Geodesic
Local dynamics of a second order equation with large exponentially distributed delay and considerable friction
Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 1, pp. 65-73.

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

We study local dynamics of a nonlinear second order differential equation with a large exponentially distributed delay in the vicinity of the zero solution under the condition $\gamma>\sqrt{2}$. The parameter $\gamma$ can be interpreted as a friction coefficient. We find such parameter values that critical cases in the stability problem are realized. We show that the characteristic equation for zero solution stability can have arbitrary many roots in the vicinity of imaginary axis. So, the critical case of an infinite dimension is realized. We construct normal forms analogues to describe dynamics of the origin equation. We formulate results about the correspondence of solutions of received PDE and second order DDE with a large exponentially distributed delay. The received asymptotic formulas allow us to evidently find characteristics of origin problem local regimes that are close to the zero solution and also to obtain domains of parameters and initial conditions, where the appearance of any given-type solution is possible.
Keywords: local dynamics, delay, normal form, asymptotic formula, small parameter. 
@article{MAIS_2015_22_1_a3,
     author = {D. V. Glazkov},
     title = {Local dynamics of a second order equation with large exponentially distributed delay and considerable friction},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {65--73},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a3/}
}
TY  - JOUR
AU  - D. V. Glazkov
TI  - Local dynamics of a second order equation with large exponentially distributed delay and considerable friction
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2015
SP  - 65
EP  - 73
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a3/
LA  - ru
ID  - MAIS_2015_22_1_a3
ER  - 
%0 Journal Article
%A D. V. Glazkov
%T Local dynamics of a second order equation with large exponentially distributed delay and considerable friction
%J Modelirovanie i analiz informacionnyh sistem
%D 2015
%P 65-73
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a3/
%G ru
%F MAIS_2015_22_1_a3
D. V. Glazkov. Local dynamics of a second order equation with large exponentially distributed delay and considerable friction. Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 1, pp. 65-73. http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a3/

[1] Landa P. S., Avtokolebaniya v raspredelennyh sistemah, Nauka, M., 1983 (in Russian)

[2] T. Kilias, K. Kutzer, A. Moegel, W. Schwarz, “Electronic chaos generators design and applications”, International Journal of Electronics, 79:6 (1995), 737–753 | DOI

[3] Glazkov D. V., Kaschenko I. S., Laser dynamics equations, YarSU, Yaroslavl, 2012, 128 pp. (in Russian)

[4] Kaschenko I. S., “Normalization of equation with linear distributed delay”, Modeling and analysis of information systems, 16:4 (2009), 109–-116 (in Russian)

[5] Kaschenko I. S., “Local dynamics of an equation with large exponential distributed delay”, Modeling and analysis of information systems, 18:3 (2011), 42–-49 (in Russian)

[6] Kashchenko S. A., “Normalization Techniques as Applied to the Investigation of Dynamics of DifferenceDifferential Equations with a Small Parameter Multiplying the Derivative”, Differ. Uravn., 25:8 (1989), 1448–-1451 (in Russian)

[7] S. A. Kashchenko, “Asymptotic behaviour of rapidly oscillating contrasting spatial structures”, Computational Mathematics and Mathematical Physics, 30:1 (1990), 186–-197 | DOI

[8] S. A. Kashchenko, “The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay”, Computational Mathematics and Mathematical Physics, 38:3 (1998), 443–-451

[9] Kaschenko S. A., “Local dynamics of nonlinear singularly perturbed delay systems”, Differ. Uravneniya, 35:10 (1999), 1343–1355 (in Russian)

[10] Glazkov D. V., “Local dynamics of an equation with long delay feedback”, Modeling and analysis of information systems, 18:1 (2011), 75–-85 (in Russian)