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

Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 140-152

Résumé

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.