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Instability of equilibria in dimension three. (English) Zbl 0930.37025

The author studies analytic vector fields defined on a neighborhood of \(0\) in \(n\) dimensional Euclidean space, which have isolated singular point at \(0\). Such a vector field is called stable, if there exists a fundamental system of neighborhoods of \(0\) which are invariant by \(v\). There are no known examples for stable vector fields in odd dimension, whereas many examples exist in even dimension. The author proves that if \(M\) is an 3-dimensional irreducible analytic subvariety of some open neighborhood \(0\), whose link has nonvanishing Euler characteristic, then \(0\) is an unstable point.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
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References:

[1] [AI] , , Ordinary differential equations, Enc. Math. Sci. vol. 1, Springer Verlag (1988). · Zbl 0718.34070
[2] [BD] , , Smooth invariant curves for germs of vector fields in R3 whose linear part generates a rotation, Jour. Diff. Eq., 102 (1980), 95-116. · Zbl 0584.58009
[3] [Ca1] , Desingularization strategies for three-dimensional vector fields, Springer Lecture Notes 1259 (1987). · Zbl 0645.14005
[4] [Ca2] , Final forms for a three dimensional vector field under blowing-up, Ann. Inst. Fourier, 37-2 (1987), 151-193. · Zbl 0607.58027
[5] [Pa] , Stability of equilibria in a potential field, Funct. Anal. Appl., 11 (1977), 42-55. · Zbl 0415.70006
[6] [Sa] , Desingularization, residuos y solucion de las ecuaciones diferenciales, Thesis, Univ. de Salamanca (1996).
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