19-13 A. Gonzalez-Enriquez, A. Haro, R. de la Llave

Efficient and reliable algorithms for the computation of non-twist invariant circles (26684K, PDF) Jan 29, 19

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Abstract. This paper presents a methodology to study non\hyph twist invariant circles a nd their bifurcations for area preserving maps. We recall that non\hyph twist in variant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with e xtra stability properties (e.g. against external noise) and hence, they appear as design goals in some applications. Our methodology leads to efficient algorithms to compute and continue, with respect to parameters, non\hyph twist invariant circles. The algorithms are quadratically convergent, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous \emph{a\hyph posteriori} theorems which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. H ence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer assisted proofs. Our algorithms are also guaranteed to converge up to the breakdown of the invariant circles, then they are suitable to compute regions of parameters where the non\hyph twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, they do not require symmetries and can run without continuous manual adjustments. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include estimates for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. These numerical explorations lead to some new mathematical conjectures.

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