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

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
$$\dot x=v(x),\qquad x\in R^n.$$

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

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
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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
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
3. N. B. Medvedeva, “Problema razlicheniya tsentra i fokusa v klasse rostkov s dvumya rebrami diagrammy Nyutona”, Vestnik ChelGU, 2003, no. 9, 86–110
4. Medvedeva N., “On an Analytic Solvability of the Center-Focus Problem”, Dokl. Math., 69:1 (2004), 120–122
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
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
7. N. B. Medvedeva, “On analytic insolubility of the stability problem on the plane”, Russian Math. Surveys, 68:5 (2013), 923–949
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