An equilibrium has asymptotic stability when it is Lyapunov stable and every trajectory starting sufficiently nearby converges to it. Stability controls how far a small perturbation can wander; attraction supplies the eventual return. The definition does not require an exponential decay rate.
For , the solution from is . Its magnitude never increases and tends to zero, so the origin is asymptotically stable. The decay is algebraic rather than exponential, showing why exponential stability is a stronger condition. A center equilibrium provides a different comparison: neighboring trajectories stay close but circulate indefinitely instead of returning to the origin. These distinctions help interpret a measured response whose amplitude changes slowly or whose observation interval is too short to establish its eventual limit.
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