Stability and periodic solutions for a price model with delay

We consider the delay differential equation: \begin{equation}\label{H1} x'(t) = a\left[ {x(t) - x(t - 1)} \right] - \left| {x(t - \tau )} \right|x(t - \tau ), \end{equation} with $a > 0$, $\tau > 0$. This is a modification of a model for exchange rates introduced by Brunovský, Erdélyi, Walther (2004) with $\tau=0.$ If $a\in \left( {0,1} \right)$ and $\tau =0$ then $x=0$ is globally attracting (Balázs, Krisztin, 2019). If $a>1$ and $\tau =0$ then there is a stable slowly oscillating periodic solution (Brunovský, Erdélyi, Walther, 2004).
If $\tau = \frac{1}{{4n}}$ for some $n\in \mathbb{N}$ then equation (\ref{H1}) has a $\frac{1}{n}$-periodic solution and global attractivity of $x=0$ is not satisfied. We estimate the region of attraction $$A\left( {a,\tau } \right) = \left\{ {\varphi \in C\left( {\left[ { - 1,0} \right],\mathbb{R}} \right): x^\varphi\left( t \right) \to 0\text{ as }t \to \infty } \right\}$$ of $0$ in case $a \in \left( {0,1} \right),$ and show that $A\left( {a,\tau } \right)$ approaches $A\left( {a,0} \right) = C\left( {\left[ { - 1,0} \right],\mathbb{R}} \right)$ as $\tau \to {0^ + }.$ Joint work with Tibor Krisztin.