Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Differentiable conjugacy of the Poincaré type vector fields on $\mathbf{R}^3$


Author: Jiazhong Yang
Journal: Proc. Amer. Math. Soc. 131 (2003), 2715-2720
MSC (2000): Primary 34K17, 37C15
Published electronically: April 23, 2003
MathSciNet review: 1974327
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that on ${\mathbf R}^3$, except for those germs of vector fields whose linear parts are conjugated to $\lambda x\partial/\partial x +\lambda y \partial/\partial y +2\lambda z \partial/\partial z$, any two Poincaré type vector fields are at least $C^3$ conjugated to each other provided their linear approximations have the same eigenvalues and the nonlinear parts are generic.


References [Enhancements On Off] (What's this?)

  • 1. D. V. Anosov and V. I. Arnol′d (eds.), Dynamical systems. I, Encyclopaedia of Mathematical Sciences, vol. 1, Springer-Verlag, Berlin, 1988. Ordinary differential equations and smooth dynamical systems; Translated from the Russian [ MR0823488 (86i:58037)]. MR 970793
  • 2. Alexander D. Bruno, Local methods in nonlinear differential equations, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1989. Part I. The local method of nonlinear analysis of differential equations. Part II. The sets of analyticity of a normalizing transformation; Translated from the Russian by William Hovingh and Courtney S. Coleman; With an introduction by Stephen Wiggins. MR 993771
  • 3. Kuo-Tsai Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693–722. MR 0160010
  • 4. Philip Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2) 5 (1960), 220–241. MR 0141856
  • 5. Fumio Ichikawa, Finitely determined singularities of formal vector fields, Invent. Math. 66 (1982), no. 2, 199–214. MR 656620, 10.1007/BF01389391
  • 6. Shlomo Sternberg, On the structure of local homeomorphisms of euclidean 𝑛-space. II., Amer. J. Math. 80 (1958), 623–631. MR 0096854
  • 7. Floris Takens, Normal forms for certain singularities of vectorfields, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 163–195 (English, with French summary). Colloque International sur l’Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972). MR 0365620
  • 8. Jiazhong Yang, Polynomial normal forms for vector fields on 𝐑³, Duke Math. J. 106 (2001), no. 1, 1–18. MR 1810364, 10.1215/S0012-7094-01-10611-X

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34K17, 37C15

Retrieve articles in all journals with MSC (2000): 34K17, 37C15


Additional Information

Jiazhong Yang
Affiliation: MAM, Institute of Mathematics, Peking University, Beijing, 100871, People’s Republic of China
Email: yang@sxx0.math.pku.edu.cn, jyang@math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07140-5
Keywords: Conjugacy, Poincar\'e type vector fields, moduli, normal form, resonance
Received by editor(s): August 20, 2000
Published electronically: April 23, 2003
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2003 American Mathematical Society