Exponential dichotomies and basic properties

Abstract

The classical notion of exponential dichotomy, which we call uniform exponential dichotomy, is a considerable restriction for the dynamics and it is important to look for more general types of hyperbolic behavior. These generalized notions can be much more typical than the notion of uniform exponential dichotomy. This is precisely the case of the notion of nonuniform exponential dichotomy, that we introduce in Section 2.1. In particular, we show that essentially any (nonautonomous) linear differential equation admits such a dichotomy (see Section 2.3). We note that each uniform exponential dichotomy is also a nonuniform exponential dichotomy.