On the bifurcations of the phase portrait of gyrostat

The global dynamics of an asymmetrical rigid body with an axisymmetric rotor are discussed. Compared with a free rigid body, a gyrostat has an extra degree of freedom, and for its integrability, an additional first integral is required. Two cases are considered: either the relative angular velocity of the rotor is constant, or the projection of angular momentum of the rotor onto its axis is constant. In both cases, phase portraits are formed by curves lying at the intersection of an ellipsoid with a family of concentric spheres offset relative to the center of the ellipsoid. The structurally stable phase portraits with differing numbers and types of singular points are classified. Rearrangements of these portraits for the first type of gyrostat occur when the rotor rotation speed changes. For the second type, a discontinuous bifurcation is possible when the rotor is released; in this case, the inertia tensor changes abruptly. In previous studies, the stability of stationary gyrostat motions and their local bifurcations were studied in detail. Global rearrangements were apparently not considered. The obtained results can be used to control the orientations of satellites and mobile robots.