Diffusion limit for a slow-fast standard map

Consider the map \((x, y) \mapsto (x + \epsilon^{-\alpha} \sin (2\pi x) + \epsilon^{-1-\alpha}z, z + \epsilon \sin(2\pi x))\), which is conjugate to the Chirikov standard map with a large parameter. The parameter value \(\alpha = 1\) is related to "scattering by resonance" phenomena. For s...

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Bibliographic Details
Published in:arXiv.org 2018-09
Main Authors: Blumenthal, Alex, De Simoi, Jacopo, Zhang, Ke
Format: Article
Language:English
Subjects:
Diffusion rate
Parameters
Resonance scattering
Theorems
Online Access:Get full text
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Summary:Consider the map \((x, y) \mapsto (x + \epsilon^{-\alpha} \sin (2\pi x) + \epsilon^{-1-\alpha}z, z + \epsilon \sin(2\pi x))\), which is conjugate to the Chirikov standard map with a large parameter. The parameter value \(\alpha = 1\) is related to "scattering by resonance" phenomena. For suitable \(\alpha\), we obtain a central limit theorem for the slow variable \(z\) for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a "finite-time" decay of correlations result.
ISSN:2331-8422
DOI:10.48550/arxiv.1806.06398