In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D 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 quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.