asymptotic stability

Lyapunov stability promises only that you stay near a resting state if you start near it. Asymptotic stability promises something stronger and more satisfying: not only do you stay near, you actually come home. Drop a marble into a real bowl with a touch of friction, and it not only rocks nearby — it slowly spirals down and stops at the bottom. That eventual return is the extra ingredient.

Precisely, an equilibrium x* is asymptotically stable when it is both Lyapunov stable (nearby trajectories stay nearby) and attracting (every trajectory that starts close enough actually converges to x* as time goes to infinity). Both halves are needed. Attraction alone is not enough — there are odd systems where trajectories eventually return but first swing wildly far away, which is not what 'asymptotically stable' should allow. A Lyapunov function whose rate V' is strictly negative away from x* certifies asymptotic stability, because the energy keeps draining until the trajectory sits at the bottom.

This is the verdict engineers usually want: a thermostat returning the room to its set temperature, a self-righting boat, a control system that recovers from a bump. For a hyperbolic equilibrium, asymptotic stability is read straight off the Jacobian — it holds exactly when every eigenvalue has strictly negative real part (a sink). 'Asymptotically stable' specifies only the eventual behaviour, not how fast; if the convergence is exponential one sometimes says exponentially stable, a still stronger guarantee with a definite rate.