[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Error estimates for approximating non-hyperbolic heteroclinic orbits of maps

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

In this paper we consider heteroclinic orbits in discrete time dynamical systems that connect a hyperbolic fixed point to a non-hyperbolic fixed point with a one-dimensional center direction. A numerical method for approximating the heteroclinic orbit by a finite orbit sequence is introduced and a detailed error analysis is presented. The loss of hyperbolicity requires special tools for proving the error estimate – the polynomial dichotomy of linear difference equations and a (partial) normal form transformation near the non-hyperbolic fixed point. This situation appears, for example, when one fixed point undergoes a flip bifurcation. For this case, the approximation method and the validity of the error estimate is illustrated by an example.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I., Afraimovich, V., Ilyashenko, Yu., Shilnikov, L.P.: Dynamical systems. V, volume 5 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 1994

  2. Beyn, W.-J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10, 379–405 (1990)

    MATH  Google Scholar 

  3. Beyn, W.-J., Hüls, Th., Kleinkauf, J.-M., Zou, Y.: Numerical analysis of degenerate connecting orbits for maps. Technical report, DFG-Research Group ‘Spectral Analysis Asymptotic Distribution and Stochastic Dynamics’, University of Bielefeld 2003. To appear in Internat. J. Bifur. Chaos Appl. Sci. Engrg.

  4. Beyn, W.-J., Kleinkauf, J.-M.: Numerical approximation of homoclinic chaos. Numer. Algorithms 14(1–3), 25–53 (1997); Dynamical numerical analysis (Atlanta GA 1995)

    Google Scholar 

  5. Beyn, W.-J., Kleinkauf, J.-M.: The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal. 34(3), 1207–1236 (1997)

    Article  MATH  Google Scholar 

  6. Henry, D.: Geometric theory of semilinear parabolic equations. Springer-Verlag, Berlin, 1981

  7. Hüls, T.: Heterokline Orbits in diskreten dynamischen Systemen. Masters’s thesis, Universität Bielefeld, 1998

  8. Hüls, T.: Numerische Approximation nicht-hyperbolischer heterokliner Orbits. PhD thesis, Universität Bielefeld, 2003 Shaker Verlag, Aachen

  9. Hüls, T., Zou, Y.: Polynomial estimates and discrete saddle-node homoclinic orbits. J. Math. Anal. Appl. 256(1), 115–126 (2001)

    Article  Google Scholar 

  10. Ilyashenko, Y., Li, W.: Nonlocal bifurcations. American Mathematical Society, Providence RI, 1999

  11. Kleinkauf, J.-M.: Numerische Berechnung diskreter homokliner Orbits. Masters’s thesis, Universität Bielefeld, 1994

  12. Kleinkauf, J.-M.: Numerische Analyse tangentialer homokliner Orbits. PhD thesis, Universität Bielefeld, 1998, Shaker Verlag, Aachen

  13. Kuznetsov, Y.A.: Elements of applied bifurcation theory. Springer-Verlag, New York, second edition, 1998

  14. Kuznetsov, Y.A., Levitin, V.V.: Content – integrated environment for analysis of dynamical systems. 1998, http://www.cwi.nl/ftp/CONTENT

  15. Lani-Wayda, B.: Hyperbolic sets shadowing and persistence for noninvertible mappings in Banach spaces. Volume 334 of Pitman Research Notes in Mathematics Series. Longman, Harlow, 1995

  16. López-Fenner, J., Pinto, M.: (h,k)-trichotomies and asymptotics of nonautonomous difference systems. Comput. Math. Appl. 33(10), 105–124 (1997)

    Google Scholar 

  17. Palis, J., Takens, F.: Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Volume 35 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993, Fractal dimensions and infinitely many attractors

  18. Palmer, K.J.: Exponential dichotomies the shadowing lemma and transversal homoclinic points. In: Dynamics reported Vol. 1, Teubner, Stuttgart, 1988, pp. 265–306

  19. Pinto, M.: Discrete dichotomies. Comput. Math. Appl. 28(1–3), 259–270 (1994); Advances in difference equations

  20. Sandstede, B.: Verzweigungstheorie homokliner Verdopplungen. PhD thesis, Universität Stuttgart, 1993

  21. Sandstede, B.: Convergence estimates for the numerical approximation of homoclinic solutions. IMA J. Numer. Anal. 17(3), 437–462 (1997)

    MATH  Google Scholar 

  22. Schecter, S.: Numerical computation of saddle-node homoclinic bifurcation points. SIAM J. Numer. Anal. 30(4), 1155–1178 (1993)

    MATH  Google Scholar 

  23. Shub, M., Fathi, A., Langevin, R.: Global stability of dynamical systems. Springer-Verlag, New York, 1987, Translated from the French by Joseph Christy

  24. Steinlein, H., Walther, H.-O.: Hyperbolic sets transversal homoclinic trajectories and symbolic dynamics for C1-maps in Banach spaces. J. Dynam. Differential Equations 2(3), 325–365 (1990)

    MATH  Google Scholar 

  25. Szekeres, G.: Regular iteration of real and complex functions. Acta Math. 100, 203–258 (1958)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  27. Vainikko, G.: Funktionalanalysis der Diskretisierungsmethoden. B.G. Teubner Verlag Leipzig, 1976, Mit Englischen und Russischen Zusammenfassungen Teubner-Texte zur Mathematik

  28. Zou, Y.-K., Beyn ,W.-J.: Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete Contin. Dynam. Systems 2(3), 351–365 (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Wolf-Jürgen Beyn & Thorsten Hüls

Authors
  1. Wolf-Jürgen Beyn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thorsten Hüls
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolf-Jürgen Beyn.

Additional information

Supported by the DFG Research Group ‘Spectral analysis, asymptotic distributions and stochastic dynamics’

Mathematics Subject Classification (2000): 37C29, 65P30

Acknowledgement The authors thank the referees for several helpful suggestions which improved the first version of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beyn, WJ., Hüls, T. Error estimates for approximating non-hyperbolic heteroclinic orbits of maps. Numer. Math. 99, 289–323 (2004). https://doi.org/10.1007/s00211-004-0563-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0563-4

Keywords