Existence of a homoclinic solution for a delay differential equation

We consider the delay differential equation \begin{equation} \label{eq1.1} y'(t)=-ay(t)+b\begin{cases} {{y}^{2}}(t-1) & \text{if } y(t-1)\in [0,1) \ 0 & \text{if } y(t-1)\ge 1 \end{cases} \end{equation} with $b>a>0$. Equation \eqref{eq1.1} is the limit version (as $n\to\infty)$ of the Mackey-Glass type equation $x'(t)=-ax(t)+bx^2(t-1)/[1+x^n(t-1)]$. Considering a local unstable manifold of the equilibrium point $\xi_* = \frac{a}{b}$ we get a solution $y:\mathbb{R} \to \mathbb{R}$ of Equation \eqref{eq1.1} such that $$\underset{t\to -\infty }{\mathop{\lim }}\,y(t)={{\xi }_{*}}, \quad y(0) = 1, \quad y(s) > 1, \text{ for } s \in [-1,0).$$ If $\underset{t\to +\infty }{\mathop{\lim }}\,y(t)={{\xi }_{*}}$ then the solution $y$ of Equation (1) is homoclinic to $\xi_*$. The transform $u(t) = by(t) - a$ leads to the equation \begin{equation} \label{eq1.2} u'(t) = -au(t) + 2au(t-1) + u^2(t-1). \end{equation} There is a unique $b^* > a$ so that $u:[-1,\infty) \to \mathbb{R}$ with $u^*(s) = b^*e^{-a(s+1)}-a,$ $-1 \le s \le 0$, oscillates. Choosing $b = b^*$ in Equation (1), $\underset{t\to +\infty }{\mathop{\lim }}\,y(t)={{\xi }_{*}}$ is satisfied if $u^*(t) \to 0$ as $t \to +\infty$. We give a $\rho = \rho(a) > 0$ such that $${u_t^*} \in {B_\rho} = \{ \varphi \in C([-1,0],\mathbb{R}): \left\| \varphi \right\| < \rho \}$$ for all large $t$, and ${B_\rho}$ does not contain periodic orbits. This step requires a careful choice of the exponential dichotomy constants at the equilibrium $u=0$, and a computer-assisted estimation of $u^*$ on a finite interval. A consequence is that $u^*(t) \to 0$, and therefore $y(t) \to \xi_*$ as $t \to +\infty$. The technique works for $a \in (0,a_*]$ so that near $a_*$ the spectral condition at $\xi_*$, required for Shilnikov chaos, is satisfied.
This is a joint work with Tibor Krisztin and Mónika Polner.