Canalis-Durand, Mireille; Schäfke, Reinhard On the normal form of a system of differential equations with nilpotent linear part. (English. Abridged French version) Zbl 1036.34105 C. R., Math., Acad. Sci. Paris 336, No. 2, 129-134 (2003). Summary: We consider prenormal forms associated to generic perturbations of the system \(\dot x= 2y\), \(\dot y= 3x^2\). It is known that they have a formal normal form \(\dot x= 2y+ 2x\Delta^*\), \(\dot y= 3x^2+ 3y\Delta^*\), with \(\Delta^*= x+ A_0(y^2- x^3)\) [see F. Loray, J. Differ. Equations 158, 152–173 (1999; Zbl 0985.37014)]. We show that the series \(A_0\) and the normalizing transformations are divergent, but 1-summable. Cited in 1 ReviewCited in 1 Document MSC: 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Keywords:generic perturbations Citations:Zbl 0985.37014 PDFBibTeX XMLCite \textit{M. Canalis-Durand} and \textit{R. Schäfke}, C. R., Math., Acad. Sci. Paris 336, No. 2, 129--134 (2003; Zbl 1036.34105) Full Text: DOI References: [1] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1964), Dover: Dover New York) · Zbl 0171.38503 [2] Canalis-Durand, M.; Michel, F.; Teisseyre, M., Algorithms for formal reduction of vector fields singularities, J. Dynamical Control Systems, 7, 1, 101-125 (2001) · Zbl 1029.34029 [3] Cerveau, D.; Moussu, R., Groupes d’automorphismes de \((C,0)\) et équations différentielles \(y dy+⋯=0\), Publ. Soc. Math. France, 116, 459-488 (1988) · Zbl 0696.58011 [4] Ecalle, J., Les fonctions résurgentes. III : L’équation du pont et la classification analytique des objets locaux, (Publ. Math. Orsay 85-05 (1985)) · Zbl 0602.30029 [5] Loray, F., Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, Differential Equations, 158, 1, 152-173 (1999) · Zbl 0985.37014 [6] Olver, F. W.J., Asymptotics and Special Functions (1974), Academic Press: Academic Press New York · Zbl 0303.41035 [7] Ramis, J.-P., Les séries \(k\)-sommables et leurs applications, (Complex Analysis, Microlocal Calcul and Relativistic Quantum Theory. Complex Analysis, Microlocal Calcul and Relativistic Quantum Theory, Lecture Notes in Phys., 126 (1980)), 178-199 · Zbl 1251.32008 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.