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Period doubling in area-preserving maps: an associated one-dimensional problem

Published online by Cambridge University Press:  02 September 2010

DENIS GAIDASHEV and
HANS KOCH
DENIS GAIDASHEV
Affiliation:
Department of Mathematics, University of Uppsala, Uppsala, Sweden (email: gaidash@math.uu.se)
HANS KOCH
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas, USA (email: koch@math.utexas.edu)
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Abstract

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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: \[ H_*(x,u)=(\phi (x)-u,x-\phi (\phi (x)-u )), \] where ϕ solves \[ \phi (x)=\frac {2}{\lambda } \phi (\phi (\lambda x))-x. \] 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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011, pp. 1193 - 1228
Copyright
Copyright © Cambridge University Press 2010

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