Global structural stability of flows on open surfaces

Kotus, Janina; Krych, Michał; Nitecki, Zbigniew

doi:10.1090/memo/0261

Cited by 24 publications

(27 citation statements)

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“…There are known conditions which imply global structural stability of vector fields on open surfaces (see [3]K It is interesting to know if these conditions are also necessary. So far the answer is known only p £o* E , In this paper we prove the necessity of one of these conditions for vector fields defined on some class of open surfaces.…”

Section: A Generic Property Of Vector Fields On Open Surfacesmentioning

confidence: 99%

A generic property of vector fields on open surfaces

Kotus¹

1983

Demonstratio Mathematica

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Section: A Generic Property Of Vector Fields On Open Surfacesmentioning

confidence: 99%

A generic property of vector fields on open surfaces

Kotus¹

1983

Demonstratio Mathematica

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“…In [2] it was proved the theorem formulating the sufficient conditions for global C (r>\) structural stability of vector fields on open surfaces (dim M = 2) . These conditions are also necessary for global C structural stability on the plane if r > 1 (see [2]) and for r = 1 on any open surface of finite genus [1]. Here we will generalize it for C (r > 1) vector fields defined on open orientable surface with finite genus and countable space of ends E .…”

Section: Introductionmentioning

confidence: 99%

“…With these definitions, we can formulate sufficient conditions for global structural stability proved in [2]. Proof of the theorem.…”

Section: Introductionmentioning

confidence: 99%

Global 𝐶^{𝑟} structural stability of vector fields on open surfaces with finite genus

Kotus¹

1990

Proc. Amer. Math. Soc.

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Abstract.A vector field X on the open manifold M is globally C structurally stable if X has a neighborhood U in the Whitney C topology such that the trajectories of every vector field Y e U can be mapped onto trajectories of A' by a homeomorphism h: M -» M which is in a preassigned compact-open neighborhood of the identity. In The ideal boundary F of a surface M is the topological space having the equivalence classes of boundary components of M as elements and endowed with such a topology that E with it is homeomorphic to a subset of a Cantor set.By Hr(M) we denote the space of complete Cr vector fields on M with the Cr Whitney topology (r > 1). X, Y denote elements of Hr(M), 4>x denotes the flow induced by X. For x € M, Ox(x) (Ox(x) ,Ox(x)) is the trajectory of x (the positive semitrajectory, the negative semitrajectory) under 4>x . By Ox[x,y] we denote the closed X-trajectory segment from x to y .

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“…These problems have arisen from description of globally structurally stable vector fields on open surfaoes (see [l], [2]). Ve prove that limit sets of oscillating trajectories of Kupka-Smale vector fields contains a saddle at infinity.…”

Section: Introductionmentioning

confidence: 99%

Global structural stability of flows on open surfaces

Cited by 24 publications

References 0 publications

A generic property of vector fields on open surfaces

A generic property of vector fields on open surfaces

Global 𝐶^{𝑟} structural stability of vector fields on open surfaces with finite genus

The Oscillating Trajectories and Saddles at Infinity of Vector Fields on Open Surfaces

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