Аннотация:
We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
We thank A.P. Kuznetsov for useful discussions. The work was carried out as part of research at
the Udmurt State University within the framework of the Program of Government of the Russian
Federation for state support of scientific research carried out under supervision of leading scientists
at Russian institutions of higher professional education (contract No 11.G34.31.0039).
Поступила в редакцию: 09.09.2012 Принята в печать: 06.09.2012
Образец цитирования:
Alexey V. Borisov, Alexey Yu. Jalnine, Sergey P. Kuznetsov, Igor R. Sataev, Yulia V. Sedova, “Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback”, Regul. Chaotic Dyn., 17:6 (2012), 512–532
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\by Alexey V.~Borisov, Alexey Yu.~Jalnine, Sergey P.~Kuznetsov, Igor R.~Sataev, Yulia V.~Sedova
\paper Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 6
\pages 512--532
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\crossref{https://doi.org/10.1134/S1560354712060044}
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\zmath{https://zbmath.org/?q=an:1263.74021}
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https://www.mathnet.ru/rus/rcd351
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Gianni Arioli, Hans Koch, “Some Reversing Orbits for a Rattleback Model”, J Nonlinear Sci, 32:3 (2022)
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
Kuznetsov S.P. Kruglov V.P. Borisov A.V., “Chaplygin Sleigh in the Quadratic Potential Field”, EPL, 132:2 (2020), 20008
S. P. Kuznetsov, “Complex Dynamics in Generalizations of the Chaplygin Sleigh”, Rus. J. Nonlin. Dyn., 15:4 (2019), 551–559
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, “Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators”, Regul. Chaotic Dyn., 24:6 (2019), 725–738
Borisov V A. Vetchanin E.V. Mamaev I.S., “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
Jones S., Hunt H.E.M., “Rattlebacks For the Rest of Us”, Am. J. Phys., 87:9 (2019), 699–713
Bizyaev I.A. Borisov V A. Kozlov V.V. Mamaev I.S., “Fermi-Like Acceleration and Power-Law Energy Growth in Nonholonomic Systems”, Nonlinearity, 32:9 (2019), 3209–3233
Bizyaev I.A. Borisov A.V. Kuznetsov S.P., “The Chaplygin Sleigh With Friction Moving Due to Periodic Oscillations of An Internal Mass”, Nonlinear Dyn., 95:1 (2019), 699–714
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
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
Bizyaev I.A. Borisov A.V. Mamaev I.S., “Dynamics of the Chaplygin Ball on a Rotating Plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433
Chistyakov V.V., “On Kinetics of a Dynamically Unbalanced Rotator With Sliding Friction in Supports”, AIP Conference Proceedings, 1959, eds. Kustova E., Leonov G., Morosov N., Yushkov M., Mekhonoshina M., Amer Inst Physics, 2018, UNSP 030005
А. В. Борисов, И. С. Мамаев, И. А. Бизяев, “Динамические системы с неинтегрируемыми связями: вакономная механика, субриманова геометрия и неголономная механика”, УМН, 72:5(437) (2017), 3–62; 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
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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
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