Non-Quasi-Periodic Normal Form Theory

We review a recent application of the ideas of normal form theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference from the standard case consists in the non-uniqueness of the normal form and the total absence of the small divisors problem. The exposition is quite general, so as to allow extensions to the case of more non-periodic coordinates, and more functional settings. Here, for simplicity, we work in the real-analytic class.