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Periodic orbits and associated bifurcations of singularly perturbedstate-dependent delay differential equations (DDEs) are studied when theprofiles of the periodic orbits contain jump discontinuities in the singularlimit. A definition of singular solution is introduced which is based on acontinuous 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 detailand periodic singular solutions are constructed with one and two local maximaper period. A complete characterisation of the conditions on the parameters forthese singular solutions to exist facilitates an investigation of bifurcationstructures in the singular case revealing folds and possible cusp bifurcations.Sophisticated boundary value techniques are used to numerically compute thebifurcation diagram of the state-dependent DDE when the perturbation parameteris close to zero. This confirms that the solutions and bifurcations constructedin the singular case persist when the perturbation parameter is nonzero, andhence demonstrates that the solutions constructed using our singular solutiondefinition are useful and relevant to the singularly perturbed problem. Foldand cusp bifurcations are found very close to the parameter values predicted bythe singular solution theory, and we also find period-doubling bifurcations aswell as periodic orbits with more than two local maxima per period, and explainthe alignment between the folds on different bifurcation branches.

Periodic Solutions of a Singularly Perturbed Delay Differential Equation With Two State-Dependent Delays