Singular perturbations of ordinary differential equations and nonstandard analysis
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 39 (1984) no. 2 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

@article{RM_1984_39_2_a46,
     author = {P. Cartier},
     title = {Singular perturbations of ordinary differential equations and nonstandard analysis},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     year = {1984},
     volume = {39},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/RM_1984_39_2_a46/}
}
TY  - JOUR
AU  - P. Cartier
TI  - Singular perturbations of ordinary differential equations and nonstandard analysis
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1984
VL  - 39
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/RM_1984_39_2_a46/
LA  - ru
ID  - RM_1984_39_2_a46
ER  - 
%0 Journal Article
%A P. Cartier
%T Singular perturbations of ordinary differential equations and nonstandard analysis
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1984
%V 39
%N 2
%U http://geodesic.mathdoc.fr/item/RM_1984_39_2_a46/
%G ru
%F RM_1984_39_2_a46
P. Cartier. Singular perturbations of ordinary differential equations and nonstandard analysis. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 39 (1984) no. 2. http://geodesic.mathdoc.fr/item/RM_1984_39_2_a46/

[1] A. A. Andronov, A. A. Vitt, S. E. Xaikin, Teoriya kolebanii, Fizmatgiz, M., 1959

[2] F. Xartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR

[3] S. Lefshets, Geometricheskaya teoriya differentsialnykh uravnenii, IL, M., 1961

[4] M. Minorsky, Théorie des oscillations, Gauthier-Villars, Paris, 1967 | MR | Zbl

[5] R. E. O'Malley, Introduction to singular perturbations, Acad. Press, New York, 1974 | MR

[6] L. S. Pontryagin, Obyknovennye differentsialnye uravneniya, Nauka, M., 1965 | MR | Zbl

[7] V. Vazov, Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[8] D. A. Flanders, J. J. Stocker, “The limit case of relaxation oscillations”, Studies in non-linear vibration theory, Inst. for Math. and Mech., New York University, New York, 1964

[9] J. Haag, “Étude asymptotique des oscillations de relaxation”, Ann. Sci. École Norm. Sup., 60 (1943), 35–64 | MR | Zbl

[10] J. Haag, “Examples concrets d'étude asymptotique d'oscillations de relaxation”, Ann. Sci. École Norm. Sup., 61 (1944), 65–111 | MR

[11] N. Levinson, “An ordinary differential equation with an interval of stability, a separation point, and an interval of instability”, J. of Math. and Phys., 28 (1949) | MR

[12] A. Liénard, “Étude des oscillations entretenues”, Rev. Gén. Electr., 23 (1928), 901-912, 946-954

[13] B. van der Pol, “Forced oscillations in a system with nonlinear resistance”, Philos. Mag., 1927

[14] Th. Vogel, Théorie des systèmes évolutifs, Gauthier-Villars, Paris, 1965 | MR | Zbl

[15] E. C. Zeeman, “Differential equations for heart beat and nerve impulse”, Dynamical Systems, eds. Peixoto, Acad. Press, New York, 1973, 683–748 | MR

[16] K. Hrbacek, “Non standard set theory”, Amer. Math. Monthly, 86:8 (1979), 659–677 | DOI | MR | Zbl

[17] H. J. Keisler, Foundations of infinitesimal calculus, Prindle, Weber and Schmidt, Boston, 1976

[18] D. Laugwitz, G. Schmieden, Kontinuum und Zahlen–Neue mathematische Überlegungen zum Endlichen, Darmstadt, 1980

[19] R. Lutz, M. Goze, “Nonstandard Analysis, a practical guide with applications”, Lect. Notes in Math., 881 (1981) | MR | Zbl

[20] E. Nelson, “Internal set theory”, Bull. Amer. Math. Soc., 83:6 (1977), 1165–1198 | DOI | MR | Zbl

[21] A. Robinson, Nonstandard Analysis, North Holland, Amsterdam, 1966 | MR | Zbl

[22] K. D. Stroyan, W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Acad. Press, New York, 1976 | MR | Zbl

[23] J.-L. Callot, Bifuracations du portrait de phase pour équations différentielles linéaire du second ordre ayant pour type l'équation d'Hermite, Diss. 24.06.81, Strasbourg

[24] F. Diener, Méthode du plan d'observabilité; Développements en $\varepsilon$-ombres, Diss. 25.11.81., Strasbourg

[25] M. Diener, Étude générique des canards, Diss. 25.11.81., Strasbourg

[26] J. Harthong, Vision macroscopique de phénomènes périodiques, Diss. 27.11.81., Strasbourg

[27] A. Troesch, Étude qualitative de systèmes differentiels: une approche basée sur l'analyse non-standard, Diss. 27.03.81., Strasbourg

[28] E. Urlacher, Oscillations de relaxation et analyse non-standard, Diss. 23.10.81., Strasbourg

[29] E. Benoit, J.-L. Callot, F. Diener, M. Diener, “Ghasse au canard”, Collectanea Mathematica, 31:3 (1980)

[30] F. Diener, “Les equations $\ddot x+(x^2-1)\dot x^{[s]}+x=a$”, Collectanea Mathematica, 29:3 (1978), 217–247 | MR | Zbl

[31] M. Diener, I. van den Berg, “Halos et galaxies. Une extension du lemme de Robinson”, C. R. Acad. Sci. Paris, sér. 1, 293:8 (1981), 385–388 | MR

[32] A. Troesch, Étude macroscopique de systèmes différentiels, Publications IRMA, Strasbourg, 1980

[33] A. Troesch, E. Urlacher, Analyse non standard et équation de van der Pol, Publications IRMA, Strasbourg, 1977

[34] A. Troesch, E. Urlacher, Perturbations singulières et analyse non-standard: $C^k$-convergence et crépitement des solutions, Publications IRMA, Strasbourg, 1977 | MR