Evgeniya Mikishanina

The problem of rolling a nonholonomic bundle of two bodies is considered: a spherical shell with a rigid body rotating along the axis of symmetry, on which rotors spinning relative to this body are fastened. This problem can be regarded as a distant generalization of the Chaplygin ball problem. The reduced system is studied by analyzing Poincaré maps constructed in Andoyer – Deprit variables. A classification of Poincaré maps of the reduced system is carried out, the behavior of the contact point is studied, and the cases of chaotic oscillations of the system are examined in detail. To study the nature of the system’s chaotic behavior, a map of dynamical regimes is constructed. The Feigenbaum type of attractor is shown.

A generalization of the Suslov problem with changing parameters is considered. The physical interpretation is a Chaplygin sleigh moving on a sphere. The problem is reduced to the study of a two-dimensional system describing the evolution of the angular velocity of a body. The system without viscous friction and the system with viscous friction are considered. Poincaré maps are constructed, attractors and noncompact attracting trajectories are found. The presence of noncompact trajectories in the Poincaré map suggests that acceleration is possible in this nonholonomic system. In the case of a system with viscous friction, a chart of dynamical regimes and a bifurcation tree are constructed to analyze the transition to chaos. The classical scenario of transition to chaos through a cascade of period doubling is shown, which may indicate attractors of Feigenbaum type.