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Uspekhi Mat. Nauk, 2010, Volume 65, Issue 2(392), Pages 71–132 (Mi umn9346)  
This article is cited in 77 scientific papers (total in 78 papers)

Topology and stability of integrable systems

A. V. Bolsinovab, A. V. Borisovc, I. S. Mamaevc

a M. V. Lomonosov Moscow State University
b School of Mathematics, Loughborough University, UK
c Institute of Computer Science, Izhevsk

Abstract: In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler–Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.
Bibliography: 82 titles.

Keywords: topology, stability, periodic trajectory, critical set, bifurcation set, bifurcation diagram.

DOI: https://doi.org/10.4213/rm9346

Full text: PDF file (1586 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2010, 65:2, 259–318

Bibliographic databases:

UDC: 517.925+517.938.5
MSC: Primary 37-02; Secondary 37J05, 37J20, 37J25, 37J35, 70E40, 70E50, 70G40, 7
Received: 19.01.2010

Citation: A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Uspekhi Mat. Nauk, 65:2(392) (2010), 71–132; Russian Math. Surveys, 65:2 (2010), 259–318

Citation in format AMSBIB
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    This publication is cited in the following articles:
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    30. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859  mathnet  crossref  mathscinet  zmath
    31. G. E. Smirnov, “Fokusnye osobennosti v klassicheskoi mekhanike”, Nelineinaya dinam., 10:1 (2014), 101–112  mathnet
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    34. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Superintegrable Generalizations of the Kepler and Hook Problems”, Regul. Chaotic Dyn., 19:3 (2014), 415–434  mathnet  crossref  mathscinet  zmath
    35. Clémence Labrousse, Jean-Pierre Marco, “Polynomial Entropies for Bott Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 19:3 (2014), 374–414  mathnet  crossref  mathscinet  zmath
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    40. P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317  mathnet
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    42. V. Dragović, M. Radnović, “Topological invariants for elliptical billiards and geodesics on ellipsoids in the Minkowski space”, J. Math. Sci., 223:6 (2017), 686–694  mathnet  crossref  mathscinet  elib
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    47. Ivan A. Bizyaev, Alexey V. Borisov, Alexey O. Kazakov, “Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors”, Regul. Chaotic Dyn., 20:5 (2015), 605–626  mathnet  crossref  mathscinet  zmath  elib
    48. E. O. Kantonistova, “Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution”, Sb. Math., 207:3 (2016), 358–399  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    49. A. V. Borisov, P. E. Ryabov, S. V. Sokolov, “Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid”, Math. Notes, 99:6 (2016), 834–839  mathnet  crossref  crossref  mathscinet  isi  elib
    50. Rasoul Akbarzadeh, “Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 21:1 (2016), 1–17  mathnet  crossref  mathscinet  zmath
    51. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the Kowalevski–Sokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65  mathnet  crossref  mathscinet  zmath
    52. Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov, “The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram”, Regul. Chaotic Dyn., 21:5 (2016), 581–592  mathnet  crossref  mathscinet  zmath
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