|
This article is cited in 66 scientific papers (total in 66 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:
     
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
\Bibitem{BolBorMam10}
\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
\mathnet{http://mi.mathnet.ru/umn9346}
\crossref{https://doi.org/10.4213/rm9346}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2668801}
\zmath{https://zbmath.org/?q=an:1202.37077}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010RuMaS..65..259B}
\elib{http://elibrary.ru/item.asp?id=20425353}
\transl
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 2
\pages 259--318
\crossref{https://doi.org/10.1070/RM2010v065n02ABEH004672}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281639500002}
\elib{http://elibrary.ru/item.asp?id=16976135}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77958527892}
Linking options:
http://mi.mathnet.ru/eng/umn9346https://doi.org/10.4213/rm9346 http://mi.mathnet.ru/eng/umn/v65/i2/p71
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Remarks
This publication is cited in the following articles:
-
A. V. Borisov, I. S. Mamaev, S. M. Ramodanov, “Dinamicheskaya advektsiya”, Nelineinaya dinam., 6:3 (2010), 521–530
 -
V. V. Vaskin, N. N. Erdakova, “Dinamika dvukh tochechnykh vikhrei v koltsevoi oblasti”, Nelineinaya dinam., 6:3 (2010), 531–548
-
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
-
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
-
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
-
A. V. Borisov, I. S. Mamaev, T. B. Ivanova, “Ustoichivost zhidkogo samogravitiruyuschego ellipticheskogo tsilindra s vnutrennim vrascheniem”, Nelineinaya dinam., 6:4 (2010), 807–822
-
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
-
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Obobschenie preobrazovaniya Chaplygina i yavnoe integrirovanie sharovogo podvesa”, Nelineinaya dinam., 7:2 (2011), 313–338
-
A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Bifurkatsionnyi analiz i indeks Konli v mekhanike”, Nelineinaya dinam., 7:3 (2011), 649–681
-
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Dinamika vikhrevykh kolets: chekharda, khoreografii i problema ustoichivosti”, Nelineinaya dinam., 8:1 (2012), 113–147
-
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov”, Nelineinaya dinam., 8:2 (2012), 289–307
-
Labrousse C., “Flat metrics are strict local minimizers for the polynomial entropy”, Regul. Chaotic Dyn., 17:6 (2012), 479–491
-
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
-
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
-
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
-
A. V. Borisov, I. S. Mamaev, “Topologicheskii analiz odnoi integriruemoi sistemy, svyazannoi s kacheniem shara po sfere”, Nelineinaya dinam., 8:5 (2012), 957–975
-
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
-
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
-
P. Lubowiecki, H. Żołądek, “The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity”, JGM, 4:4 (2013), 443–467
-
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov. II”, Nelineinaya dinam., 9:1 (2013), 59–76
-
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
-
A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640
-
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Problema dreifa i vozvraschaemosti pri kachenii shara Chaplygina”, Nelineinaya dinam., 9:4 (2013), 721–754
-
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
-
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
-
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
-
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
-
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
-
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
-
G. E. Smirnov, “Fokusnye osobennosti v klassicheskoi mekhanike”, Nelineinaya dinam., 10:1 (2014), 101–112
-
A. Izosimov, “Stability of relative equilibria of multidimensional rigid body”, Nonlinearity, 27:6 (2014), 1419–1443
-
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
-
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
-
Clémence Labrousse, Jean-Pierre Marco, “Polynomial Entropies for Bott Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 19:3 (2014), 374–414
-
Erdakova N.N., Mamaev I.S., “On the Dynamics of Point Vortices in An Annular Region”, Fluid Dyn. Res., 46:3 (2014), 031420
-
A. Izosimov, “Algebraic geometry and stability for integrable systems”, Phys. D, 291 (2015), 74–82
-
Rosemann S. Schoebel K., “Open Problems in the Theory of Finite-Dimensional Integrable Systems and Related Fields”, J. Geom. Phys., 87 (2015), 396–414
-
Rasoul Akbarzadeh, Ghorbanali Haghighatdoost, “The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 20:3 (2015), 317–344
-
P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317
-
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
-
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
-
Gidea M., “Global Diffusion on a Tight Three-Sphere”, 14, no. 2, SI, 2015, 227–263
-
Radnovic M., “Topology of the Elliptical Billiard With the Hooke'S Potential”, 42, no. 1, 2015, 1–9
-
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
-
Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604
-
E. O. Kantonistova, “Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution”, Sb. Math., 207:3 (2016), 358–399
-
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
-
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
-
Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the Kowalevski–Sokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65
-
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
-
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
-
Borisov A.V. Mamaev I.S., “Rigid body dynamics in non-Euclidean spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454
-
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
-
Ioan Caşu, Cristian Lăzureanu, “Stability and Integrability Aspects for the Maxwell–Bloch Equations with the Rotating Wave Approximation”, Regul. Chaotic Dyn., 22:2 (2017), 109–121
-
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
-
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
-
Sokolov S.V., “New Invariant Relations For One Critical Subsystem of a Generalized Two-Field Gyrostat”, Dokl. Phys., 62:12 (2017), 567–570
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
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
-
Bizyaev I.A., Borisov A.V., Mamaev I.S., “Dynamics of the Chaplygin Ball on a Rotating Plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433
|
| Number of views: |
| This page: | 949 | | Full text: | 243 | | References: | 94 | | First page: | 58 |
|
Terms of Use