Fractional Lindstedt series

The parametric equations of the surfaces on which highly resonant quasiperiodic motions develop lower-dimensional tori cannot be analytically continued, in general, in the perturbation parameter, i.e., they are not analytic functions of the perturbation parameter. However rather generally quasiperiodic motions whose frequencies satisfy only one rational relation “resonances of order 1” admit formal perturbation expansions in terms of a fractional power of the perturbation parameter depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.