Nonlinear dynamics of the rattleback: a nonholonomic model
a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russian Federation
b Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141700, Russian Federation
c Institute of Computer Science, ul. Universitetskaya 1, Izhevsk, 426034, Russian Federation
d Lobachevsky State University of Nizhny Novgorod, Faculty of Computational Mathematics and Cybernetics, pr. Gagarina 23, Nizhny Novgorod, 603950, Russian Federation
e Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch, ul. Zelenaya 38, Saratov, 410019, Russian Federation
For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.