High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps

An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bézier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

► We present a new numerical method for computing two-dimensional invariant manifolds of maps.
► We use high-order approximation tools developed for the field of computer aided geometric design.
► The method incorporates quasi-interpolation schemes based on triangular Bézier patches.
► The method achieves significant improvements in the accuracy of the calculation.
► Numerical experiments show the method handles the problems where the other methods have failed.