Lyapunov Stability
If all solutions of the dynamical system that start out near an equilibrium point x0 stay near x0 forever, then x0 is Lyapunov stable.
Asymptotic Stability
More strongly, if x0 is Lyapunov stable and all solutions that start out near x0 converge to x0, then x0 is asymptotically stable.
http://en.wikipedia.org/wiki/Lyapunov_stability
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