The Case of Differential Equations

This chapter presents a version of some results in the former chapters for differential equations, that is, for a dynamics with continuous time. In particular, we introduce the notions of tempered exponential dichotomy and of tempered spectrum for a nonautonomous linear equation \(x'=A(t)x\) on \(\mathbb {R}^d\) . Then we describe all possible forms of the tempered spectrum and its relation to the lower and upper Lyapunov exponents. We also give explicit examples of differential equations for all possible forms of the spectrum. Finally, we detail the construction of normal forms for the perturbations of a nonautonomous linear equation. We avoid repeating the details that were already given before.