A geometrical method for the approximation of invariant tori |
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Authors: | Bryan Rasmussen Luca Dieci |
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Institution: | 1. Los Alamos National Laboratory, MS T080 Los Alamos, NM 87545, USA;2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA |
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Abstract: | We consider a numerical method based on the so-called “orthogonality condition” for the approximation and continuation of invariant tori under flows. The basic method was originally introduced by Moore Computation and parameterization of invariant curves and tori, SIAM J. Numer. Anal. 15 (1991) 245–263], but that work contained no stability or consistency results. We show that the method is unconditionally stable and consistent in the special case of a periodic orbit. However, we also show that the method is unstable for two-dimensional tori in three-dimensional space when the discretization includes even numbers of points in both angular coordinates, and we point out potential difficulties when approximating invariant tori possessing additional invariant sub-manifolds (e.g., periodic orbits). We propose some remedies to these difficulties and give numerical results to highlight that the end method performs well for invariant tori of practical interest. |
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Keywords: | 37M99 65P99 |
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