Algorithms for computing normally hyperbolic invariant manifolds

  • H.W. Broer;
  • H.M. Osinga;
  • G. Vegter;

Abstract.

An efficient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.

Key words. Dynamical systems, invariant manifolds, normal hyperbolicity, stable and unstable manifolds, graph transform, constructive proofs, algorithms, Newton's method, numerical experiments. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel, 1997

Authors and Affiliations

  • H.W. Broer;
    • 1
  • H.M. Osinga;
    • 2
  • G. Vegter;
    • 3
  1. 1.Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, e-mail: broer@math.rug.nlNL
  2. 2.Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, e-mail: osinga@cs.rug.nlNL
  3. 3.Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, e-mail: vegter@cs.rug.nlNL

Personalised recommendations