Linearization of resonant vector fields

Basto-Gonçalves, J.

doi:10.1090/S0002-9947-2010-04978-5

Linearization of resonant vector fields
HTML articles powered by AMS MathViewer

by J. Basto-Gonçalves PDF

Trans. Amer. Math. Soc. 362 (2010), 6457-6476 Request permission

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.

References

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, DOI 10.1007/978-3-642-61551-1
V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786
J. Basto-Gonçalves and I. Cruz, Analytic linearizability of some resonant vector fields, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2473–2481. MR 1823934, DOI 10.1090/S0002-9939-00-05796-8
J. Basto-Gonçalves and A. C. Ferreira, Normal forms and linearization of resonant vector fields with multiple eigenvalues, J. Math. Anal. Appl. 301 (2005), no. 1, 219–236. MR 2105931, DOI 10.1016/j.jmaa.2004.06.065
G. R. Belitskiĭ, Smooth equivalence of germs of vector fields with one zero or a pair of purely imaginary eigenvalues, Funktsional. Anal. i Prilozhen. 20 (1986), no. 4, 1–8, 96 (Russian). MR 878039
A. D. Brjuno, Analytic form of differential equations. I, II, Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR 0377192
César Camacho, Nicolaas H. Kuiper, and Jacob Palis, The topology of holomorphic flows with singularity, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 5–38. MR 516913
M. Chaperon, $C^k$-conjugacy of holomorphic flows near a singularity, Publ. Math. IHES 64 (1986), 693-722.
Kuo-Tsai Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693–722. MR 160010, DOI 10.2307/2373115
John Guckenheimer, Hartman’s theorem for complex flows in the Poincaré domain, Compositio Math. 24 (1972), 75–82. MR 301765
Fumio Ichikawa, Finitely determined singularities of formal vector fields, Invent. Math. 66 (1982), no. 2, 199–214. MR 656620, DOI 10.1007/BF01389391
George R. Sell, Smooth linearization near a fixed point, Amer. J. Math. 107 (1985), no. 5, 1035–1091. MR 805804, DOI 10.2307/2374346
Shlomo Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623–631. MR 96854, DOI 10.2307/2372774
J. Yang, Polynomial normal forms of vector fields, Ph.D. thesis, Technion - Israel Institute of Technology, 1997.
Jiazhong Yang, Polynomial normal forms for vector fields on $\mathbf R^3$, Duke Math. J. 106 (2001), no. 1, 1–18. MR 1810364, DOI 10.1215/S0012-7094-01-10611-X
M. Zhitomirskii, Smooth Local Normal Forms for Vector Fields and Diffeomorphisms, lecture notes, 1992.