Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback
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 , ∞ ) → [ 0 , 1 ] . It is assumed that the nonlinear f is close to a function g:[0...
Elmentve itt :
| Szerzők: | |
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2024
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| Sorozat: | JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
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| Tárgyszavak: | |
| doi: | 10.1007/s10884-024-10399-y |
| mtmt: | 35635558 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/36471 |
| Tartalmi kivonat: | 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 , ∞ ) → [ 0 , 1 ] . It is assumed that the nonlinear f is close to a function g:[0,\infty )\rightarrow [0,1] g : [ 0 , ∞ ) → [ 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 ) = 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 ∈ [ 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 ∈ ( 0 , 1 ) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures. |
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| Terjedelem/Fizikai jellemzők: | 35 |
| ISSN: | 1040-7294 |