This article is cited in 8 scientific papers (total in 9 papers)
Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations
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
˙x=v(x),x∈Rn.
This means that there does not exist an analytic criterion that, from a finite segment vN(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.
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”, Math. USSR-Sb., 28:2 (1976), 140–152
\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 Math. USSR-Sb.
\yr 1976
\vol 28
\issue 2
\pages 140--152
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\crossref{https://doi.org/10.1070/SM1976v028n02ABEH001644}
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Linking options:
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https://doi.org/10.1070/SM1976v028n02ABEH001644
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This publication is cited in the following 9 articles:
T. N. Bakiev, A. I. Bufetov, V. A. Vasilev, S. M. Voronin, A. A. Glutsyuk, A. S. Gorodetskii, A. V. Dukov, V. Yu. Kaloshin, A. V. Klimenko, V. V. Kozlov, S. B. Kuksin, S. K. Lando, V. S. Oganesyan, G. I. Olshanskii, S. Yu. Pilyugin, O. V. Pochinka, Ya. G. Sinai, A. S. Skripchenko, A. L. Skubachevskii, I. A. Taimanov, V. A. Timorin, V. M. Tikhomirov, D. V. Treschev, D. A. Filimonov, K. M. Khanin, Kh. Khedenmalm, A. G. Khovanskii, M. A. Tsfasman, A. I. Shafarevich, I. S. Shilin, S. Yu. Yakovenko, “K 80-letiyu Yuliya Sergeevicha Ilyashenko”, UMN, 80:2(482) (2025), 171–183
N. B. Medvedeva, “On analytic insolubility of the stability problem on the plane”, Russian Math. Surveys, 68:5 (2013), 923–949
Valery V. Kozlov, Stanislav D. Furta, Springer Monographs in Mathematics, Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations, 2013, 77
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
N. B. Medvedeva, “On the Analytic Solvability of the Problem of Distinguishing between Center and Focus”, Proc. Steklov Inst. Math., 254 (2006), 7–93
Medvedeva N., “On an Analytic Solvability of the Center-Focus Problem”, Dokl. Math., 69:1 (2004), 120–122
N. B. Medvedeva, “Problema razlicheniya tsentra i fokusa v klasse rostkov
s dvumya rebrami diagrammy Nyutona”, Vestnik ChelGU, 2003, no. 9, 86–110
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
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