Geodesic
Features of Oscillations in Adiabatic Oscillators with Delay
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 5, pp. 25-44.

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

In this paper, we describe the features of oscillations in adiabatic oscillators when the delay is introduced into the equation. We give a short description of the method of asymptotic integration of one class of linear delay differential systems in the neighborhood of infinity. This method is based on the idea of transforming the initial system in order to reduce it to the system that is close in some sense to the system of ordinary differential equations. When applying this method, we need to extend the phase space of the initial system. The averaging changes of variables are also used to simplify the procedure of constructing the asymptotic formulas. Finally, we apply the functional differential analog of the Levinson theorem. We use this method to get the asymptotic formulas for adiabatic oscillators with delay under a monotonely and also oscillatory tending to zero perturbations. In conclusion, we study the transformation of the parametric resonance zone of one adiabatic oscillator when the delay is varied.
Keywords: adiabatic oscillator, delay differential equation, resonance, method of averaging, asymptotics. 
@article{MAIS_2013_20_5_a1,
     author = {P. N. Nesterov and E. N. Agafonchikov},
     title = {Features of {Oscillations} in {Adiabatic} {Oscillators} with {Delay}},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {25--44},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2013_20_5_a1/}
}
TY  - JOUR
AU  - P. N. Nesterov
AU  - E. N. Agafonchikov
TI  - Features of Oscillations in Adiabatic Oscillators with Delay
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2013
SP  - 25
EP  - 44
VL  - 20
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2013_20_5_a1/
LA  - ru
ID  - MAIS_2013_20_5_a1
ER  - 
%0 Journal Article
%A P. N. Nesterov
%A E. N. Agafonchikov
%T Features of Oscillations in Adiabatic Oscillators with Delay
%J Modelirovanie i analiz informacionnyh sistem
%D 2013
%P 25-44
%V 20
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2013_20_5_a1/
%G ru
%F MAIS_2013_20_5_a1
P. N. Nesterov; E. N. Agafonchikov. Features of Oscillations in Adiabatic Oscillators with Delay. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 5, pp. 25-44. http://geodesic.mathdoc.fr/item/MAIS_2013_20_5_a1/

[1] R. Bellman, Stability theory of differential equations, McGraw-Hill, New York, 1953 | MR | Zbl

[2] V. Sh. Burd, V. A. Karakulin, “On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients”, Math. Notes, 64:5 (1998), 571–578 | DOI | DOI | MR | Zbl

[3] Demidovich B. P., Lekcii po matematicheskoy teorii ustoychivosti, Nauka, M., 1967, 472 pp. (in Russian) | MR

[4] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955 | MR | Zbl

[5] Mayorov V. V., “Issledovanie ustoychivosti resheniy odnogo lineynogo differentsial'nogo uravneniya s posledeystviem, vstrechayushchegosya v prilozheniyah”, Vestnik Yaroslavskogo universiteta. Issledovaniya po ustoychivosti i teorii kolebaniy, 5 (1973), 86–93 (in Russian) | MR

[6] P. N. Nesterov, “Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients”, Differ. Equ., 43:6 (2007), 745–756 | DOI | MR | Zbl

[7] J. K. Hale, “Theory of functional differential equations”, Springer-Verlag, New York, 1977 | MR | MR | Zbl

[8] V. Burd, P. Nesterov, “Parametric resonance in adiabatic oscillators”, Results Math., 58:1–2 (2010), 1–15 | DOI | MR | Zbl

[9] J. S. Cassel, Z. Hou, “Asymptotically diagonal linear differential equations with retardation”, J. Lond. Math. Soc., 47 (1993), 473–483 | DOI | MR | Zbl

[10] M. S. P. Eastham, The asymptotic solution of linear differential systems, London Math. Soc. Monographs, Clarendon Press, Oxford, 1989 | MR | Zbl

[11] W. A. Jr. Harris, D. A. Lutz, “On the asymptotic integration of linear differential systems”, J. Math. Anal. Appl., 48:1 (1974), 1–16 | DOI | MR | Zbl

[12] W. A. Jr. Harris, D. A. Lutz, “A Unified Theory of Asymptotic Integration”, J. Math. Anal. Appl., 57:3 (1977), 571–586 | DOI | MR | Zbl

[13] P. Nesterov, “Asymptotic integration of functional differential systems with oscillatory decreasing coefficients”, Monatsh. Math., 171:2 (2013), 217–240 | DOI | MR | Zbl

[14] A. Wintner, “The adiabatic linear oscillator”, Amer. J. Math., 68 (1946), 385–397 | DOI | MR | Zbl

[15] A. Wintner, “Asymptotic integration of the adiabatic oscillator”, Amer. J. Math., 69 (1946), 251–272 | DOI | MR