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 Mat. Zametki, 2002, Volume 71, Issue 5, Pages 742–750 (Mi mz382)

Extremal Points of Integral Curves of Second-Order Ordinary Differential Equations and Their Local Stability

I. P. Pavlotsky, M. Strianese

Università degli Studi di Napoli Federico II

Abstract: In [1–3] an extension of the solution of the equation $a(x,\dot x)\ddot x=1$, $x\in \mathbb R$, $a(x,\dot x)\in C^1$, to the singular set $S=\{(x,y)\in \mathbb R^2:a(x,y)=0\}$, $y=\dot x$, is defined in terms of the first integral. In this case all stationary points and all local extrema of the integral curve $x(y)$ such that the function $x(y)$ has a derivative at the extreme point belong to a set $S\cup Y$, where $Y$ is the line $y=0$. We study the local stability of local extrema of different types in the families of equations $[a(x,y)+\varepsilon b(x,y)]\dot y=1$, $b(x,y)\in C^1$ for $|\varepsilon |$ small enough. Introduce the notation $S^*=\{(x,y)\in \mathbb R^2:a(x,y)+\varepsilon b(x,y)=0\}$. By abuse of language, we talk about the stability of local extrema when $S$ is replaced with $S^*$. Some sufficient conditions for stability and instability are found.

DOI: https://doi.org/10.4213/mzm382

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English version:
Mathematical Notes, 2002, 71:5, 676–683

Bibliographic databases:

UDC: 517

Citation: I. P. Pavlotsky, M. Strianese, “Extremal Points of Integral Curves of Second-Order Ordinary Differential Equations and Their Local Stability”, Mat. Zametki, 71:5 (2002), 742–750; Math. Notes, 71:5 (2002), 676–683

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz382
• https://doi.org/10.4213/mzm382
• http://mi.mathnet.ru/eng/mz/v71/i5/p742

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This publication is cited in the following articles:
1. I. P. Pavlotsky, M. Strianese, “Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order”, Math. Notes, 75:3 (2004), 352–359
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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