Structurally stable discontinuous vector fields in the plane.
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.
scientific article; zbMATH DE number 2015880
scientific article