Periodic motions of a gyrostat satellite with a large gyrostatic moment about the centre of mass

An axisymmetric gyrostat satellite, whose centre of mass moves in a fixed circular orbit about an attractive centre is considered. The satellite moves about the centre of mass under the action of the gravitational moment. Under the assumption that the gyrostatic moment of the satellite is large and is directed along its axis of symmetry, the periodic motions of this axis are investigated in a small vicinity of a fixed direction, lying in the orbital plane in absolute space. Such motions are described by a system of fourth-order differential equations with periodic coefficients that contains a large parameter. A theorem for the existence and uniqueness of a symmetric periodic solution of this system that has the required properties is proved using methods for constructing periodic solutions of differential equations with a large parameter.