Solving the Babylonian problem of quasiperiodic rotation rates

Abstract

A trajectory θ n := F n (θ 0 ), n = 0, 1, 2, . . . is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus T d for which F has the form F (θ) = θ + ρ mod 1 for all θ ∈ T d and for some ρ ∈ T d . (For d > 1 we always interpret mod 1 as being applied to each coordinate.) There is an ancient literature on computing three rotation rates for the Moon. However, for d > 1, the choice of coordinates that yields the form F (θ) = θ + ρ mod 1 is far from unique and the different choices yield a huge choice of coordinatizations (ρ 1 , · · · , ρ d ) of ρ, and these coordinations are dense in T d . Therefore instead one defines the rotation rate ρ φ from the perspective of a map φ : T d → S 1 . This is in effect the approach taken by the Babylonians and we refer to this approach as the "Babylonian Problem". However, even in the case d = 1 there has been no general method for computing ρ φ given only the sequence φ(θ n ), though there is a literature dealing with special cases. Here we present our Embedding continuation method for computing ρ φ from the image φ(θ n ) of a trajectory.It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem.The goal of this paper is to show how to compute a rotation rate of a quasiperiodic discrete-time trajectory. We begin with a motivating historical example, followed by a broad overview of our approach to determining rotation rates.Rotation rates and quasiperiodicity have been studied for millennia; namely, the Moon's orbit has three periods whose approximate values were found 2500 years ago by the Babylonians [1]. Although computation of the periods of the Moon is an easy problem today, we use it to give context to the problems we investigate. The Babylonians found that the periods of the Moon -measured relative to the distant stars -are approximately 27.3 days (the sidereal month), 8.85 years for the rotation of the apogee (the local maximum distance from the Earth), and 18.6 years for the rotation of the intersection of the Earth-Sun plane with the Moon-Earth plane. They also measured the variation in the speed of the Moon through the field of stars, and the speed is inversely correlated with the distance of the Moon. They used their results to predict eclipses of the Moon, which occur only when the Sun, Earth and Moon are sufficiently aligned to allow the Moon to pass through the shadow of the Earth. How they obtained their estimates is not fully understood but it was through observations of the trajectory of the Moon through the distant stars in the sky. In essence they viewed the Moon projected onto the two-dimensional space of distant stars. We too work with quasiperiodic motions which have been projected into one or two dimensions.The Moon has three periods because the Moon's orbit is basically three-dimensionally quasiperiodic, traveling on a three-dimensional torus T 3 that is embedded in six (position+velocity) dimensions. The torus is topologically the product of three circles, and th...

Quantitative quasiperiodicity

“…and ρ = ( √ 5 − 1)/2 [9,37]. It is found (see figure 3 in [36]) that the invariant curves get more irregular near the boundary of the disk. While typical smooth quasiperioidc curves require about 70 Fourier coefficients for 30-digit precision, near the boundary of the Siegel disk the curves are much more irregular and can require 24 000 coefficients (or more) for the same precision.…”

“…Speed of convergence of Fourier series for a conjugacy. In a separate report [36], we examine conjugacies of the quasiperiodic curves in the Siegel disk. The map is a simple onedimensional complex dynamical system z n+1 = f (z n ), where…”

Computing the Invariant Circle and the Foliation by Stable Manifolds for a 2-D Map by the Parameterization Method: Numerical Implementation and Results

“…where x ∈ T. By computing such rotation number with the help of the Birkhoff average, one can show that such sequence admits O( 1 M ) convergence rate [Kre78]. In this paper, we follow the approach in [DSSY16] and [DSSY19], which use the weighted Birkhoff average instead of the regular Birkhoff sum.…”