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 t hat the system is essentially isochronous. In contrast with previous p roofs based on the so called quadratically convergent method, the pres ent proof uses only classical expansions in a parameter. This allows u s to bring into light some mechanisms of accumulation of small divisor s, which can be useful in more complicated and interesting cases. Thes e same mechanisms allow us to prove the theorem with the Bruno conditi on on the frequencies in a very natural way.