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Mat. Sb., 2007, Volume 198, Number 3, Pages 91–136 (Mi msb1484)  

This article is cited in 14 scientific papers (total in 15 papers)

Fractional monodromy in the case of arbitrary resonances

N. N. Nekhoroshevab

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b University of Milan

Abstract: The existence of fractional monodromy is proved for the compact Lagrangian fibration on a symplectic 4-manifold that corresponds to two oscillators with arbitrary non-trivial resonant frequencies. Here one means by the monodromy corresponding to a loop in the total space of the fibration the transformation of the fundamental group of a regular fibre, which is diffeomorphic to the 2-torus. In the example under consideration the fibration is defined by a pair of functions in involution, one of which is the Hamiltonian of the system of two oscillators with frequency ratio $m_1:(-m_2)$, where $m_1$, $m_2$ are arbitrary coprime positive integers distinct from the trivial pair $m_1=m_2=1$. This is a generalization of the result on the existence of fractional monodromy in the case $m_1=1$, $m_2=2$ published before.
Bibliography: 39 titles.

DOI: https://doi.org/10.4213/sm1484

Full text: PDF file (875 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2007, 198:3, 383–424

Bibliographic databases:

UDC: 514.7+517.925
MSC: 37J35, 58K10
Received: 22.12.2005

Citation: N. N. Nekhoroshev, “Fractional monodromy in the case of arbitrary resonances”, Mat. Sb., 198:3 (2007), 91–136; Sb. Math., 198:3 (2007), 383–424

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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. Giacobbe A., “Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems”, Reg. Chaot. Dyn., 12:6 (2007), 717–731  crossref  mathscinet  zmath  isi  elib  scopus
    2. Sugny D., Mardešić P., Pelletier M., Jebrane A., Jauslin H.R., “Fractional Hamiltonian monodromy from a Gauss–Manin monodromy”, J. Math. Phys., 49:4 (2008), 042701, 35 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Nekhoroshev N., “Fuzzy fractional monodromy and the section-hyperboloid”, Milan J. Math., 76:1 (2008), 1–14  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. M. Abramov, V. I. Arnol'd, A. V. Bolsinov, A. N. Varchenko, L. Galgani, B. I. Zhilinskii, Yu. S. Il'yashenko, V. V. Kozlov, A. I. Neishtadt, V. I. Piterbarg, A. G. Khovanskii, V. V. Yashchenko, “Nikolai Nikolaevich Nekhoroshev (obituary)”, Russian Math. Surveys, 64:3 (2009), 561–566  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Efstathiou K., Sugny D., “Integrable Hamiltonian systems with swallowtails”, J. Phys. A, 43:8 (2010), 085216, 25 pp.  crossref  mathscinet  zmath  adsnasa  isi
    6. Efstathiou K., Sadovskií D., “Normalization and global analysis of perturbations of the hydrogen atom”, Rev. Mod. Phys., 82:3 (2010), 2099–2154  crossref  adsnasa  isi  elib  scopus
    7. Lukina O., Efstathiou K., Lukina O., “A geometric fractional monodromy theorem”, Discrete Contin. Dyn. Syst. Ser. S, 3:4 (2010), 517–532  crossref  mathscinet  zmath  elib  scopus
    8. Schmidt S., Dullin H.R., “Dynamics near the $p{:}-q$ resonance”, Phys. D, 239:19 (2010), 1884–1891  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. K. Efstathiou, H. W. Broer, “Uncovering Fractional Monodromy”, Commun. Math. Phys, 324:2 (2013), 549  crossref  mathscinet  zmath  isi  elib  scopus
    10. D. I. Tonkonog, “A simple proof of the “geometric fractional monodromy theorem””, Moscow University Mathematics Bulletin, 68:2 (2013), 118–121  mathnet  crossref  mathscinet
    11. Dmitrií A. Sadovskií, “Nekhoroshev’s Approach to Hamiltonian Monodromy”, Regul. Chaotic Dyn., 21:6 (2016), 720–758  mathnet  crossref  mathscinet
    12. N. N. Nekhoroshev, “Monodromiya sloya s ostsillyatornoi osoboi tochkoi tipa $1:(-2)$”, Nelineinaya dinam., 12:3 (2016), 413–541  mathnet  crossref  elib
    13. N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russian Math. Surveys, 72:3 (2017), 513–546  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. Martynchuk N. Efstathiou K., “Parallel Transport Along Seifert Manifolds and Fractional Monodromy”, Commun. Math. Phys., 356:2 (2017), 427–449  crossref  mathscinet  zmath  isi  scopus
    15. Reshetikhin N., “Semiclassical Geometry of Integrable Systems”, J. Phys. A-Math. Theor., 51:16 (2018), 164001  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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