Stable periodic solutions in scalar periodic differential delay equations

Anatoli Ivanov; Sergiy Shelyag

Archivum Mathematicum (2023)

  • Issue: 1, page 69-76
  • ISSN: 0044-8753

Abstract

top
A class of nonlinear simple form differential delay equations with a T -periodic coefficient and a constant delay τ > 0 is considered. It is shown that for an arbitrary value of the period T > 4 τ - 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.

How to cite

top

Ivanov, Anatoli, and Shelyag, Sergiy. "Stable periodic solutions in scalar periodic differential delay equations." Archivum Mathematicum (2023): 69-76. <http://eudml.org/doc/298970>.

@article{Ivanov2023,
abstract = {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.},
author = {Ivanov, Anatoli, Shelyag, Sergiy},
journal = {Archivum Mathematicum},
keywords = {delay differential equations; nonlinear negative feedback; periodic coefficients; periodic solutions; stability},
language = {eng},
number = {1},
pages = {69-76},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Stable periodic solutions in scalar periodic differential delay equations},
url = {http://eudml.org/doc/298970},
year = {2023},
}

TY - JOUR
AU - Ivanov, Anatoli
AU - Shelyag, Sergiy
TI - Stable periodic solutions in scalar periodic differential delay equations
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 1
SP - 69
EP - 76
AB - 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.
LA - eng
KW - delay differential equations; nonlinear negative feedback; periodic coefficients; periodic solutions; stability
UR - http://eudml.org/doc/298970
ER -

References

top
  1. Berezansky, L., Braverman, E., Idels, L., 10.1016/j.apm.2009.08.027, Appl. Math. Model. 34 (2010), 1405–1417. (2010) MR2592579DOI10.1016/j.apm.2009.08.027
  2. Berezansky, L., Braverman, E., Idels, L., 10.1016/j.amc.2012.10.052, Appl. Math. Comput. 219 (9) (2013), 4892–4907. (2013) MR3001538DOI10.1016/j.amc.2012.10.052
  3. Diekmann, O., van Gils, S., Verdyn Lunel, S., Walther, H.-O., Delay Equations: Complex, Functional, and Nonlinear Analysis, Springer-Verlag, New York, 1995. (1995) 
  4. Erneux, T., Applied delay differential equations, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3, Springer Verlag, 2009. (2009) MR2498700
  5. Györy, I., Ladas, G., Oscillation Theory of Delay Differential Equations, Oxford Science Publications, Clarendon Press, Oxford, 1991, 368 pp. (1991) 
  6. Hale, J.K., Verduyn Lunel, S.M., Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993. (1993) Zbl0787.34002
  7. Ivanov, A.F., Sharkovsky, A.N., Oscillations in singularly perturbed delay equations, Dynamics Reported (New Series) 1 (1991), 165–224. (1991) 
  8. Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press Inc., 1993, p. 398. (1993) Zbl0777.34002
  9. Mackey, M.C., Glass, L., 10.1126/science.267326, Science 197 (1977), 287–289. (1977) DOI10.1126/science.267326
  10. Smith, H., An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, vol. 57, Springer-Verlag, 2011. (2011) Zbl1227.34001MR2724792
  11. Walther, H.-O., 10.1016/0362-546X(81)90052-3, Nonlinear Anal. 5 (7) (1981), 775–788. (1981) DOI10.1016/0362-546X(81)90052-3
  12. Wazewska-Czyzewska, M., Lasota, A., Mathematical models of the red cell system, Matematyka Stosowana 6 (1976), 25–40, in Polish. (1976) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.