Skip to main content
SpringerLink
Log in

Smooth Normal Linearization of Vector Fields Near Lines of Singularities

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The paper deals with smooth three-dimensional vector fields exhibiting a line of singularities. For a large class of such vector fields we prove a theorem of smooth \({({\mathcal{C}}^\infty)}\) normal linearization along the line of singularities, near well chosen points. The vector fields can depend on a multiparameter and if they are subject to certain symmetries then the same symmetries can be imposed on the coordinate changes. The study is motivated by problems in geometric singular perturbation theory near planar turning points.

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. Bonckaert P.: Partially hyperbolic fixed points with constraints. Trans. Am. Math. Soc. 348, 997–1011 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. De Maesschalk P., Dumortier F.: Canard solutions at generic turning points. Trans. Am. Math. Soc. 358, 2291–2334 (2006)

    Article  Google Scholar 

  3. De Maesschalk P., Dumortier F.: Time analysis and entry–exit relation near planar turning point. J. Differ. Equ. 215, 225–267 (2005)

    Article  Google Scholar 

  4. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin (2006). ISBN: 3-540-32893-9

  5. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Memoirs Am. Math. Soc. 121(577), 1–100 (1996)

    Google Scholar 

  6. Dumortier F., Roussarie R.: Multiple Canard cycles in generalized Liénard equations. J. Differ. Equ. 174, 1–29 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dumortier F., Roussarie R.: Bifurcation of relaxation oscillations in dimension two. Discrete Contin. Dyn. Syst. 19, Ser. A 4, 631–674 (2007)

    MathSciNet  Google Scholar 

  8. Dumortier F., Roussarie R.: Canard cycles with two breaking parameters. Discrete Contin. Dyn. Syst. 17, Ser. A 4, 787–806 (2007)

    MathSciNet  Google Scholar 

  9. Dumortier F., Roussarie R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Il’yashenko, Y., Yakovenko, S.: Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Uspekhi Mat. Nauk., 46, 3–39 (1991). Translation in Russian Math. Surveys 46, n°1, 1–43 (1991)

  11. Malgrange, B.: Ideals of differentiable functions. Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, 106 pp. Oxford University Press, London (1967)

  12. Takens F.: Partially hyperbolic fixed points. Topology 10, 133–147 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universiteit Hasselt, Campus Diepenbeek, Agoralaan, Gebouw D, 3590, Diepenbeek, Belgium

    Freddy Dumortier

  2. Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S, Université de Bourgogne, B.P. 47 870, 21078, Dijon Cedex, France

    Robert Roussarie

Authors
  1. Freddy Dumortier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Roussarie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Freddy Dumortier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dumortier, F., Roussarie, R. Smooth Normal Linearization of Vector Fields Near Lines of Singularities. Qual. Theory Dyn. Syst. 9, 39–87 (2010). https://doi.org/10.1007/s12346-010-0020-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-010-0020-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation