×

Analytical singularity and singular perturbation in dimension 2. (Singularité analytique et perturbation singulière en dimension 2.) (French) Zbl 0799.34062

Summary: Given a family of ordinary differential equations \(({\mathcal E}_ \alpha): \varepsilon du/dz= f(z,u,\alpha)\) in \(\mathbb{C}^ 2\) with a small parameter \(\varepsilon\) and a control parameter \(\alpha\), we are interested in the local existence of a pair \((\alpha_ *, u_ *)\) which have an asymptotic expansion in powers of \(\varepsilon\) and such that \(u_ *\) is a solution of \(({\mathcal E}_{\alpha_ *})\). The main result of our study establishes a main connection between the existence of such a pair and a property of an unfolding of the singularity of the first approximation of the function \(f\).

MSC:

34E15 Singular perturbations for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
26E35 Nonstandard analysis
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] AFRAJMOVICH (V.S.) , ARNOLD (V.I.) , IL’YASHENKO (Yu. S.) and SHIL’NIKOV (L.P.) . - Theory of bifurcations , V.I. ARNOLD ed., Dynamical Systems, vol. 5, Encyclopedia in Mathematical Sciences, Springer, 1990 . · Zbl 0711.01008
[2] BENOĨT (E.) , CALLOT (J.-L.) , DIENER (F.) et DIENER (M.) . - Chasse aux canards , Collect. Math., t. 31-32 (1-3), 1981 , p. 37-119. Zbl 0529.34046 · Zbl 0529.34046
[3] CANDELPERGHER (B.) , DIENER (F.) et DIENER (N.) . - Retard à la bifurcation : du local au global , dans Bifurcations of planar vector fields, J.P. Françoise et R. Roussarie éd., Lecture Notes in Mathematics 1455, Springer-Verlag, 1990 . MR 92k:58188 | Zbl 0739.34021 · Zbl 0739.34021
[4] CALLOT (J.-L.) . - Champs lents-rapides complexes à une dimension lente . - Prépublication, Mulhouse, 1991 .
[5] CALLOT (J.-L.) . - Sur la piste des canards imaginaires . - Prépublication, Mulhouse, 1992 .
[6] CALLOT (J.-L.) et FRUCHARD (A.) . - Observons un polynôme de Lagrange . - Prépublication , Mulhouse-Strasbourg, 1992 . · Zbl 0974.58506
[7] CANALIS-DURAND (M.) . - Caractère Gevrey de solutions formelles pour un système analytique de \Bbb C \times \Bbb C . - Prépublications de l’Université de Nice, 1990 .
[8] DELCROIX (A.) . - Propriétés asymptotiques des champs de vecteurs lents-rapides . - Thèse, Université de Poitiers, 1989 .
[9] DELCROIX (A.) et DIENER (M.) . - Variables locales pour les équations différentielles lentes-rapides , C.R. Acad. Sci. Paris, Séries I, t. 309, 1989 , p. 277-282. MR 91j:58117 | Zbl 0695.34027 · Zbl 0695.34027
[10] DIENER (F.) . - Développements en \?-ombres , Outils et modèles mathématiques pour l’automatique, l’analyse des systèmes et le traitement du signal, tome 3, éditions du CNRS, Paris 1983 . MR 783844
[11] DIENER (F.) and DIENER (M.) . - Some asymptotic results in ordinary differential equations . Non standard analysis and its applications. - Cambridge University Press, 1988 . MR 971071 | Zbl 0658.03046 · Zbl 0658.03046
[12] DIENER (F.) and DIENER (M.) . - Maximal delay , dans Dynamic bifurcations, E. Benoît éd., Lecture Notes in Mathematics 1493, Springer Verlag, 1991 , p. 71-86. MR 93m:34055 | Zbl 0875.34017 · Zbl 0875.34017
[13] DIENER (F.) et REEB (G.) . - Analyse non standard . - Hermann, Collection Enseignement des Sciences, Paris, 1989 . MR 91k:03157 | Zbl 0682.26010 · Zbl 0682.26010
[14] DIENER (M.) . - Étude générique des canards , thèse d’Etat, Strasbourg 1983 .
[15] DIENER (M.) . - Regularizing microscopes, and rivers . - Preprint 1991 . · Zbl 0794.34048
[16] ECKHAUS (W.) . - Asymptotic analysis of singular perturbations , Studies in Mathematics and its applications, vol. 9, North-Holland, Amsterdam 1979 . MR 81a:34048 | Zbl 0421.34057 · Zbl 0421.34057
[17] MISHCHENKO (E.F.) and ROZOV (N.Kh.) . - Differential equations with small parameter and relaxation oscillations . - Plenum Press, 1980 .
[18] NEISHTADT (A.I.) . - Persistence of stability loss for dynamical bifurcations I , Differentsial’nye Uravneniya, t. 23, 12, 1987 , p. 2060-2067. MR 89f:34074 | Zbl 0716.34064 · Zbl 0716.34064
[19] NEISHTADT (A.I.) . - Persistence of stability loss for dynamical bifurcations II , Differential’nye Uravneniya, t. 24, 2, 1988 , p. 226-233. MR 90a:34107 | Zbl 0677.58035 · Zbl 0677.58035
[20] NELSON (E.) . - Internal set theory : a new approach to nonstandard analysis , Bull. Amer. Math. Soc., t. 83, 6, 1977 , p. 1165-1198. Article | MR 57 #9544 | Zbl 0373.02040 · Zbl 0373.02040 · doi:10.1090/S0002-9904-1977-14398-X
[21] ROBINSON (A.) . - Non-standard analysis . - North-Holland Publishing Company, Amsterdam, 1974 .
[22] SIBUYA (Y.) . - Gevrey Property of formal solutions in a parameter . - Preprint, 1989 .
[23] VAN DEN BERG (I.) . - Nonstandard asymptotic analysis , Lecture Notes in Mathematics 1249, Springer Verlag. MR 89g:03097 | Zbl 0633.41001 · Zbl 0633.41001
[24] WALLET (G.) . - Surstabilité pour une équation différentielle analytique complexe en dimension un , Ann. Inst. Fourier (Grenoble), t. 40, 3, 1990 , p. 557-595. Numdam | MR 92c:34061 | Zbl 0695.34055 · Zbl 0695.34055 · doi:10.5802/aif.1224
[25] WALLET (G.) . - Overstability in arbitrary dimension , dans Dynamic bifurcations, E. Benoît éd., Lecture Notes in Mathematics 1493, Springer Verlag, 1991 , p. 57-70. MR 93g:34085 | Zbl 0875.34019 · Zbl 0875.34019
[26] WASOW (W.) . - Asymptotic expansions of ordinary differential equations . - Interscience Publishers, New York, 1965 . MR 34 #3041 | Zbl 0133.35301 · Zbl 0133.35301
[27] ZVONKIN (A.K.) and SHUBIN (M.A.) . - Non-standard analysis and singular perturbations of ordinary differential equations , Russian Math. Surveys, t. 39, 2, 1984 , p. 69-131. MR 85j:34119 | Zbl 0549.34055 · Zbl 0549.34055 · doi:10.1070/RM1984v039n02ABEH003091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.