  It has been recently pointed out that dynamical systems depending on future
values of the unknowns may be useful in different areas of knowledge. We
explore in this context the extension of the concept of order reduction that
has been useful with singular and delay differential equations in
electrodynamics and general relativity. We discuss some general properties of
order reductions in this new context and explore a method of successive
approximations, which among other things is used to check and improve the
``extrapolate prediction'' and ``fixed rate prediction'' methods.
