Geodesic
The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 818-831 
A. S. Bobkova; A. Yu. Kolesov; N. Kh. Rozov. The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 818-831. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a2/
@article{MZM_2002_71_6_a2,
     author = {A. S. Bobkova and A. Yu. Kolesov and N. Kh. Rozov},
     title = {The {{\textquotedblleft}Duck} {Survival{\textquotedblright}} {Problem} in {Three-Dimensional} {Singularly} {Perturbed} {Systems} with {Two} {Slow} {Variables}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {818--831},
     year = {2002},
     volume = {71},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a2/}
}
TY  - JOUR
AU  - A. S. Bobkova
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables
JO  - Matematičeskie zametki
PY  - 2002
SP  - 818
EP  - 831
VL  - 71
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a2/
LA  - ru
ID  - MZM_2002_71_6_a2
ER  - 
%0 Journal Article
%A A. S. Bobkova
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables
%J Matematičeskie zametki
%D 2002
%P 818-831
%V 71
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a2/
%G ru
%F MZM_2002_71_6_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the system of ordinary differential equations $\dot x = f(x,y)$, $\varepsilon\dot y=g(x,y)$, where $x\in\mathbb R^2$, $y\in\mathbb R$, $0<\varepsilon \ll 1$ and $f,g\in C^\infty$. It is assumed that the equation $g = 0$ determines two different smooth surfaces $y=\varphi(x)$ and $y=\psi(x)$ intersecting generically along a curve $l$. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface $y=\varphi(x)$ are ducks, i.e., as time increases, they intersect the curve $l$ generically and pass from the stable part $\{y=\varphi(x), g'_y<0\}$ of this surface to the unstable part $\{y=\varphi(x), g'_y>0\}$. We seek a solution of the so-called duck survival problem, i.e., give an answer to the following question: what trajectories from the one-parameter family of duck trajectories for $\varepsilon=0$ are the limits as $\varepsilon\to 0$ of some trajectories of the original system.

[1] Mischenko E. F., Kolesov Yu. S., Kolesov A. Yu., Rozov N. Kh., Periodicheskie dvizheniya i bifurkatsionnye protsessy v singulyarno vozmuschennykh sistemakh, Fizmatlit, M., 1995 | Zbl

[2] Kolesov A. Yu., Mischenko E. F., Rozov N. Kh., “Reshenie singulyarno vozmuschennykh kraevykh zadach metodom “okhoty na utok””, Tr. MIAN, 224, Nauka, M., 1999, 187–207 | MR | Zbl

[3] Mischenko E. F., Rozov N. Kh., Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Nauka, M., 1975 | Zbl

[4] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | Zbl

[5] Sobolev V. A., Schepakina E. A., “Traektorii-utki v odnoi zadache teorii goreniya”, Differents. uravneniya, 32:9 (1996), 1175–1184 | MR | Zbl

[6] Lebovitz N. R., Schaar R. J., “Exchange of stabilities in autonomous systems”, Stud. Appl. Math., 54, 1975, 229–260 | MR | Zbl

[7] Sobolev V. A., Schepakina E. A., “Integralnye poverkhnosti so smenoi ustoichivosti i traektorii-utki”, Izv. RAEN. Ser. MMMIU, 1:3 (1997), 176–187