Global structural stability of vector fields on open surfaces with finite genus

Author:
Janina Kotus

Journal:
Proc. Amer. Math. Soc. **108** (1990), 1039-1046

MSC:
Primary 58F10; Secondary 34D30

MathSciNet review:
1004419

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 compact-open 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 .

**[1]**C. Camacho, M. Krych, R. Mañé, and Z. Nitecki,*An extension of Peixoto’s structural stability theorem to open surfaces with finite genus*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 60–87. MR**730263**, 10.1007/BFb0061410**[2]**Janina Kotus, Michał Krych, and Zbigniew Nitecki,*Global structural stability of flows on open surfaces*, Mem. Amer. Math. Soc.**37**(1982), no. 261, v+108. MR**653093**, 10.1090/memo/0261**[3]**Janina Kotus,*The oscillating trajectories and saddles at infinity of vector fields on open surfaces*, Demonstratio Math.**23**(1990), no. 1, 241–251. MR**1081451****[4]**V. V. Nemytskii and V. V. Stepanov,*Qualitative theory of differential equations*, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR**0121520****[5]**M. M. Peixoto,*On an approximation theorem of Kupka and Smale*, J. Differential Equations**3**(1967), 214–227. MR**0209602****[6]**Ian Richards,*On the classification of noncompact surfaces*, Trans. Amer. Math. Soc.**106**(1963), 259–269. MR**0143186**, 10.1090/S0002-9947-1963-0143186-0**[7]**Arthur J. Schwartz,*A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds*, Amer. J. Math. 85 (1963), 453-458; errata, ibid**85**(1963), 753. MR**0155061**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F10,
34D30

Retrieve articles in all journals with MSC: 58F10, 34D30

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-1004419-3

Article copyright:
© Copyright 1990
American Mathematical Society