Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Bonckaert, Patrick; Verstringe, Freek

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2211-2225.

Nous investiguons la convergence/divergence de la forme normale d’une singularité d’un champ de vecteurs de ${ℂ}^n$ avec une partie linéaire nilpotente. Nous prouvons que chaque champ de vecteurs Gevrey-$\alpha$ avec une partie linéaire nilpotente peut être réduit à une forme normale Gevrey-$1+\alpha$ en utilisant une transformation Gevrey-$1+\alpha$. Nous prouvons également que si on arrête la procédure de normalisation à un certain ordre optimal, le reste de la forme normale devient exponentiellement petit.

We explore the convergence/divergence of the normal form for a singularity of a vector field on ${ℂ}^n$ with nilpotent linear part. We show that a Gevrey-$\alpha$ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha$ type with the use of a Gevrey-$1+\alpha$ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

[1] Canalis-Durand, Mireille; Schäfke, Reinhard Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. (6), Volume 13 (2004) no. 4, pp. 493-513 | Numdam | MR | Zbl

[2] Cushman, Richard; Sanders, Jan A. Nilpotent normal forms and representation theory of $\mathrm{sl}(2,\mathbf{R})$, Multiparameter bifurcation theory (Arcata, Calif., 1985) (Contemp. Math.), Volume 56, Amer. Math. Soc., Providence, RI, 1986, pp. 31-51 | MR | Zbl

[3] Elphick, C.; Tirapegui, E.; Brachet, M. E.; Coullet, P.; Iooss, G. A simple global characterization for normal forms of singular vector fields, Phys. D, Volume 29 (1987) no. 1-2, pp. 95-127 | DOI | MR | Zbl

[4] Humphreys, James E. Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972 (Graduate Texts in Mathematics, Vol. 9) | MR | Zbl

[5] Iooss, Gérard; Lombardi, Eric Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, Volume 212 (2005) no. 1, pp. 1-61 | DOI | MR | Zbl

[6] Lombardi, Eric; Stolovitch, Laurent Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation. (Formes normales des perturbations analytiques des champs de vecteurs quasi-homogènes: rigidité, ensembles d’invariants analytiques et approximation exponentiellement petite.), Ann. Sci. Éc. Norm. Supér. (2010), pp. 659-718 | Numdam | MR | Zbl

[7] Loray, Frank A preparation theorem for codimension-one foliations, Ann. of Math. (2), Volume 163 (2006) no. 2, pp. 709-722 | DOI | MR | Zbl

[8] Malonza, David Mumo Stanley decomposition for coupled Takens-Bogdanov systems, J. Nonlinear Math. Phys., Volume 17 (2010) no. 1, pp. 69-85 | DOI | MR | Zbl

[9] Murdock, James Normal forms and unfoldings for local dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003 | MR | Zbl

[10] Stróżyna, Ewa; Żoładek, Henryk The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, Volume 179 (2002) no. 2, pp. 479-537 | DOI | MR | Zbl

[11] Stróżyna, Ewa; Żoładek, Henryk Multidimensional formal Takens normal form, Bull. Belg. Math. Soc. Simon Stevin, Volume 15 (2008) no. 5, Dynamics in perturbations, pp. 927-934 http://projecteuclid.org/getRecord?id=euclid.bbms/1228486416 | MR | Zbl

[12] Takens, Floris Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. (1974) no. 43, pp. 47-100 | Numdam | MR | Zbl