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

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

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
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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
2. M M Dodson, J A G Vickers, J Phys A Math Gen, 19:3 (1986), 349
3. A. D. Bruno, “The normal form of a Hamiltonian system”, Russian Math. Surveys, 43:1 (1988), 25–66
4. A. D. Bruno, “Normalization of a Hamiltonian system near an invariant cycle or torus”, Russian Math. Surveys, 44:2 (1989), 53–89
5. M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco, “Sorting approach to magnetic random errors”, Phys Rev E, 52:3 (1995), 3093
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
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11. Martin Sieber, Henning Schomerus, J Phys A Math Gen, 31:1 (1998), 165
12. P. Lebœuf, A. Mouchet, “Normal Forms and Complex Periodic Orbits in Semiclassical Expansions of Hamiltonian Systems”, Annals of Physics, 275:1 (1999), 54
13. J. Kaidel, M. Brack, “Semiclassical trace formulas for pitchfork bifurcation sequences”, Phys Rev E, 70:1 (2004), 016206
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
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
16. Bryuno A.D., Varin V.P., “O raspredelenii asteroidov po srednim dvizheniyam”, Astronomicheskii vestnik, 45:4 (2011), 334–340
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