Small denominators. I. Mapping of the circumference onto itself

doi:10.1007/978-3-642-01742-1_10

Cited by 9 publications

(5 citation statements)

References 8 publications

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“…The Diophantine class is crucial in the study of quasiperiodic behavior, for example in [8,9]. In equation ( 6) is central to Arnold's study of small denominators in [10].…”

Section: Diophantine Rotationsmentioning

confidence: 99%

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Super convergence of ergodic averages for quasiperiodic orbits

Das

Yorke

2018

Nonlinearity

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The Birkhoff Ergodic Theorem asserts that time averages of a function evaluated along a trajectory of length N converge to the space average, the integral of f , as N → ∞, for ergodic dynamical systems. But that convergence can be slow. Instead of uniform averages that assign equal weights to points along the trajectory, we use an average with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint N 2. We show that in quasiperiodic dynamical systems, our weighted averages converge far faster provided f is sufficiently differentiable. This result can be applied to obtain efficient numerical computation of rotation numbers, invariant densities and conjugacies of quasiperiodic systems.

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“…The Diophantine class is crucial in the study of quasiperiodic behavior, for example in [8,9]. In equation ( 6) is central to Arnold's study of small denominators in [10].…”

Section: Diophantine Rotationsmentioning

confidence: 99%

“…Since f is a C M -function, differentiating it M times gives |a k | C f ,M k −M , for each k ∈ Z d 0 for some C f ,M > 0 that depends only on f and M. In equation (10) gives…”

Section: Lemma 32 (Poisson Summation Formula [23]) For Eachmentioning

confidence: 99%

Super convergence of ergodic averages for quasiperiodic orbits

Das

Yorke

2018

Nonlinearity

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“…We now describe an example [32][33][34] of a two-dimensional quasiperiodic torus map on T 2 . This is a two-dimensional version of Arnold's family of one-dimensional maps (see [35]). The map is given by (T 1 , T 2 ) where…”

Section: A Two-dimensional Torus Mapmentioning

confidence: 99%

“…Suppose we wish to compute the Fourier coefficients of f and determine how accurate the result is. The computed kth coefficient, denoted âk is â|k| = â±k = WB N ( f (θ)e ±2πik•θ ) = m a m ψ N,m±k (35) which has significant contributions from m ± k = 0 and |m ± k| = k * . Since k * 1, we can ignore a ±k , and we conclude that for small integers n, â|k * ±n| ≈ a |n| ψ N,k * .…”

Section: 23mentioning

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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“…In addition, the authors in [33] claimed that the coupling strength for entanglement to exist in steady state must be larger than a critical value. Consequently, the entanglement tongue appears, which is the quantum analog of the Arnold tongue [5]. Finally, we refer the reader to [41] in which the authors showed that synchronization and entanglement in quantum regime can be induced simultaneously in the presence of a random network, and [58] for giving a hint of a common mechanism between classical synchronization and quantum entanglement.…”

Section: Introductionmentioning

confidence: 99%

Emergence of synchronous behaviors for the Schrödinger–Lohe model with frustration

Kim

2019

Nonlinearity

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We introduce the Schrödinger-Lohe model (S-L) with interaction frustration which is a toy model for quantum synchronization, and study its emergent collective dynamics. Our proposed model might not capture quantum entanglement which is one of genuine quantum phenomenon as it is, however it exhibits collective synchronous behaviors and can be reduced to the Kuramoto model with frustration in the case of spatial homogeneity and unit mass. For the emergent dynamics of the model, we present two sufficient frameworks leading to the complete and practical synchronizations for identical and non-identical oscillators, respectively, in terms of frustration function, system parameters and initial data. When the frustration functions admit real diagonal perturbations, complete synchronization emerges for identical ensemble with some class of initial data, in contrast, for non-identical ensemble, we can expect practical synchronization in which the diameter for wave functions can be controlled by tuning the coupling strength. For purely imaginary valued frustrations, we analytically show that complete or practical synchronization cannot occur for generic initial data. This illustrates that the S-L model exhibits some interesting dynamic patterns different from the Kuramoto synchronization. We provide several numerical examples which manifest periodic behaviors, reminiscent of the Kuramoto model with cosine coupling.

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Small denominators. I. Mapping of the circumference onto itself

Cited by 9 publications

References 8 publications

Super convergence of ergodic averages for quasiperiodic orbits

Super convergence of ergodic averages for quasiperiodic orbits

Quantitative quasiperiodicity

Emergence of synchronous behaviors for the Schrödinger–Lohe model with frustration

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