Persistence of Invariant Objects under Delay Perturbations
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In this dissertation, we investigate functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs. Despite the singular nature of the perturbations, when the perturbations are small, we get that some invariant objects of the unperturbed equations persist and depend on the parameters with high regularity. The results apply to equations with state-dependent delay perturbations and equations arising in electrodynamics. This is based on joint work with Joan Gimeno and Rafael de la Llave.