Boundaries of Siegel disks: Numerical studies of their dynamics and regularity

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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“…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.…”

Section: A Two-dimensional Torus Mapmentioning

confidence: 92%

Quantitative quasiperiodicity

et al. 2017

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“…See Siegel [6]. Also see [5,7,8] and references therein for results on the Siegel disk including numerical computations.…”

Section: Siegel Disk and The Conjugacymentioning

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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Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map

Calleja

Figueras

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We perform a numerical study of the breakdown of hyperbolicity of quasi-periodic attractors in the dissipative standard map. In this study, we compute the quasi-periodic attractors together with their stable and tangent bundles. We observe that the loss of normal hyperbolicity comes from the collision of the stable and tangent bundles of the quasi-periodic attractor. We provide numerical evidence that, close to the breakdown, the angle between the invariant bundles has a linear behavior with respect to the perturbing parameter. This linear behavior agrees with the universal asymptotics of the general framework of breakdown of hyperbolic quasi-periodic tori in skew product systems.

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Boundaries of Siegel disks: Numerical studies of their dynamics and regularity

Cited by 3 publications

References 36 publications

Quantitative quasiperiodicity

Quantitative quasiperiodicity

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

Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map

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