Structurally stable discontinuous vector fields in the plane.
type: Item
instance of: scholarly article
title: Structurally stable discontinuous vector fields in the plane.
zbMATH Open document ID: 1047.37011
DOI: 10.1007/BF02969339
author: Jorge Sotomayor
author: Ana Lúcia F. Machado
published in: Qualitative Theory of Dynamical Systems
publication date: 2003-12-09T00:00:00Z
review text: Let \(M\) be a compact and connected region in \(\mathbb{R}^{2}\) and \(f:M\to\mathbb{R}\) be a \(C^{\infty}\) function having \(0\) as a regular value. For simplicity, it is assumed that \(D=f^{-1}(0)\) has a single connected component and thus \(M\setminus D\) has two connected components \(N=f^{-1} (0,\infty)\) and \(S=f^{-1}(-\infty,0).\) Let \(\chi^{r}\) be the space of \(C^{r}\) vector fields on \(M\) for some \(r\geq1,\) and \(\Omega^{r}=\Omega^{r}(M,f)\) be the space of vector fields \(Z=(X,Y)\) on \(M\) defined by \[ Z(q)=\begin{cases} X(q) & \text{if }f(q)\geq0,\\ Y(q) & \text{if }f(q)\leq0, \end{cases} \] where \(X,Y\in\chi^{r}.\) The authors carry out the stability and transition analysis of solutions between \(N\) and \(S\) by applying the regularization method developed by \textit{J. Sotomayor} and \textit{M. A. Teixeira} [in: L. Magalhaes (ed.) et al., International conference on differential equations. Papers from the conference, EQUADIFF 95, Lisboa, Portugal, July 24--29, 1995, Singapore: World Scientific, 207--223 (1998; Zbl 0957.37015)] which consists in defining a one parameter family of continuous vector fields \(Z_{\varepsilon}\) (which approaches \(Z\) as \(\varepsilon\rightarrow0\)) by averaging \(X\) and \(Y.\) The results of Sotomayor and Teixeira are adjusted to the case in question. Namely, conditions for the regularized vector fields to be structurally stable are extended and shown to be generic.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34D30
Mathematics Subject Classification ID: 37C20
zbMATH DE Number: 2015880
zbMATH Keywords: discontinuous vector fields
zbMATH Keywords: structural stability
zbMATH Keywords: transition
zbMATH Keywords: regularization
zbMATH Keywords: poly-trajectory
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02969339
OpenAlex ID: W1968107465
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description: scientific article; zbMATH DE number 2015880