Les derniers travaux de Jean Martinet
Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, p. 15-47

Gevrey classes theory and summability are natural generalizations of Cauchy theory. We use a little bit of Non Standard Analysis and we introduce ε-functions (for ε>0, infinitely small, fixed, and ε-function is a holomorphic function defined “up to ε”, and “not too big”). We extend Cauchy theory to ε-functions and get wild Cauchy theory. The wild analytic continuation principle is one of the central results. We interpret delays in bifurcations using Gevrey asymptotics.

On montre comment la théorie des classes de Gevrey et de la sommabilité sont des généralisations naturelles de la théorie de Cauchy. On utilise le vocabulaire de l’Analyse Non Standard et on introduit la notion d’ε-fonction (fonction analytique définie “à ε près”, pour ε>0 infiniment petit fixé, et ne prenant que des valeurs infiniment petite devant 1/ε. On étend la théorie de Cauchy aux =FDe-fonctions  : c’est la théorie de Cauchy sauvage. On interprète le phénomène de retard à la bifurcation à l’aide d’asymptoticité Gevrey

@article{AIF_1992__42_1-2_15_0,
     author = {Ramis, Jean-Pierre},
     title = {Les derniers travaux de Jean Martinet},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {42},
     number = {1-2},
     year = {1992},
     pages = {15-47},
     doi = {10.5802/aif.1285},
     zbl = {0927.01031},
     mrnumber = {94m:01035},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1992__42_1-2_15_0}
}
Les derniers travaux de Jean Martinet. Annales de l'Institut Fourier, Volume 42 (1992) no. 1-2, pp. 15-47. doi : 10.5802/aif.1285. http://www.numdam.org/item/AIF_1992__42_1-2_15_0/

[CDD] B. Candelpergher, F. Diener, M. Diener, Retard à la bifurcation: du local au global, in Bifurcations of Planar Vector Fields, J.P. Françoise and R. Roussarie, ed., Springer, 1990, p. 1-19. | MR 92k:58188 | Zbl 0739.34021

[DD] F. Diener, M. Diener, Maximal Delay in Dynamical Bifurcations, E. Benoit ed., Springer, 1991.

[MaR] B. Malgrange, J.P. Ramis, Fonctions Multisommables, Ann. Inst. Fourier, Grenoble, 42, 1-2 (1992). | Numdam | MR 93e:40007 | Zbl 0759.34007

[MR1] J. Martinet, J.P. Ramis, Théorie de Galois différentielle et resommation, Computer Algebra and Differential Equations, E. Tournier ed., Acad. Press, 1989. | Zbl 0722.12007

[MR2] J. Martinet, J.P. Ramis, Elementary acceleration and multisummability, Ann. Inst. Henri Poincaré, Physique Th., Vol. 54, n° 4 (1991), p. 331-401. | Numdam | MR 93a:32036 | Zbl 0748.12005

[N] A.I. Neishtadt, Persistence of stability loss for dynamical bifurcations 1,2 Differential'nye Uravneniya (Differential Equations), 23 (12): 2060-2067, (1385-1390), 1987/1988 and 24(2) : 226-233 (171-176), 1988 (88).

[Ol] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974. | MR 55 #8655 | Zbl 0303.41035

[S] Y. Sibuya, Gevrey property of formal solutions in a parameter. Preprint, School of Mathematics, University of Minnesota, Minneapolis, Minn. 55455 U.S.A., 1989.

[V] I. Van Den Berg, Non Standard Asymptotic Analysis, Lecture Notes in Math. 1249, Springer-Verlag, Berlin et New-York, 1987. | MR 89g:03097 | Zbl 0633.41001