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Stable equilibria and vector field index. (English) Zbl 0777.58032
A (Lyapunov) stable equilibrium of a vector field on \(\mathbb{R}^ n\) is considered. It is known [H. Amann, Gewöhnliche Differentialgleichungen (De Gruyter, Berlin, 1983); English translation (1990; Zbl 0708.34002)], that if 0 is an asymptotically stable equilibrium, then the (Hopf) index of the vector field \(F\) at 0 equals \((-1)^ n\). For \(n=2\), this is true even if 0 is only a stable equilibrium. However, for \(n>2\), this is not true for a stable equilibrium. As it turns out, for \(n\geq 4\), every integer \(k\) (for \(n=3\) only \(k\geq-1)\) can be realized as an index of a smooth vector field at stable equilibrium.

MSC:
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34D20 Stability of solutions to ordinary differential equations
55M25 Degree, winding number
57R25 Vector fields, frame fields in differential topology
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