Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

TL;DR

This paper demonstrates that under mild conditions, periodic orbits in certain functional differential equations (FDEs) close to ODEs or PDEs persist with smooth dependence on parameters, extending classical results to more complex delay systems.

Contribution

It establishes the existence and smooth parameter dependence of periodic orbits in FDEs near ODEs and PDEs, using a fixed point approach without requiring smoothness of the evolution.

Findings

Persistence of nondegenerate periodic orbits in FDEs close to ODEs/PDEs

High regularity of the dependence of orbits and frequencies on parameters

Applicability to FDEs with multiple or distributed delays

Abstract

We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity. The result also applies to 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.