Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor

  • Isaac A. GarcíaEmail author
  • Héctor Giacomini
  • Maite Grau


In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p 0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p 0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p 0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p 0 corresponds to a limit cycle γ 0. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ 0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ 0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p 0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.


Inverse integrating factor Generalized Hopf bifurcation Poincaré map Limit cycle Nilpotent focus 

Mathematics Subject Classification (2000)

37G15 37G10 34C07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Álvarez M.J., Gasull A.: Generating limit cycles from a nilpotent critical point via normal forms. J. Math. Anal. Appl. 318, 271–287 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andreev A.: Investigation on the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point. Transl. Am. Math. Soc. 8, 187–207 (1958)Google Scholar
  3. 3.
    Andreev A.F., Sadovskiĭ A.P., Tsikalyuk V.A.: The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part. Differ. Equ. 39, 155–164 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Andronov A.A. et al.: Theory of Bifurcations of Dynamic Systems on a Plane. Wiley, New York (1973)Google Scholar
  5. 5.
    Arnol’d, V.I.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Reprint of the 1980 edition. “Mir”, Moscow (1984)Google Scholar
  6. 6.
    Belitskiĭ G.: Smooth equivalence of germs of vector fields with one zero or a pair of purely imaginary eigenvalues. Funct. Anal. Appl. 20, 253–259 (1986)CrossRefGoogle Scholar
  7. 7.
    Berrone L.R., Giacomini H.: On the vanishing set of inverse integrating factors. Qual. Theory Dyn. Syst. 1, 211–230 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chavarriga J., Giacomini H., Giné J., Llibre J.: On the integrability of two-dimensional flows. J. Differ. Equ. 157, 163–182 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Chavarriga J., García I.A., Giné J.: On the integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity. Int. J. Bifurc. Chaos Appl. Sci. Eng. 11, 711–722 (2001)zbMATHCrossRefGoogle Scholar
  10. 10.
    Cima A., Gasull A., Mañosas F.: Cyclicity of a family of vector fields. J. Math. Anal. Appl. 196, 921–937 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dumortier, F., Roussarie, R.: Tracking limit cycles escaping from rescaling domains. In: Dynamical Systems and Related Topics (Nagoya, 1990). Advanced Series in Dynamical Systems, vol. 9, pp. 80–99. World Scientific Publishing, River Edge (1991)Google Scholar
  12. 12.
    Dumortier F., Llibre J., Artés J.C.: Qualitative Theory of Planar Differential Systems. Universitext, Springer, Berlin (2006)zbMATHGoogle Scholar
  13. 13.
    Enciso A., Peralta-Salas D.: Existence and vanishing set of inverse integrating factors for analytic vector fields. Bull. Lond. Math. Soc. 41, 1112–1124 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fulton W.: Algebraic Topology. A First Course. Graduate Texts in Mathematics, vol. 153. Springer, New York (1995)Google Scholar
  15. 15.
    García I.A., Grau M.: A survey on the inverse integrating factor. Qual. Theory Dyn. Syst. 9, 115–166 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    García I.A., Maza S.: A new approach to center conditions for simple analytic monodromic singularities. J. Differ. Equ. 248, 363–380 (2010)zbMATHCrossRefGoogle Scholar
  17. 17.
    García I.A., Shafer D.S.: Integral invariants and limit sets of planar vector fields. J. Differ. Equ. 217, 363–376 (2005)zbMATHCrossRefGoogle Scholar
  18. 18.
    García I.A., Giné J., Grau M.: A necessary condition in the monodromy problem for analytic differential equations on the plane. J. Symb. Comput. 41, 943–958 (2006)zbMATHCrossRefGoogle Scholar
  19. 19.
    García I.A., Giacomini H., Grau M.: The inverse integrating factor and the Poincaré map. Transl. Am. Math. Soc. 362, 3591–3612 (2010)zbMATHCrossRefGoogle Scholar
  20. 20.
    Giacomini H., Viano M.: Determination of limit cycles for two-dimensional dynamical systems. Phys. Rev. E 52, 222–228 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Giacomini H., Llibre J., Viano M.: On the nonexistence, existence, and uniqueness of limit cycles. Nonlinearity 9, 501–516 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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, 406–426 (2006)zbMATHCrossRefGoogle Scholar
  23. 23.
    Hale J.K., Koçak H.: Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3. Springer, New York (1991)Google Scholar
  24. 24.
    Liu Y.: Theory of center-focus for a class of higher-degree critical points and infinite points. Sci. China A 44, 365–377 (2001)zbMATHCrossRefGoogle Scholar
  25. 25.
    Lyapunov A.M.: Stability of Motion. Mathematics in Science and Engineering, vol. 30. Academic Press, New York (1966)Google Scholar
  26. 26.
    Mañosa V.: On the center problem for degenerate singular points of planar vector fields. Int. J. Bifurc. Chaos 12, 687–707 (2002)zbMATHCrossRefGoogle Scholar
  27. 27.
    Mattei J.F., Teixeira M.A.: Vector fields in the vicinity of a circle of critical points. Transl. Am. Math. Soc. 297, 369–381 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Moussu R.: Symétrie et forme normale des centres et foyers dégénérés. Ergod. Theory Dyn. Syst. 2, 241–251 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Narasimhan, R.: Analysis on Real and Complex Manifolds. Reprint of the 1973 edition. North-Holland Mathematical Library, vol. 35, Amsterdam (1985)Google Scholar
  30. 30.
    Strózyna E., Żoladek H.: The analytic and formal normal form for the nilpotent singularity. J. Differ. Equ. 179, 479–537 (2002)zbMATHCrossRefGoogle Scholar
  31. 31.
    Takens F.: Unfoldings of certain singularities of vector fields: generalized Hopf bifurcations. J. Differ. Equ. 14, 476–493 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Yakovenko S.Y.: Smooth normalization of a vector field near a semistable limit cycle. Ann. Inst. Fourier (Grenoble) 43, 893–903 (1993)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Żoladek H., Llibre J.: The Poincaré center problem. J. Dyn. Control Syst. 14, 505–535 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Isaac A. García
    • 1
    Email author
  • Héctor Giacomini
    • 2
  • Maite Grau
    • 1
  1. 1.Departament de MatemàticaUniversitat de LleidaLleidaSpain
  2. 2.Laboratoire de Mathématiques et Physique Théorique, C.N.R.S. UMR 6083, Faculté des Sciences et TechniquesUniversité de ToursToursFrance

Personalised recommendations