Periodic delay orbits and the polyfold implicit function theorem

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 smooth vector fields on $\mathbb{R}^n$ and $\tau$ parameter. If there (suitably non-degenerate) periodic solution $x_0$ this equation for $\tau=0$, that without delay, are good reasons to expect existence solutions all sufficiently small delays, smoothly parametrized by delay. However, it seems difficult prove using classical implicit function theorem, since above not In paper, we show how use M-polyfold theorem Hofer-Wysocki-Zehnder [HWZ09, HWZ17] overcome problem natural setup.