  Periodic orbits and associated bifurcations of singularly perturbed
state-dependent delay differential equations (DDEs) are studied when the
profiles of the periodic orbits contain jump discontinuities in the singular
limit. A definition of singular solution is introduced which is based on a
continuous parametrisation of the possibly discontinuous limiting solution.
This reduces the construction of the limiting profiles to an algebraic problem.
A model two state-dependent delay differential equation is studied in detail
and periodic singular solutions are constructed with one and two local maxima
per period. A complete characterisation of the conditions on the parameters for
these singular solutions to exist facilitates an investigation of bifurcation
structures in the singular case revealing folds and possible cusp bifurcations.
Sophisticated boundary value techniques are used to numerically compute the
bifurcation diagram of the state-dependent DDE when the perturbation parameter
is close to zero. This confirms that the solutions and bifurcations constructed
in the singular case persist when the perturbation parameter is nonzero, and
hence demonstrates that the solutions constructed using our singular solution
definition are useful and relevant to the singularly perturbed problem. Fold
and cusp bifurcations are found very close to the parameter values predicted by
the singular solution theory, and we also find period-doubling bifurcations as
well as periodic orbits with more than two local maxima per period, and explain
the alignment between the folds on different bifurcation branches.
