Smooth Linearization of Nonautonomous Differential Equations with a Nonuniform Dichotomy

Abstract: In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and H\"older continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and H\"older linearization of hyperbolic systems without any non-resonant conditions.