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

@article{Saiki2018QuasiperiodicOI, title={Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem}, author={Yoshitaka Saiki and James A. Yorke}, journal={Regular and Chaotic Dynamics}, year={2018}, volume={23}, pages={735-750} }

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… Expand

#### Figures from this paper

#### References

SHOWING 1-10 OF 30 REFERENCES

Boundaries of Siegel disks: numerical studies of their dynamics and regularity.

- Mathematics, Medicine
- Chaos
- 2008

It is obtained that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. Expand

Numerical computation of rotation numbers of quasi-periodic planar curves

- Mathematics
- 2009

Abstract Recently, a new numerical method has been proposed to compute rotation numbers of analytic circle diffeomorphisms, as well as derivatives with respect to parameters, that takes advantage of… Expand

Super convergence of ergodic averages for quasiperiodic orbits

- Mathematics
- 2015

By definition, a map quasiperiodic on a set $X$ if the map is conjugate to a pure rotation. Suppose we have a trajectory $(x_n)$ that we suspect is quasiperiodic. How do we determine if it is? In… Expand

On the numerical computation of Diophantine rotation numbers of analytic circle maps

- Mathematics
- 2006

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in… Expand

On the numerical computation of Diophantine rotation numbers of analytic circle maps

- Mathematics
- 2004

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in… Expand

Hamiltonian systems with three or more degrees of freedom

- Mathematics
- 1999

Preface. List of Participants. Lectures. Contributions. List of Authors. Subject Index. Lectures. Inflection points, extatic points and curve shortening S. Angenent. Topologically necessary… Expand

Numerical computation of the normal behaviour of invariant curves of n-dimensional maps

- Mathematics
- 2001

We describe a numerical method for computing the linearized normal behaviour of an invariant curve of a diffeomorphism of n, n≥2. In the reducible case, the method computes not only the normal… Expand

Frequency Analysis of a Dynamical System

- Mathematics
- 1993

Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system (Laskar, Icarus,… Expand

Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods

- Computer Science, Mathematics
- SIAM J. Appl. Dyn. Syst.
- 2014

A new approach to the numerical computation of the basic frequencies of a quasi-periodic signal with an arbitrary number of frequencies that can be calculated with high accuracy at a moderate computational cost, without simultaneously computing the Fourier representation of the signal. Expand

Dynamics in one complex variable

- Mathematics
- 2000

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large… Expand