Impact of quadratic non-linearity on the~dynamics
Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 93-118
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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.
@article{SM_2002_193_1_a3,
author = {A. Yu. Kolesov and N. Kh. Rozov},
title = {Impact of quadratic non-linearity on the~dynamics},
journal = {Sbornik. Mathematics},
pages = {93--118},
publisher = {mathdoc},
volume = {193},
number = {1},
year = {2002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/}
}
A. Yu. Kolesov; N. Kh. Rozov. Impact of quadratic non-linearity on the~dynamics. Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 93-118. http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/