  It has been recently found that for the near extremal Kerr black holes
appearing of Zero Damped Modes (accompanied by qusinormal mode branching)
signifies about inapplicability of the regime of small perturbations and the
onset of turbulence. Here we show that this phenomena is not limited by Kerr or
Kerr-Newman solutions only, but also takes place for rotating dilatonic black
holes for which we have found Zero Damped Modes both numerically and
analytically. We have also shown that, contrary to recent claims, there is no
instability of a charged massive scalar field in the background of the rotating
dilatonic black hole under physically adequate boundary conditions. Analytic
expression for dominant quasinormal frequencies is deduced in the regime of
large coupling qQ, where q and Q are the field and black hole charges
respectively.
