Lyapunov regularity and exponential dichotomies

We show in this chapter that any linear equation v´ = A(t)v, with A(t) in block form with blocks corresponding to the stable and center-unstable components, admits a strong nonuniform exponential dichotomy. While the extra exponentials in the notion of nonuniform exponential dichotomy substantially complicate the study of invariant manifolds in former chapters, we are able to obtain fairly general results at the expense of a careful control of the nonuniformity. In particular, we showed that if the equation v´ = A(t)v has a nonuniform exponential dichotomy with sufficiently small nonuniformity (when compared to the Lyapunov exponents), then with mild assumptions on the perturbation f there exist stable and unstable manifolds for the nonlinear equation v´ = A(t)v+f(t, v). We note that we do not need the nonuniformity to be zero, only sufficiently small. Therefore, it is important to estimate in quantitative terms how much a nonuniform exponential dichotomy can deviate from a uniform one. Fortunately, there exists a device, introduced by Lyapunov, that allows one to measure this deviation. It is the so-called notion of regularity (see Section 10.1 for the definition), introduced by Lyapunov in his doctoral thesis [57] (the expression is his own), which nowadays seems unfortunately apparently overlooked in the theory of differential equations. We emphasize that we only consider finite-dimensional spaces in this chapter. The infinite-dimensional case is considered in Chapter 11. The material in this chapter is based in [13], which in its turn is inspired in [1]. See [16] for a related study in the case of the discrete time.