Advertisement

Local Integrability for Some Degenerate Nilpotent Vector Fields

  • Antonio Algaba
  • Isabel Checa
  • Cristóbal García
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

This work is about the analytic integrability problem around the origin in a family of degenerate nilpotent vector fields. The integrability problem for planar vector fields with first Hamiltonian component having simple factors in its factorization on \(\mathbb {C}[x, y]\) is solved in Algaba et al. (Nonlinearity 22:395–420, 2009) [5]. Nevertheless, when the Hamiltonian function has multiple factors on \(\mathbb {C}[x, y]\) is an open problem. In this second case our problem is framed. More concretely, we study the following degenerate systems:
$$\begin{aligned} \dot{x} = - y (x^{2n}+ny^2)+ \cdots , \quad \dot{y}=x^{2n-1} (x^{2n}+ny^2)+\cdots , \end{aligned}$$
with \(n \in \mathbb {N}\), where its first quasi-homogeneous component has Hamiltonian function given by \((x^{2n}+ny^2)^2/(2n)\). The analytic integrability of the above system is not completely solved and only partial results are obtained. The results are applied to some particular families of degenerate vector fields for which the integrability problem is completely solved.

Keywords

Integrability problem Degenerate center problem First integral Orbital normal form Blow-up Conservative-dissipative splitting Nilpotent systems 

Notes

Acknowledgements

The authors are supported by a MINECO/FEDER grant number MTM2014-56272-C2-02 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (projects P12-FQM-1658, FQM-276).

References

  1. 1.
    Algaba, A., Checa, I., García, C., Gamero, E.: On orbital-reversibility for a class of planar dynamical systems. Commun. Nonlinear Sci. Numer. Simulat. 20, 229–239 (2015)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Algaba, A., Checa, I., García, C., Giné, J.: Analytic integrability inside a family of degenerate centers. Nonlinear Anal. Real World Appl. 31, 288–307 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Algaba, A., Freire, E., Gamero, E., García, C.: Monodromy, center-focus and integrability problems for quasi-homogeneous polynomials systems. Nonlinear Anal. 72, 1726–1736 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Algaba, A., Fuentes, N., García, C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real World Appl. 13, 419–431 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Algaba, A., Gamero, E., García, C.: The integrability problem for a class of planar systems. Nonlinearity 22, 395–420 (2009)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Algaba, A., Gamero, E., García, C.: The reversibility problem for quasi-homogeneous dinamical systems. Discrete Contin. Dyn. Syst. 33, 3225–3236 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Algaba, A., Gamero, E., García, C.: The center problem. A view from the normal form theory. J. Math. Anal. Appl. 434(1), 680–697 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Algaba, A., García, C., Giné, J.: Analytic integrability for some degenerate planar systems. Commun. Pure Appl. Anal. 6, 2797–2809 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Algaba, A., García, C., Giné, J.: Analytic integrability for some degenerate planar vector fields. J. Differ. Equ. 257, 549–565 (2014)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Algaba, A., García, C., Giné, J.: Nilpotent centers via inverse integrating factors. Euro. J. Appl. Math. 27, 781–795 (2016)CrossRefGoogle Scholar
  11. 11.
    Algaba, A., García, C., Giné, J.: The analytic integrability problem for perturbation of non-hamiltonian quasi-homogeneous nilpotent systems. Submited (2017)Google Scholar
  12. 12.
    Algaba, A., García, C., Reyes, M.: The center problem for a family of systems of differential equations having a nilpotent singular point. J. Math. Anal. Appl. 340, 32–43 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Algaba, A., García, C., Reyes, M.: Integrability of two dimensional quasi-homogeneous polynomial differential systems. Rocky Mt. J. Math. 41, 1–22 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Algaba, A., García, C., Reyes, M.: Existence of an inverse integrating factor, center problem and integability of a class of nilpotent systems. Chaos Solitons Fractals 45, 869–878 (2012)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Algaba, A., García, C., Reyes, M.: A note on analytic integrability of planar vector fields. Eur. J. Appl. Math. 23, 555–562 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Algaba, A., García, C., Teixeira, M.: Reversibility and quasi-homogeneous normal form of vector fields. Nonlinear Anal. 73, 510–525 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Andreev, A., Sadovskii, A.P., Tskialyuk, V.A.: The center-focus problem for a system with homogeneous nonlinearity in the case zero eigenvalues of linear part. J. Differ. Equ. 39(2), 155–164 (2003)CrossRefGoogle Scholar
  18. 18.
    Arnold, V.I.: Local normal forms of functions. Invent. math. 35, 87–109 (1976)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Berthier, M., Moussu, R.: Reversibilit et classification des centres nilpotents. Ann. Inst. Fourier (Grenoble) 44, 465–494 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Brunella, M., Miari, M.: Topological equivalence of a plane vector field with its principal part defined through newton polyhedra. J. Differ. Equ. 85, 338–366 (1990)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Chavarriga, J., García, I., Giné, J.: Integrability of centers perturbed by quasi-homogeneous polynomials. J. Math. Anal. Appl. 210, 268–278 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: On the integrability of two-dimensional flows. J. Differ. Equ. 157, 163–182 (1999)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Local analytic integrability for nilpotent centers. Ergod. Theory Dynam. Syst. 23, 417–428 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Dumortier, F.: Singularities of vector fields on the plane. J. Differ. Equ. 23, 53–106 (1977)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Giacomini, H., Giné, J., Llibre, J.: The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems. J. Differ. Equ. 227(2), 406–426 (2006)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Giacomini, H., Giné, J., Llibre, J.: The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems (corrigendum). J. Differ. Equ. 232, 702 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    Giné, J.: Sufficient conditions for a center at a completely degenerate critical point. Int. J. Bifur. Chaos Appl. Sci. Eng. 12, 1659–1666 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Giné, J.: On the centers of planar analytic differential systems. Int. J. Bifur. Chaos Appl. Sci. Eng. 17, 3061–3070 (2007)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Giné, J.: On the degenerate center problem. Int. J. Bifur. Chaos Appl. Sci. Eng. 21, 1383–1392 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Giné, J., Llibre, J.: A method for characterizing nilpotent centers. J. Math. Anal. Appl. 413, 537–545 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifur. Chaos Appl. Sci. Eng. 13, 47–106 (2003)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mattei, J., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 4(13), 469–523 (1980)CrossRefGoogle Scholar
  33. 33.
    Moussu, R.: Symetrie et forme normale des centres et foyers degeneres. Ergod. Theory Dynam. Syst. 2, 241–251 (1982)CrossRefGoogle Scholar
  34. 34.
    Pearson, J., Lloyd, N., Christopher, C.: Algorithmic derivation of centre conditions. SIAM Rev. 38, 691–636 (1996)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. J. de Mathématiques 37, 375–442 (1881)zbMATHGoogle Scholar
  36. 36.
    Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. J. de Mathématiques 8, 251–296 (1882)zbMATHGoogle Scholar
  37. 37.
    Poincaré, H.: Ouvres de Henri Poincaré, vol. I. Gauthier-Villars, Paris (1951)Google Scholar
  38. 38.
    Romanovski, V.G., Shafer, D.: The center and cyclicity problems: a computational algebra approach. Birkhäuser Boston (2009)Google Scholar
  39. 39.
    Strózyna, E., Zoładek, H.: The analytic and normal form for the nilpotent singularity. J. Differ. Equ. 179, 479–537 (2002)Google Scholar
  40. 40.
    Teixeira, M.A., Yang, J.: The center-focus problem and reversibility. J. Differ. Equ. 174, 237–251 (2001)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Villarini, M.: Algebraic criteria for the existence of analytic first integrals. Differ. Equ. Dynam. Syst. 5, 439–454 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Antonio Algaba
    • 1
  • Isabel Checa
    • 1
  • Cristóbal García
    • 1
  1. 1.Departamento de MatemáticasCentro de Investigación de Física Teórica y Matemática FIMAT, Universidad de HuelvaHuelvaSpain

Personalised recommendations