The index of stable critical points (Q1867184)

Given \(k\in{\mathbb{Z}}\), this paper is devoted to the construction in dimension \(n\geq 3\) of a \(C^\infty\) vector field \(X:U\to{\mathbb{R}}^n\), \(U\) open in \({\mathbb{R}}^n\), with a unique zero in \(0\), with the property that \(0\) is a stable equilibrium (in positive and negative time) for the differential equation \(\dot x=X(x)\) and the local index of \(X\) at \(0\) is equal to \(k\). Here, by local index at an isolated zero \(x_0\) of a \(C^1\) vector field \(X\) it is meant the topological degree of the \(S^{n-1}\)-valued map \(x\mapsto X(x)/\|X(x)\|\) defined on the boundary of a disk \(D\subset U\) centered at \(0\) and such that \(X^{-1}(0)\cap D=\{0\}\).