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Regul. Chaotic Dyn.:

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Regul. Chaotic Dyn., 2018, том 23, выпуск 2, страницы 178–192 (Mi rcd317)  

Эта публикация цитируется в 9 научных статьях (всего в 9 статьях)

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

Sergey P. Kuznetsovab

a Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, ul. Zelenaya 38, Saratov, 410019 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia

Аннотация: The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.

Ключевые слова: nonholonomic mechanics, Chaplygin sleigh, attractor, chaos, bifurcation, Chirikov–Taylor map

Финансовая поддержка Номер гранта
Российский научный фонд 15-12-20035
This work was supported by the Russian Science Foundation, grant № 15-12-20035.


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Тип публикации: Статья
Поступила в редакцию: 09.11.2017
Принята в печать:04.12.2017
Язык публикации: английский

Образец цитирования: Sergey P. Kuznetsov, “Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint”, Regul. Chaotic Dyn., 23:2 (2018), 178–192

Цитирование в формате AMSBIB
\by Sergey P. Kuznetsov
\paper Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 2
\pages 178--192

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    Citing articles on Google Scholar: Russian citations, English citations
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    Эта публикация цитируется в следующих статьяx:
    1. Alexey V. Borisov, Sergey 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
    2. S. P. Kuznetsov, “Complex Dynamics in Generalizations of the Chaplygin Sleigh”, Нелинейная динам., 15:4 (2019), 551–559  mathnet  crossref  elib
    3. 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
    4. 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
    5. V A. Borisov , E. V. Vetchanin, I. S. Mamaev, “Motion of a smooth foil in a fluid under the action of external periodic forces. I”, Russ. J. Math. Phys., 26:4 (2019), 412–427  crossref  mathscinet  zmath  isi  scopus
    6. A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, E. A. Samylina, “Chaotic dynamics and multistability in the nonholonomic model of a celtic stone”, Radiophys. Quantum Electron., 61:10 (2019), 773–786  crossref  isi  scopus
    7. A. O. Kazakov, “On the appearance of mixed dynamics as a result of collision of strange attractors and repellers in reversible systems”, Radiophys. Quantum Electron., 61:8-9 (2019), 650–658  crossref  isi  scopus
    8. 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
    9. Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of Rubber Chaplygin Sphere under Periodic Control”, Regul. Chaotic Dyn., 25:2 (2020), 215–236  mathnet  crossref
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