UMA NOTA SOBRE O TEOREMA DE LINEARIZAÇÃO DE GROBMAN-HARTMAN PARA CAMPOS VETORIAIS

Abstract

In mathematics, the term ``nonlinear" generally corresponds to a more difficult analysis. Since linear systems are easier to analyze, a key way to understand nonlinear systems is to find out where and when they can be well-approximated by linear systems. In this respect, we have a famous theorem in nonlinear differential equations due to Grobman [Gro59] and Hartman [Har60a], which guarantees us that a vector field of class C1, X : W ⊂ Rn → Rn (where W is an open set and p is a hyperbolic singularity) is topologically conjugated to a linear field A = DX(p) (in a neighborhood of p and 0, respectively).This work aims to prove that
the conjugation (``change of variables'') in the Grobman-Hartman theorem is always H\"{o}lder continuous. Finally, we will give an example to illustrate our result.

Keywords: Vector fields, conjugation, linearization, hyperbolic singularities.