Homoclinic orbits for the limiting case of the Mackey–Glass equation

We consider the Mackey–Glass equation and let $n \to \infty$ to obtain the limiting case, namely, \begin{equation*}\label{eq:Bartha}\tag{1} x'(t)=-ax(t)+bf(x(t-1)), \end{equation*} where $f(\xi)=\xi$ for $\xi\in[0,1)$, $f(1)=1/2$, and $f(\xi)=0$ for $\xi> 1$.
In a previous work, we established the existence of \emph{complicated} looking, orbitally asymptotically stable periodic orbits for \eqref{eq:Bartha} utilizing rigorous numerics. Now, we present how those techniques can be extended to localize an unstable periodic orbit $p$ and show the existence of a homoclinic orbit to $p$.
This is a joint work with Gabriella Vas and Tibor Krisztin.