Geodesic
Dynamics of one model with delay and a large parameter
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 5, Tome 194 (2021), pp. 115-123 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we consider a system of two differential equations with delay and a finite nonlinearity. Using a special asymptotic method, we examine the existence and stability of relaxation periodic solutions of the system under the assumption that the positive coefficient of the finite nonlinearity is sufficiently large. This method allows one to reduce the problem on the behavior of solutions whose initial conditions lie in a certain set of the phase space of the original infinite-dimensional system to the study of the dynamics of a certain three-dimensional mapping. We prove that rough cycles of this mapping correspond to relaxation periodic solutions of the original system with the same stability. By stable cycles of the mapping constructed, we find exponentially orbitally stable relaxation cycles of the original system.
Keywords: asymptotics, periodic solution, large parameter, multistability.
Mots-clés : relaxation oscillations 
@article{INTO_2021_194_a10,
     author = {A. A. Kashchenko},
     title = {Dynamics of one model with delay and a large parameter},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {115--123},
     year = {2021},
     volume = {194},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_194_a10/}
}
TY  - JOUR
AU  - A. A. Kashchenko
TI  - Dynamics of one model with delay and a large parameter
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 115
EP  - 123
VL  - 194
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_194_a10/
LA  - ru
ID  - INTO_2021_194_a10
ER  - 
%0 Journal Article
%A A. A. Kashchenko
%T Dynamics of one model with delay and a large parameter
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 115-123
%V 194
%U http://geodesic.mathdoc.fr/item/INTO_2021_194_a10/
%G ru
%F INTO_2021_194_a10
A. A. Kashchenko. Dynamics of one model with delay and a large parameter. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 5, Tome 194 (2021), pp. 115-123. http://geodesic.mathdoc.fr/item/INTO_2021_194_a10/

[1] Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Diskretnye avtovolny v neironnykh sistemakh”, Zh. vychisl. mat. mat. fiz., 52:5 (2012), 840–858 | MR | Zbl

[2] Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Modelirovanie effekta vzryva v neironnykh sistemakh”, Mat. zametki., 93:5 (2013), 684–701 | MR | Zbl

[3] Dmitriev A. S., Kislov V. Ya., Stokhasticheskie kolebaniya v radiofizike i elektronike, Nauka, M., 1989 | MR

[4] Kaschenko S. A., “Issledovanie metodami bolshogo parametra sistemy nelineinykh differentsialno-raznostnykh uravnenii, modeliruyuschikh zadachu khischnik-zhertva”, Dokl. AN SSSR., 266:4 (1982), 792–795 | MR | Zbl

[5] Kaschenko S. A., Maiorov V. V., Modeli volnovoi pamyati, Librokom, M., 2009

[6] Grigorieva E. V., Kaschenko S. A., Asymptotic Representation of Relaxation Oscillations in Lasers, Springer-Verlag, 2017 | MR | Zbl

[7] Kashchenko A. A., “A family of non-rough cycles in a system of two coupled delayed generators”, Automat. Control Comput. Sci., 51:7 (2017), 753–756 | DOI | MR

[8] Kashchenko A. A., “Multistability in a system of two coupled oscillators with delayed feedback”, J. Differ. Equations., 266:1 (2019), 562–579 | DOI | MR | Zbl

[9] Kashchenko A. A., Kaschenko S. A., “Asymptotic behavior of the solutions of a system of two weakly coupled relaxation oscillators with delayed feedback”, Radiophys. Quant. Electronics., 61:8–9 (2019), 633–639 | DOI | MR

[10] Kilias T., Kelber K., Mogel A., and Schwarz W., “Electronic chaos generators-design and applications”, Int. J. Electronics., 79:6 (1995), 737–753 | DOI

[11] Preobrazhenskaia M. M., “The impulse-refractive mode in a neural network with ring synaptic interaction”, Automat. Control Comput. Sci., 52:7 (2018), 777–789 | DOI

[12] Rabinovich M. I., Varona P., Selverston A. I., Abarbanel H. D. I., “Dynamical principles in neuroscience”, Rev. Mod. Phys., 78:4 (2006), 1213–1265 | DOI