Перейти до основного вмісту
On-axis averaging in multi-frequency systems with delay
Bigun Yaroslav Yosypovych 1 , Krasnokutska Inessa Volodymyrivna 1
1 Department of Aplied Mathematics and Information Technologies, Yuriy Fedkovych Chernivtsi National University, Chernivtsi, 58000, Ukraine
Keywords: multi-frequency systems
Abstract
We study the existence of a solution and justify the averaging method on fast variables on the axis for multifrequency systems of differential equations with constant delay. We also establish the solvability of a boundary value problem for finite interval and obtain an error estimate of the averaging method for this problem.
References

[1] Arnold, V.I. Applicability conditions and error estimation of the averaging method for systems which pass through resonance in the course of evolution / V.I. Arnold // Dokl. of the USSR Academy of Sciences. - 1965. -161, № 1. - P. 9-12.

[2] Grebenikov, E.A. Constructive methods of the analysis of nonlinear systems / E.A. Grebenikov, Yu. - M.: Nauka, 1979. - 431 p.

[3] Khapaev M.M. Averaging in stability theory / M.M. Khapaev. - M.: Nauka, 1986. - 192 p.

[4] Neishtadt, A.I. On averaging in multifrequency systems / A.I. Neishtadt // Dokl. AN SS-SR. - 1976. - 226, № 6. - P. 1295-1298.

[5] Samoilenko, A.M. To the question of substantiation of the averaging method for multifrequency oscillating systems / A.M. Samoilenko // Differential Equations. - 1987. - 23, № 2. - P. 267-278.

[6] Samoilenko A.M. Mathematical aspects of the theory of nonlinear oscillations / A.M. Samoilenko, R.I. Petryshyn - K.: Naukova Dumka, 2004. - 474 p.

[7] Petryshyn R.I. Estimation of the error of the method of averaging on a semi-sphere for a multifrequency resonant system / R.I. Petryshyn, O.M. Pokhyla // Ukr. math. journal - 1997. - 49, № 5. - P. 685-690.

[8] Bihun Y.Y. Existence of a solution and averaging of nonlinear multifrequency problems with a delay // Ukrainian Mathematical Journal - 2007. - 59, № 4. - P. 435-446.

[9] Danyliuk I.M. Boundary value problem with parameters for a nonlinear oscillatory system with gating / I.M. Danyliuk // Scientific Bulletin of Chernivtsi Univ: Collection of scientific papers. 454. Mathematics - Chernivtsi: Ruta, 2009. - P. 19-27.

[10] Berezovska I.V. Averaging in multi-frequency boundary value problems with linearly transformed arguments / I.V. Berezovska // Nonlinear Oscillations. 2013. - 16, № 2. - P. 147-156.

[11] Samoilenko, A.M. On the averaging of differential equations on an infinite interval / A.M. Samoilenko, A.N. Stanzhitsky // Differential Equations. - 2006. - 42, № 4. - P. 476-482.

[12] Samoilenko, A.M. Averaging method in some boundary value problems / A.M. Samoilenko, R.I. Petrishin // Differential Equations. - 1989. -25,  № 6. - P. 956-964.

[13] Kolesov, Y.S. Auto oscillations in systems with delay / Y.S. Kolesov, D.I. Shvitra. -Vilnius: Moscow, 1979. - 148 p.

[14] Martynyuk, A.D. Stability of motion: method of intergalic inequalities / A.D. Matrynyuk, V. Lakshmikantam, S. Lila. - K.: Naukova Dumka, 1989. - 272 p.

[15] Bigun, Ya.I. Justification of the averaging principle for multi-frequency systems of differential equations with delay / Ya.I. Bigun, A.M. Samoilenko// Differential Equations. - 1999.- 35, 1. - P. 8-14.

[16] Tyshkevich V.A. Some questions of the theory of stability of functional-differential equations. - K.: Naukova Dumka, 1981. - 80 p.

[17] Bainov Drumi Integral Inequalities and Applications / Drumi Bainov, Pavel Simeonov. – Mathematics and its applications (Kluwer Academic Publishers), East European series: v. 57, 1992. – 245 p.

[18] Bigun, Y.Y. Substantiation of the averaging method for nonlinear resonant systems with a delay / Y.Y. Bigun // Nonlinear fluctuations. - 1999. - 2, № 2. - P. 162-169.

Cite
ACS Style
Bigun, Y.Y.; Krasnokutska, I.V. On-axis averaging in multi-frequency systems with delay. Bukovinian Mathematical Journal. 2016, 1
AMA Style
Bigun YY, Krasnokutska IV. On-axis averaging in multi-frequency systems with delay. Bukovinian Mathematical Journal. 2016; 1(3-4).
Chicago/Turabian Style
Yaroslav Yosypovych Bigun, Inessa Volodymyrivna Krasnokutska. 2016. "On-axis averaging in multi-frequency systems with delay". Bukovinian Mathematical Journal. 1 no. 3-4.
Export
We use own, third-party cookies, and localStorage files to analyze web traffic and page activities. Privacy Policy Settings