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 Mat. Sb., 2002, Volume 193, Number 1, Pages 93–118 (Mi msb622)

Impact of quadratic non-linearity on the dynamics of periodic solutions of a wave equation

A. Yu. Kolesova, N. Kh. Rozovb

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

Abstract: For the non-linear telegraph equation with homogeneous Dirichlet or Neumann conditions at the end-points of a finite interval the question of the existence and the stability of time-periodic solutions bifurcating from the zero equilibrium state is considered. The dynamics of these solutions under a change of the diffusion coefficient (that is, the coefficient of the second derivative with respect to the space variable) is investigated. For the Dirichlet boundary conditions it is shown that this dynamics substantially depends on the presence – or the absence – of quadratic terms in the non-linearity. More precisely, it is shown that a quadratic non-linearity results in the occurrence, under an unbounded decrease of diffusion, of an infinite sequence of bifurcations of each periodic solution. En route, the related issue of the limits of applicability of Yu.S. Kolesov's method of quasinormal forms to the construction of self-oscillations in singularly perturbed hyperbolic boundary value problems is studied.

DOI: https://doi.org/10.4213/sm622

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English version:
Sbornik: Mathematics, 2002, 193:1, 93–118

Bibliographic databases:

UDC: 517.926
MSC: Primary 35B10, 35B40; Secondary 35L70

Citation: A. Yu. Kolesov, N. Kh. Rozov, “Impact of quadratic non-linearity on the dynamics of periodic solutions of a wave equation”, Mat. Sb., 193:1 (2002), 93–118; Sb. Math., 193:1 (2002), 93–118

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb622
• https://doi.org/10.4213/sm622
• http://mi.mathnet.ru/eng/msb/v193/i1/p93

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The mechanism of hard excitation of self-oscillations in the case of the resonance 1:2”, Comput. Math. Math. Phys., 45:11 (2005), 1923–1938
2. D. S. Glyzin, “Bimodal cycles of a nonlinear telegraph equation in the case of 1:2 resonance”, Diff Equat, 43:12 (2007), 1691
3. SongPing Zhou, Ping Zhou, DanSheng Yu, “Ultimate generalization to monotonicity for uniform convergence of trigonometric series”, Sci China Ser A, 2010
4. A. Yu. Kolesov, N. Kh. Rozov, “Invariant tori for a class of nonlinear evolution equations”, Sb. Math., 204:6 (2013), 824–868
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