Uniform stability of local extrema of an integral curve of an ODE of second order
type: Item
instance of: scholarly article
title: Uniform stability of local extrema of an integral curve of an ODE of second order
zbMATH Open document ID: 1062.34001
author: I. P. Pavlotsky
author: M. Strianese
published in: Mathematical Notes
publication date: 2004-12-06T00:00:00Z
review text: Consider the system \[ \begin{gathered} \alpha(x, y)\dot y= P(x,y),\;\dot x= y,\text{ where}\\ \alpha(x,y)\in C^{1,1},\;P(x,y)\in C^{1,1},\\ (x,y)\in M\subseteq \mathbb{R}^2,\;t\in T\subseteq \mathbb{R}.\end{gathered} \] The set \(S_1= \{(x, y)\in \mathbb{R}^2: \alpha(x, y)= 0,\,P(x,y)\neq 0\}\) is called the singular set of the first type, and the set of the form \(S_2= \{(x, y)\in\mathbb{R}^2: P(x,y)= 0,\, y\alpha(x, y)\neq 0\}\) is called the singular set of the second type. The points \((x,y)\in S_1\cup S_2\) are said to be singular. It is assumed that \(\mu(S_1\cup S_2)= 0\), where \(\mu\) is the Lebesgue measure in \(\mathbb{R}^2\). Different from previous papers of the authors, where the stability and instability of extrema on \(S_1\cup S_2\) under small perturbations of the equation were considered, in the present paper, sufficient conditions are derived for the preservation of the type of a local extremum on the finite part of \(S_1\) or \(S_2\) for the case in which the perturbation on this part does not exceed some explicitly indicated quantity.
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 2121364
zbMATH Keywords: ordinary differential equation
zbMATH Keywords: integral curve
zbMATH Keywords: local extremum
zbMATH Keywords: singular point
zbMATH Keywords: stability
zbMATH Keywords: instability
MaRDI profile type: Publication
full work available at URL: b:matn.0000023314.11781.ce
OpenAlex ID: W2030399185
DOI: 10.1023/B:MATN.0000023314.11781.CE
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description: scientific article; zbMATH DE number 2121364