Stable periodic orbits for delay differential equations with unimodal feedback

Abstract

We consider delay differential equations of the form $y^{\prime }(t)=-ay(t)+bf(y(t-1))$ with positive parameters $a,b$ and a unimodal $f:[0,\infty )\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty )\to [0,1]$ with $g(\hat{I}¾)=0$ for all $\hat{I}¾>1$. The fact $g(\hat{I}¾)=0$ for all $\hat{I}¾>1$ allows to construct stable periodic orbits for the equation $x^{\prime }(t)=-cx(t)+dg(x(t-1))$ with some parameters $d>c>0$. Then it is shown that the equation $y^{\prime }(t)=-ay(t)+bf(y(t-1))$ also has a stable periodic orbit provided $a,b,f$ are sufficiently close to $c,d,g$ in a certain sense. The examples include $f(\hat{I}¾)=\frac{\hat{I}{¾}^k}{1+\hat{I}{¾}^n}$ for parameters $k>0$ and $n>0$ together with the discontinuous $g(\hat{I}¾)=\hat{I}{¾}^k$ for $\hat{I}¾\in [0,1)$, and $g(\hat{I}¾)=0$ for $\hat{I}¾>1$. The case $k=1$ is the famous Mackey--Glass equation, the case $k>1$ appears in population models with Allee effect, and the case $k\in (0,1)$ arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.