Geodesic
Complex Dynamics in Generalizations of the Chaplygin Sleigh
Russian journal of nonlinear dynamics, Tome 15 (2019) no. 4, pp. 551-559.

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The article considers the Chaplygin sleigh on a plane in a potential well, assuming that an external potential force is supplied at the mass center. Two particular cases are studied in some detail, namely, a one-dimensional potential valley and a potential with rotational symmetry; in both cases the models reduce to four-dimensional differential equations conserving mechanical energy. Assuming the potential functions to be quadratic, various behaviors are observed numerically depending on the energy, from those characteristic to conservative dynamics (regularity islands and chaotic sea) to strange attractors. This is another example of a nonholonomic system manifesting these phenomena (similar to those for Celtic stone or Chaplygin top), which reflects a fundamental nature of these systems occupying an intermediate position between conservative and dissipative dynamics.
Keywords: Chaplygin sleigh, nonholonomic system, attractor.
Mots-clés : chaos 
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S. P. Kuznetsov. Complex Dynamics in Generalizations of the Chaplygin Sleigh. Russian journal of nonlinear dynamics, Tome 15 (2019) no. 4, pp. 551-559. http://geodesic.mathdoc.fr/item/ND_2019_15_4_a14/

[1] Neimark, Ju. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., 33, AMS, Providence, R.I., 1972, 518 pp. | MR | Zbl

[2] Bloch, A., Nonholonomic Mechanics and Control, Interdiscip. Appl. Math., 24, Springer, New York, 2003, 503 pp. | DOI | MR | Zbl

[3] Borisov, A. V. and Mamaev, I. S., “The Rolling of Rigid Body on a Plane and Sphere: Hierarchy of Dynamics”, Regul. Chaotic Dyn., 7:1 (2002), 177–200 | DOI | MR | Zbl

[4] Kozlov, V. V., “On the Theory of Integration of the Equations of Nonholonomic Mechanics”, Regul. Chaotic Dyn., 7:2 (2002), 161–176 | DOI | MR | Zbl

[5] Uspekhi Fiz. Nauk, 173:4 (2003), 407–418 (Russian) | DOI | DOI

[6] Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova, J. V., “Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback”, Regul. Chaotic Dyn., 17:6 (2012), 512–532 | DOI | MR | Zbl

[7] Uspekhi Fiz. Nauk, 184:5 (2014), 493–500 (Russian) | DOI | DOI | MR

[8] Sagdeev, R. Z., Usikov, D. A., and Zaslavsky, G. M., Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Harwood Acad. Publ., Chur, 1990, 675 pp. | MR

[9] Reichl, L. E., The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Springer, New York, 2004, xvi+551 pp. | MR

[10] Bizyaev, I. A., Borisov, A. V., and Kuznetsov, S. P., “Chaplygin Sleigh with Periodically Oscillating Internal Mass”, Europhys. Lett., 119:6 (2017), 60008, 7 pp. | DOI | MR

[11] Schuster, H. G. and Just, W., Deterministic Chaos: An Introduction, Wiley-VCH, Weinheim, 2005, 312 pp. | MR | Zbl

[12] Kuznetsov, S. P., Dynamical Chaos, 2nd ed., Fizmatlit, Moscow, 2006, 356 pp. (Russian)

[13] Pikovsky, A. and Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge Univ. Press, Cambridge, 2016, 295 pp. | MR | Zbl

[14] Spitzer, F., Principles of Random Walk, Grad. Texts in Math., 34, 2nd ed., Springer, New York, 2001, 422 pp. | MR | Zbl

[15] Kuznetsov, S. P., “Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint”, Regul. Chaotic Dyn., 23:2 (2018), 178–192 | DOI | MR | Zbl