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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
##### Keywords:
vector field index; stable equilibrium
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##### References:
 [1] Amann, H., Gewöhnliche differentialgleichungen, (1983), De Gruyter Berlin [2] Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin · Zbl 0559.47040 [3] Dieudonné, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201 [4] Dold, A., Lectures on algebraic topology, (1972), Springer Berlin · Zbl 0234.55001 [5] Milnor, J., Topology from the differentiable viewpoint, (1965), The University Press of Virginia Charlottesville · Zbl 0136.20402 [6] Thews, K., Der abbildungsgrad von vektorfeldern zu stabilen ruhelagen, Arch. math., 52, 71-74, (1989) · Zbl 0633.34027
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