Convergence of Birkhoff Normal form for Essentially Isochronous Systems

We reconsider the problem of the convergence of Birkhoff's normal form for a system of perturbed harmonic oscillators, under the condition that the system is essentially isochronous. In contrast with previous proofs based on the so called quadratically convergent method, the present proof uses only classical expansions in a parameter. This allows us to bring into light some mechanisms of accumulation of small divisors, which can be useful in more complicated and interesting cases. These same mechanisms allow us to prove the theorem with the Bruno condition on the frequencies in a very natural way.