Rigorous enclosures of rotation numbers by interval methods

Belova, Anna

doi:10.3934/jcd.2016004

Cited by 2 publications

(3 citation statements)

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“…While we used the example (CR3BP) in [7], there we used a Poincaré return map whereas here in Section 4.2 no return map is used. We discuss the connections to our work with [3,4,5] in the subsequent sections of the paper. Those papers do investigate the Babylonian Problem, starting with only a set of iterates for a single finite length forward trajectory with the goal of finding a rotation number for some projection of a torus.…”

Section: [P]mentioning

confidence: 99%

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Solving the Babylonian problem of quasiperiodic rotation rates

Das¹,

Saiki²,

Sander³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - S

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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...

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Section: [P]mentioning

confidence: 99%

“…Even in that case d = 1 there has been no general method for computing the lift in order to find ρ φ , though there is a literature dealing with special cases. See for example [3,4,5]. We have established a general method for determining the lift ∆, as summarized in the Figs.…”

Section: Introductionmentioning

confidence: 99%

Solving the Babylonian problem of quasiperiodic rotation rates

Das¹,

Saiki²,

Sander³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - S

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“…Assume that

s

is as in Theorem 3.3. The rotation number

ρ ({scriptR}_{s, d, τ})

may be computed by definition, that is, using () directly, or with some more sophisticated methods, for example, [3, 4], which improve the convergence rate of the computation process. Unfortunately, these methods require homeomorphism

{scriptR}_{s, d, τ}

to be exactly given.…”

Section: Period Estimationmentioning

confidence: 99%

Period and signal reconstruction from the curve of trains of samples

Rupniewski

2021

IET Signal Processing

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A sequence of samples of a periodic signal can be treated as a point in a multi-dimensional space. All such sequences of a given length and taken at a given sampling rate form a closed curve. We prove that this curve determines the period of the sampled signal, even if the sequences are of short length and are taken at a sub-Nyquist rate. This result is obtained with a help of the theory of rotation numbers developed by Poincar. We also prove that the curve of sample-sequences determines the sampled signal up to a time shift provided that the ratio of the sampling period to the period of the signal is irrational. Eventually, we give an example, which shows that the same is not true if this ratio is rational.

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Rigorous enclosures of rotation numbers by interval methods

Cited by 2 publications

References 10 publications

Solving the Babylonian problem of quasiperiodic rotation rates

Solving the Babylonian problem of quasiperiodic rotation rates

Period and signal reconstruction from the curve of trains of samples

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