Period doubling in area-preserving maps: an associated one-dimensional problem
2010 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 31, no 04, 1193-1228 p.Article in journal (Refereed) Published
It has been observed that the famous Feigenbaum–Coullet–Tresser period-doubling universality has a counterpart for area-preserving maps of ℝ2. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmann et al [Existence of a fixed point of the doubling transformation for area-preserving maps of the plane. Phys. Rev. A 26(1) (1982), 720–722; A computer-assisted proof of universality for area-preserving maps. Mem. Amer. Math. Soc. 47 (1984), 1–121]. As is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period-doubling universality exists to date. We argue that the period-doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Hénon-like (after a coordinate change) map:
where ϕ solves We then give a ‘proof’ of existence of solutions of small analytic perturbations of this one-dimensional problem, and describe some of the properties of this solution. The ‘proof’ consists of an analytic argument for factorized inverse branches of ϕ together with verification of several inequalities and inclusions of subsets of ℂ numerically. Finally, we suggest an analytic approach to the full period-doubling problem for area-preserving maps based on its proximity to the one-dimensional case. In this respect, the paper is an exploration of possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.
Place, publisher, year, edition, pages
2010. Vol. 31, no 04, 1193-1228 p.
universality, renormalization, period doubling
Research subject Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-146985DOI: 10.1017/S0143385710000283ISI: 000293443400010OAI: oai:DiVA.org:uu-146985DiVA: diva2:399530