BIT Numerical Mathematics

, Volume 51, Issue 3, pp 555–579 | Cite as

Detecting exponential dichotomy on the real line: SVD and QR algorithms

  • Luca Dieci
  • Cinzia Elia
  • Erik Van Vleck


In this paper we propose and implement numerical methods to detect exponential dichotomy on the real line. Our algorithms are based on the singular value decomposition and the QR factorization of a fundamental matrix solution. The theoretical justification for our methods was laid down in the companion paper: “Exponential Dichotomy on the real line: SVD and QR methods.”


Exponential dichotomy Sacker-Sell spectrum Lyapunov exponents 

Mathematics Subject Classification (2000)

34D08 34D09 65L 


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  1. 1.
    Adrianova, L.Y.: Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs, vol. 146. Amer. Math. Soc., Providence (1995) MATHGoogle Scholar
  2. 2.
    Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.: Lyapunov exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part I: Theory. …Part II: Numerical applications. Meccanica 15, 9–20, 21–30 (1980) Google Scholar
  3. 3.
    Beyn, W.J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10(3), 379–405 (1990) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Beyn, W.J.: On well-posed problems for connecting orbits in dynamical systems. In: Chaotic Numerics (Geelong, 1993). Contemp. Math., vol. 172, pp. 131–168. Amer. Math. Soc., Providence (1994) Google Scholar
  5. 5.
    Beyn, W.J., Lorenz, J.: Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals. Numer. Funct. Anal. Optim. 20(3–4), 201–244 (1999) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Broer, H.W., Osinga, H.M., Vegter, G.: Algorithms for computing normally hyperbolic invariant manifolds. Z. Angew. Math. Phys. 48(3), 480–524 (1997) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath, Boston (1965) MATHGoogle Scholar
  8. 8.
    Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978) MATHGoogle Scholar
  9. 9.
    Dieci, L., Elia, C.: The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects. J. Differ. Equ. 230(2), 502–531 (2006) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dieci, L., Elia, C.: SVD algorithms to approximate spectra of dynamical systems. Math. Comput. Simul. 79(4), 1235–1254 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dieci, L., Van Vleck, E.S.: Lyapunov and other spectra: a survey. In: Collected Lectures on the Preservation of Stability under Discretization (Fort Collins, CO, 2001), pp. 197–218. SIAM, Philadelphia (2002) Google Scholar
  12. 12.
    Dieci, L., Van Vleck, E.S.: Lyapunov spectral intervals: theory and computation. SIAM J. Numer. Anal. 40(2), 516–542 (2002) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dieci, L., Van Vleck, E.S.: On the error in computing Lyapunov exponents by QR methods. Numer. Math. 101(4), 619–642 (2005) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dieci, L., Van Vleck, E.S.: Perturbation theory for approximation of Lyapunov exponents by QR methods. J. Dyn. Differ. Equ. 18(3), 815–840 (2006) MATHCrossRefGoogle Scholar
  15. 15.
    Dieci, L., Van Vleck, E.S.: Lyapunov and Sacker-Sell spectral intervals. J. Dyn. Differ. Equ. 19(2), 265–293 (2007) MATHCrossRefGoogle Scholar
  16. 16.
    Dieci, L., Van Vleck, E.S.: On the error in QR integration. SIAM J. Numer. Anal. 46(3), 1166–1189 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Dieci, L., Elia, C., Van Vleck, E.: Exponential dichotomy on the real line: SVD and QR methods. J. Differ. Equ. 248(2), 287–308 (2010) MATHCrossRefGoogle Scholar
  18. 18.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971/1972) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Huels, T.: Computing sacker and sell spectra for discrete dynamical systems. Preprint Google Scholar
  21. 21.
    Humpherys, J., Sandstede, B., Zumbrun, K.: Efficient computation of analytic bases in Evans function analysis of large systems. Numer. Math. 103(4), 631–642 (2006) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Humpherys, J., Zumbrun, K.: An efficient shooting algorithm for Evans function calculations in large systems. Physica D 220(2), 116–126 (2006) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition Google Scholar
  24. 24.
    Palmer, K.: Shadowing in Dynamical Systems. Mathematics and its Applications, vol. 501. Kluwer Academic, Dordrecht (2000) MATHGoogle Scholar
  25. 25.
    Palmer, K.J.: Exponential dichotomies and Fredholm operators. Proc. Am. Math. Soc. 104(1), 149–156 (1988) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Pilyugin, S.Y.: Shadowing in Dynamical Systems. Lecture Notes in Mathematics, vol. 1706. Springer, Berlin (1999) MATHGoogle Scholar
  27. 27.
    Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27(3), 320–358 (1978) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Sakamoto, K.: Estimates on the strength of exponential dichotomies and application to integral manifolds. J. Differ. Equ. 107(2), 259–279 (1994) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Sandstede, B.: Stability of travelling waves. In: Handbook of Dynamical Systems, vol. 2, pp. 983–1055. North-Holland, Amsterdam (2002) CrossRefGoogle Scholar
  30. 30.
    Van Vleck, E.: On the error in the product QR decomposition. SIAM J. Matrix Anal. Appl. 31, 1775–1791 (2010) MATHCrossRefGoogle Scholar

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© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Dip. di MatematicaUniversitá di BariBariItaly
  3. 3.Department of MathematicsUniversity of KansasLawrenceUSA

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