Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics

A. P. Kuznetsov ; S. P. Kuznetsov ; Yu. V. Sedova

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 2, pp. 223-234.

Résumé

Examples of mechanical systems are discussed, where quasi-periodic motions may occur, caused by an irrational ratio of the radii of rotating elements that constitute the system. For the pendulum system with frictional transmission of rotation between the elements, in the conservative and dissipative cases we note the coexistence of an infinite number of stable fixed points, and in the case of the self-oscillating system the presence of many attractors in the form of limit cycles and of quasi-periodic rotational modes is observed. In the case of quasi-periodic dynamics the frequencies of spectral components depend on the parameters, but the ratio of basic incommensurate frequencies remains constant and is determined by the irrational number characterizing the relative size of the elements.

Détail

Keywords: dynamic system, mechanical transmission, quasi-periodic oscillations, attractor.

Bibliographie

[1] Broer H. W., Huitema G. B., Sevryuk M. B., Quasi-periodic motions in families of dynamical systems: Order amidst chaos, Lecture Notes in Math., 1645, Springer, Berlin, 2009, 200 pp. | MR

[2] Glazier J. A., Libchaber A., “Quasi-periodicity and dynamical systems: An experimentalist's view”, IEEE Trans. Circuits and Systems, 35:7 (1988), 790–809 | DOI | MR

[3] Kuznetsov A. P., Sataev I. R., Stankevich N. V., Turukina L. V., Physics of quasiperiodic oscillations, Nauka, Saratov, 2013 (Russian)

[4] Kuznetsov S. P., Dynamical chaos, 2nd ed., Fizmatlit, Moscow, 2006 (Russian)

[5] Anishchenko V. S., Astakhov V. V., Vadivasova T. E., Regular and chaotic self-oscillations. Synchronization and influence of fluctuations, Intellekt, Moscow, 2009 (Russian)

[6] Poincaré H., New methods of celestial mechanics, History of Modern Physics and Astronomy, 13, Springer, New York, 1993, 1077 pp.

[7] Arnol'd V. I., Kozlov V. V., Neĭshtadt A. I., Mathematical aspects of classical and celestial mechanics, Encyclopaedia Math. Sci., 3, 3rd ed., Springer, Berlin, 2006, xiv+518 pp. | MR | Zbl

[8] Fermi E., Pasta J., Ulam S., Studies of nonlinear problems, Los Alamos Report LA-1940, 1955

[9] Landau L. D., “On the problem of turbulence”, Dokl. Akad. Nauk SSSR, 44:8 (1944), 339–349 (Russian)

[10] Hopf E., “A mathematical example displaying the features of turbulence”, Comm. Pure Appl. Math., 1:4 (1948), 303–322 | DOI | MR | Zbl

[11] Kuznetsov A. P., Kuznetsov S. P., Turukina L. V., Sataev I. R., “Landau – Hopf scenario in the ensemble of interacting oscillators”, Nelin. Dinam., 8:5 (2012), 863–873 (Russian)

[12] Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Turukina L. V., “About Landau – Hopf scenario in a system of coupled self-oscillators”, Phys. Lett. A, 377:45–48 (2013), 3291–3295 | 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] A. V. Borisov, I. S. Mamaev (eds.), Nonholonomic dynamical systems: Integrability, chaos, strange attractors, Institute of Computer Science, Moscow – Izhevsk, 2002 (Russian) | MR

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

[16] 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

[17] Kuznetsov S. P., “On the validity of the nonholonomic model of the rattleback”, Physics-Uspekhi, 58:12 (2015), 1223–1224 | DOI | DOI

[18] Kuznetsov S. P., Jalnine A. Y., Sataev I. R., Sedova J. V., “Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback”, Nelin. Dinam., 8:4 (2012), 735–762 (Russian)

[19] Borisov A. V., Jalnin 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

[20] Landa P. S., Nonlinear oscillations and waves in dynamical systems, Math. Appl., 360, Springer, Dordrecht, 2013, xv+544 pp.

[21] Gantmacher F. R., Lectures in analytical mechanics, Mir, Moscow, 1975, 264 pp.

[22] Goldstein H., Poole Ch. P. Jr., Safko J. L., Classical mechanics, 3rd ed., Addison-Wesley, Boston, Mass., 2001, 680 pp. | MR

[23] Kuznetsov A. P., Kuznetsov S. P., Ryskin N. M., Nonlinear oscillations, Fizmatlit, Moscow, 2002 (Russian)

[24] Stein D. L., “Spin glasses”, Sci. Am., 261 (1989), 52–59 | DOI

[25] Sveshnikov A. A., Applied methods of the theory of random functions, Pergamon, New York, 1966, 321 pp. | MR | MR | Zbl

[26] Jenkins G. M., Watts D. G., Spectral analysis and its applications, Holden-Day, San Francisco, 1968, 548 pp. | MR | Zbl

[27] Grebogi C., Ott E., Pelikan S., Yorke J. A., “Strange attractors that are not chaotic”, Phys. D, 13:1–2 (1984), 261–268 | DOI | MR | Zbl

[28] Kuznetsov S. P., Pikovsky A. S., Feudel U., “Strange non-chaotic attractor”, Nonlinear waves: Proc. Sympos. (Univ. Nizhniy Novgorod, Russia, 2004), eds. A. V. Gaponov-Grekhov, V. I. Nekorkin, Institute of Applied Physics, Nizhniy Novgorod, 2005, 484–509 (Russian)