Persistence and smooth dependence on parameters of periodic orbits in functional differential equations close to an ODE or an evolutionary PDE

Abstract

We consider functional differential equations (FDEs) which are perturbations of smooth ordinary differential equations (ODEs). The FDE can involve multiple state-dependent delays, distributed delays, or implicitly defined delays (forward or backward). We show that, under some mild assumptions on the perturbation, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, when the perturbation depends on some parameters, we get smooth dependence of the periodic orbit and its frequency on the parameters with high regularity.The method can also be applied to treat equations with small delays appearing in electrodynamics and FDEs which are perturbations of some evolutionary partial differential equations (PDEs).The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed-point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs.