Super convergence of ergodic averages for quasiperiodic orbits

Das, Suddhasattwa; Yorke, James A.

doi:10.1088/1361-6544/aa99a0

Cited by 35 publications

(36 citation statements)

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“…We have recently established a method for speeding up the convergence of the Birkhoff average in Theorem 1 through introducing a C ∞ weighting function when the process is quasiperiodic and the function f is C ∞ , a method we describe in [17][18][19]. In [19] it is proved that the limit of using WB [p] N is the same as Birkhoff's limit. Define the C ∞ weighting function w as follows.…”

Section: N )mentioning

confidence: 99%

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

Saiki

Yorke

2018

Regul. Chaot. Dyn.

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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".

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Section: N )mentioning

confidence: 99%

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

Saiki

Yorke

2018

Regul. Chaot. Dyn.

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“…4 convergences much faster to the same limit ∫ f (θ)dθ. We will formalize this claim using the main result from the companion paper [3].…”

Section: Weighted Birkhoff Averaging Wb N and Its Applicationsmentioning

confidence: 99%

“…The main convergence result we are using is Theorem 3.1. It is proved in [3] and an outline of the proof is given here in Section 4. We now state a special case of the theorem that avoids unnecessary terminology and states only the C ∞ case.…”

Section: Introductionmentioning

confidence: 99%

Quantitative quasiperiodicity

et al. 2017

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The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, B N (f ) ∶= Σ N −1 n=0 f (x n ) N of a function f along a length N ergodic trajectory (x n ) of a function T converge to the space average ∫ f dµ, where µ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f (x n ) by giving very small weights to the "end" terms when n is near 0 or N − 1. When (x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our Weighted Birkhoff average (denoted WB N (f )) converges "super" fast to ∫ f dµ with respect to the number of iterates N , i.e. with error decaying faster than N −m for every integer m. Our goal is to show that our Weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy. d j=1 a j ρ j = 0, then every a j = 0. We then say such a ρ is irrational.Let T be a C ∞ quasiperiodic map. The quasiperiodicity persists for most small perturbations by the Kolmogorov-Arnold-Moser theory. We believe that quasiperiodicity is one of only three types of invariant sets with a dense trajectory that can occur in typical smooth maps. The other two types are periodic sets and chaotic sets. See [1] for the statement of our formal conjecture of this triumvirate. For example, quasiperiodicity occurs in a system of weakly coupled oscillators, in which there is an invariant smooth *

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“…We have recently developed a method for speeding up the convergence of the Birkhoff sum in Theorem 1.2 through introducing a C ∞ weighting function by orders of magnitude when the process is quasiperiodic and the function f is C ∞ , a method we describe in [7,8,9]. In [9] it is proved that the limit of using WB [p] N is the same as Birkhoff's limit. Weighted Birkhoff (WB…”

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confidence: 99%

“…Comparison to previous work. We have written previously about computation of rotation rate in the papers [7,8,9]. A complete streamlined method for the case d = 1 is provided in Section 2; the Embedding continuation method is announced in [8], but this is the first paper in which it is explained.…”

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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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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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Super convergence of ergodic averages for quasiperiodic orbits

Cited by 35 publications

References 25 publications

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

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

Quantitative quasiperiodicity

Solving the Babylonian problem of quasiperiodic rotation rates

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