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

Patrick Bonckaert[1]; Freek Verstringe[1]

  • [1] Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2211-2225
  • ISSN: 0373-0956

Abstract

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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- α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey- 1 + α type with the use of a Gevrey- 1 + α 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.

How to cite

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Bonckaert, Patrick, and Verstringe, Freek. "Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part." Annales de l’institut Fourier 62.6 (2012): 2211-2225. <http://eudml.org/doc/251148>.

@article{Bonckaert2012,
abstract = {We explore the convergence/divergence of the normal form for a singularity of a vector field on $\mathbb\{C\}^n$ 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.},
affiliation = {Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium); Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium)},
author = {Bonckaert, Patrick, Verstringe, Freek},
journal = {Annales de l’institut Fourier},
keywords = {normal forms; nilpotent linear part; representation theory; Gevrey normalization},
language = {eng},
number = {6},
pages = {2211-2225},
publisher = {Association des Annales de l’institut Fourier},
title = {Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part},
url = {http://eudml.org/doc/251148},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Bonckaert, Patrick
AU - Verstringe, Freek
TI - Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2211
EP - 2225
AB - We explore the convergence/divergence of the normal form for a singularity of a vector field on $\mathbb{C}^n$ 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.
LA - eng
KW - normal forms; nilpotent linear part; representation theory; Gevrey normalization
UR - http://eudml.org/doc/251148
ER -

References

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  8. David Mumo Malonza, Stanley decomposition for coupled Takens-Bogdanov systems, J. Nonlinear Math. Phys. 17 (2010), 69-85 Zbl1194.34066MR2647461
  9. James Murdock, Normal forms and unfoldings for local dynamical systems, (2003), Springer-Verlag, New York Zbl1014.37001MR1941477
  10. Ewa Stróżyna, Henryk Żoładek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations 179 (2002), 479-537 Zbl1005.34034MR1885678
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