A. Gonzalez-Enriquez, A. Haro, R. de la Llave
Efficient and reliable algorithms for the computation of non-twist invariant circles
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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.