Regular and Chaotic Dynamics

, Volume 23, Issue 7–8, pp 933–947 | Cite as

Local Integrability of Poincaré–Dulac Normal Forms

  • Shogo YamanakaEmail author


We consider dynamical systems in Poincaré–Dulac normal form having an equilibrium at the origin, and give a sufficient condition for them to be integrable, and prove that it is necessary for their special integrability under some condition. Moreover, we show that they are integrable if their resonance degrees are 0 or 1 and that they may be nonintegrable if their resonance degrees are greater than 1, as in Birkhoff normal forms for Hamiltonian systems. We demonstrate the theoretical results for a normal form appearing in the codimension-two fold-Hopf bifurcation.


Poincaré–Dulac normal form integrability dynamical system 

MSC2010 numbers

34M35 37J30 


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  1. 1.
    Arnol’d, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Grundlehren Math. Wiss., vol. 250, New York: Springer, 1988.Google Scholar
  2. 2.
    Ayoul, M. and Zung, N.T., Galoisian Obstructions to Non-Hamiltonian Integrability, C. R. Math. Acad. Sci. Paris, 2010, vol. 348, nos. 23–24, pp. 1323–1326.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogoyavlenski, O. I., Extended Integrability and Bi-Hamiltonian Systems, Comm. Math. Phys., 1998, vol. 196, no. 1, pp. 19–51.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brjuno, A.D., Analytic Form of Differential Equations: 1, Trans. Moscow Math. Soc., 1971, vol. 25, pp. 131–288; see also: Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119–262.MathSciNetGoogle Scholar
  5. 5.
    Brjuno, A.D., Analytic Form of Differential Equations: 2, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 199–239; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 199–239.Google Scholar
  6. 6.
    Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989.CrossRefGoogle Scholar
  7. 7.
    Christov, O., Non-Integrability of First Order Resonances in Hamiltonian Systems in Three Degrees of Freedom, Celestial Mech. Dynam. Astronom., 2012, vol. 112, no. 2, pp. 149–167.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chow, Sh.-N., Li, C. Z., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge: Cambridge Univ. Press, 1994.CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, J., Yi, Y., and Zhang, X., First Integrals and Normal Forms for Germs of Analytic Vector Fields, J. Differential Equations, 2008, vol. 245, no. 5, pp. 1167–1184.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Du, Z., Romanovski, V.G., and Zhang, X., Varieties and Analytic Normalizations of Partially Integrable Systems, J. Differential Equations, 2016, vol. 260, no. 9, pp. 6855–6871.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duistermaat, J. J., Nonintegrability of the 1: 1: 2-Resonance, Ergodic Theory Dynam. Systems, 1984, vol. 4, no. 4, pp. 553–568.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ito, H., Convergence of Birkhoff Normal Forms for Integrable Systems, Comment. Math. Helv., 1989, vol. 64, no. 3, pp. 412–461.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ito, H., Integrability of Hamiltonian Systems and Birkhoff Normal Forms in the Simple Resonance Case, Math. Ann., 1992, vol. 292, no. 3, pp. 411–444.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ito, H., Birkhoff Normalization and Superintegrability of Hamiltonian Systems, Ergodic Theory Dynam. Systems, 2009, vol. 29, no. 6, pp. 1853–1880.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kappeler, T., Kodama, Y., and Nemethi, A., On the Birkhoff Normal Form of a Completely Integrable Hamiltonian System near a Fixed Point with Resonance, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1998, vol. 26, no. 4, pp. 623–661.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kuznetsov, Yu.A., Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci., vol. 112, New York: Springer, 2004.Google Scholar
  17. 17.
    Llibre, J., Pantazi, Ch., and Walcher, S., First Integrals of Local Analytic Differential Systems, Bull. Sci. Math., 2012, vol. 136, no. 3, pp. 342–359.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Morales-Ruiz, J. J., Ramis, J.-P., and Simó, C., Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations, Ann. Sci. École Norm. Sup. (4), 2007, vol. 40, no. 6, pp. 845–884.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Siegel, C. L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtsl ösung, Nachr. Akad. Wiss. Göttingen, math.-phys. Kl., 1952, vol. 1952, pp. 21–30.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Stolovitch, L., Singular Complete Integrability, Inst. Hautes Études Sci. Publ. Math., 2000, no. 91, pp. 133–210.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stolovitch, L., Normalisation holomorphe d’algebres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. of Math. (2), 2005, vol. 161, no. 2, pp. 589–612.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vey, J., Sur certains systèmes dynamiques séparables, Amer. J. Math., 1978, vol. 100, no. 3, pp. 591–614.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vey, J., Algèbres commutatives de champs de vecteurs isochores, Bull. Soc. Math. France, 1979, vol. 107, no. 4, pp. 423–432.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Walcher, S., On Differential Equations in Normal Form, Math. Ann., 1991, vol. 291, no. 2, pp. 293–314.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Walcher, S., Symmetries and Convergence of Normal Form Transformations, in Proc. of the 6th Conf. on Celestial Mechanics, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, vol. 25, Zaragoza: Real Acad. Ci. Exact., Fís. Quím. Nat. Zar., 2004, pp. 251–268.MathSciNetGoogle Scholar
  26. 26.
    Yagasaki, K., Nonintegrability of the Unfolding of the Fold-Hopf Bifurcation, Nonlinearity, 2018, vol. 31, no. 2, pp. 341–350.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zung, Nguyen Tien, Convergence versus Integrability in Poincaré–Dulac Normal Form, Math. Res. Lett., 2002, vol. 9, nos. 2–3, pp. 217–228.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zung, Nguyen Tien, Convergence versus Integrability in Birkhoff Normal Form, Ann. of Math. (2), 2005, vol. 161, no. 1, pp. 141–156.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zung, Nguyen Tien, Non-Degenerate Singularities of Integrable Dynamical Systems, Ergodic Theory Dynam. Systems, 2015, vol. 35, no. 3, pp. 994–1008.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto University, Yoshida-Honmachi, Sakyo-kuKyotoJapan

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