Linearization of resonant vector fields

Author:
J. Basto-Gonçalves

Journal:
Trans. Amer. Math. Soc. **362** (2010), 6457-6476

MSC (2010):
Primary 32S65, 34M35

DOI:
https://doi.org/10.1090/S0002-9947-2010-04978-5

Published electronically:
August 3, 2010

MathSciNet review:
2678982

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A method allowing the formal linearization of a large class of vector fields with resonant eigenvalues is presented, the admissible nonlinearities being characterized by conditions that are easy to check. This method also gives information on the terms that are actually present in a nonlinear normal form of a given resonant vector field.

**1.**D. Anosov, V. Arnold,*Dynamical Systems I*, Encyclopædia of Mathematical Sciences, Springer-Verlag, 1988. MR**970793 (89g:58060)****2.**V. Arnold,*Geometrical Methods in the Theory of*, Springer-Verlag, 1983. MR**695786 (84d:58023)****3.**J. Basto-Gonçalves, I. Cruz,*Analytic linearizability of some resonant vector fields*, Proc. AMS 129 (2001), 2473-2481. MR**1823934 (2002e:37070)****4.**J. Basto-Gonçalves, A.C. Ferreira,*Normal forms and linearization of vector fields with multiple eigenvalues*, J. Math. Anal. Appl. 301 (2005), 219-236. MR**2105931 (2005h:37104)****5.**G. Belitskii,*Smooth equivalence of germs of vector fields with one zero or a pair of pure imaginary eigenvalues*, Funct. Anal. Appl. 20 (1986), 253-259. MR**878039 (88f:58125)****6.**A. Brjuno,*Analytical form of differential equations*, Trans. Moscow Math. Soc. 25 (1971), 131-288. MR**0377192 (51:13365)****7.**C. Camacho, N. Kuiper, J. Palis,*The topology of holomorphic flows with singularity*, Publ. Math. IHES 48 (1978), 5-38. MR**516913 (80j:58045)****8.**M. Chaperon,*-conjugacy of holomorphic flows near a singularity*, Publ. Math. IHES 64 (1986), 693-722.**9.**K. T. Chen,*Equivalence and decomposition of vector fields about an elementary critical point*, Am. J. Math. 85 (1963), 693-722. MR**0160010 (28:3224)****10.**J. Guckenheimer,*Hartman's theorem for complex flows in the Poincaré domain*, Comp. Mathematica 24 (1972), 75-82. MR**0301765 (46:920)****11.**F. Ichikawa,*Finitely determined singularities of formal vector fields*, Inv. Math 66 (1982), 199-214. MR**656620 (83j:58021)****12.**G. Sell,*Smooth linearization near a fixed point*, Am. J. Math. 107 (1985), 1035-1091. MR**805804 (87c:58095)****13.**S. Sternberg,*On the structure of local homeomorphisms of Euclidean -space II*, Am. J. Math. 80 (1958), 623-631. MR**0096854 (20:3336)****14.**J. Yang,*Polynomial normal forms of vector fields*, Ph.D. thesis, Technion - Israel Institute of Technology, 1997.**15.**-,*Polynomial normal forms for vector fields on*, Duke Math. J. 106 (2001), 1-18. MR**1810364 (2001m:34165)****16.**M. Zhitomirskii,*Smooth Local Normal Forms for Vector Fields and Diffeomorphisms*, lecture notes, 1992.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
32S65,
34M35

Retrieve articles in all journals with MSC (2010): 32S65, 34M35

Additional Information

**J. Basto-Gonçalves**

Affiliation:
Centro de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Email:
jbg@fc.up.pt

DOI:
https://doi.org/10.1090/S0002-9947-2010-04978-5

Received by editor(s):
September 22, 2008

Published electronically:
August 3, 2010

Additional Notes:
This research had financial support from the Fundação para a Ciência e a Tecnologia and the Calouste Gulbenkian Foundation

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.