  The aim of this paper is to study the normal forms of nonautonomous
differential systems. For doing so, we first investigate the nonuniform
dichotomy spectrum of the linear evolution operators that admit a nonuniform
exponential dichotomy, where the linear evolution operators are defined by
nonautonomous differential equations $\dot x=A(t)x$ in $\R^n$. Using the
nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous
linear differential equations. Finally we establish the finite jet normal forms
of the nonlinear differential systems $\dot x=A(t)x+f(t,x)$ in $\R^n$, which is
based on the nonuniform dichotomy spectrum and the normal forms of the
nonautonomous linear systems.
