American Institute of Mathematical Sciences

Advanced
2003, 2003(Special): 133-140. doi: 10.3934/proc.2003.2003.133

Numerical approximation of normally hyperbolic invariant manifolds

Henk Broer 1, , Aaron Hagen 2, and  Gert Vegter 3,

1. 

Department of Mathematics, University of Groningen, PO Box 407, 9700 AK, Groningen, Netherlands
2. 

Department of Mathematics, University of Texas, Arlington, TX 76019, United States
3. 

Department of Mathematics and Computing Science, University of Groningen, Netherlands

Received  June 2002 Revised  March 2003 Published  April 2003

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: bifurcation theory, Invariant manifolds, chaotic dynamics, normal hyperbolicity, numerical continuation, graph transform., computational geometry.
Mathematics Subject Classification: 34C28, 34C30, 34C45, 37D10, 37M99, 65D18, 65P9.
Citation: Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
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