Geodesic
Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations
Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 140-152 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations},
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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. Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 140-152. http://geodesic.mathdoc.fr/item/SM_1976_28_2_a1/

[1] V. I. Arnold, “Algebraicheskaya nerazreshimost problemy ustoichivosti po Lyapunovu i problemy topologicheskoi klassifikatsii osobykh tochek analiticheskoi sistemy differentsialnykh uravnenii”, Uspekhi matem. nauk, XXV:2 (152) (1970), 265–266 | MR

[2] V. I. Arnold, “Algebraicheskaya nerazreshimost problemy ustoichivosti po Lyapunovu i problemy topologicheskoi klassifikatsii osobykh tochek analiticheskoi sistemy differentsialnykh uravnenii”, Funk. analiz, 4:3 (1970), 1–9 | MR

[3] V. I. Arnold, “Malye znamenateli. I: Ob otobrazheniyakh okruzhnosti na sebya”, Izv. AN SSSR, seriya matem., 25 (1961), 21–86 | MR

[4] Yu. S. Ilyashenko, “Algebraicheskaya nerazreshimost i pochti algebraicheskaya razreshimost problemy tsentr—fokus”, Funkts. analiz, 6:3 (1972), 30–37 | MR