
This article is cited in 3 scientific papers (total in 3 papers)
Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order
I. P. Pavlotsky^{}, M. Strianese^{}
Abstract:
A secondorder 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 Received: 30.01.2003
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
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\paper Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order
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\pages 384391
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\transl
\jour Math. Notes
\yr 2004
\vol 75
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\pages 352359
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http://mi.mathnet.ru/eng/mz42https://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:

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

Pavlotsky, IP, “Stability of an integral curve of a secondorder ordinary differential equation at the intersection of its singular set with the axis y=0”, Doklady Mathematics, 77:2 (2008), 179

Pavlotsky, IP, “Behavior of the trajectories of a secondorder ordinary differential equation in a neighborhood of a singular point”, Doklady Mathematics, 77:2 (2008), 205

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