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Mat. Sb. (N.S.), 1970, Volume 83(125), Number 2(10), Pages 273–312 (Mi msb3513)  

This article is cited in 16 scientific papers (total in 16 papers)

Instability in a Hamiltonian system and the distribution of asteroids

A. D. Bruno


Abstract: The formal stability of periodic solutions is investigated for a Hamiltonian system in two degrees of freedom. The nature of the zones of instability is exhibited in the case of a resonance of order $q\geqslant3$. In contrast to classical theory, an isoenergetic reduction is not carried out. This permits unstable solutions close to periodic solutions to be studied in full. The results are applied to the restricted problem of three bodies, which allows us to explain qualitatively the nature of all gaps with $q\geqslant3$ in the distribution of asteroids.
Figures: 19.
Bibliography: 37 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 12:2, 271–312

Bibliographic databases:

UDC: 517.913+521.41
MSC: 70H12, 70H14, 37J25, 37J45, 37J15, 70H33
Received: 05.02.1970

Citation: A. D. Bruno, “Instability in a Hamiltonian system and the distribution of asteroids”, Mat. Sb. (N.S.), 83(125):2(10) (1970), 273–312; Math. USSR-Sb., 12:2 (1970), 271–312

Citation in format AMSBIB
\Bibitem{Bru70}
\by A.~D.~Bruno
\paper Instability in a~Hamiltonian system and the distribution of asteroids
\jour Mat. Sb. (N.S.)
\yr 1970
\vol 83(125)
\issue 2(10)
\pages 273--312
\mathnet{http://mi.mathnet.ru/msb3513}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=274867}
\zmath{https://zbmath.org/?q=an:0217.12401}
\transl
\jour Math. USSR-Sb.
\yr 1970
\vol 12
\issue 2
\pages 271--312
\crossref{https://doi.org/10.1070/SM1970v012n02ABEH000922}


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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. V. I. Arnol'd, “Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields”, Funct. Anal. Appl., 11:2 (1977), 85–92  mathnet  crossref  mathscinet  zmath
    2. M M Dodson, J A G Vickers, J Phys A Math Gen, 19:3 (1986), 349  crossref  mathscinet  zmath  adsnasa
    3. A. D. Bruno, “The normal form of a Hamiltonian system”, Russian Math. Surveys, 43:1 (1988), 25–66  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. D. Bruno, “Normalization of a Hamiltonian system near an invariant cycle or torus”, Russian Math. Surveys, 44:2 (1989), 53–89  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco, “Sorting approach to magnetic random errors”, Phys Rev E, 52:3 (1995), 3093  crossref  adsnasa  isi
    6. E Todesco, “Local analysis of formal stability and existence of fixed points in 4d symplectic mappings”, Physica D: Nonlinear Phenomena, 95:1 (1996), 1  crossref
    7. Henning Schomerus, Martin Sieber, J Phys A Math Gen, 30:13 (1997), 4537  crossref  mathscinet  zmath
    8. Martin Sieber, J Phys A Math Gen, 30:13 (1997), 4563  crossref  mathscinet  zmath
    9. Henning Schomerus, J Phys A Math Gen, 31:18 (1998), 4167  crossref  mathscinet  zmath
    10. A M Ozorio de Almeida, Wei-Mou Zheng, J Phys A Math Gen, 31:23 (1998), L441  crossref  mathscinet  zmath  adsnasa
    11. Martin Sieber, Henning Schomerus, J Phys A Math Gen, 31:1 (1998), 165  crossref  zmath  adsnasa
    12. P. Lebœuf, A. Mouchet, “Normal Forms and Complex Periodic Orbits in Semiclassical Expansions of Hamiltonian Systems”, Annals of Physics, 275:1 (1999), 54  crossref
    13. J. Kaidel, M. Brack, “Semiclassical trace formulas for pitchfork bifurcation sequences”, Phys Rev E, 70:1 (2004), 016206  crossref  isi
    14. J. P. Keating, S. D. Prado, M. Sieber, “Universal quantum signature of mixed dynamics in antidot lattices”, Phys Rev B, 72:24 (2005), 245334  crossref  adsnasa  isi
    15. A. G. Magner, K.-i. Arita, S. N. Fedotkin, “Semiclassical Approach for Bifurcations in a Smooth Finite-Depth Potential”, Progress of Theoretical Physics, 115:3 (2006), 523  crossref
    16. Bryuno A.D., Varin V.P., “O raspredelenii asteroidov po srednim dvizheniyam”, Astronomicheskii vestnik, 45:4 (2011), 334–340  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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