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Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems

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Abstract

A family of discontinuous symplectic maps arising naturally in the study of nonsmooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general description of the map and a construction of nontrivial unbounded solutions for the special case of the pinball transformation. An asymptotic expansion of the pinball map in the limit of large energy is derived and used for the construction of unbounded solutions. For the generic values of the parameters, in the large energy limit, the map behaves similarly to another one considered earlier by Kesten (Acta Arith 1966).

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080, USA

    Maxim Arnold

  2. Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Science, Moscow, Russia

    Maxim Arnold

  3. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Vadim Zharnitsky

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  1. Maxim Arnold
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  2. Vadim Zharnitsky
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Correspondence to Vadim Zharnitsky.

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Communicated by M. Lyubich

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Arnold, M., Zharnitsky, V. Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems. Commun. Math. Phys. 338, 501–521 (2015). https://doi.org/10.1007/s00220-015-2386-9

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  • DOI: https://doi.org/10.1007/s00220-015-2386-9

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