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Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 10, Pages 1812–1826 (Mi zvmmf4771)  

This article is cited in 1 scientific paper (total in 1 paper)

Dynamic effects associated with spatial discretization of nonlinear wave equations

A. Yu. Kolesova, N. Kh. Rozovb

a Faculty of Mathematics, Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000, Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

Abstract: A new phenomenon is detected that the attractors of a nonlinear wave equation can differ substantially from those of its finite-dimensional analogue obtained by replacing the spatial derivatives with corresponding difference operators (regardless of the discretization step). The presentation is based on a typical example, namely, on the boundary value problem for a Van-der-Pol-type telegraph equation with zero Neumann conditions at the ends of the unit interval. Under certain generic conditions, the problem is shown to admit only stable time-periodic motions, which are fairly numerous. When the problem is replaced by an approximating system of ordinary differential equations, the situation becomes fundamentally different: all the periodic motions (except for one or two) become unstable and, instead of them, stable two-dimensional invariant tori appear.

Key words: nonlinear telegraph equation, discretization, periodic motion, invariant torus, attractor.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:10, 1733–1747

Bibliographic databases:

UDC: 519.63
Received: 11.03.2009

Citation: A. Yu. Kolesov, N. Kh. Rozov, “Dynamic effects associated with spatial discretization of nonlinear wave equations”, Zh. Vychisl. Mat. Mat. Fiz., 49:10 (2009), 1812–1826; Comput. Math. Math. Phys., 49:10 (2009), 1733–1747

Citation in format AMSBIB
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\by A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper Dynamic effects associated with spatial discretization of nonlinear wave equations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2009
\vol 49
\issue 10
\pages 1812--1826
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\transl
\jour Comput. Math. Math. Phys.
\yr 2009
\vol 49
\issue 10
\pages 1733--1747
\crossref{https://doi.org/10.1134/S096554250910008X}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-76349091328}


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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, “Two-frequency self-oscillations in a FitzHugh–Nagumo neural network”, Comput. Math. Math. Phys., 57:1 (2017), 106–121  mathnet  crossref  crossref  isi  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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