Skip to main content
SpringerLink
Log in

Preservation of Takens–Bogdanov Bifurcations for Delay Differential Equations by Euler Discretization

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we study the discretization effects on the Takens–Bogdanov bifurcation of delay differential equations by forward Euler scheme. We show that the Takens–Bogdanov point is inherited by the discretization without any shift and turns into a 1:1 resonance point. The normal form on the center manifold near this singular point of the forward Euler method is calculated by applying a new technique, which is developed in this work for a general class of parameterized maps. The local dynamical behaviors are investigated in detail through this normal form. We show that the bifurcated Hopf point branch and the homoclinic branch of the numerical method are \(O(h)\) close to their continuous counterparts, where \(h\) is stepsize. A numerical experiment is presented to illustrate the theoretical results.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ashkenazi, M., Chow, S.N.: Normal forms near critical points for differential equations and maps. IEEE Trans. Circuits Syst. 35, 850–862 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Clarendon press, Oxford (2003)

    Book  MATH  Google Scholar 

  3. Beyn, W.J.: The effect of discretization on homoclinic orbits. In: Küpper, T. (ed.) Bifurcation, Analysis, Algorithms, Applications, pp. 1–8. Birkhäuser, Basel (1987)

    Chapter  Google Scholar 

  4. Broer, H., Roussarie, R., Simó, C.: Invariant circles in the Bogdanov–Takens bifurcation for diffeomorphisms. Ergod. Theory Dyn. Syst. 16, 1147–1172 (1996)

    Article  MATH  Google Scholar 

  5. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  MATH  Google Scholar 

  6. Chávez, J.P.: Discretizing bifurcation diagrams near codimension two singularities. Internat. J. Bifurc. Chaos Appl. Sci. Eng. 20, 1391–1403 (2010)

    Article  MATH  Google Scholar 

  7. Chávez, J.P.: Discretizing dynamical systems with generalized Hopf bifurcations. Numer. Math. 118, 229–246 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chávez, J.P.: Discretizing dynamical systems with Hopf-Hopf bifurcations. IMA J. Numer. Anal. 32, 185–201 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Farkas, G.: Unstable manifolds for RFDEs under discretization: the Euler method. Comput. Math. Appl. 42, 1069–1081 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Farkas, G.: A numerical \(C^1\)-shadowing result for retarded functional differential equations. J. Comput. Appl. Math. 145, 269–289 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995a)

    Article  MATH  Google Scholar 

  12. Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995b)

    Article  MATH  Google Scholar 

  13. Fiedler, B., Scheurle, J.: Discretization of homoclinic orbits, rapid forcing and “invisible” chaos. Mem. Amer. Math. Soc. 119, viii+79 (1996)

  14. Gelfreich, V., Naudot, V.: Width of the homoclinic zone in the parameter space for quadratic maps. Exp. Math. 18, 409–427 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hofbauer, J., Iooss, G.: A Hopf bifurcation theorem for difference equations approximating a differential equation. Monatsh. Math. 98, 99–113 (1984)

    Article  MathSciNet  Google Scholar 

  16. In’t Hout, K., Lubich, C.: Periodic orbits of delay differential equations under discretization. BIT 38, 71–91 (1998)

    MathSciNet  Google Scholar 

  17. Liu, M., Gao, J., Yang, Z.: Preservation of oscillations of the Runge–Kutta method for equation \(x^{\prime }(t)+ax(t)+a_1x([t-1])=0\). Comput. Math. Appl. 58, 1113–1125 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lóczi, L., Chávez, J.P.: Preservation of bifurcations under Runge–Kutta methods. Int. J. Qual. Theory Differ. Equ. Appl. 3, 81–98 (2009)

    MATH  Google Scholar 

  19. Wang, X., Blum, E., Li, Q.: Consistency of local dynamics and bifurcation of continuous-time dynamical systems and their numerical discretizations. J. Differ. Equ. Appl. 4, 29–57 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)

    Book  MATH  Google Scholar 

  21. Wulf, V., Ford, N.: Numerical Hopf bifurcation for a class of delay differential equations. J. Comput. Appl. Math. 115, 601–616 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Xu, Y., Huang, M.: Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity. J. Differ. Equ. 244, 582–598 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Xu, Y., Huang, M.: Preservation of Hopf bifurcation under the Euler discretization of delay differential systems. Dyn. Contin. Discret. Impuls. Syst. Ser. B 17, 347–355 (2010)

    MATH  MathSciNet  Google Scholar 

  24. Xu, Y., Zou, Y.: Preservation of homoclinic orbits under discretization of delay differential equations. Discret. Contin. Dyn. Syst. 31, 275–299 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zou, Y., Beyn, W.J.: On manifolds of connecting orbits in discretizations of dynamical systems. Nonlinear Anal 52, 1499–1529 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous referees for their valuable comments and suggestions which have greatly improved the presentation. Supported by NSFC Grants Nos. 11201061, 11271065, 11371171, 11271157, China Postdoctoral Science Foundation 2012M520657 and NSF of Jilin Province 20140520058JH.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun , 130024, China

    Yingxiang Xu

  2. School of Mathematics, Jilin University, Changchun , 130012, China

    Yongkui Zou

Authors
  1. Yingxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongkui Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongkui Zou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, Y., Zou, Y. Preservation of Takens–Bogdanov Bifurcations for Delay Differential Equations by Euler Discretization. J Dyn Diff Equat 26, 1029–1048 (2014). https://doi.org/10.1007/s10884-014-9354-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-014-9354-5

Keywords

Search

Navigation