Differentiability of Palmer's linearization theorem and converse result for density functions

We study differentiability properties in a particular case of the Palmer's linearization theorem, which states the existence of a 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 C2 preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.