General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2010, Volume 65, Issue 2(392), Pages 71–132 (Mi umn9346)  

This article is cited in 65 scientific papers (total in 65 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.


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

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

Bibliographic databases:

Document Type: Article
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
\by A.~V.~Bolsinov, A.~V.~Borisov, I.~S.~Mamaev
\paper Topology and stability of integrable systems
\jour Uspekhi Mat. Nauk
\yr 2010
\vol 65
\issue 2(392)
\pages 71--132
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 2
\pages 259--318

Linking options:

    SHARE: FaceBook Twitter Livejournal

    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. A. V. Borisov, I. S. Mamaev, S. M. Ramodanov, “Dinamicheskaya advektsiya”, Nelineinaya dinam., 6:3 (2010), 521–530  mathnet  elib
    2. V. V. Vaskin, N. N. Erdakova, “Dinamika dvukh tochechnykh vikhrei v koltsevoi oblasti”, Nelineinaya dinam., 6:3 (2010), 531–548  mathnet  elib
    3. A. V. Vaskina, “Novye statsionarnye konfiguratsii v sisteme trekh tochechnykh vikhrei v krugovoi oblasti i ikh ustoichivost”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 4, 61–70  mathnet  elib
    4. N. N. Erdakova, “Tomsonovskie konfiguratsii v dinamike dvukh vikhrei v koltsevoi oblasti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 4, 71–76  mathnet  elib
    5. T. B. Ivanova, “Postroenie bifurkatsionnoi diagrammy i analiz ustoichivosti zhidkogo samogravitiruyuschego ellipticheskogo tsilindra s vnutrennim vrascheniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 4, 77–86  mathnet  elib
    6. A. V. Borisov, I. S. Mamaev, T. B. Ivanova, “Ustoichivost zhidkogo samogravitiruyuschego ellipticheskogo tsilindra s vnutrennim vrascheniem”, Nelineinaya dinam., 6:4 (2010), 807–822  mathnet  elib
    7. A. V. Borisov, I. S. Mamaev, A. V. Vaskina, “Novye otnositelnye ravnovesiya v sisteme trekh tochechnykh vikhrei v krugovoi oblasti i ikh ustoichivost”, Nelineinaya dinam., 7:1 (2011), 119–138  mathnet  elib
    8. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Obobschenie preobrazovaniya Chaplygina i yavnoe integrirovanie sharovogo podvesa”, Nelineinaya dinam., 7:2 (2011), 313–338  mathnet  elib
    9. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Bifurkatsionnyi analiz i indeks Konli v mekhanike”, Nelineinaya dinam., 7:3 (2011), 649–681  mathnet
    10. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Dinamika vikhrevykh kolets: chekharda, khoreografii i problema ustoichivosti”, Nelineinaya dinam., 8:1 (2012), 113–147  mathnet
    11. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov”, Nelineinaya dinam., 8:2 (2012), 289–307  mathnet
    12. Labrousse C., “Flat metrics are strict local minimizers for the polynomial entropy”, Regul. Chaotic Dyn., 17:6 (2012), 479–491  crossref  mathscinet  zmath  adsnasa  isi
    13. Bardin B.S. Rudenko T.V. Savin A.A., “On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case”, Regul. Chaotic Dyn., 17:6 (2012), 533–546  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. Bolsinov A.V., Borisov A.V., Mamaev I.S., “Bifurcation analysis and the Conley index in mechanics”, Regul. Chaotic Dyn., 17:5 (2012), 451–478  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Borisov A.V., Kilin A.A., Mamaev I.S., “How to control Chaplygin's sphere using rotors”, Regul. Chaotic Dyn., 17:3-4 (2012), 258–272  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. A. V. Borisov, I. S. Mamaev, “Topologicheskii analiz odnoi integriruemoi sistemy, svyazannoi s kacheniem shara po sfere”, Nelineinaya dinam., 8:5 (2012), 957–975  mathnet
    17. Borisov A.V., Kilin A.A., Mamaev I.S., “Generalized Chaplygin's transformation and explicit integration of a system with a spherical support”, Regul. Chaotic Dyn., 17:2 (2012), 170–190  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Molero F.J., Lara M., Ferrer S., Céspedes F., “2-D duffing oscillator: elliptic functions from a dynamical systems point of view”, Qual. Theory Dyn. Syst., 12:1 (2013), 115–139  crossref  mathscinet  zmath  isi  elib
    19. P. Lubowiecki, H. Żołądek, “The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity”, JGM, 4:4 (2013), 443–467  crossref  mathscinet  isi
    20. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov. II”, Nelineinaya dinam., 9:1 (2013), 59–76  mathnet
    21. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    22. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640  mathnet
    23. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Problema dreifa i vozvraschaemosti pri kachenii shara Chaplygina”, Nelineinaya dinam., 9:4 (2013), 721–754  mathnet
    24. Hamad M. Yehia, E. G. El-Hadidy, “On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor”, Regul. Chaotic Dyn., 18:5 (2013), 539–552  mathnet  crossref  mathscinet  zmath
    25. Alexey V. Borisov, Ivan S. Mamaev, “Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere”, Regul. Chaotic Dyn., 18:4 (2013), 356–371  mathnet  crossref  mathscinet  zmath
    26. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328  mathnet  crossref  mathscinet  zmath
    27. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem”, Regul. Chaotic Dyn., 18:1-2 (2013), 33–62  mathnet  crossref  mathscinet  zmath
    28. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “How to Control the Chaplygin Ball Using Rotors. II”, Regul. Chaotic Dyn., 18:1-2 (2013), 144–158  mathnet  crossref  mathscinet  zmath
    29. 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
    30. G. E. Smirnov, “Fokusnye osobennosti v klassicheskoi mekhanike”, Nelineinaya dinam., 10:1 (2014), 101–112  mathnet
    31. A. Izosimov, “Stability of relative equilibria of multidimensional rigid body”, Nonlinearity, 27:6 (2014), 1419–1443  crossref  mathscinet  zmath  isi
    32. A. V. Borisov, A. A. Kilin, I. S. Mamaev, V. A. Tenenev, “The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid”, Fluid Dyn. Res., 46:3 (2014), 031415  crossref  mathscinet  zmath  isi  scopus
    33. 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
    34. Clémence Labrousse, Jean-Pierre Marco, “Polynomial Entropies for Bott Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 19:3 (2014), 374–414  mathnet  crossref  mathscinet  zmath
    35. Erdakova N.N., Mamaev I.S., “On the Dynamics of Point Vortices in An Annular Region”, Fluid Dyn. Res., 46:3 (2014), 031420  crossref  mathscinet  zmath  isi  scopus
    36. A. Izosimov, “Algebraic geometry and stability for integrable systems”, Phys. D, 291 (2015), 74–82  crossref  mathscinet  zmath  isi  scopus
    37. Rosemann S. Schoebel K., “Open Problems in the Theory of Finite-Dimensional Integrable Systems and Related Fields”, J. Geom. Phys., 87 (2015), 396–414  crossref  mathscinet  zmath  isi  scopus
    38. Rasoul Akbarzadeh, Ghorbanali Haghighatdoost, “The Topology of Liouville Foliation for the BorisovMamaevSokolov Integrable Case on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 20:3 (2015), 317–344  mathnet  crossref  mathscinet  zmath  adsnasa
    39. P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317  mathnet
    40. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, International Journal of Bifurcation and Chaos, 25:10 (2015), 15300–21  mathnet  crossref  isi
    41. 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
    42. Gidea M., “Global Diffusion on a Tight Three-Sphere”, 14, no. 2, SI, 2015, 227–263  crossref  mathscinet  zmath  isi  scopus
    43. Radnovic M., “Topology of the Elliptical Billiard With the Hooke'S Potential”, 42, no. 1, 2015, 1–9  crossref  mathscinet  zmath  isi
    44. 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
    45. Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604  mathnet  crossref  mathscinet  zmath
    46. 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
    47. 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
    48. Rasoul Akbarzadeh, “Topological Analysis Corresponding to the BorisovMamaevSokolov Integrable System on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 21:1 (2016), 1–17  mathnet  crossref  mathscinet  zmath
    49. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the KowalevskiSokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65  mathnet  crossref  mathscinet  zmath
    50. 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
    51. P. E. Ryabov, E. O. Biryucheva, “Diskriminantnoe mnozhestvo i bifurkatsionnaya diagramma integriruemogo sluchaya M. Adlera i P. van Merbeke”, Nelineinaya dinam., 12:4 (2016), 633–650  mathnet  crossref  elib
    52. Borisov A.V. Mamaev I.S., “Rigid body dynamics in non-Euclidean spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454  crossref  mathscinet  zmath  isi  scopus
    53. Bolsinov A., “Singularities of Bi-Hamiltonian Systems and Stability Analysis”: Bolsinov, A MoralesRuiz, JJ Zung, NT, Geometry and Dynamics of Integrable Systems, Adv. Courses Math CRM Barc., Advanced Courses in Mathematics Crm Barcelona, Birkhauser Verlag Ag, 2016, 35–84  crossref  mathscinet  isi
    54. Ioan Caşu, Cristian Lăzureanu, “Stability and Integrability Aspects for the MaxwellBloch Equations with the Rotating Wave Approximation”, Regul. Chaotic Dyn., 22:2 (2017), 109–121  mathnet  crossref
    55. Alexander A. Kilin, Elena N. Pivovarova, “The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane”, Regul. Chaotic Dyn., 22:3 (2017), 298–317  mathnet  crossref  mathscinet
    56. Oshemkov A.A., Ryabov P.E., Sokolov S.V., “Explicit Determination of Certain Periodic Motions of a Generalized Two-Field Gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525  crossref  mathscinet  zmath  isi  scopus
    57. Sokolov S.V., “New Invariant Relations For One Critical Subsystem of a Generalized Two-Field Gyrostat”, Dokl. Phys., 62:12 (2017), 567–570  crossref  isi  scopus
    58. Sergei V. Sokolov, Pavel E. Ryabov, “Bifurcation Analysis of the Dynamics of Two Vortices in a Bose Einstein Condensate. The Case of Intensities of Opposite Signs”, Regul. Chaotic Dyn., 22:8 (2017), 976–995  mathnet  crossref
    59. Ryabov P.E., “Explicit Integration of the System of Invariant Relations For the Case of M. Adler and P. Van Moerbeke”, Dokl. Math., 95:1 (2017), 17–20  crossref  mathscinet  zmath  isi  scopus
    60. Borisov A.V., Garcia-Naranjo L.C., Mamaev I.S., Montaldi J., “Reduction and Relative Equilibria For the Two-Body Problem on Spaces of Constant Curvature”, Celest. Mech. Dyn. Astron., 130:6 (2018), UNSP 43  crossref  mathscinet  isi  scopus
    61. Sokolov S.V., Ryabov P.E., “Bifurcation Diagram of the Two Vortices in a Bose–Einstein Condensate With Intensities of the Same Signs”, Dokl. Math., 97:3 (2018), 286–290  crossref  zmath  isi  scopus
    62. Kurek R., Lubowiecki P., Zoladek H., “The Hess-Appelrot System. III. Splitting of Separatrices and Chaos”, Discret. Contin. Dyn. Syst., 38:4 (2018), 1955–1981  crossref  mathscinet  zmath  isi  scopus
    63. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability”, Regul. Chaotic Dyn., 23:5 (2018), 613–636  mathnet  crossref
    64. Bolsinov A., Guglielmi L., Kudryavtseva E., “Symplectic Invariants For Parabolic Orbits and Cusp Singularities of Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170424  crossref  isi  scopus
    65. Alexander A. Kilin, Elena N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907  mathnet  crossref
  •   Russian Mathematical Surveys
    Number of views:
    This page:928
    Full text:230
    First page:58

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019