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".

arXiv comment16 pages, 10 figures

References in corpus (4)

• Dynamics in one complex variable: introductory lectures
• Super convergence of ergodic averages for quasiperiodic orbits
• Quantitative Quasiperiodicity
• Measuring quasiperiodicity

Cited by in corpus (1)

• Weighted Birkhoff averages: Deterministic and probabilistic perspectives