Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 093, 11 pages      arXiv:1204.5701

Orbital Linearization of Smooth Completely Integrable Vector Fields

Nguyen Tien Zung ab
a) School of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, P.R. China
b) Institut de Mathématiques de Toulouse, UMR5219 CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Received July 04, 2017, in final form November 30, 2017; Published online December 12, 2017

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.

Key words: integrable system; normal form; linearization; nondegenerate singularity.

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  1. Belitskii G.R., Kopanskii A.Ya., Equivariant Sternberg-Chen theorem, J. Dynam. Differential Equations 14 (2002), 349-367.
  2. Chaperon M., A forgotten theorem on ${\bf Z}^k\times{\bf R}^m$-action germs and related questions, Regul. Chaotic Dyn. 18 (2013), 742-773.
  3. Chen K.-T., Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693-722.
  4. Dufour J.-P., Zung N.T., Poisson structures and their normal forms, Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Basel, 2005.
  5. Golubitsky M., Stewart I., Schaeffer D.G., Singularities and groups in bifurcation theory, Vol. II, Applied Mathematical Sciences, Vol. 69, Springer-Verlag, New York, 1988.
  6. Jiang K., Local normal forms of smooth weakly hyperbolic integrable systems, Regul. Chaotic Dyn. 21 (2016), 18-23.
  7. Maksymenko S.I., Symmetries of center singularities of plane vector fields, Nonlinear Oscil. 13 (2010), 196-227, arXiv:0907.0359.
  8. Malgrange B., Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 3, Tata Institute of Fundamental Research, Bombay, Oxford University Press, London, 1967.
  9. Schwarz G.W., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68.
  10. Sternberg S., On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623-631.
  11. Ziglin S.L., Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funct. Anal. Appl. 16 (1982), 181-189.
  12. Zung N.T., Convergence versus integrability in Poincaré-Dulac normal form, Math. Res. Lett. 9 (2002), 217-228, math.DS/0105193.
  13. Zung N.T., Non-degenerate singularities of integrable dynamical systems, Ergodic Theory Dynam. Systems 35 (2015), 994-1008, arXiv:1108.3551.
  14. Zung N.T., Geometry of integrable non-Hamiltonian systems, in Geometry and Dynamics of Integrable Systems, Editors E. Miranda, V. Matveev, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser/Springer, Cham, 2016, 85-140, arXiv:1407.4494.
  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.

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