On the computation of invariant manifolds of fixed points

  • A. J. Homburg
  • H. M. Osinga
  • G. Vegter
Original Papers


We present a method for the numerical computation of invariant manifoids of hyperbolic and pseudohyperbolic fixed points of diffeomorphisms. The derivation of this algorithm is based on well-known properties of (almost) invariant foliations. Numerical results illustrate the performance of our method.


Manifold Numerical Computation Mathematical Method Invariant Manifold Invariant Foliation 
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]
    H. W. Broer and G. Vegter,Bifurcational Aspects of Parametric Resonance, vol. 1 ofDynamics Reported (New Series), chapt. 1, pp. 1–53. Springer Verlag, Berlin-Heidelberg-New York 1992.Google Scholar
  2. [2]
    R. L. Devaney,An Introduction to Chaotic Dynamical Systems. Addison-Wesley Publ Co., Redwood City, CA 1987.Google Scholar
  3. [3]
    G. H. Golub and C. F. Van Loan,Matrix Computations. Johns Hopkins University Press, Baltimore 1983.Google Scholar
  4. [4]
    J. Hadamard,Sur l'itération et les solutions asymptotiques des equations différentielles. Bull. Soc. Math. France,29, 224–228 (1901).Google Scholar
  5. [5]
    M. W. Hirsch, C. C. Pugh, and M. Shub,Invariant Manifolds. Springer-Verlag, Berlin 1977.Google Scholar
  6. [6]
    A. J. Homburg,On the Computation of Hyperbolic Sets and their Invariant Manifolds. Technical Report 68, Institut für Angewandte Analysis und Stochastik, Berlin 1993.Google Scholar
  7. [7]
    F. Ma and T. Küpper,Numerical calculation of invariant manifolds for maps. J. Num. Linear Alg. Appl.,1,2, 141–150 (1994).Google Scholar
  8. [8]
    H. E. Nusse and J. A. Yorke,A procedure for finding numerical trajectories on chaotic saddles. Physica D,36, 137–156 (1989).Google Scholar
  9. [9]
    J. Palis and W. de Melo,Geometric Theory of Dynamical Systems. Springer-Verlag, New York 1982.Google Scholar
  10. [10]
    J. Palis and F. Takens,Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations, vol. 35 of Cambridge studies in advanced math. Cambridge University Press 1993.Google Scholar
  11. [11]
    T. S. Parker and L. O. Chua,Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, Berlin 1989.Google Scholar
  12. [12]
    O. Perron,Ueber Stabilität und Asymptotisches Verhalten der Lösungen eines Systemes endlicher Differenzengleichungen. J. Reine Angew. Math.,161, 41–46 (1929).Google Scholar
  13. [13]
    M. Shub,Global Stability of Dynamical Systems. Springer-Verlag, Berlin 1977.Google Scholar
  14. [14]
    S. van Gils and A. Vanderbauwhede,Center manifolds and contractions on a scale of Banach spaces. J. Funct. Analysis, pp 209–224 (1987).Google Scholar
  15. [15]
    A. Vanderbauwhede,Center Manifolds, Normal Forms and Elementary Bifurcations. In Dynamics Reported, vol. 2, pp 89–170. John Wiley & Sons Ltd and B.G. Teubner, Stuttgart 1989.Google Scholar
  16. [16]
    H. E. S. Westerveld, Numerieke Bepaling van Invariante Variëteiten (in Dutch). Master's thesis, University of Twente 1990.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • A. J. Homburg
    • 1
  • H. M. Osinga
    • 1
  • G. Vegter
    • 1
  1. 1.Dept of Mathematics and Computing ScienceAV GroningenThe Netherlands

Personalised recommendations