Periodic solutions with long period for the Mackey–Glass equation

Abstract

The limiting version of the Mackey–Glass delay differential equation $x'(t)=-ax(t)+bf(x(t-1))$ is considered where $a,b$ are positive reals, and $f(\xi)=\xi$ for $\xi\in[0,1)$, $f(1)=1/2$, and $f(\xi)=0$ for $\xi> 1$. For every $a>0$ we prove the existence of an $\varepsilon_0=\varepsilon_0(a)>0$ so that for all $b\in (a,a+\varepsilon_0)$ there exists a periodic solution $p=p(a,b):\mathbb{R}\to (0,\infty)$ with minimal period $\omega(a,b)$ such that $\omega(a,b)\to \infty $ as $b\to a+$. A consequence is that, for each $a>0$, $b\in (a,a+\varepsilon_0(a))$ and sufficiently large $n$, the classical Mackey–Glass equation $y'(t)=-ay(t)+by(t-1)/[1+y^n(t-1)]$ has an orbitally asymptotically stable periodic orbit, as well, close to the periodic orbit of the limiting equation.

Keywords

• mackey–glass equation
• periodic solution
• limiting nonlinearity
• discontinuous right-hand side
• long period