Geodesic
Duck trajectories of relaxation systems connected with violation of the normal switching conditions
Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 291-301 
A. Yu. Kolesov. Duck trajectories of relaxation systems connected with violation of the normal switching conditions. Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 291-301. http://geodesic.mathdoc.fr/item/SM_1991_68_1_a14/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Assume that for $x\in R$ and $y\in R^2$ at an isolated point of discontinuity of the relaxation system $$ \varepsilon\dot x=f(x,y),\quad\dot y=g(x,y),\qquad0<\varepsilon\ll1, $$ the so-called normal switching condition is violated generically. Under this assumption a theorem on the existence and the asymptotic properties of two structurally stable duck trajectories is proved. Their role in the dynamics of relaxation systems is stressed. Bibliography: 6 titles.

[1] Kolesov A. Yu., Kolesov Yu. S, Mischenko E. F., Rozov N. X., “Asimptoticheskoe integrirovanie sistemy v variatsiyakh mnogomernogo relaksatsionnogo tsikla. I, II”, Differents. uravneniya, 23:11 (1987), 1881–1889; 2036–2047 | MR

[2] Mischenko E. F., Rozov N. X., Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Nauka, M., 1975 | MR

[3] Zvonkii A. K., Shubin M. A., “Nestandartnyi analiz i singulyarnye vozmuscheniya”, UMN, 39:2 (1984), 77–129 | MR

[4] Kolesov A. Yu., Kolesov Yu. S, Mischenko E. F., Rozov N. X., “Relaksatsionnaya sistema v okrestnosti tochki sryva: svedenie k regulyarnomu sluchayu”, UMN, 43:2 (1988), 141–142 | MR

[5] Mitropolskii Yu. A., Lykova O. B., Integralnye mnogoobraziya v nelineinoi mekhanike, Nauka, M., 1973 | MR

[6] Kolesov A. Yu., Mischenko E. F., “Asimptotika relaksatsionnykh kolebanii”, Matem. sb., 137(179) (1988), 3–18