Skip to main content
Springer Nature Link
Log in

On the error in computing Lyapunov exponents by QR Methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider the error introduced using QR methods to approximate Lyapunov exponents. We give a backward error statement for linear non-autonomous systems, and further discuss nonlinear autonomous problems. In particular, for linear systems we show that one approximates a ``nearby'' discontinuous problem where how nearby is measured in terms of local errors and a measure of non-normality. For nonlinear problems we use a type of shadowing result.

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
$34.99 /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 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. Adrianova, L.Y.: Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs Vol. 146, AMS, Providence, R.I. 1995

  2. Arnold, L., Wihstutz, V.: Lyapunov Exponents. Proceedings, Bremen 1984. Springer-Verlag, Berlin, Lecture Notes in Mathematics 1186, 1986

  3. Arnold, L., Crauel, H., Eckmann J.P.: Lyapunov Exponents. Proceedings, Oberwolfach 1990. Springer-Verlag, Berlin, Lecture Notes in Mathematics 1486, 1991

  4. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: ``Lyapunov Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them. Part 1: Theory '', and `` . . . Part 2: Numerical Applications ''. Meccanica 15, 9–20, 21–30 (1980)

    Article  Google Scholar 

  5. Bylov, B.F., Izobov, N.A.: ``Necessary and sufficient conditions for stability of characteristic exponents of a linear system.'' Differentsial'nye Uravneniya 5, 1794–1903 (1969)

    Google Scholar 

  6. Constantin, P., Foias, C.: ``Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations.'' Commun. Pure Appl. Mathem. 38, 1–27 (1985)

    Google Scholar 

  7. Dieci, L., Morini, B., Papini, A., Pasquali, A.: ``On real logarithms of nearby matrices and structured matrix interpolation''. Appl. Numer. Mathem. 19, 145–165 (1999)

    Article  Google Scholar 

  8. Dieci, L., Russell, R.D., Van Vleck, E.S.: ``On the computation of Lyapunov exponents for continuous dynamical systems''. SIAM J. Numer. Anal. 34, 402–423 (1997)

    Google Scholar 

  9. Dieci, L., Van Vleck, E.S.: ``Lyapunov and other spectra: a survey.'' In: Estep, D., Tavener, S. (Eds.), Preservation of Stability under Discretization SIAM Publications 2002

  10. Dieci, L., Van Vleck, E.S.: ``Lyapunov spectral intervals: theory and computation.'' SIAM J. Numer. Anal. 40 516–542 (2003)

    Google Scholar 

  11. Dieci, L., Van Vleck, E.S.: ``Lyapunov and Sacker-Sell spectral intervals.'' (2003), submitted

  12. Dieci, L., Van Vleck, E.S.: ``LESLIS and LESLIL: Codes for approximating Lyapunov exponents of linear systems''. Technical Report (2004): http://www.math.gatech.edu/~dieci.

  13. Hairer, E., Lubich, C.: ``The life-span of backward error analysis for numerical integrators.'' Numer. Math. 76, 441–462 (1997)

    Article  Google Scholar 

  14. McDonald, E., Higham, D.: ``Error analysis of QR algorithms for computing Lyapunov exponents''. ETNA 12, 234–251 (2001)

    Google Scholar 

  15. Millionshchikov, V.M.: ``Systems with integral division are everywhere dense in the set of all linear systems of differential equations.'' Differentsial'nye Uravneniya 5, 1167–1170 (1969)

    Google Scholar 

  16. Oliveira, S., Stewart, D.E.: ``Exponential splitting of products of matrices and accurately computing singular values of long products''. LAA 309, 175–190 (2000)

    Google Scholar 

  17. Oseledec, V.I.: ``A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems''. Trans. Moscow Mathem. Society 19, 197 (1968)

    Google Scholar 

  18. Palmer, K.J.: ``The structurally stable systems on the half-line are those with exponential dichotomy.'' J. Diff. Eqn. 33, 16–25 (1979)

    Article  Google Scholar 

  19. Palmer, K.J.: Shadowing in Dynamical Systems, Mathematics and Its Applications, Vol. 501, Kluwer, 2000

  20. Ruelle, D.: Chaotic evolution and strange attractors. Cambridge University Press, Cambridge, 1989

  21. Stewart, D.E.: ``A new algorithm for the SVD of a long product of matrices and the stability of products''. ETNA, 5, 29–47 (1997)

    Google Scholar 

  22. Trefethen, L.N., Bau, D.: Numerical Linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332, USA

    Luca Dieci

  2. Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045

    Erik S. Van Vleck

Authors
  1. Luca Dieci
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Erik S. Van Vleck
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Luca Dieci.

Additional information

This work was supported in part under NSF Grants DMS/FRG-0139895 and DMS/FRG-0139824

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dieci, L., Vleck, E. On the error in computing Lyapunov exponents by QR Methods. Numer. Math. 101, 619–642 (2005). https://doi.org/10.1007/s00211-005-0644-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0644-z

Mathematics Subject Classification (2000)