Numerical approximation of normally hyperbolic invariant manifolds

Pages: 133 - 140, Issue Special, July 2003

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Henk Broer - Department of Mathematics, University of Groningen, PO Box 407, 9700 AK, Groningen, Netherlands (email)
Aaron Hagen - Department of Mathematics, University of Texas, Arlington, TX 76019, United States (email)
Gert Vegter - Department of Mathematics and Computing Science, University of Groningen, Netherlands (email)

Abstract: This paper deals with the numerical continuation of invariant manifolds, regardless of the restricted dynamics. Typically, invariant manifolds make up the skeleton of the dynamics of phase space. Examples include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds in phase plus parameter space on which bifurcations occur. These manifolds are for the most part invisible to current numerical methods. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. The key issue is the discretization of the differential geometric components of the graph transform, and its consequences. Examples of computations will be given, with and without non-uniform adaptive refinement.

Keywords:  Invariant manifolds, normal hyperbolicity, chaotic dynamics, numerical continuation, bifurcation theory, computational geometry, graph transform.
Mathematics Subject Classification:  34C28, 34C30, 34C45, 37D10, 37M99, 65D18, 65P99.

Received: June 2002;      Revised: March 2003;      Published: April 2003.