  We consider several models of State Dependent Delay Differential Equations
(SDDEs), in which the delay is affected by a small parameter. This is a very
singular perturbation since the nature of the equation changes.
  Under some conditions, we construct formal power series, which solve the
SDDEs order by order. These series are quasi-periodic functions of time. This
is very similar to the Lindstedt procedure in celestial mechanics.
  Truncations of these power series can be taken as input for a-posteriori
theorems, that show that near the approximate solutions there are true
solutions. In this way, we hope that one can construct a catalogue of solutions
for SDDEs, bypassing the need of a systematic theory of existence and
uniqueness for all initial conditions.
