  We study differentiability properties in a particular case of the Palmer's
linearization Theorem, which states the existence of an homeomorphism $H$
between the solutions of a linear ODE system having exponential dichotomy and a
quasilinear system. Indeed, if the linear system is uniformly asymptotically
stable, sufficient conditions ensuring that $H$ is a $C^{2}$ preserving
orientation diffeomorphism are given. As an application, we generalize a
converse result of density functions for a nonlinear system in the
nonautonomous case.
