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Regul. Chaotic Dyn., 2013, Volume 18, Issue 6, Pages 832–859 (Mi rcd171)  

This article is cited in 52 scientific papers (total in 52 papers)

The Problem of Drift and Recurrence for the Rolling Chaplygin Ball

Alexey V. Borisovabc, Alexander A. Kilincab, Ivan S. Mamaevbac

a Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334 Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990 Russia

Abstract: We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.

Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.1248.2011
1.7734.2013
MD-2324.2013.1
Russian Foundation for Basic Research 13-01-12462-ofi_m
This research was supported by the Target Programmes for 2012–2014 (State contract 1.1248.2011, 1.7734.2013) and grant RFBR 13-01-12462-ofi_m. A. A. Kilin’s research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists–Doctors of Science (MD-2324.2013.1).


DOI: https://doi.org/10.1134/S1560354713060166

References: PDF file   HTML file

Bibliographic databases:

MSC: 70E18, 37J60, 70K43, 37J15, 37J20
Received: 19.09.2013
Accepted:11.11.2013
Language:

Citation: 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

Citation in format AMSBIB
\Bibitem{BorKilMam13}
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 6
\pages 832--859
\mathnet{http://mi.mathnet.ru/rcd171}
\crossref{https://doi.org/10.1134/S1560354713060166}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3146594}
\zmath{https://zbmath.org/?q=an:1286.70007}
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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. A. V. Tsyganov, “O share Chaplygina v absolyutnom prostranstve”, Nelineinaya dinam., 9:4 (2013), 711–719  mathnet
    2. Andrey V. Tsiganov, “On the Lie Integrability Theorem for the Chaplygin Ball”, Regul. Chaotic Dyn., 19:2 (2014), 185–197  mathnet  crossref  mathscinet  zmath
    3. Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733  mathnet  crossref  mathscinet  zmath
    4. A. V. Borisov, I. S. Mamaev, “Equations of motion of non-holonomic systems”, Russian Math. Surveys, 70:6 (2015), 1167–1169  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Andrey V. Tsiganov, “On Integrable Perturbations of Some Nonholonomic Systems”, SIGMA, 11 (2015), 085, 19 pp.  mathnet  crossref
    6. A. A. Kilin, E. V. Vetchanin, “Upravlenie dvizheniem tverdogo tela v zhidkosti s pomoschyu dvukh podvizhnykh mass”, Nelineinaya dinam., 11:4 (2015), 633–645  mathnet
    7. A. A. Kilin, Yu. L. Karavaev, “Eksperimentalnye issledovaniya dinamiki sfericheskogo robota kombinirovannogo tipa”, Nelineinaya dinam., 11:4 (2015), 721–734  mathnet
    8. 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
    9. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    10. 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
    11. Alexey V. Borisov, Ivan S. Mamaev, Alexander A. Kilin, Ivan A. Bizyaev, “Qualitative Analysis of the Dynamics of a Wheeled Vehicle”, Regul. Chaotic Dyn., 20:6 (2015), 739–751  mathnet  crossref  mathscinet  adsnasa
    12. 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
    13. 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
    14. Yury L. Karavaev, Alexander A. Kilin, “The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform”, Regul. Chaotic Dyn., 20:2 (2015), 134–152  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    15. V. V. Kozlov, “Dinamika sistem s servosvyazyami. II”, Nelineinaya dinam., 11:3 (2015), 579–611  mathnet
    16. A. V. Borisov, I. S. Mamaev, “Simmetrii i reduktsiya v negolonomnoi mekhanike”, Nelineinaya dinam., 11:4 (2015), 763–823  mathnet
    17. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. II”, Regul. Chaotic Dyn., 20:4 (2015), 401–427  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    18. Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604  mathnet  crossref  mathscinet  zmath
    19. 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
    20. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles”, Regul. Chaotic Dyn., 20:6 (2015), 752–766  mathnet  crossref  mathscinet  adsnasa
    21. 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
    22. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Vortex Sources in a Deformation Flow”, Regul. Chaotic Dyn., 21:3 (2016), 367–376  mathnet  crossref  mathscinet
    23. 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
    24. Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810  mathnet  crossref
    25. 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
    26. 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
    27. A. P. Ivanov, N. N. Erdakova, “On a mechanical lens”, Int. J. Non-Linear Mech., 79 (2016), 115–121  crossref  isi  scopus
    28. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika sanei Chaplygina na tsilindre”, Nelineinaya dinam., 12:4 (2016), 675–687  mathnet  crossref  mathscinet  zmath  elib
    29. I. A. Bizyaev, A. V. Borisov, A. O. Kazakov, “Dinamika zadachi Suslova v pole tyazhesti: revers i strannye attraktory”, Nelineinaya dinam., 12:2 (2016), 263–287  mathnet  mathscinet  elib
    30. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “O probleme Adamara–Gamelya i dinamike kolesnykh ekipazhei”, Nelineinaya dinam., 12:1 (2016), 145–163  mathnet
    31. 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
    32. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Dynamics of the Chaplygin Sleigh on a Cylinder”, Regul. Chaotic Dyn., 21:1 (2016), 136–146  mathnet  crossref  mathscinet  zmath  elib
    33. 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
    34. A. V. Borisov, I. S. Mamaev, “Neodnorodnye sani Chaplygina”, Nelineinaya dinam., 13:4 (2017), 625–639  mathnet  crossref  mathscinet  elib
    35. 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
    36. I. A. Bizyaev, “Sani Chaplygina s dvizhuscheisya tochechnoi massoi”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 583–589  mathnet  crossref  elib
    37. 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
    38. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Hamilton'S Principle and the Rolling Motion of a Symmetric Ball”, Dokl. Phys., 62:6 (2017), 314–317  mathnet  crossref  mathscinet  isi  scopus
    39. Alexey V. Borisov, Ivan S. Mamaev, “An Inhomogeneous Chaplygin Sleigh”, Regul. Chaotic Dyn., 22:4 (2017), 435–447  mathnet  crossref  mathscinet
    40. A. V. Borisov, A. O. Kazakov, E. N. Pivovarova, “Regulyarnaya i khaoticheskaya dinamika v «rezinovoi» modeli volchka Chaplygina”, Nelineinaya dinam., 13:2 (2017), 277–297  mathnet  crossref  elib
    41. 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  mathscinet
    42. V. Putkaradze, S. Rogers, “On the dynamics of a rolling ball actuated by internal point masses”, Meccanica, 53:15 (2018), 3839–3868  crossref  mathscinet  isi  scopus
    43. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of the Chaplygin ball on a rotating plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433  crossref  mathscinet  zmath  isi  scopus
    44. Alexander A. Kilin, Elena N. Pivovarova, “Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 24:2 (2019), 212–233  mathnet  crossref
    45. Ivan Yu. Polekhin, “Precession of the Kovalevskaya and Goryachev – Chaplygin Tops”, Regul. Chaotic Dyn., 24:3 (2019), 281–297  mathnet  crossref  mathscinet
    46. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582  mathnet  crossref  mathscinet
    47. Borisov A. Kilin A. Mamaev I., “Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930008  crossref  mathscinet  zmath  isi  scopus
    48. Alexey V. Borisov, Evgeniya A. Mikishanina, “Two Nonholonomic Chaotic Systems. Part II. On the Rolling of a Nonholonomic Bundle of Two Bodies”, Regul. Chaotic Dyn., 25:4 (2020), 392–400  mathnet  crossref  mathscinet
    49. A. V. Borisov, E. A. Mikishanina, “Dynamics of the Chaplygin Ball with Variable Parameters”, Rus. J. Nonlin. Dyn., 16:3 (2020), 453–462  mathnet  crossref  mathscinet
    50. Bizyaev I.A. Mamaev I.S., “Separatrix Splitting and Nonintegrability in the Nonholonomic Rolling of a Generalized Chaplygin Sphere”, Int. J. Non-Linear Mech., 126 (2020), 103550  crossref  mathscinet  isi  scopus
    51. Putkaradze V. Rogers S., “on the Optimal Control of a Rolling Ball Robot Actuated By Internal Point Masses”, J. Dyn. Syst. Meas. Control-Trans. ASME, 142:5 (2020)  crossref  isi  scopus
    52. Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane”, Regul. Chaotic Dyn., 26:6 (2021), 775–786  mathnet  crossref
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