Periodic motions of a satellite-gyrostat with a large inner angular momentum

We consider an axially symmetric satellite with a large inner angular momentum (i.e. the satellite is a gyrostat) directed along the symmetry axis. The satellite center of mass moves along a circular orbit around an attractive center, the satellite attitude motion being affected by the gravitational torque. We investigate periodic motions of the satellite symmetry axis in a vicinity of an invariable direction in the absolute space. Such motions are described by the system of differential equations of the fourth order that has periodic coefficients and contains a large parameter. We prove the unique existence of the symmetric periodic solution of the system using specific methods of large parameters.