  A common task when analysing dynamical systems is the determination of normal
forms near local bifurcations of equilibria. As most of these normal forms have
been classified and analysed, finding which particular class of normal form one
encounters in a numerical bifurcation study guides follow-up computations. This
paper builds on normal form algorithms for equilibria of delay differential
equations with constant delay that were developed and implemented in
DDE-Biftool recently. We show how one can extend these methods to
delay-differential equations with state-dependent delay (sd-DDEs). Since higher
degrees of regularity of local center manifolds are still open for sd-DDEs, we
give an independent (still only partial) argument which phenomena from the
truncated normal must persist in the full sd-DDE. In particular, we show that
all invariant manifolds with a sufficient degree of normal hyperbolicity
predicted by the normal form exist also in the full sd-DDE.
