Global structural stability of vector fields on open surfaces with finite genus
Author:
Janina Kotus
Journal:
Proc. Amer. Math. Soc. 108 (1990), 10391046
MSC:
Primary 58F10; Secondary 34D30
MathSciNet review:
1004419
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Abstract: A vector field on the open manifold is globally structurally stable if has a neighborhood in the Whitney topology such that the trajectories of every vector field can be mapped onto trajectories of by a homeomorphism which is in a preassigned compactopen neighborhood of the identity. In [2] it was proved the theorem formulating the sufficient conditions for global structural stability of vector fields on open surfaces . These conditions are also necessary for global structural stability on the plane if (see [2]) and for on any open surface of finite genus [1]. Here we will generalize it for vector fields defined on open orientable surface with finite genus and countable space of ends .
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 [1]
 C. Camacho, M. Krych, R. Mané and Z. Nitecki, An extension of Peixoto's structural stability theorem to open surfaces with finite genus, Lecture Notes in Math. 1007 (1983), 6087. MR 730263 (85i:58065)
 [2]
 J. Kotus, M. Krych and Z. Nitecki, Global structural stability of flows on open surfaces, Mem. Amer. Math. Soc. 37(261) (1982). MR 653093 (83h:58055)
 [3]
 J. Kotus, The oscillating trajectories and saddles at infinity of vector fields on open surface, Demonstratio Math. (to appear). MR 1081451 (92d:58097)
 [4]
 V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960. MR 0121520 (22:12258)
 [5]
 M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214227. MR 0209602 (35:499)
 [6]
 I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259269. MR 0143186 (26:746)
 [7]
 A. J. Schwartz, A generalisation of a PoincaréBendixon theorem to closed twodimensional manifolds, Amer. J. Math. 85 (1963), 453458. MR 0155061 (27:5003)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010044193
PII:
S 00029939(1990)10044193
Article copyright:
© Copyright 1990 American Mathematical Society
