Comptes Rendus
no. 9-10
Dynamical Systems
On the flat remainder in normal forms of families of analytic planar saddles
[Sur le reste plat de la forme normale d'une famille de points de selle analytiques dans le plan]

Présenté par : Étienne Ghys

Patrick Bonckaert1 ; Freek Verstringe1
1 Hasselt University, Agoralaan, B-3590 Diepenbeek, Belgium
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 553-558.

Nous donnons une expression explicite du reste plat d'ordre fini, obtenue après une reduction analytique en forme normale, pur une famille de difféomorphismes ou de champs de vecteurs du plan ayant un point de selle à l'origine. Nous faisons la distinction entre un rapport rationnel ou irrationnel des modules des valuers propres pour une certaine valeur du paramètre.

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.012
Patrick Bonckaert 1 ; Freek Verstringe 1

1 Hasselt University, Agoralaan, B-3590 Diepenbeek, Belgium 
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Patrick Bonckaert; Freek Verstringe. On the flat remainder in normal forms of families of analytic planar saddles. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 553-558. doi : 10.1016/j.crma.2008.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.012/

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[2] P. Bonckaert, I. Hoveijn, F. Verstringe, Local analytic reduction of families of diffeomorphisms, preprint

[3] N.N. Brušlinskaja A finiteness theorem for families of vector fields in the neighborhood of a singular point of Poincaré type, Funktsional. Anal. i Prilozhen., Volume 5 (1971) no. 3, pp. 10-15

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[6] Y.S. Ilyashenko; S.Y. Yakovenko Finitely smooth normal forms of local families of diffeomorphisms and vector fields, Russian Math. Surveys, Volume 46 (1991), pp. 1-43

[7] C.D. Olds Continued Fractions, Random House, New York, 1963

[8] K.H. Rosen Elementary Number Theory and its Applications, Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1988

[9] C. Rousseau Normal forms for germs of analytic families of planar vector fields unfolding a generic saddle-node or resonant saddle, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., vol. 48, Amer. Math. Soc., Providence, RI, 2006, pp. 227-245

[10] C.L. Siegel Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., Volume 1952 (1952), pp. 21-30

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