Stable periodic orbits for delay differential equations with unimodal feedback

We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal f:[0,∞)→ [0,1]. It is assumed that the nonlinear f is close to a function g:[0,∞)→ [0,1] with g(ξ)=0 for all ξ>1. The fact g(ξ)=0 for all ξ>1 allows to construct stable periodic orbits for the equation $x'(t)=-cx(t)+dg(x(t-1))$ with some parameters $d>c>0$. Then it is shown that the equation $ y'(t)=-ay(t)+bf(y(t-1)) $ also has a stable periodic orbit provided $a,b,f$ are sufficiently close to $c,d,g$ in a certain sense. The examples include f(ξ)=ξᵏ/1+ξⁿ for parameters $k>0$ and $n>0$ together with the discontinuous g(ξ)=ξᵏ for ξ∈[0,1), and g(ξ)=0 for ξ>1. The case $k=1$ is the famous Mackey--Glass equation, the case $k>1$ appears in population models with Allee effect, and the case k∈(0,1) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.