Abstract
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
(x) over dot + alpha(y) over dot - beta x = epsilon F-1(t, x, (x) over dot, y, (y) over dot)
(y) over dot + alpha(x) over dot - beta y = epsilon F-2(t, x, (x) over dot, y, (y) over dot)
being alpha and beta parameters satisfying Delta=alpha (2)-4 beta > 0 and , epsilon 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 epsilon=0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.