  Hartman-Grobman theorem was initially extended to the non-autonomous cases by
Palmer. Usually, dichotomy is an essential condition of Palmer's linearization
theorem. Is Palmer's linearization theorem valid for the systems with
trichotomy? In this paper, we obtain new versions of the linearization theorem
if linear system admits exponential trichotomy on $\mathbb{R}$. { Furthermore,
the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of
our linearization theorems are H\"{o}lder continuous}. In addition, if a system
is periodic, we find the equivalent function $\mathscr H(t,x)$ and its inverse
$\mathscr L(t,y)$ of our linearization theorems do not have periodicity or
asymptotical periodicity. To the best of our knowledge, this is the first paper
studying the linearization with exponential trichotomy.
