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Mat. Sb. (N.S.), 1976, Volume 99(141), Number 2, Pages 162–175 (Mi msb2744)  

This article is cited in 7 scientific papers (total in 7 papers)

Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations

Yu. S. Il'yashenko


Abstract: In this paper the analytic unsolvability of the Ljapunov stability problem and the problem of topological classification of the singular points is proved for the analytic system of differential equations
\begin{equation} \dot x=v(x),\qquad x\in R^n. \end{equation}

This means that there does not exist an analytic criterion that, from a finite segment $v_N(x)$ of the Taylor series of the field $v(x)$ at the origin, would permit one to say whether the singular point $0$ of equation (1) is stable or unstable, or that the stability investigation must consider a longer segment of the Taylor series. In other words, there does not exist an analytic criterion permitting one to distinguish stable, unstable and neutral jets of analytic vector fields with singular point $0$.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1976, 28:2, 140–152

Bibliographic databases:

UDC: 517.9
MSC: 34C05, 34D20, 34D30
Received: 09.10.1974

Citation: Yu. S. Il'yashenko, “Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations”, Mat. Sb. (N.S.), 99(141):2 (1976), 162–175; Math. USSR-Sb., 28:2 (1976), 140–152

Citation in format AMSBIB
\Bibitem{Ily76}
\by Yu.~S.~Il'yashenko
\paper Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations
\jour Mat. Sb. (N.S.)
\yr 1976
\vol 99(141)
\issue 2
\pages 162--175
\mathnet{http://mi.mathnet.ru/msb2744}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=405509}
\zmath{https://zbmath.org/?q=an:0332.34042}
\transl
\jour Math. USSR-Sb.
\yr 1976
\vol 28
\issue 2
\pages 140--152
\crossref{https://doi.org/10.1070/SM1976v028n02ABEH001644}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1976EM69100002}


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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. Kosterin A., “A Stability-Criterion in Case of 1-3 Resonance”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1990, no. 3, 80–83  mathscinet  zmath  isi
    2. Novickov M., “An Investigation Into Stability of Conservative Mechanical Systems Using Analytic Calculations”, Casc'99: Computer Algebra in Scientific Computing, eds. Ganzha V., Mayr E., Vorozhtsov E., Springer-Verlag Berlin, 1999, 317–322  mathscinet  zmath  isi
    3. N. B. Medvedeva, “Problema razlicheniya tsentra i fokusa v klasse rostkov s dvumya rebrami diagrammy Nyutona”, Vestnik ChelGU, 2003, no. 9, 86–110  mathnet
    4. Medvedeva N., “On an Analytic Solvability of the Center-Focus Problem”, Dokl. Math., 69:1 (2004), 120–122  mathnet  zmath  isi
    5. N. B. Medvedeva, “On the Analytic Solvability of the Problem of Distinguishing between Center and Focus”, Proc. Steklov Inst. Math., 254 (2006), 7–93  mathnet  crossref  mathscinet  elib
    6. L. G. Kurakin, “O kriteriyakh ustoichivosti raboty A. M. Lyapunova «Issledovanie odnogo iz osobennykh sluchaev zadachi ob ustoichivosti dvizheniya»”, Vladikavk. matem. zhurn., 11:3 (2009), 28–37  mathnet  mathscinet
    7. 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
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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