  We construct and analyze families of periodic delay orbits for a class of
delay differential equations in two dimensions depending on two real-valued
functions. These families are parametrized by the delay parameter. It is
possible to represent the dependency of these periodic delay orbits on the
delay parameter by a curve in the plane, without loss of information. It turns
out that the singularities of these curves necessarily are cusps in the
non-degenerate case. After discussing degenerate situations in general, we
explain how to glue different families of periodic delay orbits at degeneracies
in the delay parameter.
