Uma condição de injetividade e a estabilidade assintótica global no plano / A injectividade condition and the global asymptotic estability on the plane

AUTOR(ES)

Wender José de Souza

DATA DE PUBLICAÇÃO

2010

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.

ASSUNTO(S)

estabilidade global; atrator global; conjectura jacobiana; componentes reeb; condições de injetividade atrator global componentes reeb geometria e topologia global estability global attractor injective condition condições de injetividade estabilidade global conjectura jacobiana jacobian conjecture reeb component

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