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Regul. Chaotic Dyn.:

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Regul. Chaotic Dyn., 2017, том 22, выпуск 8, страницы 955–975 (Mi rcd302)  

Эта публикация цитируется в 18 научных статьях (всего в 18 статьях)

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

Ivan A. Bizyaeva, Alexey V. Borisovb, Ivan S. Mamaevc

a Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

Аннотация: This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

Ключевые слова: nonholonomic mechanics, Fermi acceleration, Chaplygin sleigh, parametric oscillator, tensor invariants, involution, strange attractor, Lyapunov exponents, reversible systems, chaotic dynamics

Финансовая поддержка Номер гранта
Министерство образования и науки Российской Федерации 1.2404.2017/4.6

The work of A.V.Borisov (Sections 1 and 2) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2404.2017/4.6). The work of I.A.Bizyaev (Section 3) was carried out at MIPT under project 5-100 for state support for leading universities of the Russian Federation. The work of I. S.Mamaev (Section 4) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2405.2017/4.6).


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Тип публикации: Статья
MSC: 37J60, 34A34
Поступила в редакцию: 06.11.2017
Принята в печать:07.12.2017
Язык публикации: английский

Образец цитирования: 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

Цитирование в формате AMSBIB
\by Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev
\paper The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 8
\pages 955--975

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    Эта публикация цитируется в следующих статьяx:
    1. V. Fedonyuk, Ph. Tallapragada, “Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh”, Nonlinear Dyn., 93:2 (2018), 835–846  crossref  isi  scopus
    2. Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502  mathnet  crossref  mathscinet
    3. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control”, Regul. Chaotic Dyn., 23:7-8 (2018), 983–994  mathnet  crossref
    4. I. S. Mamaev, V. A. Tenenev, E. V. Vetchanin, “Dynamics of a Body with a Sharp Edge in a Viscous Fluid”, Нелинейная динам., 14:4 (2018), 473–494  mathnet  crossref  elib
    5. A. V. Borisov, S. P. Kuznetsov, “Comparing Dynamics Initiated By An Attached Oscillating Particle For the Nonholonomic Model of a Chaplygin Sleigh and For a Model With Strong Transverse and Weak Longitudinal Viscous Friction Applied At a Fixed Point on the Body”, Regul. Chaotic Dyn., 23:7-8 (2018), 803–820  mathnet  crossref  mathscinet  zmath  isi  scopus
    6. 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
    7. A. A. Kilin, E. N. Pivovarova, “Chaplygin top with a periodic gyrostatic moment”, Russ. J. Math. Phys., 25:4 (2018), 509–524  crossref  mathscinet  zmath  isi  scopus
    8. E. V. Vetchanin, “The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque”, Нелинейная динам., 15:1 (2019), 41–57  mathnet  crossref  elib
    9. Yu. L. Karavaev, A. A. Kilin, “The Dynamics of a Spherical Robot of Combined Type by Periodic Control Actions”, Нелинейная динам., 15:4 (2019), 497–504  mathnet  crossref  mathscinet  elib
    10. 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
    11. Andrey A. Ardentov, Yury L. Karavaev, Kirill S. Yefremov, “Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization”, Regul. Chaotic Dyn., 24:3 (2019), 312–328  mathnet  crossref
    12. 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
    13. А. В. Борисов, А. В. Цыганов, “Влияние эффектов Барнетта-Лондона и Эйнштейна-де Гааза на движение неголономной сферы Рауса”, Вестн. Удмуртск. ун-та. Матем. Мех. Компьют. науки, 29:4 (2019), 583–598  mathnet  crossref
    14. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of a Chaplygin sleigh with an unbalanced rotor: regular and chaotic motions”, Nonlinear Dyn., 98:3 (2019), 2277–2291  crossref  mathscinet  zmath  isi  scopus
    15. I. A. Bizyaev, V A. Borisov , V. V. Kozlov, I. S. Mamaev, “Fermi-like acceleration and power-law energy growth in nonholonomic systems”, Nonlinearity, 32:9 (2019), 3209–3233  crossref  mathscinet  zmath  isi  scopus
    16. A. V. Borisov, A. A. Kilin, E. N. Pivovarova, “Speedup of the Chaplygin TOP By Means of Rotors”, Dokl. Phys., 64:3 (2019), 120–124  mathnet  crossref  isi  scopus
    17. I. A. Bizyaev, A. V. Borisov, S. P. Kuznetsov, “The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass”, Nonlinear Dyn., 95:1 (2019), 699–714  crossref  isi  scopus
    18. Gzenda V., Putkaradze V., “Integrability and Chaos in Figure Skating”, J. Nonlinear Sci., 30:3 (2020), 831–850  crossref  mathscinet  zmath  isi  scopus
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