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Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 735–750 | Cite as

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

  • Yoshitaka SaikiEmail author
  • James A. Yorke
Article

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”.

Keywords

quasi-periodic orbits rotation rates weighted Birkhoff averaging Siegel disk Siegel ball 

MSC2010 numbers

37F50 37C55 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Graduate School of Business AdministrationHitotsubashi UniversityTokyoJapan
  2. 2.JST PRESTOSaitamaJapan
  3. 3.University of MarylandCollege ParkUSA

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