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The index of stable critical points. (English) Zbl 1013.37016
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 \(X^{-1}(0)\cap D=\{0\}\).

MSC:
37C10 Dynamics induced by flows and semiflows
34D20 Stability of solutions to ordinary differential equations
55M25 Degree, winding number
57R25 Vector fields, frame fields in differential topology
57R70 Critical points and critical submanifolds in differential topology
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