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Mat. Sb., 1994, Volume 185, Number 6, Pages 113–124 (Mi msb906)  

This article is cited in 24 scientific papers (total in 25 papers)

Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi–Ulam problem

L. D. Pustyl'nikov


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.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:1, 231–241

Bibliographic databases:

UDC: 517.928.7+517.938.5
MSC: Primary 58F10, 58F13, 58F05; Secondary 82C05
Received: 16.01.1993

Citation: L. D. Pustyl'nikov, “Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi–Ulam problem”, Mat. Sb., 185:6 (1994), 113–124; Russian Acad. Sci. Sb. Math., 82:1 (1995), 231–241

Citation in format AMSBIB
\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 Mat. Sb.
\yr 1994
\vol 185
\issue 6
\pages 113--124
\mathnet{http://mi.mathnet.ru/msb906}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1280400}
\zmath{https://zbmath.org/?q=an:0854.58028}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 1
\pages 231--241
\crossref{https://doi.org/10.1070/SM1995v082n01ABEH003561}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RR54800012}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. L. D. Pustyl'nikov, “Poincaré models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism”, Russian Math. Surveys, 50:1 (1995), 145–189  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Rafael Ortega, Nonlinearity, 10:1 (1997), 195  crossref  mathscinet  zmath  isi
    3. Loskutov A., Ryabov A., Akinshin L., “Mechanism of Fermi Acceleration in Dispersing Billiards with Time-Dependent Boundaries”, J. Exp. Theor. Phys., 89:5 (1999), 966–974  crossref  adsnasa  isi
    4. Loskutov A., Ryabov A., Akinshin L., “Properties of Some Chaotic Billiards with Time-Dependent Boundaries”, J. Phys. A-Math. Gen., 33:44 (2000), 7973–7986  crossref  mathscinet  zmath  adsnasa  isi
    5. L. D. Pustylnikov, “Dinamicheskie sistemy s uprugimi otrazheniyami i mekhanizm uskoreniya Fermi”, Matem. prosv., ser. 3, 8, Izd-vo MTsNMO, M., 2004, 164–180  mathnet
    6. Edson Leonel, P. McClintock, J. da Silva, “Fermi-Ulam Accelerator Model under Scaling Analysis”, Phys Rev Letters, 93:1 (2004), 014101  crossref  mathscinet  isi  elib
    7. Deryabin M. Pustyl'nikov L., “Exponential Attractors in Generalized Relativistic Billiards”, Commun. Math. Phys., 248:3 (2004), 527–552  crossref  mathscinet  zmath  adsnasa  isi
    8. Edson D Leonel, P V E McClintock, “A hybrid Fermi–Ulam-bouncer model”, J Phys A Math Gen, 38:4 (2005), 823–839  crossref  zmath  isi  elib
    9. Edson D Leonel, P V E McClintock, “A crisis in the dissipative Fermi accelerator model”, J Phys A Math Gen, 38:23 (2005), L425–L430  crossref  mathscinet  zmath  isi
    10. 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  crossref  adsnasa  isi
    11. A. Yu. Loskutov, “Dynamical chaos: systems of classical mechanics”, Phys. Usp., 50:9 (2007), 939–964  mathnet  crossref  crossref  adsnasa  isi  elib
    12. 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  crossref
    13. F. Lenz, F. K. Diakonos, P. Schmelcher, “Tunable Fermi Acceleration in the Driven Elliptical Billiard”, Phys Rev Letters, 100:1 (2008), 014103  crossref  adsnasa  isi
    14. 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  crossref
    15. A. Yu. Loskutov, A. B. Ryabov, A. K. Krasnova, O. A. Chichigina, “Bilyardy s vozmuschaemymi granitsami i nekotorye ikh svoistva”, Nelineinaya dinam., 6:3 (2010), 573–604  mathnet  elib
    16. 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  crossref
    17. 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  crossref
    18. V. Gelfreich, V. Rom-Kedar, D. Turaev, “Fermi acceleration and adiabatic invariants for non-autonomous billiards”, Chaos, 22:3 (2012), 033116  crossref
    19. Oliveira D.F.M., Robnik M., “Scaling Invariance in a Time-Dependent Elliptical Billiard”, Int. J. Bifurcation Chaos, 22:9 (2012), 1250207  crossref  zmath  isi
    20. de Simoi J., Dolgopyat D., “Dynamics of Some Piecewise Smooth Fermi-Ulam Models”, Chaos, 22:2 (2012), 026124  crossref  adsnasa  isi
    21. L. D. Pustylnikov, M. V. Deryabin, “Chërnye dyry i obobschënnye relyativistskie billiardy”, Preprinty IPM im. M. V. Keldysha, 2013, 054, 36 pp.  mathnet
    22. De Simoi J., “Fermi Acceleration in Anti-Integrable Limits of the Standard Map”, Commun. Math. Phys., 321:3 (2013), 703–745  crossref  isi
    23. V Gelfreich, V Rom-Kedar, D Turaev, “Oscillating mushrooms: adiabatic theory for a non-ergodic system”, J. Phys. A: Math. Theor, 47:39 (2014), 395101  crossref
    24. 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  crossref
    25. T. Pereira, D. Turaev, D.C. Wunsch, G. Fridman, J. Levesley, I. Tyukin, “Fast Fermi Acceleration and Entropy Growth”, Math. Model. Nat. Phenom, 10:3 (2015), 31  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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