A Geometrical Method for the Approximation of Invariant Tori

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 [1], 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 also 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.