Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motion being α and β parameters satisfying Δ = α 2 −4 β >0 and , ε a small parameter and, F 1 and F 2 smooth periodic maps in the variable t in resonance p : q with some of the periodic solutions of the system for ε =0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.