Critical behavior of period doubling in coupled area-preserving maps

AbstractIt 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 Eckmannet 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 solvesWe 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.

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PERIOD DOUBLING RENORMALIZATION FOR AREA-PRESERVING MAPS AND MILD COMPUTER ASSISTANCE IN CONTRACTION MAPPING PRINCIPLE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411030477 ◽

2011 ◽

Vol 21 (11) ◽

pp. 3217-3230 ◽

Cited By ~ 3

Author(s):

DENIS G. GAIDASHEV

Keyword(s):

Contraction Mapping ◽

Period Doubling ◽

Contraction Mapping Principle ◽

Real Numbers ◽

Computer Assistance ◽

Computer Aided ◽

Area Preserving Maps ◽

Renormalization Operator ◽

Renormalization Approach ◽

Area Preserving

A universal period doubling cascade analogous to the famous Feigenbaum–Coullet–Tresser period doubling has been observed in area-preserving maps of ℝ2. The existence of the "universal" map with orbits of all binary periods has been proved via a renormalization approach in [Eckmann et al., 1984] and [Gaidashev et al., 2011]. These proofs use "hard" computer assistance.In this paper, we attempt to reduce computer assistance in the argument, and present a mild computer aided proof of the analyticity and compactness of the renormalization operator in a neighborhood of a renormalization fixed point: that is, a proof that does not use generalizations of interval arithmetics to functional spaces — but rather relies on interval arithmetics on real numbers only to estimate otherwise explicit expressions. The proof relies on several instances of the Contraction Mapping Principle, which is, again, verified via mild computer assistance.

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