Diffusion Limit for a Slow-Fast Standard Map

  • Alex BlumenthalEmail author
  • Jacopo De Simoi
  • Ke Zhang


Consider the map \((x, z) \mapsto (x + \epsilon ^{-\alpha } \sin (2\pi x) + \epsilon ^{-(1+\alpha )}z, z + \epsilon \sin (2\pi x))\), 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.



The authors wish to thank the anonymous referees for their comments and suggestions which substantially improved the readability of this paper.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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