Periodic perturbations with delay of autonomous differential equations on manifolds

TL;DR

This paper uses topological methods to analyze harmonic solutions of periodically perturbed autonomous differential equations on manifolds, incorporating fixed delays and defining a novel infinite-dimensional Poincaré translation operator.

Contribution

It introduces a new approach to handle delays in differential equations on manifolds by defining an infinite-dimensional Poincaré translation operator and deriving a fixed point index formula.

Findings

Established a fixed point index formula for the Poincaré translation operator with delay.

Extended topological methods to analyze solutions of delayed differential equations on manifolds.

Provided a framework for studying harmonic solutions in the presence of periodic perturbations and delays.

Abstract

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincar\'e $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.