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 Mat. Zametki, 2004, Volume 75, Issue 3, Pages 384–391 (Mi mz42)

Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

I. P. Pavlotsky, M. Strianese

Abstract: A second-order equation can have singular sets of first and second type, $S_1$ and $S_2$ (see the introduction), where the integral curve $x(y)$ does not exist in the ordinary sense but where it can be extended by using the first integral [1–5]. Denote by $Y$ the Cartesian axis $y=0$. If the function $x(y)$ has a derivative at a point of local extremum of this function, then this point belongs to $S_1\cup Y$. The extrema at which $y'(x)$ does not exist can be placed on $S_2$. In [5–8], the stability and instability of extrema on $S_1\cup S_2$ under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of x(y) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of $S_1$ or $S_2$ are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.

DOI: https://doi.org/10.4213/mzm42

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English version:
Mathematical Notes, 2004, 75:3, 352–359

Bibliographic databases:

UDC: 517.925.5

Citation: I. P. Pavlotsky, M. Strianese, “Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order”, Mat. Zametki, 75:3 (2004), 384–391; Math. Notes, 75:3 (2004), 352–359

Citation in format AMSBIB
\Bibitem{PavStr04} \by I.~P.~Pavlotsky, M.~Strianese \paper Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order \jour Mat. Zametki \yr 2004 \vol 75 \issue 3 \pages 384--391 \mathnet{http://mi.mathnet.ru/mz42} \crossref{https://doi.org/10.4213/mzm42} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2068801} \zmath{https://zbmath.org/?q=an:1062.34001} \transl \jour Math. Notes \yr 2004 \vol 75 \issue 3 \pages 352--359 \crossref{https://doi.org/10.1023/B:MATN.0000023314.11781.ce} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000221289900006} 

• http://mi.mathnet.ru/eng/mz42
• https://doi.org/10.4213/mzm42
• http://mi.mathnet.ru/eng/mz/v75/i3/p384

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This publication is cited in the following articles:
1. Pavlotskii, IP, “Properties of the trajectory of an ordinary differential equation in a neighborhood of a singular point of the second type”, Doklady Mathematics, 75:3 (2007), 440
2. Pavlotsky, IP, “Stability of an integral curve of a second-order ordinary differential equation at the intersection of its singular set with the axis y=0”, Doklady Mathematics, 77:2 (2008), 179
3. Pavlotsky, IP, “Behavior of the trajectories of a second-order ordinary differential equation in a neighborhood of a singular point”, Doklady Mathematics, 77:2 (2008), 205
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