Renormalization for breakup of invariant tori

On reconnection phenomena in the standard nontwist map

“…We aim to deepen the understanding of the behavior of the SNM in different regions in (a, b)-parameter space by extending the discussions of Refs. [20,21,14,19,18].…”

Renormalization group operators for maps and universal scaling of universal scaling exponents

“…The eigenvalues and phase-space scalings of the fixed points of RGO are independent of the head of the continued fraction expansion. This is seen explicitly for the phase-space scalings of the simple twist fixed points and non-twist two-cycles found in [15] for numbers with periodicity n = 1. Hence, for the purposes of this paper, it suffices to study the operators and fixed points for numbers of the form…”

Renormalization of vector fields

“…In the case of the golden mean and related rotation numbers, numerical investigations of specific two-parameter families [30,31,3,4] reveal self-similarity phenomena, with asymptotic scaling ratios (both in parameter space and phase space) that seem to be independent of the family considered. The self-similarity transformation involves as 12-step shift in the sequence of continued fraction approximants for ϑ, as opposed to the ordinary 1-step shift describing the similarity of periodic orbits in the twist case.…”

“…There is no proof for the existence of such an orbit; nor are there any accurate numerical results, not even from experiments on specific families [30,31,3,4], which usually yield very good values for the universal constants (but only anecdotal evidence for universality). The main problem is that only a few levels of self-similarity can be examined, since each of them involves 12 steps in the continued fraction expansion.…”