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We present and analyze rigorously a quadratically convergent algorithm tocompute an invariant circle for 2-dimensional maps along with the correspondingfoliation by stable manifolds. We prove that when the algorithm starts from aninitial guess that satisfies the invariance equation very approximately(depending on some condition numbers, evaluated on the approximate solution),then the algorithm converges to a true solution which is close to the initialguess. The convergence is faster than exponential in smooth norms. The distancefrom the exact solution and the approximation is bounded by the initial error.This allows validating the numerical approximations (a-posteriori results). Italso implies the usual persistence formulations since the exact solutions ofthe invariance equation for a model are approximate solutions for a similarmodel. The algorithm we present works irrespective of whether the dynamics onthe invariant circle is a rotation or it is phase-locked. The condition numbersrequired do not involve any global qualitative properties of the map. They areobtained by evaluating derivatives of the initial guess, derivatives of the mapin a neighborhood of the guess, performing algebraic operations, and takingsuprema. The proof of the convergence is based on a general Nash-Moser implicitfunction theorem specially tailored for this problem. The Nash-Moser procedurehas unusual properties. As it turns out, the regularity requirements are notvery severe (only 2 derivatives suffice). We hope that this implicit functiontheorem may be of independent interest and have presented it in aself-contained appendix. The algorithm in this paper is very practical since itconverges quadratically, and it requires moderate storage and operation count.Details of the implementation and results of the runs are described in acompanion paper [YdlL21].

Computing the Invariant Circle and its Stable Manifolds for a 2-D Map by  the Parameterization Method: Effective Algorithms and Rigorous Proofs of  Convergence