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We present and implement an algorithm for computing the invariant circle andthe corresponding stable manifolds for 2-dimensional maps. The algorithm isbased on the parameterization method, and it is backed up by an a-posterioritheorem established in [YdlL21]. The algorithm works irrespective of whetherthe internal dynamics in the invariant circle is a rotation or it isphase-locked. The algorithm converges quadratically and the number ofoperations and memory requirements for each step of the iteration is linearwith respect to the size of the discretization. We also report on the result ofrunning the implementation in some standard models to uncover new phenomena. Inparticular, we explored a bundle merging scenario in which the invariant circleloses hyperbolicity because the angle between the stable directions and thetangent becomes zero even if the rates of contraction are separated. We alsodiscuss and implement a generalization of the algorithm to 3 dimensions, andimplement it on the 3-dimensional Fattened Arnold Family (3D-FAF) map withnon-resonant eigenvalues and present numerical results.

Computing the Invariant Circle and the Foliation by Stable Manifolds for  a 2-D Map by the Parameterization Method: Numerical Implementation and  Results