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Multiple Purpose Algorithms for Invariant Manifolds

By Gert Vegter, Aaron Hagen and Henk Broer


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. Examples of computations of both attracting and saddle-type (1D and 2D) manifolds will be given, with and without non-uniform adaptive refinement. A convergence result for the algorithm will be sketched.

Year: 2003
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