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arXiv · 1806.06398

Diffusion limit for a slow-fast standard map

Alex Blumenthal·Jacopo De Simoi·Ke Zhang
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Abstract

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.

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Alex Blumenthal, Jacopo De Simoi, Ke Zhang. 2018-06-17. Diffusion limit for a slow-fast standard map. https://doi.org/10.1007/s00220-019-03600-7

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