  We study the differentiability properties of the topological equivalence
between a uniformly asymptotically stable linear nonautonomous system and a
perturbed system with suitable nonlinearities. For this purpose, we construct a
uniformly continuous homeomorphism inspired in the Palmer's one restricted to
the positive half line, providing sufficient conditions ensuring its
$C^{r}$--smoothness. Additionally, we study the preservation of the uniform
stability properties by this homeomorphism.
