A nonautonomous Grobman–Hartman theorem

A fundamental problem in the study of the local behavior of a dynamical system is whether the linearization of the system along a given solution approximates well the solution itself in some open neighborhood. In other words, we look for an appropriate local change of variables, called a conjugacy, that can transform the system into a linear one. Moreover, as a means to distinguish the dynamics in a neighborhood of the solution further than in the topological category (such as, for example, to distinguish different types of nodes), the change of variables should be as regular as possible. The problem goes back to the pioneering work of Poincaré, that can be interpreted today as looking for an analytic change of variables which transforms the initial system into a linear one. The work of Sternberg [89, 90] showed that there are algebraic obstructions, expressed in terms of resonances between the eigenvalues of the linear approximation, that prevent the existence of conjugacies with a prescribed high regularity (see also [19, 20, 87, 61] for further related work). The main purpose of this chapter is to establish a nonautonomous and nonuniform version of the Grobman–Hartman theorem in Banach spaces. In addition, we show that the conjugacies are always Hölder continuous, with Hölder exponent expressed in terms of ratios of Lyapunov exponents. We follow closely [17, 10].