Stable planar polynomial vector fields
Let \(\chi_ n\) be the vector space of polynomial vector fields in \(R^ 2\) with coefficients of degree \(\leq n\). Every \(X\in \chi_ n\) transported to the upper hemisphere \(S^ 2_+\) by the central projection and normalized suitably extends to an analytic vector field \({\mathcal P}(X)\) on \(S^ 2\). For \(r=1,2,...,\infty,\omega\) denote by \(\Sigma^ r_ n\) the set of all \(X\in \chi_ n\) such that \({\mathcal P}(X)\) is r-stable on \(S^ 2\). Let \(\Sigma_ n\) be the topologically stable fields. Consider \({\mathcal S}_ n=\{X\in \chi_ n:\) all zeros and periodic orbits of \({\mathcal P}(X)\) are hyperbolic and \({\mathcal P}(X)\) has no saddle connection in \(S^ 2\setminus S^ 1\}\). The main theorem says that for \(r=1,2,...,\infty,\omega\) the set \(\Sigma^ r_ n\) coincides with \({\mathcal S}_ n\). Furthermore the \(C^ r\)-bifurcation set \(\chi_ n^{r,1}=\chi_ n\setminus \Sigma^ r_ n\) is contained in the union of countably many analytic submanifolds of codimension \(\geq 1\) in \(\chi_ n\). It follows that \(\chi_ n^{r,1}\) and therefore the topological bifurcation set \(\chi_ n\) \(1=\chi_ n\setminus \Sigma_ n\) have null Lebesgue measure in \(\chi_ n\). Also the generic one- parameter \(C^ 1\)-family of elements in \(\chi_ n\) has at most countably many bifurcations.
scientific article; zbMATH DE number 4050582