Abstract:
The Ulam model is studied in this paper: a small elastic ball moves vertically between two infinitely heavy horizontal walls, each of which moves in the vertical direction according to a periodic law. It is proved that the velocity of the ball is always bounded. The proof is based on a generalization of Moser's theorem on the existence of invariant curves under an area preserving mapping of an annulus.
Citation:
L. D. Pustyl'nikov, “Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi–Ulam problem”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 231–241
\Bibitem{Pus94}
\by L.~D.~Pustyl'nikov
\paper Existence of invariant curves for maps close to degenerate maps, and a~solution of the~Fermi--Ulam problem
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 1
\pages 231--241
\mathnet{http://mi.mathnet.ru/eng/sm906}
\crossref{https://doi.org/10.1070/SM1995v082n01ABEH003561}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1280400}
\zmath{https://zbmath.org/?q=an:0854.58028}
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Linking options:
https://www.mathnet.ru/eng/sm906
https://doi.org/10.1070/SM1995v082n01ABEH003561
https://www.mathnet.ru/eng/sm/v185/i6/p113
This publication is cited in the following 29 articles:
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Danilo S. Rando, Arturo C. Martí, Edson D. Leonel, “Bifurcations, relaxation time, and critical exponents in a dissipative or conservative Fermi model”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 33:2 (2023)
Joelson D.V. Hermes, Marcelo A. dos Reis, Iberê L. Caldas, Edson D. Leonel, “Break-up of invariant curves in the Fermi-Ulam model”, Chaos, Solitons & Fractals, 162 (2022), 112410
D.F..M. Oliveira, Mario Roberto Silva, E.D.. Leonel, “A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping”, Physica A: Statistical Mechanics and its Applications, 2015
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L. D. Pustylnikov, M. V. Deryabin, “Chërnye dyry i obobschënnye relyativistskie billiardy”, Preprinty IPM im. M. V. Keldysha, 2013, 054, 36 pp.
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Diego F. M. Oliveira, Edson D. Leonel, “In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas”, Chaos, 22:2 (2012), 026123
V. Gelfreich, V. Rom-Kedar, D. Turaev, “Fermi acceleration and adiabatic invariants for non-autonomous billiards”, Chaos, 22:3 (2012), 033116
Oliveira D.F.M., Robnik M., “Scaling Invariance in a Time-Dependent Elliptical Billiard”, Int. J. Bifurcation Chaos, 22:9 (2012), 1250207
de Simoi J., Dolgopyat D., “Dynamics of Some Piecewise Smooth Fermi-Ulam Models”, Chaos, 22:2 (2012), 026124
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Alexander Loskutov, Alexei Ryabov, E.D.. Leonel, “Separation of particles in time-dependent focusing billiards”, Physica A: Statistical Mechanics and its Applications, 389:23 (2010), 5408
F. Lenz, F. K. Diakonos, P. Schmelcher, “Tunable Fermi Acceleration in the Driven Elliptical Billiard”, Phys Rev Letters, 100:1 (2008), 014103
Denis Gouvêa Ladeira, Jafferson Kamphorst Leal da Silva, “Scaling of dynamical properties of the Fermi–Ulam accelerator”, Physica A: Statistical Mechanics and its Applications, 387:23 (2008), 5707
A. Yu. Loskutov, “Dynamical chaos: systems of classical mechanics”, Phys. Usp., 50:9 (2007), 939–964
Edson D. Leonel, Diego F.M. Oliveira, R. Egydio de Carvalho, “Scaling properties of the regular dynamics for a dissipative bouncing ball model”, Physica A: Statistical Mechanics and its Applications, 386:1 (2007), 73
Denis Gouvêa Ladeira, Jafferson Kamphorst Leal da Silva, “Time-dependent properties of a simplified Fermi-Ulam accelerator model”, Phys Rev E, 73:2 (2006), 026201
Edson D Leonel, P V E McClintock, “A hybrid Fermi–Ulam-bouncer model”, J Phys A Math Gen, 38:4 (2005), 823–839
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