbert's Problem one asks for a bound on the number of limit cycles which appear after a polynomial perturbation of a planar polynomial Hamiltonian vector field. It is known that this number is finite for an individual vector field. In the multidimensional generalization of this problem one considers polynomial perturbation of a polynomial vector field with an invariant plane supporting a Hamiltonian dynamics. We present an explicit example of such perturbation with an infinite number of limit cycles which accumulate at some separatrix loop.

Key words and phrases: Polynomial vector field, limit cycle, invariant manifold, Abelian integral

DOI: https://doi.org/10.17323/1609-4514-2007-7-1-1-20

Full text: http://www.ams.org/.../abst7-1-2007.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 34C07, 34C08
Received: January 19, 2006; in revised form June 7, 2006
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Citation: M. Bobieński, H. Żołądek, “A counterexample to a multidimensional version of the weakened Hilbert's 16th problem”, Mosc. Math. J., 7:1 (2007), 1–20

Citation in format AMSBIB
\Bibitem{Bobij}
\by M.~Bobie{\'n}ski, H.~{\.Z}o\l {\k a}dek
\paper A counterexample to a~multidimensional version of the weakened Hilbert's 16th problem
\jour Mosc. Math.~J.
\yr 2007
\vol 7
\issue 1
\pages 1--20
\mathnet{http://mi.mathnet.ru/mmj268}
\crossref{https://doi.org/10.17323/1609-4514-2007-7-1-1-20}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2324554}
\zmath{https://zbmath.org/?q=an:05202833}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261708300001}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Lubowiecki P., Zoladek H., “The Hess-Appelrot system. II. Perturbation and limit cycles”, J Differential Equations, 252:2 (2012), 1701–1722  crossref  mathscinet  zmath  isi  scopus
    2. Lubowiecki P., Zoladek H., “The Hess-Appelrot System. I. Invariant Torus and its Normal Hyperbolicit”, J. Geom. Mech., 4:4 (2012), 443–467  crossref  mathscinet  zmath  isi  elib  scopus
    3. Coll B., Gasull A., Prohens R., “Periodic Orbits for Perturbed Non-Autonomous Differential Equations”, Bull. Sci. Math., 136:7 (2012), 803–819  crossref  mathscinet  zmath  isi  elib  scopus
    4. Caubergh M., “Hilbert's Sixteenth Problem for Polynomial Lienard Equations”, Qual. Theor. Dyn. Syst., 11:1, SI (2012), 3–18  crossref  mathscinet  zmath  isi  elib  scopus
  • Moscow Mathematical Journal
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