Uma condição de injetividade e a estabilidade assintótica global no plano
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Data
2010-03-29
Autores
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Editor
Universidade Federal de Goiás
Resumo
In this work we are interested in the solution of the following problem: Let Y = ( f ,g)
be a vector field of class C1 in R2. Suppose that (x, y) = (0,0) is a singular point
of Y and assume that for any q ∈ R2, the eigenvalues of DY have negative real part,
this is, det(DY) > 0 and tr(DY) < 0. Then, the solution (x, y) = (0,0) of Y is globally
asymptotically stable.
To this end, we show that this problema is equivalent to the following: Let Y : R2 →R2
be a C1 vector field. If det(DY) > 0 and tr(DY) < 0, then Y is globally injective. This
equivalence was proved by C. Olech [1].
So we show the injectivity of the vector field Y under the conditions det(DY) > 0 and
tr(DY)<0. In fact, we present a more stronger result, which was obtained by C. Gutierrez
and can be found in [4]. This result is given by: Any planar vector field X of class C2
satisfying the r-eigenvalue condition for some r ∈ [0,¥) is injective.
Descrição
Palavras-chave
Estabilidade global , Atrator global , Conjectura jacobiana , Componentes Reeb , Condições de injetividade , Global estability , Global attractor , Jacobian conjecture , Reeb component , Injective condition , Estabilidade global; Atrator global; Conjectura jacobiana; Componentes Reeb; Condições de injetividade
Citação
SOUZA, Wender José de. A injectividade condition and the global asymptotic estability on the plane. 2010. 99 f. Dissertação (Mestrado em Ciências Exatas e da Terra) - Universidade Federal de Goiás, Goiânia, 2010.