Nonuniform Dichotomy Spectrum and Normal Forms for Nonautonomous Differential Systems

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.