Quasiperiodic Orbits in Siegel Disks/Balls and the Babylonian Problem
Abstract
We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic 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 quasiperiodic 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 quasiperiodic 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".
 Publication:

Regular and Chaotic Dynamics
 Pub Date:
 November 2018
 DOI:
 10.1134/S1560354718060084
 arXiv:
 arXiv:1811.03148
 Bibcode:
 2018RCD....23..735S
 Keywords:

 Mathematics  Dynamical Systems;
 37F50;
 37C55
 EPrint:
 16 pages, 10 figures