Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors

I. A. Bizyaev ; A. V. Borisov ; A. O. Kazakov

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 2, pp. 263-287.

Résumé

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

Détail

Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor.

Bibliographie

[1] Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M., “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them: P. 1: Theory; P. 2: Numerical application”, Meccanica, 15 (1980), 9–30 | DOI

[2] Bizyaev I. A., Borisov A. V., Mamaev I. S., “The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside”, Regul. Chaotic Dyn., 19:2 (2014), 198–213 | DOI | MR | Zbl

[3] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Rolling of a ball without spinning on a plane: The absence of an invariant measure in a system with a complete set of integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579 | DOI | MR | Zbl

[4] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Geometrisation of Chaplygin's reducing multiplier theorem”, Nonlinearity, 28:7 (2015), 2307–2318 | DOI | MR | Zbl

[5] Borisov A. V., Mamaev I. S., Bizyaev I. A., “The Jacobi integral in nonholonomic mechanics”, Regul. Chaotic Dyn., 20:3 (2015), 383–400 | DOI | MR | Zbl

[6] Borisov A. V., Kilin A. A., Mamaev I. S., “Hamiltonicity and integrability of the Suslov problem”, Regul. Chaotic Dyn., 16:1–2 (2011), 104–116 | DOI | MR | Zbl

[7] Borisov A. V., Mamaev I. S., Bizyaev I. A., “The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere”, Regul. Chaotic Dyn., 8:3 (2013), 277–328 | DOI | MR

[8] Borisov A. V., Kazakov A. O., Sataev I. R., “The reversal and chaotic attractor in the nonholonomic model of Chaplygin's top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733 | DOI | MR | Zbl

[9] Borisov A. V., Kazakov A. O., Kuznetsov S. P., “Nonlinear dynamics of the rattleback: A nonholonomic model”, Physics-Uspekhi, 57:5 (2014), 453–460 | DOI | DOI

[10] Borisov A. V., Jalnine A. Yu., Kuznetsov S. P., Sataev I. R., 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

[11] Borisov A. V., Mamaev I. S., Kilin A. A., “The rolling motion of a ball on a surface: New integrals and hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 201–219 | DOI | MR | Zbl

[12] Borisov A. V., Mamaev I. S., “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490 | DOI | MR | Zbl

[13] Borisov A. V., Mamaev I. S., “Rolling of a rigid body on plane and sphere: Hierarchy of dynamics”, Regul. Chaotic Dyn., 7:2 (2002), 177–200 | DOI | MR | Zbl

[14] Borisov A. V., Kilin A. A., Mamaev I. S., “The problem of drift and recurrence for the rolling Chaplygin ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859 | DOI | MR | Zbl

[15] García-Naranjo L. C., Maciejewski A. J., Marrero J. C., Przybylska M., “The inhomogeneous Suslov problem”, Phys. Lett. A, 378:32–33 (2014), 2389–2394 | MR | Zbl

[16] Gonchenko A. S., Gonchenko S. V., Kazakov A. O., “Richness of chaotic dynamics in nonholonomic models of a celtic stone”, Regul. Chaotic Dyn., 18:5 (2013), 521–538 | DOI | MR | Zbl

[17] Fedorov Yu. N., Maciejewski A. J., Przybylska M., “Suslov problem: Integrability, meromorphic and hypergeometric solutions”, Nonlinearity, 22:9 (2009), 2231–2259 | DOI | MR | Zbl

[18] Feigenbaum M. J., “Universal behavior in nonlinear systems”, Phys. D, 7:1–3 (1983), 16–39 | DOI | MR

[19] Fassò F., Sansonetto N., Conservation of energy and momenta in nonholonomic systems with affine constraints, 2015, arXiv: math/1505.01172v1 [math.DS] | MR

[20] Fernandez O. E., Bloch A. M., Zenkov D. V., “The geometry and integrability of the Suslov problem”, J. Math. Phys., 55:11 (2014), 112704, 14 pp. | DOI | MR | Zbl

[21] Mahdi A., Valls C., “Analytic non-integrability of the Suslov problem”, J. Math. Phys., 53:12 (2012), 122901, 8 pp. | DOI | MR | Zbl

[22] Maciejewski A. J., Przybylska M., “Nonintegrability of the Suslov problem”, J. Math. Phys., 45:3 (2004), 1065–1078 | DOI | MR | Zbl

[23] Kazakov A. O., “Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane”, Regul. Chaotic Dyn., 18:5 (2013), 508–520 | DOI | MR | Zbl

[24] Kane T. R., Levinson D. A., “A realistic solution of the symmetric top problem”, J. Appl. Mech., 45:4 (1978), 903–909 | DOI | MR

[25] Kozlov V. V., “The phenomenon of reversal in the Euler – Poincaré – Suslov nonholonomic systems”, J. Dyn. Control Syst., 2015, 12 pp.

[26] Rocard Y., L'instabilité en mécanique: Automobiles, avions, ponts suspendus, Masson, Paris, 1954, 239 pp.

[27] Pikovsky A., Topaj D., “Reversibility vs. synchronization in oscillator latties”, Phys. D, 170 (2002), 118–130 | DOI | MR | Zbl

[28] Walker G. T., “On a curious dynamical property of celts”, Proc. Cambridge Phil. Soc., 8:5 (1895), 305–306

[29] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318 | DOI | DOI | MR | Zbl

[30] Borisov A. V., Mamaev I. S., “Strange attractors in rattleback dynamics”, Physics-Uspekhi, 46:4 (2003), 393–403 | DOI | DOI

[31] Borisov A. V., Kilin A. A., Mamaev I. S., “New effects in dynamics of rattlebacks”, Dokl. Phys., 51:5 (2006), 272–275 | DOI | MR | Zbl

[32] Vagner V. V., “A geometric interpretation of nonholonomic dynamical systems”, Tr. Semin. Vectorn. Tenzorn. Anal., 5, 1941, 301–327 (Russian) | MR | Zbl

[33] Suslov G. K., Theoretical mechanics, Gostekhizdat, Moscow, 1946 (Russian)

[34] Kozlova Z. P., “The Suslov problem”, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1989, no. 1, 13–16 (Russian)

[35] Kozlov V. V., “On the existence of an integral invariant of a smooth dynamic system”, J. Appl. Math. Mech., 51:4 (1987), 420–426 | DOI | MR | Zbl

[36] Kozlov V. V., “On the theory of integration of the equations of nonholonomic mechanics”, Uspekhi Mekh., 8:3 (1985), 85–107 (Russian) | MR

[37] Markeev A. P., “The dynamics of a rigid body on an absolutely rough plane”, J. Appl. Math. Mech., 47:4 (1983), 473–478 | DOI | Zbl

[38] Ziglin S. L., “On the absence of an additional first integral in the special case of the G. K. Suslov problem”, Russian Math. Surveys, 52:2 (1997), 434–435 | DOI | DOI | MR | Zbl

[39] Kharlamova-Zabelina E. I., “Rigid body motion about a fixed point under nonholonomic constraint”, Tr. Donetsk. Industr. Inst., 20:1 (1957), 69–75 (Russian)

[40] Tatarinov Ya. V., “Separation of variables and new topological phenomena in holonomic and nonholonomic systems”, Tr. Sem. Vektor. Tenzor. Anal., 23, 1988, 160–174 (Russian) | MR | Zbl

[41] Neimark Ju. I., Fufaev N. A., Dynamics of nonholonomic systems, Trans. Math. Monogr., 33, AMS, Providence, R.I., 1972, 518 pp. | Zbl