Geodesic
A family of non-rough cycles in a system of two coupled delayed generators
Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 5, pp. 649-654.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, finite and smooth. The main assumption in the problem is that the coupling between the generators is sufficiently small. With the help of asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. For this purpose, a special set is constructed in the phase space of the original system. Then we build an asymptotics of the solutions of the given system with initial conditions from this set. Using this asymptotics, a special mapping is constructed. Dynamics of this map describes the dynamics of the original problem in general. It is proved that all solutions of this mapping are non-rough cycles of period two. As a result, we formulate conditions for the coupling parameter such that the initial system has a two-parameter family of non-rough inhomogeneous relaxation periodic asymptotic (with respect to the residual) solutions.
Keywords: large parameter, periodic solution, asymptotics, delay.
Mots-clés : relaxation oscillation 
@article{MAIS_2017_24_5_a8,
     author = {A. A. Kashchenko},
     title = {A family of non-rough cycles in a system of two coupled delayed generators},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {649--654},
     publisher = {mathdoc},
     volume = {24},
     number = {5},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2017_24_5_a8/}
}
TY  - JOUR
AU  - A. A. Kashchenko
TI  - A family of non-rough cycles in a system of two coupled delayed generators
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2017
SP  - 649
EP  - 654
VL  - 24
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2017_24_5_a8/
LA  - ru
ID  - MAIS_2017_24_5_a8
ER  - 
%0 Journal Article
%A A. A. Kashchenko
%T A family of non-rough cycles in a system of two coupled delayed generators
%J Modelirovanie i analiz informacionnyh sistem
%D 2017
%P 649-654
%V 24
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2017_24_5_a8/
%G ru
%F MAIS_2017_24_5_a8
A. A. Kashchenko. A family of non-rough cycles in a system of two coupled delayed generators. Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 5, pp. 649-654. http://geodesic.mathdoc.fr/item/MAIS_2017_24_5_a8/

[1] Kilias T. et al., “Electronic chaos generators-design and applications”, International journal of electronics, 79:6 (1995), 737–753 | DOI

[2] Kilias T., Mogel A., Schwarz W., “Generation and application of broadband signals using chaotic electronic systems”, Nonlinear Dynamics: New Theoretical and Applied Results, Akademie Verlag, Berlin, 1995, 92–111 | Zbl

[3] Balachandran B., Kalmar-Nagy T., Gilsinn D. E., Delay differential equations, Springer, Berlin, 2009 | MR | Zbl

[4] Dmitriev A. S., Kislov V. Ya., Stokhasticheskie kolebaniya v radiotekhnike, Nauka, Moskva, 1989 (in Russian) | MR

[5] Dmitriev A. S., Kaschenko S. A., “Dinamika generatora s zapazdyvayushchey obratnoy svyazyu i nizkodobrotnym filtrom vtorogo poryadka”, Radiotekhnika i elektronika, 34:12 (1989), 24–39 (in Russian)

[6] Kaschenko S. A., “Asymptotics of relaxation oscillations in systems of differential-difference equations with a compactly supported nonlinearity. I”, Differential Equations, 31:8 (1995), 1275–1285 | MR | MR

[7] Kaschenko S. A., “Asymptotics of relaxation oscillations in systems of differential-difference equations with a compactly supported nonlinearity. II”, Differential Equations, 31:12 (1995), 1938–1946 | MR

[8] Kashchenko A. A., “Dynamics of a system of two simplest oscillators with finite non-linear feedbacks”, Modeling and Analysis of Information Systems, 23:6 (2016), 841–849 (in Russian) | MR

[9] Kaschenko S. A., “Investigation, by large parameter methods, of a system of nonlinear differential-difference equations modeling a predator-prey problem”, Soviet Mathematics. Doklady, 26:2 (1982), 420–423 | MR

[10] Grigorieva E. V., Kashchenko S. A., “Regular and chaotic pulsations in laser diode with delayed feedback”, International Journal of Bifurcation and Chaos, 3:6 (1993), 1515–1528 | DOI | Zbl

[11] Bestehorn M., Grigorieva E. V., Kaschenko S. A., “Spatiotemporal structures in a model with delay and diffusion”, Physical Review E, 70:2 (2004), 026202 | DOI | MR

[12] Grigorieva E. V., Kashchenko S. A., “Dynamics of spikes in delay coupled semiconductor lasers”, Regular and Chaotic Dynamics, 15:2/3 (2010), 319–327 | DOI | MR | Zbl

[13] Kaschenko D. , Kaschenko S., Schwarz W., “Dynamics of First Order Equations with Nonlinear Delayed Feedback”, International Journal of Bifurcation and Chaos, 22:8 (2012), 1250184 | DOI | MR | Zbl

[14] Kaschenko S. A., “Relaxation oscillations in a system with delays modeling the predator-prey problem”, Automatic Control and Computer Sciences, 49:7 (2015), 547–581 | DOI