Skip to main content
SpringerLink
Log in

Local Integrability for Some Degenerate Nilpotent Vector Fields

  • Chapter
  • First Online:
Nonlinear Systems, Vol. 1

Part of the book series: Understanding Complex Systems ((UCS))

  • 1206 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Algaba, A., Fuentes, N., García, C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real World Appl. 13, 419–431 (2012)

    Article  MathSciNet  Google Scholar 

  5. Algaba, A., Gamero, E., García, C.: The integrability problem for a class of planar systems. Nonlinearity 22, 395–420 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  6. Algaba, A., Gamero, E., García, C.: The reversibility problem for quasi-homogeneous dinamical systems. Discrete Contin. Dyn. Syst. 33, 3225–3236 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Algaba, A., García, C., Giné, J.: Analytic integrability for some degenerate planar systems. Commun. Pure Appl. Anal. 6, 2797–2809 (2013)

    Article  MathSciNet  Google Scholar 

  9. Algaba, A., García, C., Giné, J.: Analytic integrability for some degenerate planar vector fields. J. Differ. Equ. 257, 549–565 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  10. Algaba, A., García, C., Giné, J.: Nilpotent centers via inverse integrating factors. Euro. J. Appl. Math. 27, 781–795 (2016)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  13. Algaba, A., García, C., Reyes, M.: Integrability of two dimensional quasi-homogeneous polynomial differential systems. Rocky Mt. J. Math. 41, 1–22 (2011)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  15. Algaba, A., García, C., Reyes, M.: A note on analytic integrability of planar vector fields. Eur. J. Appl. Math. 23, 555–562 (2012)

    Article  MathSciNet  Google Scholar 

  16. Algaba, A., García, C., Teixeira, M.: Reversibility and quasi-homogeneous normal form of vector fields. Nonlinear Anal. 73, 510–525 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  18. Arnold, V.I.: Local normal forms of functions. Invent. math. 35, 87–109 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  19. Berthier, M., Moussu, R.: Reversibilit et classification des centres nilpotents. Ann. Inst. Fourier (Grenoble) 44, 465–494 (1994)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  21. Chavarriga, J., García, I., Giné, J.: Integrability of centers perturbed by quasi-homogeneous polynomials. J. Math. Anal. Appl. 210, 268–278 (1997)

    Article  MathSciNet  Google Scholar 

  22. Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: On the integrability of two-dimensional flows. J. Differ. Equ. 157, 163–182 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  23. Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Local analytic integrability for nilpotent centers. Ergod. Theory Dynam. Syst. 23, 417–428 (2003)

    Article  MathSciNet  Google Scholar 

  24. Dumortier, F.: Singularities of vector fields on the plane. J. Differ. Equ. 23, 53–106 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  28. Giné, J.: On the centers of planar analytic differential systems. Int. J. Bifur. Chaos Appl. Sci. Eng. 17, 3061–3070 (2007)

    Article  MathSciNet  Google Scholar 

  29. Giné, J.: On the degenerate center problem. Int. J. Bifur. Chaos Appl. Sci. Eng. 21, 1383–1392 (2011)

    Article  MathSciNet  Google Scholar 

  30. Giné, J., Llibre, J.: A method for characterizing nilpotent centers. J. Math. Anal. Appl. 413, 537–545 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  32. Mattei, J., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 4(13), 469–523 (1980)

    Article  Google Scholar 

  33. Moussu, R.: Symetrie et forme normale des centres et foyers degeneres. Ergod. Theory Dynam. Syst. 2, 241–251 (1982)

    Article  Google Scholar 

  34. Pearson, J., Lloyd, N., Christopher, C.: Algorithmic derivation of centre conditions. SIAM Rev. 38, 691–636 (1996)

    Article  MathSciNet  Google Scholar 

  35. Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. J. de Mathématiques 37, 375–442 (1881)

    MATH  Google Scholar 

  36. Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. J. de Mathématiques 8, 251–296 (1882)

    MATH  Google Scholar 

  37. Poincaré, H.: Ouvres de Henri Poincaré, vol. I. Gauthier-Villars, Paris (1951)

    Google Scholar 

  38. Romanovski, V.G., Shafer, D.: The center and cyclicity problems: a computational algebra approach. Birkhäuser Boston (2009)

    Google Scholar 

  39. Strózyna, E., Zoładek, H.: The analytic and normal form for the nilpotent singularity. J. Differ. Equ. 179, 479–537 (2002)

    Google Scholar 

  40. Teixeira, M.A., Yang, J.: The center-focus problem and reversibility. J. Differ. Equ. 174, 237–251 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  41. Villarini, M.: Algebraic criteria for the existence of analytic first integrals. Differ. Equ. Dynam. Syst. 5, 439–454 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, 21071, Huelva, Spain

    Antonio Algaba, Isabel Checa & Cristóbal García

Authors
  1. Antonio Algaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Isabel Checa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cristóbal García
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isabel Checa .

Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Algaba, A., Checa, I., García, C. (2018). Local Integrability for Some Degenerate Nilpotent Vector Fields. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation