General information
Latest issue
Impact factor

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Regul. Chaotic Dyn.:

Personal entry:
Save password
Forgotten password?

Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 718–733 (Mi rcd194)  

This article is cited in 30 scientific papers (total in 30 papers)

The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top

Alexey V. Borisovab, Alexey O. Kazakovc, Igor R. Sataevd

a Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
d Saratov Branch of Kotelnikovís Institute of Radio-Engineering and Electronics of RAS ul. Zelenaya 38, Saratov, 410019 Russia

Abstract: In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
Russian Science Foundation 14-12-00811
The work of A. V. Borisov was supported by the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the state assignment to institutions of higher education. The work of A. O. Kazakov on Section 3 was supported by the grant of the Russian Scientific Foundation No 14-12- 00811, the work of Section 4.1 was partially supported by the grant of the Russian Scientific Foundation 14-41- 00044 and by the grant of the President of the Russian Federation for support of young doctors of science MD-2324.2013.1. The remaining part of the work of A. O. Kazakov was supported by the Ministry of Education and Science (project No 2000). The work of I. R. Sataev was supported by the grant of the President of the Russian Federation for support of leading scientific schools NSh-1726.2014.2.


References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
MSC: 37J60, 37N15, 37G35, 70E18, 70F25, 70H45
Received: 26.08.2014
Language: English

Citation: Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733

Citation in format AMSBIB
\by Alexey~V.~Borisov, Alexey~O.~Kazakov, Igor~R.~Sataev
\paper The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 718--733

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. Gonchenko, S. Gonchenko, A. Kazakov, D. Turaev, “Simple scenarios of onset of chaos in three-dimensional maps”, Int. J. Bifurcation Chaos, 24:8 (2014), 1440005  crossref  mathscinet  zmath  isi  scopus
    2. 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
    3. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. I”, Regul. Chaotic Dyn., 20:3 (2015), 205–224  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    4. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “The Jacobi Integral in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:3 (2015), 383–400  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    5. Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    6. Sergey P. Kuznetsov, “Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models”, Regul. Chaotic Dyn., 20:3 (2015), 345–382  mathnet  crossref  mathscinet  zmath  adsnasa
    7. Nikolay A. Kudryashov, “Analytical Solutions of the Lorenz System”, Regul. Chaotic Dyn., 20:2 (2015), 123–133  mathnet  crossref  mathscinet  zmath  adsnasa
    8. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172  mathnet  crossref  mathscinet  zmath  adsnasa
    9. Alexander P. Kuznetsov, Natalia A. Migunova, Igor R. Sataev, Yuliya V. Sedova, Ludmila V. Turukina, “From Chaos to Quasi-Periodicity”, Regul. Chaotic Dyn., 20:2 (2015), 189–204  mathnet  crossref  mathscinet  zmath  adsnasa
    10. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453  crossref  mathscinet  zmath  isi  scopus
    11. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    12. V. A. Tenenev, E. V. Vetchanin, L. F. Ilaletdinov, “Khaoticheskaya dinamika v zadache o padenii tela vintovoi formy v zhidkosti”, Nelineinaya dinam., 12:1 (2016), 99–120  mathnet
    13. I. R. Sataev, A. O. Kazakov, “Stsenarii perekhoda k khaosu v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 12:2 (2016), 235–250  mathnet  elib
    14. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248  mathnet  crossref  mathscinet  zmath  elib
    15. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period”, Regul. Chaotic Dyn., 21:4 (2016), 455–476  mathnet  crossref
    16. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.  mathnet  crossref
    17. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    18. Alexey V. Borisov, Sergey P. Kuznetsov, “Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts”, Regul. Chaotic Dyn., 21:7-8 (2016), 792–803  mathnet  crossref
    19. Alexey V. Borisov, Alexey O. Kazakov, Elena N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 885–901  mathnet  crossref
    20. Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “Spiral Chaos in the Nonholonomic Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 939–954  mathnet  crossref
    21. A. S. Gonchenko, S. V. Gonchenko, “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps”, Physica D, 337 (2016), 43–57  crossref  mathscinet  zmath  isi  scopus
    22. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724  crossref  mathscinet  zmath  isi  scopus
    23. G. M. Rozenblat, “On the choice of physically realizable parameters when studying the dynamics of spherical and ellipsoidal rigid bodies”, Mech. Sol., 51:4 (2016), 415–423  crossref  isi  scopus
    24. A. Nanda, P. Singla, M. A. Karami, “Energy harvesting using rattleback: theoretical analysis and simulations of spin resonance”, J. Sound Vibr., 369 (2016), 195–208  crossref  isi  scopus
    25. E. V. Vetchanin, A. O. Kazakov, “Bifurcations and chaos in the dynamics of two point vortices in an acoustic wave”, Int. J. Bifurcation Chaos, 26:4 (2016), 1650063  crossref  mathscinet  zmath  isi  scopus
    26. T. A. Levanova, A. O. Kazakov, G. V. Osipov, J. Kurths, “Dynamics of ensemble of inhibitory coupled Rulkov maps”, Eur. Phys. J.-Spec. Top., 225:1 (2016), 147–157  crossref  isi  scopus
    27. Stefan Rauch-Wojciechowski, Maria Przybylska, “Understanding Reversals of a Rattleback”, Regul. Chaotic Dyn., 22:4 (2017), 368–385  mathnet  crossref
    28. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    29. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975  mathnet  crossref
    30. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684  mathnet  crossref  mathscinet
  • Number of views:
    This page:63

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