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Mat. Sb., 2004, Volume 195, Number 7, Pages 3–20 (Mi msb832)  

This article is cited in 10 scientific papers (total in 10 papers)

Poincaré map for some polynomial systems of differential equations

V. P. Varin

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: One approach to the classical problem of distinguishing between a centre and a focus for a system of differential equations with polynomial right-hand sides in the plane is discussed. For a broad class of such systems necessary and sufficient conditions for a centre are expressed in terms of equations in variations of higher order. By contrast with the existing methods of investigation, attention is concentrated on the explicit calculation of the asymptotic behaviour of the Poincaré map rather than on finding sufficient centre conditions as such; this also enables one to study bifurcations of birth of arbitrarily strongly degenerate cycles.

DOI: https://doi.org/10.4213/sm832

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English version:
Sbornik: Mathematics, 2004, 195:7, 917–934

Bibliographic databases:

UDC: 517.5
MSC: 34C05
Received: 18.06.2003 and 15.01.2004

Citation: V. P. Varin, “Poincaré map for some polynomial systems of differential equations”, Mat. Sb., 195:7 (2004), 3–20; Sb. Math., 195:7 (2004), 917–934

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. Algaba A., Gamero E., García C., “The integrability problem for a class of planar systems”, Nonlinearity, 22:2 (2009), 395–420  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Varin V., “Analysis of Degenerate Solutions to ODEs with Variational Equations of Higher Order”, Numerical Analysis and Applied Mathematics, AIP Conf. Proc., 1168, 2009, 1300–1303  crossref  adsnasa  isi
    3. V. P. Varin, “Ploskie razlozheniya reshenii ODU vblizi osobennosti”, Preprinty IPM im. M. V. Keldysha, 2010, 064, 12 pp.  mathnet
    4. G. A. Leonov, N. V. Kuznetsov, E. V. Kudryashova, O. A. Kuznetsova, “Sovremennye metody simvolnykh vychislenii: lyapunovskie velichiny i 16-aya problema Gilberta”, Tr. SPIIRAN, 16 (2011), 5–36  mathnet
    5. A. S. Voronin, N. B. Medvedeva, “Asymptotics of the monodromy transformation in certain classes of monodromy germs”, Izv. Math., 77:2 (2013), 253–270  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. N. B. Medvedeva, “On analytic insolubility of the stability problem on the plane”, Russian Math. Surveys, 68:5 (2013), 923–949  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. J. Giné, J. Llibre, “A method for characterizing nilpotent centers”, J. Math. Anal. Appl., 413:1 (2014), 537–545  crossref  mathscinet  zmath  isi
    8. N. B. Medvedeva, “Asimptoticheskoe razlozhenie preobrazovaniya monodromii”, Chelyab. fiz.-matem. zhurn., 1:1 (2016), 59–72  mathnet  elib
    9. N. B. Medvedeva, M. A. Sosnovskaya, “Vychislenie koeffitsientov stepennogo ryada preobrazovaniya monodromii v sisteme Maple”, Chelyab. fiz.-matem. zhurn., 2:2 (2017), 181–192  mathnet  mathscinet  elib
    10. N. B. Medvedeva, “Problema razlicheniya tsentra i fokusa v prostranstve vektornykh polei s fiksirovannoi diagrammoi Nyutona”, Matem. sb., 211:10 (2020), 50–97  mathnet  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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