Linearization for Nonautonomous Differential Equations

In the first part of the chapter, we show that a hyperbolic nonautonomous differential equation can be smoothly linearized provided that the associated Sacker-Sell spectrum of the linear system satisfies a non-resonance condition. This result extends the classical Sternberg theorem to nonautonomous differential equations. The second part of the chapter is devoted to a partial linearization theorem for planar nonautonomous differential equations with one center-like direction and one hyperbolic direction. Gap conditions are formulated in terms of the dichotomy spectral intervals. The content of this chapter is a part of Cuong et al. (J Dyn Differ Equ 31(4):1279–1299, 2019) and Bonckaert et al. (J Differ Equ 258:1618–1652, 2015).