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 Mat. Sb., 2014, Volume 205, Number 6, Pages 21–86 (Mi msb8271)

The theory of nonclassical relaxation oscillations in singularly perturbed delay systems

S. D. Glyzina, A. Yu. Kolesova, N. Kh. Rozovb

a P. G. Demidov Yaroslavl State University
b M. V. Lomonosov Moscow State University

Abstract: Some special classes of relaxation systems are introduced, with one slow and one fast variable, in which the evolution of the slow component $x(t)$ in time is described by an ordinary differential equation, while the evolution of the fast component $y(t)$ is described by a Volterra-type differential equation with delay $y(t-h)$, $h=\mathrm{const}>0$, and with a small parameter $\varepsilon>0$ multiplying the time derivative. Questions relating to the existence and stability of impulse-type periodic solutions, in which the $x$-component converges pointwise to a discontinuous function as $\varepsilon\to 0$ and the $y$-component is shaped like a $\delta$-function, are investigated. The results obtained are illustrated by several examples from ecology and laser theory.
Bibliography: 11 titles.

Keywords: nonclassical relaxation oscillations, singularly perturbed delay systems, asymptotic behaviour, stability.
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DOI: https://doi.org/10.4213/sm8271

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English version:
Sbornik: Mathematics, 2014, 205:6, 781–842

Bibliographic databases:

UDC: 517.926
MSC: Primary 34C26, 34C10; Secondary 37N20, 37N25

Citation: S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The theory of nonclassical relaxation oscillations in singularly perturbed delay systems”, Mat. Sb., 205:6 (2014), 21–86; Sb. Math., 205:6 (2014), 781–842

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8271
• https://doi.org/10.4213/sm8271
• http://mi.mathnet.ru/eng/msb/v205/i6/p21

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This publication is cited in the following articles:
1. Bobodzhanov A., Safonov V., Kachalov V., “Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov'S Regularization Method”, Axioms, 8:1 (2019), 27
2. M. M. Preobrazhenskaya, “A relay Mackey–Glass model with two delays”, Theoret. and Math. Phys., 203:1 (2020), 524–534
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