scholarly journals On the Coercivity of Continuously Differentiable Vector Fields

2020 ◽  
Vol 19 (2) ◽  
Author(s):  
Răzvan M. Tudoran
Keyword(s):  
Vector Fields ◽  
Differentiable Vector

scholarly journals Global asymptotic stability for differentiable vector fields of R2

2004 ◽  
Vol 206 (2) ◽  
pp. 470-482 ◽  
Author(s):  
Alexandre Fernandes ◽  
Carlos Gutierrez ◽  
Roland Rabanal
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Vector Fields ◽  
Differentiable Vector

scholarly journals Zero sets of Lie algebras of analytic vector fields on real and complex two-dimensional manifolds

2017 ◽  
Vol 39 (4) ◽  
pp. 954-979 ◽  
Author(s):  
MORRIS W. HIRSCH ◽  
F.-J. TURIEL
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Fixed Point Theorems ◽  
Zero Set ◽  
Analytic Vector ◽  
Ground Field ◽  
Zero Sets ◽  
Finite Dimensional ◽  
Differentiable Vector ◽  
Analytic Vector Field

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

scholarly journals Asymptotic stability at infinity for differentiable vector fields of the plane

2006 ◽  
Vol 231 (1) ◽  
pp. 165-181 ◽  
Author(s):  
Carlos Gutierrez ◽  
Benito Pires ◽  
Roland Rabanal
Keyword(s):  
Asymptotic Stability ◽  
Vector Fields ◽  
Differentiable Vector
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