Nonlinear dynamics of the rattleback: a nonholonomic model

For a solid convex body moving on a rough horizontal plane — known in mechanics as a rattleback — numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to the nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics„ evolution. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as the presence of stable equilibrium points relevant to stationary rotations; limit cycles (rotations with oscillations), and strange chaotic attractors. A chart of dynamical regimes is plotted in a plane of parameters whose axes are the total mechanical energy and the angle of relative rotation of 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.