Periodic delay orbits and the polyfold implicit function theorem

TL;DR

This paper demonstrates how the polyfold implicit function theorem can be used to establish the existence of smoothly parametrized families of periodic solutions in delay differential equations, overcoming classical smoothness limitations.

Contribution

It applies the M-polyfold implicit function theorem to delay differential equations, providing a novel approach to prove existence of solutions with small delays.

Findings

Existence of periodic solutions for small delays established

Polyfold theory applied to delay differential equations

Overcomes classical smoothness limitations in implicit function theorem

Abstract

We consider differential delay equations of the form $\partial_tx(t) = X_{t}(x(t - \tau))$ in $\mathbb{R}^n$, where $(X_t)_{t\in S^1}$ is a time-dependent family of smooth vector fields on $\mathbb{R}^n$ and $\tau$ is a delay parameter. If there is a (suitably non-degenerate) periodic solution $x_0$ of this equation for $\tau=0$, that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by delay. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder [HWZ09, HWZ17] to overcome this problem in a natural setup.