The stability behavior of a synchronous machine, connected to an in$nite bus, is considered. Direct and quadrature axis damper-windings are taken into account and the input torque is said to be zero. The behavior of this machine is characterized by a system of $l%e $&-order nonlinear ordinary differential equations containing periodic nonlinear terms. These equations are shown to have two different types of equilibrium points. By means of lirae,arized analysis it is shoum that all points of the first type are (locally) asymptotically stable, whereas all points of the second type are (locally) unstable. &ability behavior of the system ?:s analyzed for all initial conditions. This analysis is performed by a combination sf various techniques: the second method of Liapunov, the topological methods of Waiewski, extension theory and results of the topological properties of the regions of attraction. These techniques, combined, allow a complete characterization of the stability behavior of the machine. Both global results (the set of all equilibrium points is a global attractor), as well as local results (the trajectory leaving one unstable equilibrium point tends to an asymptotically stable equilibrium point), are proved.