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
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.
nonlinear telegraph equation, discretization, periodic motion, invariant torus, attractor.
PDF file (1941 kB)
Computational Mathematics and Mathematical Physics, 2009, 49:10, 1733–1747
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
\by A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper Dynamic effects associated with spatial discretization of nonlinear wave equations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
Citing articles on Google Scholar:
Related articles on Google Scholar:
This publication is cited in the following articles:
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
|Number of views:|