Stable periodic orbits for delay differential equations with unimodal feedback
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arXiv
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| Format: | Preprint |
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2024
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| author | Benedek, Gábor Krisztin, Tibor Szczelina, Robert |
| author_facet | Benedek, Gábor Krisztin, Tibor Szczelina, Robert |
| contents | We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal $f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty)\to [0,1]$ 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 $f(ξ)=\frac{ξ^k}{1+ξ^n}$ for parameters $k>0$ and $n>0$ together with the discontinuous $g(ξ)=ξ^k$ for $ξ\in[0,1)$, 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 $k\in(0,1)$ arises in some economic growth models. The obtained stable periodic orbits may have complicated structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable periodic orbits for delay differential equations with unimodal feedback Benedek, Gábor Krisztin, Tibor Szczelina, Robert Dynamical Systems 34K30, 34K39, 65G30 We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal $f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty)\to [0,1]$ 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 $f(ξ)=\frac{ξ^k}{1+ξ^n}$ for parameters $k>0$ and $n>0$ together with the discontinuous $g(ξ)=ξ^k$ for $ξ\in[0,1)$, 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 $k\in(0,1)$ arises in some economic growth models. The obtained stable periodic orbits may have complicated structures. |
| title | Stable periodic orbits for delay differential equations with unimodal feedback |
| topic | Dynamical Systems 34K30, 34K39, 65G30 |
| url | https://arxiv.org/abs/2407.18016 |