Global asymptotic stability for differentiable vector fields of ℝ2

Alexandre Fernandes; Carlos Gutierrez; Roland Rabanal

doi:10.1016/j.jde.2004.04.015

Global asymptotic stability for differentiable vector fields of ℝ²

Alexandre Fernandes
, Carlos Gutierrez
, Roland Rabanal

Universidade Federal do Ceará
Universidade de São Paulo

Research output: Contribution to journal › Article › peer-review

47 Scopus citations

Abstract

(a) Let X: ℝ²→ℝ² be a differentiable map (not necessarily C¹) and let Spec(X) be the set of (complex) eigenvalues of the derivative DX_p when p varies in ℝ². If, for some ε>0, Spec(X)∩[0,ε)=∅ then X is injective. (b) Let X: ℝ²→ℝ² be a differentiable vector field such that X(0)=0 and Spec(X)⊂{z∈ ℂ: R(z)<0}. Then, for all p∈ℝ², there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}.

Original language	English
Pages (from-to)	470-482
Number of pages	13
Journal	Journal of Differential Equations
Volume	206
Issue number	2
DOIs	https://doi.org/10.1016/j.jde.2004.04.015
State	Published - 15 Nov 2004
Externally published	Yes

Keywords

Asymptotic stability
Global injectivity
Planar vector fields

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10.1016/j.jde.2004.04.015

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title = "Global asymptotic stability for differentiable vector fields of ℝ2",

abstract = "(a) Let X: ℝ2→ℝ2 be a differentiable map (not necessarily C1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DXp when p varies in ℝ2. If, for some ε>0, Spec(X)∩[0,ε)=∅ then X is injective. (b) Let X: ℝ2→ℝ2 be a differentiable vector field such that X(0)=0 and Spec(X)⊂\{z∈ ℂ: R(z)<0\}. Then, for all p∈ℝ2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to \{0\}.",

keywords = "Asymptotic stability, Global injectivity, Planar vector fields",

author = "Alexandre Fernandes and Carlos Gutierrez and Roland Rabanal",

year = "2004",

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T1 - Global asymptotic stability for differentiable vector fields of ℝ2

AU - Fernandes, Alexandre

AU - Gutierrez, Carlos

AU - Rabanal, Roland

PY - 2004/11/15

Y1 - 2004/11/15

N2 - (a) Let X: ℝ2→ℝ2 be a differentiable map (not necessarily C1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DXp when p varies in ℝ2. If, for some ε>0, Spec(X)∩[0,ε)=∅ then X is injective. (b) Let X: ℝ2→ℝ2 be a differentiable vector field such that X(0)=0 and Spec(X)⊂{z∈ ℂ: R(z)<0}. Then, for all p∈ℝ2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}.

AB - (a) Let X: ℝ2→ℝ2 be a differentiable map (not necessarily C1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DXp when p varies in ℝ2. If, for some ε>0, Spec(X)∩[0,ε)=∅ then X is injective. (b) Let X: ℝ2→ℝ2 be a differentiable vector field such that X(0)=0 and Spec(X)⊂{z∈ ℂ: R(z)<0}. Then, for all p∈ℝ2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}.

KW - Asymptotic stability

KW - Global injectivity

KW - Planar vector fields

UR - https://www.scopus.com/pages/publications/4944242961

U2 - 10.1016/j.jde.2004.04.015

DO - 10.1016/j.jde.2004.04.015

M3 - Article

AN - SCOPUS:4944242961

SN - 0022-0396

VL - 206

SP - 470

EP - 482

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

Global asymptotic stability for differentiable vector fields of ℝ²

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Keywords

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