Regular and Chaotic Dynamics

, Volume 21, Issue 4, pp 410–436 | Cite as

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

  • Laurent Stolovitch
  • Freek Verstringe


We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Stróżyna–Żołladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.


local analytic dynamics fixed point normal form Belitskii normal form small divisors Newton method analytic invariant manifold complete integrability 

MSC2010 numbers

34M35 34C20 37J40 37F50 58C15 34C45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, V. I., Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Moscow: Mir, 1980.MATHGoogle Scholar
  2. 2.
    Belitskii, G. R., Invariant Normal Forms of Formal Series, Funct. Anal. Appl., 1979, vol. 13, no. 1, pp. 46–47; see also: Funktsional. Anal. i Prilozhen., 1979, vol. 13, no. 1, pp. 59–60.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Belitskii, G. R., Normal Forms in Relation to the Filtering Action of a Group, Trans. Moscow Math. Soc., 1981, no. 2, pp. 1–39; see also: Tr. Mosk. Mat. Obs., 1979, vol. 40, pp. 3–46.MathSciNetMATHGoogle Scholar
  4. 4.
    Bourbaki, N., Groupes et algèbres de Lie: Chapitres 7 et 8, Paris: Hermann, 1975.Google Scholar
  5. 5.
    Bruno, A.D., Analytic Form of Differential Equations: 1, Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119–262 (Russian); see also: Trans. Moscow Math. Soc., 1971, vol. 25, pp. 131–288; Bruno, A.D., Analytic Form of Differential Equations: 2, Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 199–239 (Russian); see also: Trans. Moscow Math. Soc., 1972, vol. 26, pp. 199–239.MathSciNetGoogle Scholar
  6. 6.
    Bonckaert, P. and Verstringe, F., Normal Forms with Exponentially Small Remainder and Gevrey Normalization for Vector Fields with a Nilpotent Linear Part, Ann. Inst. Fourier (Grenoble), 2012, vol. 62, no. 6, pp. 2211–2225.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Canalis-Durand, M. and Schäfke, R., Divergence and Summability of Normal Forms of Systems of Differential Equations with Nilpotent Linear Part, Ann. Fac. Sci. Toulouse Math. (6), 2004, vol. 13, no. 4, pp. 493–513.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cairns, G. and Ghys, E., The Local Linearization Problem for Smooth SL(n)-Actions, Enseign. Math. (2), 1997, vol. 43, nos. 1–2, pp. 133–171.MathSciNetMATHGoogle Scholar
  9. 9.
    Cushman, R. and Sanders, J.A., Nilpotent Normal Forms and Representation Theory of sl(2,R), in Multiparameter Bifurcation Theory (Arcata,Calif., 1985), Contemp. Math., vol. 56, Providence, R.I.: AMS, 1986, pp. 31–51.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fischer, E., Über die Differentiationsprozesse der Algebra, J. Reine Angew. Math., 1918, vol. 148, pp. 1–78.MathSciNetMATHGoogle Scholar
  11. 11.
    Gong, X., Integrable Analytic Vector Fields with a Nilpotent Linear Part, Ann. Inst. Fourier (Grenoble), 1995, vol. 45, no. 5, pp. 1449–1470.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grauert, H. and Remmert, R., Analytische Stellenalgebren, Grundlehren Math. Wiss., vol. 176, New York: Springer, 1971.Google Scholar
  13. 13.
    Guillemin, V. W. and Sternberg, Sh., Remarks on a Paper of Hermann, Trans. Amer. Math. Soc., 1968, vol. 130, pp. 110–116.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gazor, M. and Yu, P., Spectral Sequences and Parametric Normal Forms, J. Differential Equations, 2012, vol. 252, no. 2, pp. 1003–1031.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hermann, R., The Formal Linearization of a Semisimple Lie Algebra of Vector Fields about a Singular Point, Trans. Amer. Math. Soc., 1968, vol. 130, pp. 105–109.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Iooss, G. and Lombardi, E., Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields, J. Differential Equations, 2005, vol. 212, no. 1, pp. 1–61.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ito, H., Convergence of Birkhoff Normal Forms for Integrable Systems, Comment. Math. Helv., 1989, vol. 64, no. 3, pp. 412–461.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ito, H., Birkhoff Normalization and Superintegrability of Hamiltonian Systems, Ergodic Theory Dynam. Systems, 2009, vol. 29, no. 6, pp. 1853–1880.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kokubu, H., Oka, H., and Wang, D., Linear Grading Function and Further Reduction of Normal Forms, J. Differential Equations, 1996, vol. 132, no. 2, pp. 293–318.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kushnirenko, A.G., Linear-Equivalent Action of a Semisimple Lie Group in the Neighborhood of a Stationary Point, Funct. Anal. Appl., 1967, vol. 1, no. 1, pp. 89–90; see also: Funktsional. Anal. i Prilozhen., 1967, vol. 1, no. 1, pp. 103–104.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Loray, F., A Preparation Theorem for Codimension-One Foliations, Ann. of Math. (2), 2006, vol. 163, no. 2, pp. 709–722.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lombardi, E. and Stolovitch, L., Normal Forms of Analytic Perturbations of Quasihomogeneous Vector Fields: Rigidity, Invariant Analytic Sets and Exponentially Small Approximation, Ann. Sci. Éc. Norm. Supér. (4), 2010, vol. 43, no. 4, pp. 659–718.MathSciNetMATHGoogle Scholar
  23. 23.
    Martinet, J., Normalisation des champs de vecteurs holomorphes (d’après A.-D.Brjuno), in Bourbaki Seminar: Vol. 1980/81, Lecture Notes in Math., vol. 901, New York: Springer, 1981, pp. 55–70.MathSciNetGoogle Scholar
  24. 24.
    Moussu, R. and Cerveau, D., Groupes d’automorphismes de (C, 0) et équations différentielles ydy +· · · = 0, Bull. Soc. Math. France, 1988, vol. 116, no. 4, pp. 459–488.MathSciNetMATHGoogle Scholar
  25. 25.
    Moser, J., Stable and Random Motions in Dynamical Systems, Princeton,N.J.: Princeton Univ. Press, 2001.CrossRefMATHGoogle Scholar
  26. 26.
    Murdock, J., Normal Forms and Unfoldings for Local Dynamical Systems, Springer Monogr. Math., Berlin: Springer, 2003.CrossRefMATHGoogle Scholar
  27. 27.
    Murdock, J., Hypernormal Form Theory: Foundations and Algorithms, J. Differential Equations, 2004, vol. 205, no. 2, pp. 424–465.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nowicki, A., Polynomial Derivations and Their Rings of Constants, Torun: Uniwersytet Mikolaja Kopernika, 1994.MATHGoogle Scholar
  29. 29.
    Rüssmann, H., Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., 1967, vol. 169, pp. 55–72.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sanders, J.A., Normal Form Theory and Spectral Sequences, J. Differential Equations, 2003, vol. 192, no. 2, pp. 536–552.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Serre, J.-P., Complex Semisimple Lie Algebras, New York: Springer, 1987.CrossRefMATHGoogle Scholar
  32. 32.
    Siegel, C. L., Iteration of Analytic Functions, Ann. of Math. (2), 1942, vol. 43, pp. 607–612.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Siegel, C. L. and Moser, J. K., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.Google Scholar
  34. 34.
    Stolovitch, L., Singular Complete Integrabilty, Publ. Math. Inst. Hautes Études Sci., 2000, vol. 91, no. 1, pp. 133–210.MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Stolovitch, L., Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. of Math. (2), 2005, vol. 161, no. 2, pp. 589–612.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Strżyna, E. and Zoladek, H., The Analytic and Formal Normal Form for the Nilpotent Singularity, J. Differential Equations, 2002, vol. 179, no. 2, pp. 479–537.MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Strżyna, E. and Zoladek, H., Multidimensional Formal Takens Normal Form, Bull. Belg. Math. Soc. Simon Stevin, 2008, vol. 15, no. 5, pp. 927–934.MathSciNetMATHGoogle Scholar
  38. 38.
    Strżyna, E. and Zoladek, H., Divergence of the Reduction to the Multidimensional Nilpotent Takens Normal Form, Nonlinearity, 2011, vol. 24, no. 11, pp. 3129–3141.MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Takens, F., Singularities of Vector Fields, Publ. Math. Inst. Hautes Études Sci., 1974, no. 43, pp. 47–100.MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Vey, J., Sur certains systèmes dynamiques séparables, Amer. J. Math., 1978, vol. 100, no. 3, pp. 591–614.MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Vey, J., Algèbres commutatives de champs de vecteurs isochores, Bull. Soc. Math. France, 1979, vol. 107, no. 4, pp. 423–432.MathSciNetMATHGoogle Scholar
  42. 42.
    Weitzenbock, R., Über die Invarianten von linearen Gruppen, Acta Math., 1932, vol. 58, no. 1, pp. 231–293.MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zung, Nguyen Tien, Convergence versus Integrability in Poincaré–Dulac Normal Form, Math. Res. Lett., 2002, vol. 9, nos. 2–3, pp. 217–228.MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zung, Nguyen Tien, Convergence versus Integrability in Birkhoff Normal Form, Ann. of Math. (2), 2005, vol. 161, no. 1, pp. 141–156.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621Université de Nice — Sophia Antipolis, Parc ValroseNice Cedex 02France
  2. 2.Royal Observatory of BelgiumBrusselsBelgium

Personalised recommendations