On the Minimal Sets of Non-Singular Vector Fields

1966 ◽  
Vol 84 (3) ◽  
pp. 529 ◽  
Author(s):  
F. Wesley Wilson
Keyword(s):  
Vector Fields ◽  
Singular Vector ◽  
Minimal Sets

Chaos and non-trivial minimal sets in discontinuous vector fields

2015 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Rodrigo D. Euzébio ◽  
Tiago De Carvalho
Keyword(s):  
Vector Fields ◽  
Minimal Sets ◽  
Discontinuous Vector Fields

A simple global characterization for normal forms of singular vector fields

1987 ◽  
Vol 29 (1-2) ◽  
pp. 95-127 ◽  
Author(s):  
C. Elphick ◽  
E. Tirapegui ◽  
M.E. Brachet ◽  
P. Coullet ◽  
G. Iooss
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Singular Vector

scholarly journals On Poincaré-Bendixson Theorem and non-trivial minimal sets in planar nonsmooth vector fields

2018 ◽  
Vol 62 ◽  
pp. 113-131 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Tiago Carvalho ◽  
Rodrigo D. Euzébio
Keyword(s):  
Vector Fields ◽  
Minimal Sets

scholarly journals Extended Fuller index, sky catastrophes and the Seifert conjecture

2018 ◽  
Vol 29 (13) ◽  
pp. 1850096 ◽  
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
General Class ◽  
Smooth Manifold ◽  
Vector Fields ◽  
Singular Vector ◽  
Natural Question ◽  
Reeb Vector ◽  
Fuller Index

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

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