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Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem
Authors:Yoshitaka Saiki  James A. Yorke
Affiliation:1.Graduate School of Business Administration,Hitotsubashi University,Tokyo,Japan;2.JST PRESTO,Saitama,Japan;3.University of Maryland,College Park,USA
Abstract:We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.
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