On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization

  • Christian Lubich


This article reviews recent results on long-time behaviour, invariant sets and bifurcations of evolution equations under discretization by numerical methods. The emphasis is on time discretization. Finite-time error bounds of low order for non-smooth data, of high order for smooth data, and attractive invariant manifolds are tools that pervade large parts of the article. To illustrate the mechanisms, the following combinations of dynamics/equations have been selected for a detailed discussion:
  • Shadowing near hyperbolic equilibria of singularly perturbed ODEs

  • Hyperbolic periodic orbits of delay differential equations

  • Hopf bifurcation of semilinear parabolic equations

  • Inertial manifolds of semilinear parabolic equations

  • Attractors of damped wave equations.


Periodic Orbit Hopf Bifurcation Error Bound Invariant Manifold Delay Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Alouges, A. Debussche, On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point. Numer. Funct. Anal. Optimiz. 12 (1991), 253–269.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    F. Alouges, A. Debussche, On the discretization of a partial differential equation in the neighborhood of a periodic orbit. Numer. Math. 65 (1993), 143–175.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    G. Benettin, A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Statist. Phys. 74 (1994), 1117–1143.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    W.-J. Beyn, On the numerical approximation of phase portraits near stationary points. SIAM J. Numer. Anal. 24 (1987), 1095–1113.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    W.-J. Beyn, On invariant closed curves for one-step methods. Numer. Math. 51 (1987), 103–122.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    W.-J. Beyn, Numerical methods for dynamical systems, in Advances in Numerical Analysis, Vol. I, Nonlinear partial differential equations and dynamical systems, W. Light, ed., Clarendon Press, Oxford, 1991, 175–236.Google Scholar
  7. 7.
    M. Braun, J. Hershenov, Periodic solutions of finite difference equations. Quart. Appl. Math. 35 (1977), 139–147.MathSciNetMATHGoogle Scholar
  8. 8.
    F. Brezzi, S. Ushiki, H. Fujii, Real and ghost bifurcation dynamics in difference schemes for ordinary differential equations. In: T. Küpper, H.D. Mittelmann, H. Weber, eds., Numerical Methods for Bifurcation Problems. Birkhäuser, Boston, 1984, 79–104.Google Scholar
  9. 9.
    S.-N. Chow, X.-B. Lin, K.J. Palmer, A shadowing lemma with applications to semilinear parabolic equations. SIAM J. Math. Anal. 20 (1989), 547–557.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    F. Demengel, J.-M. Ghidaglia, Inertial manifolds for partial differential evolution equations under time discretization: existence, convergence, and applications. J. Math. Anal. Appl. 155 (1991), 177–225.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J.L.M. van Dorsselaer, Ch. Lubich, Inertial manifolds of parabolic differential equations under higher-order discretizations. IMA J. Numer. Anal. 19 (1999), 455–471.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    T. Eirola, Invariant curves of one-step methods. BIT 28 (1988), 113–122.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    C.M. Elliott, A.M. Stuart, The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993), 1622–1663.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq. 31 (1979), pp. 53–98.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    C. Foiaş, G.R. Sell, R. Temam, Inertial manifolds for nonlinear evolutionary equations. J. Diff. Eq. 73 (1988), 309–353.MATHCrossRefGoogle Scholar
  16. 16.
    E. Hairer, Ch. Lubich, The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997), 441–462.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    E. Hairer, Ch. Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations. Preprint, 1999Google Scholar
  18. 18.
    E. Hairer, Ch. Lubich, M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential-algebraic equations. BIT 28 (1988), 678–700.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, 2nd ed., 1993.Google Scholar
  20. 20.
    E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, 2nd ed., 1996.Google Scholar
  21. 21.
    J. Hale, Theory of Functional Differential Equations. Springer-Verlag, New York, 1977.MATHCrossRefGoogle Scholar
  22. 22.
    J.K. Hale, X.-B. Lin, G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comp. 50 (1988), 89–123.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    J.K. Hale, G. Raugel, Lower semicontinuity of attractors of gradient systems and applications. Ann. Mat. Pura Appl. 154 (1989), 281–326.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    D. Henry, Geometric Theory of Semilinear Parabolic Equations. LNM 840, Springer-Verlag, 1981.Google Scholar
  25. 25.
    J.G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem II. Stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23 (1986), 750–777.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    A.T. Hill, E. Süli, Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations. Math. Comp. 64 (1995), 1097–1122.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    A.T. Hill, E. Süli, Set convergence for discretizations of the attractor. IMA J. Numer. Anal. 16 (1996), 289–296.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    M. Hochbruck, Ch. Lubich, H. Selhofer, Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. 19 (1998), 1552–1574.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    K. in’ t Hout, Ch. Lubich, Periodic orbits of delay differential equations under discretization. BIT 38 (1998), 72–91.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    D.A. Jones, A.M. Stuart, Attractive invariant manifolds under approximation. Inertial manifolds. J. Diff. Eq. 123 (1995), 588–637.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    D.A. Jones, A.M. Stuart, E.S. Titi, Persistence of invariant sets for dissipative evolution equations. J. Math. Anal. Appl. 219 (1998), 479–502.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    U. Kirchgraber, F. Lasagni, K. Nipp, D. Stoffer, On the application of invariant manifold theory, in particular to numerical analysis. Internat. Ser. Numer. Math. 97, Birkhäuser, Basel, 1991, 189–197.Google Scholar
  33. 33.
    Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory. Springer-Verlag, New York, 1995.MATHCrossRefGoogle Scholar
  34. 34.
    S. Larsson, The long-time behaviour of finite element approximations of solutions to semilinear parabolic problems. SIAM J. Numer. Anal. 26 (1989), 348–365MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    S. Larsson, Nonsmooth data error estimates with applications to the study of the long-time behavior of finite element solutions of semilinear parabolic problems. Report 1992-36, Dept. of Mathematics, Chalmers Univ. Göteborg, 1992. (
  36. 36.
    S. Larsson, S. Yu. Pilyugin, Numerical shadowing near the global attractor for a semilinear parabolic equation. Preprint 1998-21, Department of Mathematics, Chalmers University of Technology.Google Scholar
  37. 37.
    S. Larsson, J.M. Sanz-Serna, The behaviour of finite element solutions of semilinear parabolic problems near stationary points. SIAM J. Numer. Anal. 31 (1994), 1000–1018.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    S. Larsson, J.M. Sanz-Serna, A shadowing result with applications to finite element approximation of reaction-diffusion equations. Math. Comp. 68 (1999), 55–72.MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    G.J. Lord, Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg-Landau equation. SIAM J. Numer. Anal. 34 (1997), 1483–1512.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Ch. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations. Numer. Math. 58 (1991), 839–853.MathSciNetGoogle Scholar
  41. 41.
    Ch. Lubich, K. Nipp, D. Stoffer, Runge-Kutta solutions of stiff differential equations near stationary points. SIAM J. Numer. Anal. 32 (1995), 1296–1307.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Ch. Lubich, A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature. Math. Comp. 60 (1993), 105–131.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Ch. Lubich, A. Ostermann, Runge-Kutta time discretization of reactiondiffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour. Appl. Numer. Math. 22 (1996), 279–292.MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ch. Lubich, A. Ostermann, Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization. Numer. Math. 81 (1998), 53–84.MathSciNetCrossRefGoogle Scholar
  45. 45.
    K. Matthies, Time averaging of parabolic partial differential equations: exponential estimates. Doctoral dissertation, FU Berlin, 1999.Google Scholar
  46. 46.
    K. Nipp, Invariant manifolds of singularly perturbed ordinary differential equations. ZAMP 36 (1985), 309–320.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    K. Nipp, D. Stoffer, Attractive invariant manifolds for maps: Existence, smoothness and continuous dependence on the map. Report 92-11, SAM, ETH Zürich, 1992. (
  48. 48.
    K. Nipp, D. Stoffer, Invariant manifolds of numerical integration schemes applied to stiff systems of singular perturbation type. I: RK-methods. Numer. Math. 70 (1995), 245–257.MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    K. Nipp, D. Stoffer, Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type. II: Linear multistep methods. Numer. Math. 74 (1996), 305–323MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    R.E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York, 1991.MATHCrossRefGoogle Scholar
  51. 51.
    A. Ostermann, C. Palencia, Shadowing for nonautonomous parabolic problems with applications to long-time error bounds. Preprint 2-1999, IMG, Univ. Innsbruck. To appear in SIAM J. Numer. Anal.Google Scholar
  52. 52.
    K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations. Nonl. Anal., Theory, Methods and Appl.ic. 16 (1991), 959–980.MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    J.M. Sanz-Serna, A.M. Stuart, A note on uniform in time error estimates for approximations to reaction-diffusion equations. IMA J. Numer. Anal. 12 (1992), 457–462MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    T. Shardlow, Inertial manifolds and linear multi-step methods. Numer. Algor. 14 (1997), 189–209.MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    A.M. Stuart, Perturbation theory for infinite dimensional dynamical systems. In: M. Ainsworth, J. Levesley, W.A. Light, M. Marietta (eds.), Theory and Numerics of Ordinary and Partial Differential Equations. Advances in Numerical Analysis IV, Clarendon Press, Oxford 1995, 181–290.Google Scholar
  56. 56.
    A.M. Stuart, A.R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge Univ. Press, 1996.Google Scholar
  57. 57.
    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 2nd ed., 1997.MATHGoogle Scholar
  58. 58.
    A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations. In: U. Kirchgraber, H.O. Walther, eds., Dynamics Reported 2, Teubner, Stuttgart, and J. Wiley, Chichester, 1989.Google Scholar
  59. 59.
    A. Vanderbauwhede, G. Iooss, Center manifold theory in infinite dimensions. In: C.K.R.T. Jones, U. Kirchgraber, H.O. Walther, eds., Dynamics Reported 1, New series, Springer-Verlag, Berlin, 1992.Google Scholar
  60. 60.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christian Lubich
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

Personalised recommendations