  This work concerns the dynamics of a certain class of delay differential
equations (DDEs) which we refer to as state dependent delay maps. These maps
are generated by delay differential equations where the derivative of the
current state depends only on delayed variables, and not on the un-delayed
state. However, we allow that the delay is itself a function of the state
variable. A delay map with constant delays can be rewritten explicitly as a
discrete time dynamical system on an appropriate function space, and a delay
map with small state dependent terms can be viewed as a ``non-autonomous''
perturbation. We develop a fixed point formulation for the Cauchy problem of
such perturbations, and under appropriate assumptions obtain the existence of
forward iterates of the map.
  The proof is constructive and leads to numerical procedures which we
implement for illustrative examples, including the cubic Ikeda and Mackey-Glass
systems with constant and state-dependent delays. After proving a local
convergence result for the method, we study more qualitative/global convergence
issues using data analytic tools for time series analysis (dimension and
topological measures derived from persistent homology). Using these tools we
quantify the convergence of the dynamics in the finite dimensional projections
to the dynamics of the infinite dimensional system.
