Geodesic
Dynamic effects associated with spatial discretization of nonlinear wave equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 10, pp. 1812-1826 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{ZVMMF_2009_49_10_a7,
     author = {A. Yu. Kolesov and N. Kh. Rozov},
     title = {Dynamic effects associated with spatial discretization of nonlinear wave equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1812--1826},
     year = {2009},
     volume = {49},
     number = {10},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_10_a7/}
}
TY  - JOUR
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - Dynamic effects associated with spatial discretization of nonlinear wave equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 1812
EP  - 1826
VL  - 49
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_10_a7/
LA  - ru
ID  - ZVMMF_2009_49_10_a7
ER  - 
%0 Journal Article
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T Dynamic effects associated with spatial discretization of nonlinear wave equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 1812-1826
%V 49
%N 10
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_10_a7/
%G ru
%F ZVMMF_2009_49_10_a7
A. Yu. Kolesov; N. Kh. Rozov. Dynamic effects associated with spatial discretization of nonlinear wave equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 10, pp. 1812-1826. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_10_a7/

[1] Kolesov A. Yu., Rozov N. Kh., Invariantnye tory nelineinykh volnovykh uravnenii, Fizmatlit, M., 2004

[2] Mischenko E. F., Sadovnichii V. A., Kolesov A. Yu., Rozov N. Kh., Avtovolnovye protsessy v nelineinykh sredakh s diffuziei, Fizmatlit, M., 2005

[3] Bogolyubov H. H., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[4] Bibikov Yu. N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, Izd-vo LGU, L., 1991 | MR