Skip to main content
SpringerLink
Log in

Practical computation of normal forms of the Bogdanov–Takens bifurcation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we study the Bogdanov–Takens (double-zero) bifurcations for any autonomous ODEs system, and derive simple computational formulae for both critical normal forms and generic norm forms. These formulae involve only coefficients of the Taylor expansions of its right-hand sides at the equilibrium. They are equally suitable for both numerical and symbolic evaluations and conveniently allow us to classify codimension 2 Bogdanov–Takens bifurcations. Furthermore, we can use them to check whether there exist the original parameters such that a system presents the Bogdanov–Takens bifurcation, and compute the corresponding bifurcation curves with high precision. Two known models are used as test examples to demonstrate the advantages of this method.

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. Poincaré, H.: Thesis (1879). Also Oeuvres I, pp. 59–129. Gauthier Villars, Paris (1928)

  2. Lyapunov, A.: Probleme General de la Stabilite du Mouvement. Princeton University Press, Princeton (1947)

    MATH  Google Scholar 

  3. Birkhoff, G.D.: Dynamical Systems. Am. Math. Soc., New York (1927)

    MATH  Google Scholar 

  4. Siegel, C.L.: Über die Normalform Dnalytischer Differential gleichungen in der Nahe einer Gleichgewichtslösung. Math. Phys. Kl., IIa 5, 21–36 (1952)

    Google Scholar 

  5. Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space, I. Am. J. Math. 80, 623–631 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space, II. Am. J. Math. 81, 578–605 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arnold, V.I.: Ordinary Differential Equations, Dynamical Systems II. Springer, New York (1988)

    Google Scholar 

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

    Google Scholar 

  10. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, II. Springer, New York (1988)

    Google Scholar 

  11. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  12. Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Springer, New York (1973)

    Google Scholar 

  13. Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972)

    Article  Google Scholar 

  14. Bogdanov, R.I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Tr. Semin. Petrovsk. 2, 37–65 (1976) (in Russian)

    MathSciNet  Google Scholar 

  15. Bogdanov, R.I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Sel. Math. Sov. 1, 389–421 (1981) (in English)

    Google Scholar 

  16. Takens, F.: Singularities of vector fields. Publ. Math. IHÉS 43, 47–100 (1974)

    MathSciNet  Google Scholar 

  17. Kopell, N., Howard, L.N.: Bifurcations and trajectories connecting critical points. Adv. Math. 18, 306–358 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bogdanov, R.I.: Bifurcations of the limit cycle of a family of plane vector fields. Tr. Semin. Petrovsk. 2, 23–35 (1976). (in Russian)

    MathSciNet  Google Scholar 

  19. Bogdanov, R.I.: Bifurcations of the limit cycle of a family of plane vector fields Sel. Math. Sov. 1, 373–387 (1981) (in English)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Takens, F.: Forced oscillations and bifurcations. Commun. Math. Inst., Rijkuniversiteit Utrecht 3, 1–59 (1974)

    MathSciNet  Google Scholar 

  22. Chow, S.N., Li, C.Z., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Knobloch, E.: Normal forms for bifurcations at a double zero eigenvalue. Phys. Lett. A 115, 199–201 (1986)

    Article  MathSciNet  Google Scholar 

  25. Gamero, E., Freire, E., Ponce, E.: Normal forms for planar systems with nilpotent linear part. Int. Ser. Numer. Math. 97, 123–127 (1990)

    MathSciNet  Google Scholar 

  26. Kuznetsov, Y.A.: Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36, 1104–1124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Springer, New York (1973)

    MATH  Google Scholar 

  28. Kelley, A.: The stable, center stable, center, center unstable and unstable manifolds. J. Differ. Equ. 3, 546–570 (1981)

    Article  MathSciNet  Google Scholar 

  29. Carr, J.: Applications of Center Manifold Theory. Springer, New York (1981)

    Google Scholar 

  30. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 3–60 (1965)

    Google Scholar 

  31. Smith, J.M.: Models in Ecology. Cambridge University Press, Cambridge (1974)

    MATH  Google Scholar 

  32. Yodzis, P.: Predator–prey theory and management of multispecies fisheries. Ecol. Appl. 4, 51–58 (2004)

    Article  Google Scholar 

  33. Peng, G.J., Jiang, Y.L., Li, C.P.: Bifurcations of a Holling-type II predator–prey system with constant rate harvesting. Int. J. Bifurc. Chaos Appl. Sci. Eng. 19, 2499–2514 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)

    MATH  Google Scholar 

  35. Algaba, A., Freire, E., Gamero, E., Rodriguez-Luis, A.J.: On the Takens–Bogdanov bifurcation in the Chua’s equation. IEICE Trans. Fundam. 82, 1722–1728 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Guojun Peng & Yaolin Jiang

Authors
  1. Guojun Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaolin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guojun Peng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peng, G., Jiang, Y. Practical computation of normal forms of the Bogdanov–Takens bifurcation. Nonlinear Dyn 66, 99–132 (2011). https://doi.org/10.1007/s11071-010-9914-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9914-0

Keywords

Search

Navigation