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Regul. Chaotic Dyn., 2013, Volume 18, Issue 5, Pages 508–520 (Mi rcd136)  

This article is cited in 23 scientific papers (total in 23 papers)

Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane

Alexey O. Kazakovab

a The Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia
b Institute of computer science, ul. Universitetskaya 1, Izhevsk, 426034, Russia

Abstract: We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.

Keywords: mixed dynamics, strange attractor, unbalanced ball, rubber rolling, reversibility, two-dimensional Poincaré map, bifurcation, focus, saddle, invariant manifolds, homoclinic tangency, Lyapunov’s exponents

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00589
Ministry of Education and Science of the Russian Federation 14.B37.21.0361
This work was supported by the RFBR grants No. 13-01-00589 and 13-01-97028-povolzhye, the Federal Target Program “Personnel” No.14.B37.21.0361, and by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (Contract No. 14.B37.21.0863).


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Document Type: Article
MSC: 37J60, 37N15, 37G35
Received: 30.05.2013
Language: English

Citation: Alexey O. Kazakov, “Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane”, Regul. Chaotic Dyn., 18:5 (2013), 508–520

Citation in format AMSBIB
\by Alexey O. Kazakov
\paper Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 5
\pages 508--520

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    This publication is cited in the following articles:
    1. Alexander S. Gonchenko, Sergey V. Gonchenko, Alexey O. Kazakov, “Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone”, Regul. Chaotic Dyn., 18:5 (2013), 521–538  mathnet  crossref  mathscinet  zmath
    2. 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
    3. I. A. Bizyaev, “Nonintegrability and obstructions to the Hamiltonianization of a nonholonomic Chaplygin top”, Dokl. Math., 90:2 (2014), 631–634  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. 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
    5. 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
    6. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    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. 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
    11. 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
    12. 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
    13. A. Delshams, M. Gonchenko, S. Gonchenko, “On dynamics and bifurcations of area-preserving maps with homoclinic tangencies”, Nonlinearity, 28:9 (2015), 3027–3071  crossref  mathscinet  zmath  isi  scopus
    14. A. S. Gonchenko, S. V. Gonchenko, “Retracted: Lorenz-like attractors in a nonholonomic model of a rattleback”, Nonlinearity, 28:9 (2015), 3403–3417; retracted article, 30 (2017), c3  crossref  mathscinet  zmath  isi  scopus
    15. 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
    16. I. R. Sataev, A. O. Kazakov, “Stsenarii perekhoda k khaosu v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 12:2 (2016), 235–250  mathnet  elib
    17. E. V. Vetchanin, A. A. Kilin, “Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid”, Proc. Steklov Inst. Math., 295 (2016), 302–332  mathnet  crossref  crossref  mathscinet  isi  elib
    18. 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
    19. 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
    20. 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
    21. S. V. Gonchenko, D. V. Turaev, “On three types of dynamics and the notion of attractor”, Proc. Steklov Inst. Math., 297 (2017), 116–137  mathnet  crossref  crossref  mathscinet  isi  elib
    22. A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, D. V. Turaev, “On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators”, Physica D, 350 (2017), 45–57  crossref  mathscinet  zmath  isi  scopus
    23. 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
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