Dynamics of the Chaplygin ball on a rotating plane

J. Alves and J. Dias, “Design and Control of a Spherical Mobile Robot,” J. Syst. Control Eng. 217, 457–467 (2003). P. Balseiro and L. C. García-Naranjo, “Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems,” Arch. Ration. Mech. Anal. 205, 267–310 (2012). I. A. Bizyaev, A. V. Borisov and I. S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration,” Regul. Chaotic Dyn. 22, 955–975 (2017). A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, “Topology and Stability of Integrable Systems,” Russian Math. Surveys 65, 259–318 see also: Uspekhi Mat. Nauk 65, 71–132 (2010). A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, “Geometrisation of Chaplygin’s Reducing Multiplier Theorem,” Nonlinearity 28, 2307–2318 (2015). A. V. Borisov, A. Yu. Jalnine, S. P. Kuznetsov, I. R. Sataev and J. V. Sedova, “Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback,” Regul. Chaotic Dyn. 17, 512–532 (2012). A. V. Borisov, A. O. Kazakov and I. R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top,” Regul. Chaotic Dyn. 19, 718–733 (2014). A. V. Borisov, A. O. Kazakov and E. N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top,” Regul. Chaotic Dyn. 21, 885–901 (2016). A. V. Borisov, A. A. Kilin and I. S. Mamaev, “Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support,” Regul. Chaotic Dyn. 17, 170–190 (2012). A. V. Borisov, A. A. Kilin and I. S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball,” Regul. Chaotic Dyn. 18, 832–859 (2013). A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, “The Jacobi Integral in Nonholonomic Mechanics,” Regul. Chaotic Dyn. 20, 383–400 (2015). A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, “Historical and Critical Review of the Development of Nonholonomic Mechanics: The Classical Period,” Regul. Chaotic Dyn. 21, 455–476 (2016). A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, “Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics,” Russian Math. Surveys bf 72 783–840 (2017); see also: Uspekhi Mat. Nauk, 72 3–62. A. V. Borisov, I. S. Mamaev and A. A. Kilin, “Rolling of a Ball on a Surface: New Integrals and Hierarchy of Dynamics,” Regul. Chaotic Dyn. 7, 201–219 (2002). A. V. Borisov and I. S. Mamaev, “The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics,” Regul. Chaotic Dyn. 7, 177–200 (2002). A. V. Borisov and I. S. Mamaev, “Strange Attractors in Rattleback Dynamics,” PhysicsUspekhi 46, 393–403 (2003); see also: Uspekhi Fiz. Nauk 173, 407–418. A. V. Borisov and I. S. Mamaev, “Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere,” Regul. Chaotic Dyn. 18 356–371 (2013). A. V. Borisov and I. S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics,” Regul. Chaotic Dyn. 20, 553–604 (2015). C. Camicia, F. Conticelli and A. Bicch, “Nonholonomic Kinematics and Dynamics of the Sphericle,” in Proc. of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS, Takamatsu, Japan) 1, 805–810 (2000). J. H. E. Cartwright, M. Feingold, and O. Piro, “An Introduction to Chaotic Advection,” in Mixing: Chaos and Turbulence H Chaté, E Villermaux end J-M Chomaz (Eds.), 1999 NATO ASI Series (Series B: Physics) 373 (Boston, Mass.: Springer), 307–342 (1999). S. A. Chaplygin, “On a Ball’s Rolling on a Horizontal Plane,” Regul. Chaotic Dyn. 7 131–148 (2002); see also: S. A. Chaplygin, “On a Ball’s Rolling on a Horizontal Plane,” Math. Sb. 24 139–168 (1903). Ch. Q. Cheng and Y. S. Sun, “Existence of Invariant Tori in Three-Dimensional Measure-Preserving Mappings,” Celest. Mech. Dynam. Astronom. 47 275–292 (1989/90). H. R. Dullin and J. D. Meiss, “Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations,” SIAM J. Appl. Dyn. Syst. 8, 76–128 (2009). S. Earnshaw, Dynamics, or an Elementary Treatise on Motion 3rd ed. (Cambridge: Deighton, 1844). F. Fassò and N. Sansonetto, “Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints,” Regul. Chaotic Dyn. 20, 449–462 (2015). F. Fassò and N. Sansonetto, “Conservation of Moving Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces,” J. Nonlinear Sci. 26, 519–544 (2016). F. Fassò, L. C. García-Naranjo and N. Sansonetto, “Moving Energies As First Integrals of Nonholonomic Systems with Affine Constraints,” Nonlinearity 31, 755–782 (2018). J. Gersten, H. Soodak and M. S. Tiersten, “Ball Moving on Stationary or Rotating Horizontal Surface,” Am. J. Phys. 60, 43–47 (1992). A. P. Ivanov, “The ANAIS Billiard Experiment,” Dokl. Phys. 61, 285–287 (2016), 61 (6), 285–287; see also: Dokl. Akad. Nauk 468, 401–402 (2016). J. D. Mireles James, “Quadratic Volume-Preserving Maps: (Un)stableManifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations,” J. Nonlinear Sci. 23, 585–615 (2013). V. V. Kozlov, “The Phenomenon of Reversal in the Euler–Poincaré–Suslov Nonholonomic Systems,” J. Dyn. Control Syst. 22, 713–724 (2016). J-M. Levy-Leblond, “The ANAIS Billiard Table,” Eur. J. Phys. 7, 252–258 (1986). J. D. Meiss, N. Miguel, C. Simó and A. Vieiro, “Accelerator Modes and Anomalous Diffusion in 3D Volume-Preserving Maps,” arXiv:1802.10484 (2018). E. A. Milne, Vectorial Mechanics (New York: Interscience, 1948). E. J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject 6th ed. (New York: Dover, 1955). I. Tzénoff, “Quelques formes différentes des équations générales du mouvement des systémes matériels,” Bull. Soc. Math. France 53, 80–105 (1925). I. Tzénoff, “Sur les équations générales du mouvement des systèmes matériels non holonomes,” J. Math. Pures Appl. (8) 3, 245–263 (1920). A. V. Borisov and Yu. N. Fedorov, “On two modified integrable problems in dynamics,” Mosc. Univ. Mech. Bull. 50 16–18 (1995); see also: Vestnik Moskov. Univ. Ser. 1. Mat. Mekh. 6, 102–105 (1995). A. V. Borisov and I. S. Mamaev, “Chaplygin’s Ball Rolling Problem Is Hamiltonian,” Math. Notes 70, 720–723 (2001) see also: Mat. Zametki 70, 793–795. V. V. Kozlov, “On the Theory of Integration of the Equations of Nonholonomic Mechanics,” Uspekhi Mekh. 8, 85–107 (1985)(Russian). Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems Trans. Math. Monogr. 33 (Providence, R.I.: AMS, 1972).