Skip to main content

Conclusions and discussion of remaining problems

  • Part II: Rescalings And Analytic Treatment
  • Chapter
  • First Online:
Bifurcations of Planar Vector Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1480))

  • 582 Accesses

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. Arnol'd Chapitres supplémentaires de la théorie des equations différentielles ordinaires, Ed. Mir, Moscow, 1980

    Google Scholar 

  2. V. Arnol'd Lectures on bifurcations in versal families, Russian Math. Surveys V, 26, 1971

    Google Scholar 

  3. A. Andronov, E. Leontovich, et al. Theory of bifurcations of Dynamical Systems on a Plane, I.P.S.T., Jerusalem, 1971

    Google Scholar 

  4. A.D. Basikin, Yu. Kuznietzov, A.I. Khibnik Bifurcational diagrams of dynamical systems on the plane, Computer Center Acad. Sciences URSS, Puschino, 1985

    Google Scholar 

  5. R. Bogdanov-Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. (R) Seminar Petrovski, 1976, (E) Selecta Mathematica Sovietica, vol. 1, 4, 1981, 389–421.

    MATH  Google Scholar 

  6. -Bifurcation of a limit cycle for a family of vector fields on the plane, (R) Seminar Petrovski, 1976, (E) Selecta Math. Sov., vol. 1, 4, 1981, 373–388.

    MATH  Google Scholar 

  7. F. Dumortier-Singularities of vector fields on the plane, J. Diff. Equat., vol 23, 1 (1977), 53–106

    Article  MathSciNet  MATH  Google Scholar 

  8. -Singularities of vector fields. Monografias de Matemática 32 IMPA, Rio de Janeiro, 1978

    MATH  Google Scholar 

  9. G. Dangelmayer, J. Guckenheimer On a four parameter family of planar vector fields, Arch. Rat. Mech. Anal., 97, 1987, 321–352.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Dumortier, C. Rousseau Cubic Liénard equations with linear damping, Nonlinearity, to appear.

    Google Scholar 

  11. F. Dumortier, R. Roussarie, J. Sotomayor Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case. Ergodic theory and dynam. systems, 7, 1987, 375–413

    MathSciNet  MATH  Google Scholar 

  12. J. Guckenheimer, P. Holmes Non-linear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sc. 42, Springer-Verlag, 1983

    Google Scholar 

  13. J.K. Hale, S.-N. Chow Methods of bifurcation theory, Springer-Verlag, Berlin, 1982

    MATH  Google Scholar 

  14. R. Roussarie On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Mat., Vol. 17, 2, 1986, 67–101

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Schecter The Saddle-node separatrix-loop bifurcation, SIAM Journ. Math. Anal., Vol. 18, 4, 1987, 1142–56.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Seidenberg A New decision method for elementary algebra, Ann. of Math. 60, 1954, 365–374

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Sotomayor Generic one-parameter families of vector fields on two-dimensional manifolds, Publ. Math. I.H.E.S., Vol. 43, 1974

    Google Scholar 

  18. J. Sotomayor Curvas definidas por equações diferenciais no plano, IMPA, Rio de Janeiro, 1981

    Google Scholar 

  19. D. Stowe Linearization in two dimensions, Journ. of Diff. Equat. 63, 1986, 183–226

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Takens Unfoldings of certain singularities of vector fields. Generalized Hopf bifurcations. Journ. of Diff. Equat. 14, 1973, 476–493

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Takens Forced oscillations and bifurcations. In: Applications of Global Analysis I, Communications of Math. Inst. Rijksuniv. Utrecht, 3, 1974

    Google Scholar 

  22. M.A. Teixeira Generic bifurcation in manifolds with boundary, J. Diff. Equat., vol. 25, 1, 65–89, 1977

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Zoladek Abelian integrals in unfoldings of cod. 3 singular planar vector fields, Part II. The saddle and elliptic case. This volume

    Google Scholar 

  24. H. Zoladek Abelian integrals in unfoldings of cod. 3 singular planar vector fields, Part III. The focus case. This volume

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this chapter

Cite this chapter

Dumortier, F., Roussarie, R., Sotomayor, J., Żaładek, H. (1991). Conclusions and discussion of remaining problems. In: Bifurcations of Planar Vector Fields. Lecture Notes in Mathematics, vol 1480. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098360

Download citation

  • DOI: https://doi.org/10.1007/BFb0098360

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54521-7

  • Online ISBN: 978-3-540-38433-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics