Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback

Sergey P. Kuznetsov ; Alexey Yu. Jalnine ; Igor R. Sataev ; Julia V. Sedova

Russian journal of nonlinear dynamics, Tome 8 (2012) no. 4, pp. 735-762.

Résumé

We perform a numerical study of the motion of the rattleback, a rigid body with a convex surface on a rough horizontal plane in dependence on the parameters, applying the methods used previously for the treatment of dissipative dynamical systems, and adapted for the nonholonomic model. Charts of dynamical regimes are presented on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body. Presence of characteristic structures in the parameter space, previously observed only for dissipative systems, is demonstrated. A method of calculating for the full spectrum of Lyapunov exponents is developed and implemented. It is shown that analysis of the Lyapunov exponents of chaotic regimes of the nonholonomic model reveals two classes, one of which is typical for relatively high energies, and the second for the relatively small energies. For the model reduced to a three-dimensional 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 the quasi-conservative type, with close in magnitude positive and negative Lyapunov exponents, and the rest one about zero. The transition to chaos through a sequence of period-doubling bifurcations is illustrated, and the observed scaling corresponds to that intrinsic to the dissipative systems. A study of strange attractors is provided, in particularly, phase portraits are presented as well as the Lyapunov exponents, the Fourier spectra, the results of calculating the fractal dimensions.

Détail

Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents
Mots-clés : bifurcation, fractal dimension.

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