Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Systems, Vol. 1 pp 37–65Cite as

Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part

  • Chapter
  • First Online:

  • 1205 Accesses

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

Abstract

We present the basic ideas of the Normal Form Theory by using quasi-homogeneous expansions of the vector field, where the structure of the normal form is determined by the principal part of the vector field. We focus on a class of tridimensional systems whose principal part is the coupling of a Hamiltonian planar system and an unidimensional system, in such a way that the quoted principal part does not depend on the last variable and has free divergence. Our study is based on several decompositions of quasi-homogeneous vector fields. An application, corresponding to the coupling of a Takens-Bogdanov and a saddle-node singularities, (in fact, it is a triple-zero singularity with geometric multiplicity two), that falls into the class considered, is analyzed.

This is a preview of subscription content, 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

Learn about institutional subscriptions

References

  1. Algaba, A., Freire, E., Gamero, E.: Hypernormal form for the Hopf-zero bifurcation. Int. J. Bifurc. Chaos 8, 1855–1887 (1998)

    Google Scholar 

  2. Algaba, A., Freire, E., Gamero, E.: Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies. Rocky Mt. J. Math. 29, 13–45 (1999)

    Article  MathSciNet  Google Scholar 

  3. Algaba, A., Freire, E., Gamero, E.: Characterizing and computing normal forms using lie transforms: a survey. Computation of normal forms and applications. Dyn. Contin. Discret. Impulsive Syst. 8(4), 449–475 (2001)

    Google Scholar 

  4. Algaba, A., Freire, E., Gamero, E.: Computing simplest normal forms for the Takens-Bogdanov singularity. Qual. Theory Dyn. Syst. 3(2), 377–435 (2002)

    Article  MathSciNet  Google Scholar 

  5. Algaba, A., Freire, E., Gamero, E., García, C.: Quasi-homogeneous normal forms. J. Comput. Appl. Math. 150, 193–216 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  6. Algaba, A., Freire, E., Gamero, E., García, C.: Quasi-homogeneous normal forms for null linear part. Dyn. Contin. Discret. Impulsive Syst. 10, 247–261 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Algaba, A., Freire, E., Gamero, E., García, C.: An algorithm for computing quasi-homogeneous formal normal forms under equivalence. Acta Apl. Math. 80, 335–359 (2004)

    Article  MathSciNet  Google Scholar 

  8. Algaba, A., Fuentes, N., Gamero, E., García, C.: Normal forms for a class of three-dimensional suspended hamiltonian planar systems. Preprint. (2016)

    Google Scholar 

  9. 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 

  10. Algaba, A., García, C., Giné, J.: Analytic integrability for some degenerate planar systems. Commun. Pure Appl. Anal. 12, 2797–2809 (2013)

    Article  MathSciNet  Google Scholar 

  11. 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 

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

    Article  MathSciNet  Google Scholar 

  13. Baider, A.: Unique normal forms for vector fields and Hamiltonians. J. Differ. Equ. 78(1), 33–52 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  14. Baider, A., Sanders, J.A.: Further reduction of the Takens-Bogdanov normal form. J. Differ. Equ. 99(2), 205–244 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  15. Basov, V., Slutskaya, V.: Generalized normal forms of two-dimensional real systems of ordinary differential equations with a quasi-homogeneous polynomial in the unperturbatid part. Differ. Uravn Protsessy Upr. 4, 108–133 (2010)

    Google Scholar 

  16. Chen, G., Wang, D., Yang, J.: Unique normal forms for Hopf-zero vector fields. C. R. Acad. Sci. Paris (Ser. I), 345–348 (2003)

    Article  MathSciNet  Google Scholar 

  17. Chen, G., Wang, D., Yang, J.: Unique orbital normal forms for vector fields of Hopf-Zero singularity. J. Dyn. Diff. Equ. 17, 3–20 (2005)

    Article  MathSciNet  Google Scholar 

  18. Chow, S., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    Book  Google Scholar 

  19. Chow, S., Li, C., Wang, D.: Normal Forms and Bifurcations of Planar Vector Fields. Cambridge University Press (1994)

    Google Scholar 

  20. Chua, L.O., Kokubu, H.: Normal forms of nonlinear vector fields—part I: theory and algorithm. IEEE Trans. Circuits Syst. CAS 35, 863–880 (1988)

    Article  Google Scholar 

  21. Elphick, C., Tirapegui, E., Brachet, M.E., Coullet, P., Iooss, G.: A simple global characterization for normal forms of singular vector fields. Phys. D 29, 95–127 (1987)

    Article  MathSciNet  Google Scholar 

  22. Gazor, M., Mokhtari, F.: Volume-preserving normal forms for Hopf-zero singularity. Nonlinearity 26, 2809–2832 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  23. Golubitsky, M., Shaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, New York (1985)

    Book  Google Scholar 

  24. Guckenheimer, J., Holmes, P.: Nonlinear Oscilations, Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)

    Google Scholar 

  25. Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications. World Scientific, Singapore (1992)

    Book  Google Scholar 

  26. Kokubu, H., Oka, H., Wang, D.: Linear grading function and further reduction of normal forms. J. Differ. Equ. 132, 293–318 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  27. Lombardi, E., Stolovich, L.: Normal forms of analytic perturbations of quasi-homogeneous vector fields: rigidity, invariant analytic set and exponentially small approximation. Ann. Sci. Ec. Norm. Sup. pp. 659–718 (2010)

    Article  MathSciNet  Google Scholar 

  28. Strozyna, E.: Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case. J. Differ. Equ. 259, 6718–6748 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  29. Strozyna, E., Zoladek, H.: The complete formal normal form for the Bogdanov-takens singularity. Mosc. Math. J. 15, 141–178 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Takens, F.: Singularities of vector fields. Publ. Math. Inst. Hautes Éstudes Sci 43, 47–100 (1974)

    Article  MathSciNet  Google Scholar 

  31. Ushiki, S.: Normal forms for singularities of vector fields. Jpn. J. Appl. Math. 1, 1–37 (1984)

    Article  MathSciNet  Google Scholar 

  32. Vanderbauwhede, A.: Centre manifolds, normal forms and elementay bifurcations. Dyn. Rep. 2, 89–169 (1989)

    Google Scholar 

  33. Wang, D., Li, J., Huang, M., Jiang, Y.: Unique normal form of Bogdanov-Takens singularities. J. Differ. Equ. 163, 223–238 (2000)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work has been partially supported by Ministerio de Ciencia y Tecnología, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the project MTM2014-56272-C2-02, and by Consejería de Educación y Ciencia de la Junta de Andalucía (FQM-276 and P12-FQM-1658).

Author information

Authors and Affiliations

  1. Department of Integrated Sciences, Investigation Center of Theoretical Physics and Mathematic FIMAT, Huelva University, Av. Fuerzas Armadas s/n, 21071, Huelva, Spain

    Antonio Algaba, Natalia Fuentes & Cristóbal García

  2. Department of Applied Mathematic II, E.T.S.I. Sevilla University, Avd. Camino de los Descubridores s/n, Sevilla, Spain

    Estanislao Gamero

Authors
  1. Antonio Algaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natalia Fuentes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Estanislao Gamero
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cristóbal García
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Estanislao Gamero .

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., Fuentes, N., Gamero, E., García, C. (2018). Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part. 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_2

Download citation

Publish with us

Policies and ethics

Search

Navigation