Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback

期刊论文
出版模式：

开放获取
发表日期：

2024年12月10日

查看全文

Gábor Benedek,
Tibor Krisztin,
Robert Szczelina

摘要

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