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

Expansions in the delay of quasi-periodic solutions for state dependent delay equations