Skip to main content
SpringerLink
Log in

Normal forms for singularities of vector fields

  • Published:
Japan Journal of Applied Mathematics Aims and scope Submit manuscript

Abstract

A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser: Asymptotic chaos. preprint.

  2. V. I. Arnold: On matrices depending on a parameter. Russian Math. Surveys,26 (1971), 29–43.

    Article  Google Scholar 

  3. V. I. Arnold: Lectures on bifurcation in versal families. Russian Math. Surveys,27 (1972), 54–123.

    Article  Google Scholar 

  4. V. I. Arnold: Loss of stability of self oscillations close to resonance and versal deformations of equivariant vector fields. Functional Anal. Appl.,11 (1977), 85–92.

    Article  Google Scholar 

  5. V. I. Arnold: Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordinaires. Editions Mir, Moscow, 1980.

  6. G. D. Birkhoff: Dynamical Systems. Amer. Math. Soc. Colloq. Publ., New York, 1927.

  7. R. I. Bogdanov: Versal deformations of a singular point of a vector field on a plane in the case of zero eigenvalues. Proceedings of the I.G. Petrovskii Seminar,2 (1976), 37–65.

    MathSciNet  Google Scholar 

  8. R. I. Bogdanov: Orbital equivalence of singular points of vector fields on the plane. Functional Anal. Appl.,10 (1976), 316–317.

    Article  Google Scholar 

  9. H. Broer: Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case. Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Math. 898. Springer, Berlin-Heidelberg-New York, 1981, 54–74.

    Google Scholar 

  10. F. Dumortier: Singularities of vector fields in the plane. J. Differential Equations,23 (1977), 53–106.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Guckenheimer: On a codimension two bifurcation. Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Math. 898, Springer, Berlin-Heidelberg-New York, 1981, 99–142.

    Book  Google Scholar 

  12. P. Hartman: A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc.,11 (1960), 610–620.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Hartman: Ordinary Differential Equations. Wiley & Sons, New York, 1964.

    MATH  Google Scholar 

  14. P. J. Holmes: Center manifolds, normal forms and bifurcation of vector fields. Physica,2D (1981), 449–481.

    Google Scholar 

  15. P. J. Holmes: A strange family of three-dimensional vector fields near a degenerate singularity. J. Differential Equations,37 (1980), 382–403.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Ichikawa: Finitely determined singularities of formal vector fields. Invent. Math.,66 (1982), 199–214.

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Ichikawa: On finite determinacy of formal vector fields. preprint.

  18. F. Ichikawa: Classification of finitely determined singularities of formal vector fields on a plane. preprint.

  19. G. Iooss and D. D. Joseph: Elementary stability and bifurcation theory. Springer, New York, 1980.

    MATH  Google Scholar 

  20. W. Langford: Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math.,37 (1979), 22–48.

    Article  MATH  MathSciNet  Google Scholar 

  21. W. Langford and G. Iooss: Interactions of Hopf and pictchfork bifurcations. Workshop on Bifurcation Problems, Intern. Ser. Numer. Math. 54, Birkhäuser, Basel, 1980.

  22. H. Poincaré: Thèse (1879). Oeuvre I, Gauthier-Villars, Paris, 1928, 69–129.

    Google Scholar 

  23. H. Poincaré: Mémoire sur les courbes définies par une équation différentielle, I, II, III, and IV. J. Math. Pures Appl., (3)7 (1881), 375–422, (3)8 (1882), 251–286, (4)1 (1885), 167–244, (4)2 (1886), 151–217.

    Google Scholar 

  24. H. Poincaré: Les Méthodes Nouvelles de la Mécanique Céleste, I, III, Gauthier-Villars. Paris, 1892, 1899.

    Google Scholar 

  25. C. L. Siegel: Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtlösung. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl., 1952, 21–30.

  26. S. Sternberg: Local contractions and a theorem of Poincaré. Amer. J. Math.,79 (1957), 578–604.

    Google Scholar 

  27. S. Sternberg: On the structure of local homeomorphisms of Euclideann-space. Amer. J. Math.,80 (1958), 623–631.

    Article  MATH  MathSciNet  Google Scholar 

  28. F. Takens: Singularities of vector fields. Publ. Math. I.H.E.S.,43 (1973), 47–100.

    MATH  Google Scholar 

  29. F. Takens: Normal forms for certain singularities of vector fields. Ann. Inst. Fourier,23 (1973), 163–195.

    MATH  MathSciNet  Google Scholar 

  30. F. Takens: Forced oscillations and bifurcations: Applications of global analysis. Comm. Math. Inst. Rijksuniv, Utrecht,3 (1974), 1–59.

    MathSciNet  Google Scholar 

  31. S. Ushiki: Normal forms for degenerate singularities of ordinary differential equations, (Japanese). Technical Research Report of Association of Electronic Communication, Vol. 82, No. 207, CAS82-115, 1982.

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Yoshida College, Kyoto University, 606, Kyoto, Japan

    Shigehiro Ushiki

Authors
  1. Shigehiro Ushiki
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Ushiki, S. Normal forms for singularities of vector fields. Japan J. Appl. Math. 1, 1–37 (1984). https://doi.org/10.1007/BF03167860

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167860

Key words

Search

Navigation