  In this paper we give a characterization of the Gromov hyperbolicity of
trains (a large class of Denjoy domains which contains the flute surfaces) in
terms of the behavior of a real function. This function describes somehow the
distances between some remarkable geodesics in the train. This theorem has
several consequences; in particular, it allows to deduce a result about
stability of hyperbolicity, even though the original surface and the modified
one are not quasi-isometric.
