FAQ: Does a stable system have a stable equilibrium point? a limit cycle?
Strictly speaking, we never say that a system is stable or unstable. Rather, we talk about an equilibrium point (or limit cycle) being stable. Thus, you first find the equilibrium points of a system and then you determine whether those equilibrium points are stable or not.
In the case that a system has a single equilibrium point that is globally attractive (meaning that solutions converge to that equilibrium point for all initial conditions), we sometimes get sloppy and say that the "system" is stable. But we should really say that the equibrium point is globally stable.
--Ling Shi, 9 Oct 2006