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Funktsional. Anal. i Prilozhen., 2010, Volume 44, Issue 1, Pages 27–43 (Mi faa2980)  

This article is cited in 6 scientific papers (total in 6 papers)

The Monodromy Problem and the Tangential Center Problem

C. Christophera, P. Mardešicb

a School of Mathematics and Statistics, University of Plymouth
b Institut de Mathématiques de Bourgogne, Unité mixte de recherche 5584 du C.N.R.S., Université de Bourgogne

Abstract: We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center–focus problem asks for conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both of these questions for the case in which the Hamiltonian is hyperelliptic. As a by-product, we solve the corresponding problems for the $0$-dimensional Abelian integrals defined by Gavrilov and Movasati.

Keywords: tangential center, Abelian integral, composition, monodromy

DOI: https://doi.org/10.4213/faa2980

Full text: PDF file (295 kB)
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English version:
Functional Analysis and Its Applications, 2010, 44:1, 22–35

Bibliographic databases:

UDC: 517.9
Received: 08.10.2008

Citation: C. Christopher, P. Mardešic, “The Monodromy Problem and the Tangential Center Problem”, Funktsional. Anal. i Prilozhen., 44:1 (2010), 27–43; Funct. Anal. Appl., 44:1 (2010), 22–35

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Francoise J.P., Pakovich F., Yomdin Y., Zhao W., “Moment vanishing problem and positivity: Some examples”, Bull. Sci. Math., 135:1 (2011), 10–32  crossref  mathscinet  zmath  isi  scopus
    2. Alvarez A., Bravo J.L., Mardesic P., “Vanishing Abelian Integrals on Zero-Dimensional Cycles”, Proc. London Math. Soc., 107:6 (2013), 1302–1330  crossref  mathscinet  zmath  isi  scopus
    3. A. Álvarez, J. L. Bravo, P. Mardešić, “Inductive solution of the tangential center problem on zero-cycles”, Mosc. Math. J., 13:4 (2013), 555–583  mathnet  crossref  mathscinet
    4. A. Álvarez, J.L. Bravo, C. Christopher, “On the Trigonometric Moment Problem”, Ergod. Theory Dyn. Syst., 34:1 (2014), 1–20  crossref  mathscinet  zmath  isi  scopus
    5. L. Gavrilov, F. Pakovich, “Moments on Riemann surfaces and hyperelliptic Abelian integrals”, Comment. Math. Helv., 89:1 (2014), 125–155  crossref  mathscinet  zmath  isi  scopus
    6. Pontigo-Herrera J., “Tangential Center Problem For a Family of Non-Generic Hamiltonians”, J. Dyn. Control Syst., 23:3 (2017), 597–622  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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