Geodesic
On Tykhonov's theorem for convergence of solutions of slow and fast systems
Electronic journal of differential equations, Tome 1998 (1998)
Slow and fast systems gain their special structure from the presence of two time scales. Their analysis is achieved with the help of Singular Perturbation Theory. The fundamental tool is Tykhonov's theorem which describes the limiting behaviour, for compact interval of time, of solutions of the perturbed system which is a one-parameter deformations of the so-called unperturbed system. Our aim here is to extend this description to the solutions of all systems that belong to a small neighbourhood of the unperturbed system. We investigate also the behaviour of solutions on the infinite time interval. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic approach to Nonstandard Analysis.
Classification : 34D15, 34E15, 03H05
Keywords: singular perturbations, deformations, asymptotic stability, nonstandard analysis 
@article{EJDE_1998__1998__a16,
     author = {Lobry,  Claude and Sari,  Tewfik and Touhami,  Sefiane},
     title = {On {Tykhonov's} theorem for convergence of solutions of slow and fast systems},
     journal = {Electronic journal of differential equations},
     year = {1998},
     volume = {1998},
     zbl = {0897.34052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a16/}
}
TY  - JOUR
AU  - Lobry,  Claude
AU  - Sari,  Tewfik
AU  - Touhami,  Sefiane
TI  - On Tykhonov's theorem for convergence of solutions of slow and fast systems
JO  - Electronic journal of differential equations
PY  - 1998
VL  - 1998
UR  - http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a16/
LA  - en
ID  - EJDE_1998__1998__a16
ER  - 
%0 Journal Article
%A Lobry,  Claude
%A Sari,  Tewfik
%A Touhami,  Sefiane
%T On Tykhonov's theorem for convergence of solutions of slow and fast systems
%J Electronic journal of differential equations
%D 1998
%V 1998
%U http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a16/
%G en
%F EJDE_1998__1998__a16
Lobry,  Claude; Sari,  Tewfik; Touhami,  Sefiane. On Tykhonov's theorem for convergence of solutions of slow and fast systems. Electronic journal of differential equations, Tome 1998 (1998). http://geodesic.mathdoc.fr/item/EJDE_1998__1998__a16/