Geodesic
Orbital Linearization of Smooth Completely Integrable Vector Fields
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg–Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.
Keywords: integrable system; normal form; linearization; nondegenerate singularity. 
@article{SIGMA_2017_13_a92,
     author = {Nguyen Tien Zung},
     title = {Orbital {Linearization} of {Smooth} {Completely} {Integrable} {Vector} {Fields}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a92/}
}
TY  - JOUR
AU  - Nguyen Tien Zung
TI  - Orbital Linearization of Smooth Completely Integrable Vector Fields
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a92/
LA  - en
ID  - SIGMA_2017_13_a92
ER  - 
%0 Journal Article
%A Nguyen Tien Zung
%T Orbital Linearization of Smooth Completely Integrable Vector Fields
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a92/
%G en
%F SIGMA_2017_13_a92
Nguyen Tien Zung. Orbital Linearization of Smooth Completely Integrable Vector Fields. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a92/

[1] Belitskii G. R., Kopanskii A. Ya., “Equivariant Sternberg–Chen theorem”, J. Dynam. Differential Equations, 14 (2002), 349–367 | DOI | MR | Zbl

[2] Chaperon M., “A forgotten theorem on $\mathbf{ Z}^k\times\mathbf{ R}^m$-action germs and related questions”, Regul. Chaotic Dyn., 18 (2013), 742–773 | DOI | MR | Zbl

[3] Chen K.-T., “Equivalence and decomposition of vector fields about an elementary critical point”, Amer. J. Math., 85 (1963), 693–722 | DOI | MR | Zbl

[4] Dufour J.-P., Zung N. T., Poisson structures and their normal forms, Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005 | DOI | MR | Zbl

[5] Golubitsky M., Stewart I., Schaeffer D. G., Singularities and groups in bifurcation theory, v. II, Applied Mathematical Sciences, 69, Springer-Verlag, New York, 1988 | DOI | MR | Zbl

[6] Jiang K., “Local normal forms of smooth weakly hyperbolic integrable systems”, Regul. Chaotic Dyn., 21 (2016), 18–23 | DOI | MR | Zbl

[7] Maksymenko S. I., “Symmetries of center singularities of plane vector fields”, Nonlinear Oscil., 13 (2010), 196–227, arXiv: 0907.0359 | DOI | MR | Zbl

[8] Malgrange B., Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967 | MR

[9] Schwarz G. W., “Smooth functions invariant under the action of a compact Lie group”, Topology, 14 (1975), 63–68 | DOI | MR | Zbl

[10] Sternberg S., “On the structure of local homeomorphisms of euclidean $n$-space. II”, Amer. J. Math., 80 (1958), 623–631 | DOI | MR | Zbl

[11] Ziglin S. L., “Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I”, Funct. Anal. Appl., 16 (1982), 181–189 | DOI | MR

[12] Zung N. T., “Convergence versus integrability in Poincaré–Dulac normal form”, Math. Res. Lett., 9 (2002), 217–228, arXiv: math.DS/0105193 | DOI | MR | Zbl

[13] Zung N. T., “Non-degenerate singularities of integrable dynamical systems”, Ergodic Theory Dynam. Systems, 35 (2015), 994–1008, arXiv: 1108.3551 | DOI | MR | Zbl

[14] Zung N. T., “Geometry of integrable non-Hamiltonian systems”, Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics, CRM Barcelona, eds. E. Miranda, V. Matveev, Birkhäuser/Springer, Cham, 2016, 85–140, arXiv: 1407.4494 | DOI | MR | Zbl

[15] Zung N. T., Minh N. V., “Geometry of integrable dynamical systems on 2-dimensional surfaces”, Acta Math. Vietnam., 38 (2013), 79–106, arXiv: 1204.1639 | DOI | MR | Zbl