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A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems

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Abstract

In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43, 1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.

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Authors and Affiliations

  1. Universitat Politècnica de Catalunya, Barcelona, Spain

    M. Guardia

  2. University of Maryland at College Park, College Park, USA

    V. Kaloshin

  3. University of Toronto, Toronto, Canada

    J. Zhang

Authors
  1. M. Guardia
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  2. V. Kaloshin
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  3. J. Zhang
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Correspondence to V. Kaloshin.

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Communicated by C. Liverani

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Guardia, M., Kaloshin, V. & Zhang, J. A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems. Commun. Math. Phys. 348, 321–361 (2016). https://doi.org/10.1007/s00220-016-2705-9

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  • DOI: https://doi.org/10.1007/s00220-016-2705-9

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