| Title:
|
Stable periodic solutions in scalar periodic differential delay equations (English) |
| Author:
|
Ivanov, Anatoli |
| Author:
|
Shelyag, Sergiy |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
59 |
| Issue:
|
1 |
| Year:
|
2023 |
| Pages:
|
69-76 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A class of nonlinear simple form differential delay equations with a $T$-periodic coefficient and a constant delay $\tau >0$ is considered. It is shown that for an arbitrary value of the period $T>4\tau -d_0$, for some $d_0>0$, there is an equation in the class such that it possesses an asymptotically stable $T$-period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The periodic solutions and their stability properties are shown to persist when the nonlinearities are “smoothed” at the discontinuity points. (English) |
| Keyword:
|
delay differential equations |
| Keyword:
|
nonlinear negative feedback |
| Keyword:
|
periodic coefficients |
| Keyword:
|
periodic solutions |
| Keyword:
|
stability |
| MSC:
|
34K13 |
| MSC:
|
34K20 |
| MSC:
|
34K39 |
| idZBL:
|
Zbl 07675575 |
| idMR:
|
MR4563017 |
| DOI:
|
10.5817/AM2023-1-69 |
| . |
| Date available:
|
2023-02-22T14:27:28Z |
| Last updated:
|
2023-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151551 |
| . |
| Reference:
|
[1] Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: Main results and open problems.Appl. Math. Model. 34 (2010), 1405–1417. MR 2592579, 10.1016/j.apm.2009.08.027 |
| Reference:
|
[2] Berezansky, L., Braverman, E., Idels, L.: Mackey-Glass model of hematopoiesis with monotone feedback revisited.Appl. Math. Comput. 219 (9) (2013), 4892–4907. MR 3001538, 10.1016/j.amc.2012.10.052 |
| Reference:
|
[3] Diekmann, O., van Gils, S., Verdyn Lunel, S., Walther, H.-O.: Delay Equations: Complex, Functional, and Nonlinear Analysis.Springer-Verlag, New York, 1995. |
| Reference:
|
[4] Erneux, T.: Applied delay differential equations.Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3, Springer Verlag, 2009. MR 2498700 |
| Reference:
|
[5] Györy, I., Ladas, G.: Oscillation Theory of Delay Differential Equations.Oxford Science Publications, Clarendon Press, Oxford, 1991, 368 pp. |
| Reference:
|
[6] Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations.Applied Mathematical Sciences, Springer-Verlag, New York, 1993. Zbl 0787.34002 |
| Reference:
|
[7] Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations.Dynamics Reported (New Series) 1 (1991), 165–224. |
| Reference:
|
[8] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics.Mathematics in Science and Engineering, vol. 191, Academic Press Inc., 1993, p. 398. Zbl 0777.34002 |
| Reference:
|
[9] Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems.Science 197 (1977), 287–289. 10.1126/science.267326 |
| Reference:
|
[10] Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences.Texts in Applied Mathematics, vol. 57, Springer-Verlag, 2011. Zbl 1227.34001, MR 2724792 |
| Reference:
|
[11] Walther, H.-O.: Homoclinic solution and chaos in $\dot{x}(t)=f(x(t-1))$.Nonlinear Anal. 5 (7) (1981), 775–788. 10.1016/0362-546X(81)90052-3 |
| Reference:
|
[12] Wazewska-Czyzewska, M., Lasota, A.: Mathematical models of the red cell system.Matematyka Stosowana 6 (1976), 25–40, in Polish. |
| . |
